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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [315,3,Mod(37,315)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("315.37"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 3, 4])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 315.ca (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,2,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.58312832735\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 172.6
Character \(\chi\) \(=\) 315.172
Dual form 315.3.ca.a.163.6

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.792728 - 2.95850i) q^{2} +(-4.66022 - 2.69058i) q^{4} +(0.307372 + 4.99054i) q^{5} +(3.94249 + 5.78419i) q^{7} +(-2.99127 + 2.99127i) q^{8} +(15.0082 + 3.04678i) q^{10} +(3.12159 - 5.40675i) q^{11} +(11.7345 - 11.7345i) q^{13} +(20.2378 - 7.07857i) q^{14} +(-4.28389 - 7.41992i) q^{16} +(4.56805 - 1.22400i) q^{17} +(11.4064 - 6.58549i) q^{19} +(11.9950 - 24.0840i) q^{20} +(-13.5213 - 13.5213i) q^{22} +(34.8444 + 9.33654i) q^{23} +(-24.8110 + 3.06791i) q^{25} +(-25.4144 - 44.0190i) q^{26} +(-2.81005 - 37.5631i) q^{28} +28.5683i q^{29} +(10.1948 - 17.6579i) q^{31} +(-41.6924 + 11.1715i) q^{32} -14.4849i q^{34} +(-27.6544 + 21.4530i) q^{35} +(-6.06849 + 22.6479i) q^{37} +(-10.4410 - 38.9664i) q^{38} +(-15.8475 - 14.0086i) q^{40} +45.1077 q^{41} +(-21.9185 + 21.9185i) q^{43} +(-29.0946 + 16.7978i) q^{44} +(55.2443 - 95.6860i) q^{46} +(-2.79810 + 10.4426i) q^{47} +(-17.9136 + 45.6081i) q^{49} +(-10.5920 + 75.8356i) q^{50} +(-86.2582 + 23.1128i) q^{52} +(-19.2094 - 71.6904i) q^{53} +(27.9421 + 13.9165i) q^{55} +(-29.0951 - 5.50902i) q^{56} +(84.5194 + 22.6469i) q^{58} +(-14.6115 - 8.43595i) q^{59} +(16.7743 + 29.0540i) q^{61} +(-44.1593 - 44.1593i) q^{62} +97.9319i q^{64} +(62.1686 + 54.9548i) q^{65} +(-95.8731 + 25.6891i) q^{67} +(-24.5814 - 6.58656i) q^{68} +(41.5465 + 98.8221i) q^{70} -66.2415 q^{71} +(-26.7635 - 99.8826i) q^{73} +(62.1933 + 35.9073i) q^{74} -70.8751 q^{76} +(43.5805 - 3.26020i) q^{77} +(36.1253 - 20.8569i) q^{79} +(35.7127 - 23.6596i) q^{80} +(35.7582 - 133.451i) q^{82} +(-7.39450 + 7.39450i) q^{83} +(7.51254 + 22.4208i) q^{85} +(47.4706 + 82.2215i) q^{86} +(6.83554 + 25.5106i) q^{88} +(-19.2257 + 11.0999i) q^{89} +(114.138 + 21.6115i) q^{91} +(-137.262 - 137.262i) q^{92} +(28.6764 + 16.5563i) q^{94} +(36.3712 + 54.9000i) q^{95} +(-73.6717 - 73.6717i) q^{97} +(120.731 + 89.1523i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 2 q^{2} + 4 q^{5} - 6 q^{7} + 36 q^{8} + 14 q^{10} + 24 q^{11} - 8 q^{13} + 20 q^{16} + 48 q^{17} + 72 q^{20} + 104 q^{22} + 86 q^{23} - 16 q^{25} - 140 q^{26} + 186 q^{28} + 120 q^{31} - 130 q^{32}+ \cdots - 482 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.792728 − 2.95850i 0.396364 − 1.47925i −0.423080 − 0.906092i \(-0.639051\pi\)
0.819444 − 0.573159i \(-0.194282\pi\)
\(3\) 0 0
\(4\) −4.66022 − 2.69058i −1.16505 − 0.672645i
\(5\) 0.307372 + 4.99054i 0.0614745 + 0.998109i
\(6\) 0 0
\(7\) 3.94249 + 5.78419i 0.563212 + 0.826312i
\(8\) −2.99127 + 2.99127i −0.373909 + 0.373909i
\(9\) 0 0
\(10\) 15.0082 + 3.04678i 1.50082 + 0.304678i
\(11\) 3.12159 − 5.40675i 0.283781 − 0.491523i −0.688532 − 0.725206i \(-0.741744\pi\)
0.972313 + 0.233683i \(0.0750778\pi\)
\(12\) 0 0
\(13\) 11.7345 − 11.7345i 0.902657 − 0.902657i −0.0930085 − 0.995665i \(-0.529648\pi\)
0.995665 + 0.0930085i \(0.0296484\pi\)
\(14\) 20.2378 − 7.07857i 1.44556 − 0.505612i
\(15\) 0 0
\(16\) −4.28389 − 7.41992i −0.267743 − 0.463745i
\(17\) 4.56805 − 1.22400i 0.268709 − 0.0720003i −0.121949 − 0.992536i \(-0.538914\pi\)
0.390657 + 0.920536i \(0.372248\pi\)
\(18\) 0 0
\(19\) 11.4064 − 6.58549i 0.600337 − 0.346605i −0.168837 − 0.985644i \(-0.554001\pi\)
0.769174 + 0.639039i \(0.220668\pi\)
\(20\) 11.9950 − 24.0840i 0.599751 − 1.20420i
\(21\) 0 0
\(22\) −13.5213 − 13.5213i −0.614606 − 0.614606i
\(23\) 34.8444 + 9.33654i 1.51498 + 0.405936i 0.918084 − 0.396386i \(-0.129736\pi\)
0.596891 + 0.802322i \(0.296402\pi\)
\(24\) 0 0
\(25\) −24.8110 + 3.06791i −0.992442 + 0.122716i
\(26\) −25.4144 − 44.0190i −0.977475 − 1.69304i
\(27\) 0 0
\(28\) −2.81005 − 37.5631i −0.100359 − 1.34154i
\(29\) 28.5683i 0.985114i 0.870280 + 0.492557i \(0.163938\pi\)
−0.870280 + 0.492557i \(0.836062\pi\)
\(30\) 0 0
\(31\) 10.1948 − 17.6579i 0.328865 − 0.569611i −0.653422 − 0.756994i \(-0.726667\pi\)
0.982287 + 0.187383i \(0.0600006\pi\)
\(32\) −41.6924 + 11.1715i −1.30289 + 0.349108i
\(33\) 0 0
\(34\) − 14.4849i − 0.426026i
\(35\) −27.6544 + 21.4530i −0.790126 + 0.612944i
\(36\) 0 0
\(37\) −6.06849 + 22.6479i −0.164013 + 0.612106i 0.834151 + 0.551537i \(0.185958\pi\)
−0.998164 + 0.0605694i \(0.980708\pi\)
\(38\) −10.4410 − 38.9664i −0.274763 − 1.02543i
\(39\) 0 0
\(40\) −15.8475 − 14.0086i −0.396188 − 0.350216i
\(41\) 45.1077 1.10019 0.550094 − 0.835103i \(-0.314592\pi\)
0.550094 + 0.835103i \(0.314592\pi\)
\(42\) 0 0
\(43\) −21.9185 + 21.9185i −0.509733 + 0.509733i −0.914445 − 0.404711i \(-0.867372\pi\)
0.404711 + 0.914445i \(0.367372\pi\)
\(44\) −29.0946 + 16.7978i −0.661241 + 0.381767i
\(45\) 0 0
\(46\) 55.2443 − 95.6860i 1.20096 − 2.08013i
\(47\) −2.79810 + 10.4426i −0.0595339 + 0.222184i −0.989283 − 0.146010i \(-0.953357\pi\)
0.929749 + 0.368193i \(0.120024\pi\)
\(48\) 0 0
\(49\) −17.9136 + 45.6081i −0.365584 + 0.930779i
\(50\) −10.5920 + 75.8356i −0.211840 + 1.51671i
\(51\) 0 0
\(52\) −86.2582 + 23.1128i −1.65881 + 0.444477i
\(53\) −19.2094 − 71.6904i −0.362441 − 1.35265i −0.870856 − 0.491537i \(-0.836435\pi\)
0.508415 − 0.861112i \(-0.330232\pi\)
\(54\) 0 0
\(55\) 27.9421 + 13.9165i 0.508039 + 0.253028i
\(56\) −29.0951 − 5.50902i −0.519556 − 0.0983754i
\(57\) 0 0
\(58\) 84.5194 + 22.6469i 1.45723 + 0.390464i
\(59\) −14.6115 − 8.43595i −0.247652 − 0.142982i 0.371036 − 0.928618i \(-0.379002\pi\)
−0.618689 + 0.785636i \(0.712336\pi\)
\(60\) 0 0
\(61\) 16.7743 + 29.0540i 0.274989 + 0.476295i 0.970132 − 0.242576i \(-0.0779924\pi\)
−0.695143 + 0.718871i \(0.744659\pi\)
\(62\) −44.1593 − 44.1593i −0.712247 − 0.712247i
\(63\) 0 0
\(64\) 97.9319i 1.53019i
\(65\) 62.1686 + 54.9548i 0.956440 + 0.845459i
\(66\) 0 0
\(67\) −95.8731 + 25.6891i −1.43094 + 0.383420i −0.889352 − 0.457223i \(-0.848844\pi\)
−0.541590 + 0.840643i \(0.682177\pi\)
\(68\) −24.5814 − 6.58656i −0.361491 − 0.0968612i
\(69\) 0 0
\(70\) 41.5465 + 98.8221i 0.593521 + 1.41174i
\(71\) −66.2415 −0.932979 −0.466489 − 0.884527i \(-0.654481\pi\)
−0.466489 + 0.884527i \(0.654481\pi\)
\(72\) 0 0
\(73\) −26.7635 − 99.8826i −0.366623 − 1.36825i −0.865208 − 0.501414i \(-0.832813\pi\)
0.498585 − 0.866841i \(-0.333853\pi\)
\(74\) 62.1933 + 35.9073i 0.840450 + 0.485234i
\(75\) 0 0
\(76\) −70.8751 −0.932567
\(77\) 43.5805 − 3.26020i 0.565980 − 0.0423402i
\(78\) 0 0
\(79\) 36.1253 − 20.8569i 0.457282 − 0.264012i −0.253619 − 0.967304i \(-0.581621\pi\)
0.710901 + 0.703292i \(0.248288\pi\)
\(80\) 35.7127 − 23.6596i 0.446408 − 0.295745i
\(81\) 0 0
\(82\) 35.7582 − 133.451i 0.436075 − 1.62745i
\(83\) −7.39450 + 7.39450i −0.0890904 + 0.0890904i −0.750247 − 0.661157i \(-0.770066\pi\)
0.661157 + 0.750247i \(0.270066\pi\)
\(84\) 0 0
\(85\) 7.51254 + 22.4208i 0.0883828 + 0.263774i
\(86\) 47.4706 + 82.2215i 0.551984 + 0.956064i
\(87\) 0 0
\(88\) 6.83554 + 25.5106i 0.0776766 + 0.289893i
\(89\) −19.2257 + 11.0999i −0.216019 + 0.124718i −0.604105 − 0.796904i \(-0.706469\pi\)
0.388087 + 0.921623i \(0.373136\pi\)
\(90\) 0 0
\(91\) 114.138 + 21.6115i 1.25426 + 0.237489i
\(92\) −137.262 − 137.262i −1.49198 − 1.49198i
\(93\) 0 0
\(94\) 28.6764 + 16.5563i 0.305068 + 0.176131i
\(95\) 36.3712 + 54.9000i 0.382855 + 0.577894i
\(96\) 0 0
\(97\) −73.6717 − 73.6717i −0.759502 − 0.759502i 0.216730 − 0.976232i \(-0.430461\pi\)
−0.976232 + 0.216730i \(0.930461\pi\)
\(98\) 120.731 + 89.1523i 1.23195 + 0.909717i
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 315.3.ca.a.172.6 24
3.2 odd 2 35.3.l.a.32.1 yes 24
5.3 odd 4 inner 315.3.ca.a.298.1 24
7.2 even 3 inner 315.3.ca.a.37.1 24
15.2 even 4 175.3.p.c.18.1 24
15.8 even 4 35.3.l.a.18.6 yes 24
15.14 odd 2 175.3.p.c.32.6 24
21.2 odd 6 35.3.l.a.2.6 ✓ 24
21.5 even 6 245.3.m.b.177.6 24
21.11 odd 6 245.3.g.c.197.1 12
21.17 even 6 245.3.g.b.197.1 12
21.20 even 2 245.3.m.b.67.1 24
35.23 odd 12 inner 315.3.ca.a.163.6 24
105.2 even 12 175.3.p.c.93.6 24
105.23 even 12 35.3.l.a.23.1 yes 24
105.38 odd 12 245.3.g.b.148.1 12
105.44 odd 6 175.3.p.c.107.1 24
105.53 even 12 245.3.g.c.148.1 12
105.68 odd 12 245.3.m.b.128.1 24
105.83 odd 4 245.3.m.b.18.6 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.3.l.a.2.6 ✓ 24 21.2 odd 6
35.3.l.a.18.6 yes 24 15.8 even 4
35.3.l.a.23.1 yes 24 105.23 even 12
35.3.l.a.32.1 yes 24 3.2 odd 2
175.3.p.c.18.1 24 15.2 even 4
175.3.p.c.32.6 24 15.14 odd 2
175.3.p.c.93.6 24 105.2 even 12
175.3.p.c.107.1 24 105.44 odd 6
245.3.g.b.148.1 12 105.38 odd 12
245.3.g.b.197.1 12 21.17 even 6
245.3.g.c.148.1 12 105.53 even 12
245.3.g.c.197.1 12 21.11 odd 6
245.3.m.b.18.6 24 105.83 odd 4
245.3.m.b.67.1 24 21.20 even 2
245.3.m.b.128.1 24 105.68 odd 12
245.3.m.b.177.6 24 21.5 even 6
315.3.ca.a.37.1 24 7.2 even 3 inner
315.3.ca.a.163.6 24 35.23 odd 12 inner
315.3.ca.a.172.6 24 1.1 even 1 trivial
315.3.ca.a.298.1 24 5.3 odd 4 inner