def l_function_maass_browse_page():
info = {"bread": get_bread(2, [("Maass Form", url_for('.l_function_maass_browse_page'))])}
info["contents"] = [processMaassNavigation()]
info["gl2spectrum0"] = [paintSvgMaass(1, 10, 0, 10, L="/L")]
info["colorminus1"] = rgbtohex(signtocolour(-1))
info["colorplus1"] = rgbtohex(signtocolour(1))
return render_template("MaassformGL2.html", title='L-functions of GL(2) Maass Forms of Weight 0', **info)
def browse_graph(min_level, max_level, min_R, max_R):
r"""
Render a page with a graph with clickable dots for all
with min_R <= R <= max_R and levels in the similar range.
"""
info = {}
info['contents'] = [paintSvgMaass(min_level, max_level, min_R, max_R)]
info['min_level'] = min_level
info['max_level'] = max_level
info['min_R'] = min_R
info['max_R'] = max_R
info['coloreven'] = rgbtohex(signtocolour(1))
info['colorodd'] = rgbtohex(signtocolour(-1))
bread = bread_prefix() + [('Browse graph', '')]
info['bread'] = bread
info['learnmore'] = learnmore_list()
return render_template("maass_browse_graph.html", title='Browsing graph of Maass forms', **info)
def render_maass_browse_graph(min_level, max_level, min_R, max_R):
r"""
Render a page with a graph with clickable dots for all
with min_R <= R <= max_R and levels in the similar range.
"""
info = {}
info['contents'] = [paintSvgMaass(min_level, max_level, min_R, max_R)]
info['min_level'] = min_level
info['max_level'] = max_level
info['min_R'] = min_R
info['max_R'] = max_R
info['coloreven'] = rgbtohex(signtocolour(1))
info['colorodd'] = rgbtohex(signtocolour(-1))
bread = [('Modular forms', url_for('mf.modular_form_main_page')),
('Maass waveforms', url_for('.render_maass_waveforms'))]
info['bread'] = bread
return render_template("mwf_browse_graph.html", title='Browsing graph of Maass forms', **info)
def render_maass_browse_graph(min_level, max_level, min_R, max_R):
r"""
Render a page with a graph with clickable dots for all
with min_R <= R <= max_R and levels in the similar range.
"""
info = {}
info['contents'] = [paintSvgMaass(min_level, max_level, min_R, max_R)]
info['min_level'] = min_level
info['max_level'] = max_level
info['min_R'] = min_R
info['max_R'] = max_R
info['coloreven'] = rgbtohex(signtocolour(1))
info['colorodd'] = rgbtohex(signtocolour(-1))
bread = [('Modular Forms', url_for('mf.modular_form_main_page')),
('Maass Waveforms', url_for('.render_maass_waveforms'))]
info['bread'] = bread
return render_template("mwf_browse_graph.html", title='Browsing graph of Maass forms', **info)
def test_signtocolour(self):
r"""
Checking utility: signtocolour
"""
self.assertEqual(signtocolour(0), 'rgb(63,63,255)')
self.assertEqual(signtocolour(1 + 2j), 'rgb(197,0,184)')
def test_signtocolour(self):
r"""
Checking utility: signtocolour
"""
self.assertEqual(signtocolour(0), 'rgb(63,63,255)')
self.assertEqual(signtocolour(1+2j), 'rgb(197,0,184)')
def paintSvgMaass(min_level, max_level, min_R, max_R, weight=0, char=1,
width=1000, heightfactor=20, L=""):
''' Returns the contents (as a string) of the svg-file for
all Maass forms in the database.
Takes all levels from min_level to max_level
Spectral parameter in [min_R, max_R]
Set L="/L" to make link go to the L-function
'''
xMax = int(max_R)
yMax = int(max_level)
xMin = int(min_R)
yMin = int(min_level)
extraSpace = 40
length_R = xMax - xMin
length_level = yMax - yMin + 1
if length_level < 15:
heightfactor = heightfactor * 2
height = length_level * heightfactor + extraSpace
xfactor = (width - extraSpace)/length_R
yfactor = (height - extraSpace)/length_level
ticlength = 4
radius = 3
xshift = extraSpace
# Start of file and add coordinate system
ans = "<svg xmlns='http://www.w3.org/2000/svg'"
ans += " xmlns:xlink='http://www.w3.org/1999/xlink'"
ans += " height='{0}' width='{1}'>\n".format(height + 20, width + 20)
ans += paintCSMaass(width, height, xMin, xMax, yMin, yMax,
xfactor, yfactor, ticlength, xshift)
# Fetch Maass forms from database
# NB although base.getDBConnection().PORT works it gives the
# default port number of 27017 and not the actual one!
if pymongo.version_tuple[0] < 3:
host = base.getDBConnection().host
port = base.getDBConnection().port
else:
host, port = base.getDBConnection().address
db = MaassDB(host=host, port=port)
search = {'level1': yMin, 'level2': yMax, 'char': char,
'R1': xMin, 'R2': xMax, 'Newform' : None, 'weight' : weight}
fields = {'Eigenvalue', 'Level', 'Symmetry'}
forms = db.get_Maass_forms(search, fields,
do_sort=False, limit=10000)
# Loop through all forms and add a clickable dot for each
for f in forms:
linkurl = L + "/ModularForm/GL2/Q/Maass/{0}".format(f['_id'])
x = (f['Eigenvalue'] - xMin) * xfactor + xshift
y = (f['Level'] - yMin + 1) * yfactor
try: # Shifting even slightly up and odd slightly down
if f['Symmetry'] == 0 or f['Symmetry'] == 'even':
y -= 1
color = signtocolour(1)
elif f['Symmetry'] == 1 or f['Symmetry'] == 'odd':
y += 1
color = signtocolour(-1)
else:
color = signtocolour(0)
except Exception:
color = signtocolour(0)
ans += "<a xlink:href='{0}' target='_top'>".format(linkurl)
ans += "<circle cx='{0}' cy='{1}' ".format(str(x)[0:6],str(y))
ans += "r='{0}' style='fill:{1}'>".format(str(radius),color)
ans += "<title>{0}</title></circle></a>\n".format(f['Eigenvalue'])
ans += "</svg>"
return ans
def paintSvgMaass(min_level,
max_level,
min_R,
max_R,
weight=0,
char=1,
width=1000,
heightfactor=20,
L=""):
''' Returns the contents (as a string) of the svg-file for
all Maass forms in the database.
Takes all levels from min_level to max_level
Spectral parameter in [min_R, max_R]
Set L="/L" to make link go to the L-function
'''
xMax = int(max_R)
yMax = int(max_level)
xMin = int(min_R)
yMin = int(min_level)
extraSpace = 40
length_R = xMax - xMin
length_level = yMax - yMin + 1
if length_level < 15:
heightfactor = heightfactor * 2
height = length_level * heightfactor + extraSpace
xfactor = (width - extraSpace) / length_R
yfactor = (height - extraSpace) / length_level
ticlength = 4
radius = 3
xshift = extraSpace
# Start of file and add coordinate system
ans = "<svg xmlns='http://www.w3.org/2000/svg'"
ans += " xmlns:xlink='http://www.w3.org/1999/xlink'"
ans += " height='{0}' width='{1}'>\n".format(height + 20, width + 20)
ans += paintCSMaass(width, height, xMin, xMax, yMin, yMax, xfactor,
yfactor, ticlength, xshift)
# Fetch Maass forms from database
search = {
'level1': yMin,
'level2': yMax,
'char': char,
'R1': xMin,
'R2': xMax,
'Newform': None,
'weight': weight
}
projection = ['maass_id', 'Eigenvalue', 'Level', 'Symmetry']
forms = maass_db.get_Maass_forms(search, projection, sort=[], limit=10000)
# Loop through all forms and add a clickable dot for each
for f in forms:
linkurl = L + "/ModularForm/GL2/Q/Maass/{0}".format(f['maass_id'])
x = (f['Eigenvalue'] - xMin) * xfactor + xshift
y = (f['Level'] - yMin + 1) * yfactor
try: # Shifting even slightly up and odd slightly down
if f['Symmetry'] == 0 or f['Symmetry'] == 'even':
y -= 1
color = signtocolour(1)
elif f['Symmetry'] == 1 or f['Symmetry'] == 'odd':
y += 1
color = signtocolour(-1)
else:
color = signtocolour(0)
except Exception:
color = signtocolour(0)
ans += "<a xlink:href='{0}' target='_top'>".format(linkurl)
ans += "<circle cx='{0}' cy='{1}' ".format(str(x)[0:6], str(y))
ans += "r='{0}' style='fill:{1}'>".format(str(radius), color)
ans += "<title>{0}</title></circle></a>\n".format(f['Eigenvalue'])
ans += "</svg>"
return ans
def paintSvgHoloGeneral(Nmin, Nmax, kmin, kmax, imagewidth, imageheight):
# the import must be here to avoid circular import
from lmfdb.classical_modular_forms.web_newform import WebNewform
from lmfdb.classical_modular_forms.web_space import WebGamma1Space
xfactor = 90
yfactor = 30
extraSpace = 20
ticlength = 4
radius = 3.3
xdotspacing = 0.30 # horizontal spacing of dots
ydotspacing = 0.11 # vertical spacing of dots
# colourplus = signtocolour(1) # not used
# colourminus = signtocolour(-1) # not used
maxdots = 5 # max number of dots to display
ans = svgBegin()
xMax = int(Nmax)
yMax = int(kmax)
width = xfactor * xMax + extraSpace
height = yfactor * yMax + extraSpace
# make the coordinate system
ans += paintCSHoloTMP(width, height, xMax, yMax, xfactor, yfactor,
ticlength)
alphabet = ['a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i', 'j']
# create appearanceinfo, common to all points
appearanceinfo = []
# loop over levels and weights, using plotsector to put the appropriate dots at each lattice point
for x in range(int(Nmin), int(Nmax) + 1): # x is the level
for y in range(int(kmin), int(kmax) + 1, 2): # y is the weight
# lid = "(" + str(x) + "," + str(y) + ")" # not used
# linkurl = "/L/ModularForm/GL2/Q/holomorphic/" + str(y) + "/" + str(x) + "/1/" # not used
WS = WebGamma1Space(
level=x,
weight=y) # space of modular forms of weight y, level x
galois_orbits = WS.decomp # make a list of Galois orbits
numlabels = len(galois_orbits) # one label per Galois orbit
thelabels = alphabet[
0:
numlabels] # list of labels for the Galois orbits for weight y, level x
# countplus = 0 # count how many Galois orbits have sign Plus (+ 1) (not used)
# countminus = 0 # count how many Galois orbits have sign Minus (- 1) (not used)
# ybaseplus = y # baseline y-coord for plus cases (not used)
# ybaseminus = y # baseline y-coord for minus cases (not used)
# numpluslabels = 0 # not used
# numminuslabels = 0 # not used
# plotsector requires three dictionaries: dimensioninfo, appearanceinfo, and urlinfo
# create dimensioninfo
dimensioninfo = {}
dimensioninfo['offset'] = [0, height]
dimensioninfo['scale'] = [xfactor, -1 * yfactor]
dimensioninfo['vertexlocation'] = [x, y]
dimensioninfo['maxdots'] = maxdots
dimensioninfo['dotspacing'] = [xdotspacing, ydotspacing]
dimensioninfo['edge'] = [[0, 1], [1, 0]
] # unit vectors defining edges of sector
# dimensioninfo['edgelength'] = [float(dimensioninfo['scale'][0])/float(Nmax), float(dimensioninfo['scale'][1])/float(kmax)] #add comment
dimensioninfo['edgelength'] = [0.5, 0.5]
dimensioninfo['dotradius'] = radius
dimensioninfo[
'connectinglinewidth'] = dimensioninfo['dotradius'] / 1.5
dimensioninfo['firstdotoffset'] = [0.0, 0.0]
#
appearanceinfo = {}
# appearanceinfo['edgewidth'] = dimensioninfo['dotspacing'][0]/1.0 #just a guess
appearanceinfo['edgewidth'] = [0, 0] # remove the sector edges
appearanceinfo['edgestyle'] = 'stroke-dasharray:3,3'
appearanceinfo['edgecolor'] = 'rgb(202,202,102)'
appearanceinfo['fontsize'] = 'font-size:11px'
appearanceinfo['fontweight'] = ""
#
urlinfo = {'base': '/L/ModularForm/GL2/Q/holomorphic?'}
urlinfo['space'] = {'weight': y}
urlinfo['space']['level'] = x
urlinfo['space']['character'] = 0
#
# scale = 1 # not used
# Symmetry types: +1 or -1
symmetrytype = [1, -1]
for signtmp in symmetrytype:
# urlinfo['space']['orbits'] = [ [] for label in thelabels ] # initialise
# an empty list for each orbit
urlinfo['space']['orbits'] = []
for label in thelabels: # looping over Galois orbit: one label per orbit
# do '+' case first
MF = WebNewform.by_label(
label=label
) # one of the Galois orbits for weight y, level x
numberwithlabel = MF.degree(
) # number of forms in the Galois orbit
if x == 1: # For level 1, the sign is always plus
signfe = 1
else:
# signfe = -1
frickeeigenvalue = prod(MF.atkin_lehner_eigenvalues(
).values()) # gives Fricke eigenvalue
signfe = frickeeigenvalue * (-1)**float(
y / 2) # sign of functional equation
if signfe == signtmp: # we find an orbit with sign of "signtmp"
if signfe == 1:
dimensioninfo['edge'] = [[0, 1], [1, 0]]
# unit vectors defining edges of sector for signfe positive
else:
# dimensioninfo['edge'] = [[0,1],[-1,0]] # unit vectors defining edges
# of sector for signfe negative
dimensioninfo['edge'] = [[0, -1], [-1, 0]]
# unit vectors defining edges of sector for signfe negative
dimensioninfo['dotspacing'] = [
signfe * xdotspacing, ydotspacing
]
dimensioninfo['firstdotoffset'] = [
0.5 * (dimensioninfo['dotspacing'][0] *
dimensioninfo['edge'][0][0] +
dimensioninfo['dotspacing'][1] *
dimensioninfo['edge'][1][0]), 0
]
signcolour = signtocolour(signfe)
appearanceinfo['edgecolor'] = signcolour
orbitdescriptionlist = []
for n in range(numberwithlabel):
orbitdescriptionlist.append({
'label': label,
'number': n,
'color': signcolour
})
urlinfo['space']['orbits'].append(orbitdescriptionlist)
# urlinfo['space']['orbits'][0][0]['color'] = signtocolour(-1)
# appearanceinfo['orbitcolor'] = 'rgb(102,102,102)'
ans += plotsector(dimensioninfo, appearanceinfo, urlinfo)
ans += svgEnd()
return (ans)
def paintSvgHolo(Nmin, Nmax, kmin, kmax):
# the import must be here to avoid circular import
from lmfdb.classical_modular_forms.web_space import WebGamma1Space
xfactor = 90
yfactor = 30
extraSpace = 20
ticlength = 4
radius = 3.3
xdotspacing = 0.11 # horizontal spacing of dots
ydotspacing = 0.28 # vertical spacing of dots
colourplus = signtocolour(1)
colourminus = signtocolour(-1)
maxdots = 5 # max number of dots to display
ans = svgBegin()
xMax = int(Nmax)
yMax = int(kmax)
width = xfactor * xMax + extraSpace
height = yfactor * yMax + extraSpace
ans += paintCSHolo(width, height, xMax, yMax, xfactor, yfactor, ticlength)
# alphabet = ['a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i', 'j']
# loop over levels and weights
for x in range(int(Nmin), int(Nmax) + 1): # x is the level
for y in range(int(kmin), int(kmax) + 1): # y is the weight
# lid = "(" + str(x) + "," + str(y) + ")" # not used
linkurl = "/L/ModularForm/GL2/Q/holomorphic/" + str(x) + "/" + str(
y) + "/1/"
try:
WS = WebGamma1Space(level=x, weight=y)
except ValueError:
continue
newspaces = WS.decomp
# numlabels = len(WS.decomp) # one label per Galois orbit
# thelabels = alphabet[0:numlabels] # list of labels for the Galois orbits for weight y, level x
# countplus = 0 # count how many Galois orbits have sign Plus (+ 1) # not used
# countminus = 0 # count how many Galois orbits have sign Minus (- 1) # not used
ybaseplus = y # baseline y-coord for plus cases
ybaseminus = y # baseline y-coord for minus cases
numpluslabels = 0
numminuslabels = 0
for newsp in newspaces: # looping over Galois orbit
for MF in newsp[1]:
print(MF)
linkurl = "/L/ModularForm/GL2/Q/holomorphic/%d/%d/%s/%s/" % (
x, y, MF['char_orbit_label'],
cremona_letter_code(MF['hecke_orbit'] - 1))
numberwithlabel = MF[
'dim'] # number of forms in the Galois orbit
# frickeeigenvalue = prod(MF.atkin_lehner_eigenvalues().values()) # gives Fricke eigenvalue
self_dual = MF['char_is_real'] * (-1)**float(
y / 2) # sign of functional equation
xbase = x - self_dual * (xdotspacing / 2.0)
if self_dual > 0: # go to right in BLUE if plus
ybase = ybaseplus
ybaseplus += ydotspacing
thiscolour = colourplus
numpluslabels += 1
else: # go to the left in RED of minus
ybase = ybaseminus
ybaseminus += ydotspacing
thiscolour = colourminus
numminuslabels += 1
if numberwithlabel > maxdots: # if more than maxdots in orbit, use number as symbol
xbase += 1.5 * self_dual * xdotspacing
if self_dual < 0: # move over more to position numbers on minus side.
xbase += self_dual * xdotspacing
ybase += -0.5 * ydotspacing
if (self_dual > 0 and numpluslabels > 1) or (
self_dual < 0 and numminuslabels > 1):
ybase += ydotspacing
ans += "<a xlink:href='" + url_for(
'not_yet_implemented') + "' target='_top'>\n"
# TODO: Implement when there is more than maxdots forms
ans += ("<text x='" +
str(float(xbase) * xfactor)[0:7] + "' y='" +
str(height - float(ybase) * yfactor)[0:7] +
"' style='fill:" + thiscolour +
";font-size:14px;font-weight:bold;'>" +
str(numberwithlabel) + "</text>\n")
ans += "</a>\n"
if self_dual < 0:
ybaseminus += 1.5 * ydotspacing
else:
ybaseplus += 1.5 * ydotspacing
else: # otherwise, use one dot per form in orbit, connected with a line
if numberwithlabel > 1: # join dots if there are at least two
# add lines first and then dots to prevent line from hiding link
firstcenterx = xbase + self_dual * xdotspacing
firstcentery = ybase
lastcenterx = xbase + (numberwithlabel *
self_dual * xdotspacing)
lastcentery = ybase
ans += "<line x1='%s' " % str(
float(firstcenterx) * xfactor)[0:7]
ans += "y1='%s' " % str(
float(height - firstcentery * yfactor))[0:7]
ans += "x2='%s' " % str(
float(lastcenterx) * xfactor)[0:7]
ans += "y2='%s' " % str(
float(height - lastcentery * yfactor))[0:7]
ans += "style='stroke:%s;stroke-width:2.4'/>" % thiscolour
for number in range(0, numberwithlabel):
xbase += self_dual * xdotspacing
ans += "<a xlink:href='" + linkurl + str(
number + 1) + "/' target='_top'>\n"
ans += "<circle cx='" + str(
float(xbase) * xfactor)[0:7]
ans += "' cy='" + str(height -
float(ybase) * yfactor)[0:7]
ans += "' r='" + str(radius)
ans += "' style='fill:" + thiscolour + "'>"
ans += "<title>" + str((x, y)).replace(
"u", "").replace("'", "") + "</title>"
ans += "</circle></a>\n"
ans += svgEnd()
return ans
def paintSvgFileAll(glslist): # list of group and level
xfactor = 20
yfactor = 20
extraSpace = 20
ticlength = 4
radius = 3
ans = svgBegin()
paralist = []
xMax = 0
yMax = 0
for gls in glslist:
group = gls[0]
level = gls[1]
for l in db.lfunc_lfunctions.search(
{
'group': group,
'conductor': level
}, ['origin', 'root_number']):
splitOrigin = l['origin'].split('/')
char = splitOrigin[5]
R = splitOrigin[6]
ap_id = splitOrigin[7]
splitId = R.split('_')
paralist.append((splitId[0], splitId[1], l['origin'], group, level,
char, R, ap_id, l['root_number']))
if float(splitId[0]) > xMax:
xMax = float(splitId[0])
if float(splitId[1]) > yMax:
yMax = float(splitId[1])
xMax = int(math.ceil(xMax))
yMax = int(math.ceil(yMax))
width = xfactor * xMax + extraSpace
height = yfactor * yMax + extraSpace
ans += paintCS(width, height, xMax, yMax, xfactor, yfactor, ticlength)
for (x, y, lid, group, level, char, R, ap_id, sign) in paralist:
if float(x) > 0 and float(y) > 0: # Only one of dual pair
try:
linkurl = url_for('.l_function_maass_gln_page',
group=group,
level=level,
char=char,
R=R,
ap_id=ap_id)
except Exception: # catch when running a test
linkurl = lid
ans += "<a xlink:href='" + linkurl + "' target='_top'>\n"
ans += "<circle cx='" + str(float(x) * xfactor)[0:7]
ans += "' cy='" + str(height - float(y) * yfactor)[0:7]
ans += "' r='" + str(radius)
ans += "' style='fill:" + signtocolour(sign) + "'>"
ans += "<title>" + str(
(x, y)).replace("u", "").replace("'", "") + "</title>"
ans += "</circle></a>\n"
ans += svgEnd()
return (ans)
def paintSvgMaass(min_level,
max_level,
min_R,
max_R,
width=1000,
heightfactor=20,
L=""):
''' Returns the contents (as a string) of the svg-file for
all Maass forms in the database of the specified weight.
Takes all levels from min_level to max_level
Spectral parameter in [min_R, max_R]
Set L="/L" to make link go to the L-function
'''
xMax = int(max_R)
yMax = int(max_level)
xMin = int(min_R)
yMin = int(min_level)
extraSpace = 40
length_R = xMax - xMin
length_level = yMax - yMin + 1
if length_level < 30:
heightfactor = heightfactor * 2
height = length_level * heightfactor + extraSpace
xfactor = (width - extraSpace) / length_R
yfactor = (height - extraSpace) / length_level
ticlength = 4
radius = 3
xshift = extraSpace
# Start of file and add coordinate system
ans = "<svg xmlns='http://www.w3.org/2000/svg'"
ans += " xmlns:xlink='http://www.w3.org/1999/xlink'"
ans += " height='{0}' width='{1}'>\n".format(height + 20, width + 20)
ans += paintCSMaass(width, height, xMin, xMax, yMin, yMax, xfactor,
yfactor, ticlength, xshift)
# Fetch Maass forms from database
forms = db.maass_newforms.search(
{
'spectral_parameter': {
'$gte': xMin,
'$lte': xMax
},
'level': {
'$gte': yMin,
'$lte': yMax
}
}, ["maass_id", "spectral_parameter", "level", "symmetry"],
sort=[("level", 1), ("symmetry", -1), ("spectral_parameter", 1),
("maass_id", 1)])
# Loop through all forms and add a clickable dot for each
for f in forms:
linkurl = L + url_for("maass.by_label", label=f['maass_id'])
x = (f['spectral_parameter'] - xMin) * xfactor + xshift
y = (f['level'] - yMin + 1) * yfactor
s = f.get('symmetry', 0)
y -= s # Shifting even slightly up and odd slightly down
color = signtocolour(s)
ans += "<a xlink:href='{0}' target='_top'>".format(linkurl)
ans += "<circle cx='{0}' cy='{1}' ".format(str(x)[0:6], str(y))
ans += "r='{0}' style='fill:{1}'>".format(str(radius), color)
ans += "<title>{0}</title></circle></a>\n".format(
f['spectral_parameter'])
ans += "</svg>"
return ans
