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Properties

Label 31.8.g.a.7.3
Level $31$
Weight $8$
Character 31.7
Analytic conductor $9.684$
Analytic rank $0$
Dimension $144$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [31,8,Mod(7,31)] 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("31.7"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(31, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([28])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 31.g (of order \(15\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.68393579001\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(18\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label 7.3
Character \(\chi\) \(=\) 31.7
Dual form 31.8.g.a.9.3

$q$-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+(-5.07105 - 15.6071i) q^{2} +(91.2249 - 19.3905i) q^{3} +(-114.312 + 83.0523i) q^{4} +(-24.8635 + 43.0649i) q^{5} +(-765.235 - 1325.43i) q^{6} +(-449.966 - 200.338i) q^{7} +(176.534 + 128.260i) q^{8} +(5948.07 - 2648.25i) q^{9} +(798.203 + 169.663i) q^{10} +(-456.789 - 4346.05i) q^{11} +(-8817.65 + 9793.00i) q^{12} +(-4548.76 - 5051.91i) q^{13} +(-844.889 + 8038.58i) q^{14} +(-1433.13 + 4410.71i) q^{15} +(-4482.34 + 13795.2i) q^{16} +(-1169.39 + 11126.0i) q^{17} +(-71494.5 - 79402.7i) q^{18} +(15282.4 - 16972.9i) q^{19} +(-734.447 - 6987.80i) q^{20} +(-44932.7 - 9550.74i) q^{21} +(-65512.9 + 29168.2i) q^{22} +(49587.8 + 36027.7i) q^{23} +(18591.3 + 8277.39i) q^{24} +(37826.1 + 65516.7i) q^{25} +(-55778.6 + 96611.4i) q^{26} +(326249. - 237034. i) q^{27} +(68074.8 - 14469.8i) q^{28} +(38783.0 + 119362. i) q^{29} +76105.8 q^{30} +(-145752. + 79176.9i) q^{31} +265964. q^{32} +(-125942. - 387611. i) q^{33} +(179574. - 38169.6i) q^{34} +(19815.3 - 14396.6i) q^{35} +(-459991. + 796727. i) q^{36} +(-5042.96 - 8734.67i) q^{37} +(-342395. - 152444. i) q^{38} +(-512919. - 372657. i) q^{39} +(-9912.76 + 4413.44i) q^{40} +(437419. + 92976.3i) q^{41} +(78796.8 + 749701. i) q^{42} +(563100. - 625386. i) q^{43} +(413166. + 458868. i) q^{44} +(-33843.4 + 321998. i) q^{45} +(310825. - 956620. i) q^{46} +(-273936. + 843089. i) q^{47} +(-141405. + 1.34538e6i) q^{48} +(-388724. - 431722. i) q^{49} +(830708. - 922595. i) q^{50} +(109060. + 1.03764e6i) q^{51} +(939549. + 199707. i) q^{52} +(62931.8 - 28019.0i) q^{53} +(-5.35384e6 - 3.88979e6i) q^{54} +(198520. + 88386.8i) q^{55} +(-53739.1 - 93078.9i) q^{56} +(1.06503e6 - 1.84468e6i) q^{57} +(1.66622e6 - 1.21058e6i) q^{58} +(-1.99135e6 + 423274. i) q^{59} +(-202496. - 623220. i) q^{60} +3.19548e6 q^{61} +(1.97484e6 + 1.87325e6i) q^{62} -3.20697e6 q^{63} +(-774980. - 2.38514e6i) q^{64} +(330658. - 70283.6i) q^{65} +(-5.41082e6 + 3.93119e6i) q^{66} +(-1.05467e6 + 1.82674e6i) q^{67} +(-790362. - 1.36895e6i) q^{68} +(5.22224e6 + 2.32509e6i) q^{69} +(-325174. - 236253. i) q^{70} +(-756642. + 336879. i) q^{71} +(1.38970e6 + 295390. i) q^{72} +(88549.2 + 842490. i) q^{73} +(-110750. + 123000. i) q^{74} +(4.72108e6 + 5.24329e6i) q^{75} +(-337325. + 3.20944e6i) q^{76} +(-665139. + 2.04709e6i) q^{77} +(-3.21506e6 + 9.89494e6i) q^{78} +(410822. - 3.90871e6i) q^{79} +(-482644. - 536030. i) q^{80} +(1.56378e7 - 1.73675e7i) q^{81} +(-767086. - 7.29833e6i) q^{82} +(-2.12666e6 - 452036. i) q^{83} +(5.92955e6 - 2.64000e6i) q^{84} +(-450064. - 326990. i) q^{85} +(-1.26160e7 - 5.61699e6i) q^{86} +(5.85245e6 + 1.01367e7i) q^{87} +(476784. - 825815. i) q^{88} +(-7.87918e6 + 5.72456e6i) q^{89} +(5.19708e6 - 1.10467e6i) q^{90} +(1.03470e6 + 3.18447e6i) q^{91} -8.66065e6 q^{92} +(-1.17609e7 + 1.00491e7i) q^{93} +1.45473e7 q^{94} +(350959. + 1.08014e6i) q^{95} +(2.42626e7 - 5.15717e6i) q^{96} +(3.03864e6 - 2.20770e6i) q^{97} +(-4.76668e6 + 8.25614e6i) q^{98} +(-1.42265e7 - 2.46409e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 144 q - 6 q^{2} - 8 q^{3} - 2362 q^{4} - 75 q^{5} - 1637 q^{6} + 5518 q^{7} + 3401 q^{8} + 14542 q^{9} - 5312 q^{10} + 1361 q^{11} - 18281 q^{12} + 22457 q^{13} + 28470 q^{14} - 86918 q^{15} - 155730 q^{16}+ \cdots - 102755966 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/31\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{14}{15}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.07105 − 15.6071i −0.448222 − 1.37949i −0.878911 − 0.476985i \(-0.841730\pi\)
0.430689 − 0.902500i \(-0.358270\pi\)
\(3\) 91.2249 − 19.3905i 1.95069 − 0.414633i 0.959624 − 0.281287i \(-0.0907612\pi\)
0.991070 − 0.133346i \(-0.0425722\pi\)
\(4\) −114.312 + 83.0523i −0.893060 + 0.648846i
\(5\) −24.8635 + 43.0649i −0.0889545 + 0.154074i −0.907069 − 0.420981i \(-0.861686\pi\)
0.818115 + 0.575055i \(0.195019\pi\)
\(6\) −765.235 − 1325.43i −1.44632 − 2.50511i
\(7\) −449.966 − 200.338i −0.495834 − 0.220759i 0.143556 − 0.989642i \(-0.454146\pi\)
−0.639390 + 0.768883i \(0.720813\pi\)
\(8\) 176.534 + 128.260i 0.121903 + 0.0885676i
\(9\) 5948.07 − 2648.25i 2.71974 − 1.21091i
\(10\) 798.203 + 169.663i 0.252414 + 0.0536522i
\(11\) −456.789 − 4346.05i −0.103476 − 0.984512i −0.915890 − 0.401430i \(-0.868513\pi\)
0.812413 − 0.583082i \(-0.198153\pi\)
\(12\) −8817.65 + 9793.00i −1.47305 + 1.63599i
\(13\) −4548.76 − 5051.91i −0.574237 − 0.637755i 0.384135 − 0.923277i \(-0.374500\pi\)
−0.958372 + 0.285522i \(0.907833\pi\)
\(14\) −844.889 + 8038.58i −0.0822908 + 0.782945i
\(15\) −1433.13 + 4410.71i −0.109639 + 0.337434i
\(16\) −4482.34 + 13795.2i −0.273580 + 0.841994i
\(17\) −1169.39 + 11126.0i −0.0577280 + 0.549245i 0.926990 + 0.375087i \(0.122387\pi\)
−0.984718 + 0.174158i \(0.944280\pi\)
\(18\) −71494.5 − 79402.7i −2.88947 − 3.20909i
\(19\) 15282.4 − 16972.9i 0.511157 − 0.567698i −0.431221 − 0.902246i \(-0.641917\pi\)
0.942379 + 0.334548i \(0.108584\pi\)
\(20\) −734.447 − 6987.80i −0.0205284 − 0.195315i
\(21\) −44932.7 − 9550.74i −1.05875 − 0.225045i
\(22\) −65512.9 + 29168.2i −1.31174 + 0.584024i
\(23\) 49587.8 + 36027.7i 0.849821 + 0.617431i 0.925097 − 0.379732i \(-0.123984\pi\)
−0.0752755 + 0.997163i \(0.523984\pi\)
\(24\) 18591.3 + 8277.39i 0.274518 + 0.122223i
\(25\) 37826.1 + 65516.7i 0.484174 + 0.838614i
\(26\) −55778.6 + 96611.4i −0.622388 + 1.07801i
\(27\) 326249. − 237034.i 3.18989 − 2.31759i
\(28\) 68074.8 − 14469.8i 0.586049 − 0.124568i
\(29\) 38783.0 + 119362.i 0.295290 + 0.908809i 0.983124 + 0.182941i \(0.0585616\pi\)
−0.687834 + 0.725868i \(0.741438\pi\)
\(30\) 76105.8 0.514628
\(31\) −145752. + 79176.9i −0.878716 + 0.477345i
\(32\) 265964. 1.43482
\(33\) −125942. − 387611.i −0.610061 − 1.87758i
\(34\) 179574. − 38169.6i 0.783551 − 0.166549i
\(35\) 19815.3 − 14396.6i 0.0781199 − 0.0567574i
\(36\) −459991. + 796727.i −1.64320 + 2.84610i
\(37\) −5042.96 − 8734.67i −0.0163674 − 0.0283492i 0.857726 − 0.514108i \(-0.171877\pi\)
−0.874093 + 0.485758i \(0.838543\pi\)
\(38\) −342395. − 152444.i −1.01224 − 0.450680i
\(39\) −512919. − 372657.i −1.38459 − 1.00597i
\(40\) −9912.76 + 4413.44i −0.0244898 + 0.0109035i
\(41\) 437419. + 92976.3i 0.991184 + 0.210683i 0.674842 − 0.737962i \(-0.264212\pi\)
0.316342 + 0.948645i \(0.397545\pi\)
\(42\) 78796.8 + 749701.i 0.164110 + 1.56141i
\(43\) 563100. − 625386.i 1.08006 − 1.19952i 0.101223 − 0.994864i \(-0.467725\pi\)
0.978833 − 0.204660i \(-0.0656088\pi\)
\(44\) 413166. + 458868.i 0.731207 + 0.812088i
\(45\) −33843.4 + 321998.i −0.0553643 + 0.526756i
\(46\) 310825. − 956620.i 0.470829 − 1.44906i
\(47\) −273936. + 843089.i −0.384864 + 1.18449i 0.551715 + 0.834033i \(0.313974\pi\)
−0.936579 + 0.350457i \(0.886026\pi\)
\(48\) −141405. + 1.34538e6i −0.184553 + 1.75591i
\(49\) −388724. − 431722.i −0.472014 − 0.524225i
\(50\) 830708. − 922595.i 0.939839 − 1.04380i
\(51\) 109060. + 1.03764e6i 0.115125 + 1.09534i
\(52\) 939549. + 199707.i 0.926633 + 0.196962i
\(53\) 62931.8 − 28019.0i 0.0580637 − 0.0258516i −0.377499 − 0.926010i \(-0.623216\pi\)
0.435563 + 0.900158i \(0.356549\pi\)
\(54\) −5.35384e6 − 3.88979e6i −4.62687 − 3.36162i
\(55\) 198520. + 88386.8i 0.160892 + 0.0716338i
\(56\) −53739.1 − 93078.9i −0.0408914 − 0.0708260i
\(57\) 1.06503e6 − 1.84468e6i 0.761725 − 1.31935i
\(58\) 1.66622e6 − 1.21058e6i 1.12133 − 0.814696i
\(59\) −1.99135e6 + 423274.i −1.26231 + 0.268312i −0.789992 − 0.613118i \(-0.789915\pi\)
−0.472316 + 0.881429i \(0.656582\pi\)
\(60\) −202496. − 623220.i −0.121029 − 0.372488i
\(61\) 3.19548e6 1.80253 0.901264 − 0.433270i \(-0.142640\pi\)
0.901264 + 0.433270i \(0.142640\pi\)
\(62\) 1.97484e6 + 1.87325e6i 1.05235 + 0.998219i
\(63\) −3.20697e6 −1.61586
\(64\) −774980. − 2.38514e6i −0.369539 − 1.13732i
\(65\) 330658. − 70283.6i 0.149342 − 0.0317437i
\(66\) −5.41082e6 + 3.93119e6i −2.31665 + 1.68314i
\(67\) −1.05467e6 + 1.82674e6i −0.428404 + 0.742018i −0.996732 − 0.0807847i \(-0.974257\pi\)
0.568327 + 0.822803i \(0.307591\pi\)
\(68\) −790362. − 1.36895e6i −0.304821 − 0.527966i
\(69\) 5.22224e6 + 2.32509e6i 1.91375 + 0.852055i
\(70\) −325174. − 236253.i −0.113311 − 0.0823253i
\(71\) −756642. + 336879.i −0.250892 + 0.111704i −0.528330 − 0.849039i \(-0.677181\pi\)
0.277438 + 0.960744i \(0.410515\pi\)
\(72\) 1.38970e6 + 295390.i 0.438791 + 0.0932679i
\(73\) 88549.2 + 842490.i 0.0266413 + 0.253475i 0.999734 + 0.0230611i \(0.00734123\pi\)
−0.973093 + 0.230414i \(0.925992\pi\)
\(74\) −110750. + 123000.i −0.0317710 + 0.0352853i
\(75\) 4.72108e6 + 5.24329e6i 1.29219 + 1.43512i
\(76\) −337325. + 3.20944e6i −0.0881458 + 0.838651i
\(77\) −665139. + 2.04709e6i −0.166033 + 0.510998i
\(78\) −3.21506e6 + 9.89494e6i −0.767111 + 2.36092i
\(79\) 410822. − 3.90871e6i 0.0937473 − 0.891946i −0.842048 − 0.539403i \(-0.818650\pi\)
0.935795 − 0.352544i \(-0.114683\pi\)
\(80\) −482644. − 536030.i −0.105393 − 0.117051i
\(81\) 1.56378e7 − 1.73675e7i 3.26947 − 3.63111i
\(82\) −767086. − 7.29833e6i −0.153637 − 1.46176i
\(83\) −2.12666e6 − 452036.i −0.408249 − 0.0867760i −0.000790373 − 1.00000i \(-0.500252\pi\)
−0.407459 + 0.913224i \(0.633585\pi\)
\(84\) 5.92955e6 − 2.64000e6i 1.09155 − 0.485990i
\(85\) −450064. − 326990.i −0.0794891 − 0.0577522i
\(86\) −1.26160e7 − 5.61699e6i −2.13883 − 0.952269i
\(87\) 5.85245e6 + 1.01367e7i 0.952842 + 1.65037i
\(88\) 476784. − 825815.i 0.0745818 − 0.129179i
\(89\) −7.87918e6 + 5.72456e6i −1.18472 + 0.860750i −0.992696 − 0.120641i \(-0.961505\pi\)
−0.192024 + 0.981390i \(0.561505\pi\)
\(90\) 5.19708e6 − 1.10467e6i 0.751468 − 0.159729i
\(91\) 1.03470e6 + 3.18447e6i 0.143936 + 0.442989i
\(92\) −8.66065e6 −1.15956
\(93\) −1.17609e7 + 1.00491e7i −1.51618 + 1.29550i
\(94\) 1.45473e7 1.80649
\(95\) 350959. + 1.08014e6i 0.0419976 + 0.129255i
\(96\) 2.42626e7 − 5.15717e6i 2.79890 − 0.594925i
\(97\) 3.03864e6 − 2.20770e6i 0.338048 − 0.245606i −0.405790 − 0.913966i \(-0.633004\pi\)
0.743838 + 0.668360i \(0.233004\pi\)
\(98\) −4.76668e6 + 8.25614e6i −0.511593 + 0.886106i
\(99\) −1.42265e7 − 2.46409e7i −1.47358 − 2.55231i
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 31.8.g.a.7.3 ✓ 144
31.9 even 15 inner 31.8.g.a.9.3 yes 144
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
31.8.g.a.7.3 ✓ 144 1.1 even 1 trivial
31.8.g.a.9.3 yes 144 31.9 even 15 inner