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Properties

Label 31.10.c.a.5.13
Level $31$
Weight $10$
Character 31.5
Analytic conductor $15.966$
Analytic rank $0$
Dimension $46$
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] = [31,10,Mod(5,31)] 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("31.5"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(31, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 31.c (of order \(3\), degree \(2\), 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: \(15.9661109211\)
Analytic rank: \(0\)
Dimension: \(46\)
Relative dimension: \(23\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 5.13
Character \(\chi\) \(=\) 31.5
Dual form 31.10.c.a.25.13

$q$-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+4.02855 q^{2} +(-112.657 + 195.128i) q^{3} -495.771 q^{4} +(-618.220 - 1070.79i) q^{5} +(-453.846 + 786.084i) q^{6} +(-3386.19 + 5865.05i) q^{7} -4059.85 q^{8} +(-15541.9 - 26919.4i) q^{9} +(-2490.53 - 4313.72i) q^{10} +(-13413.8 - 23233.3i) q^{11} +(55852.3 - 96739.0i) q^{12} +(76141.8 + 131881. i) q^{13} +(-13641.4 + 23627.6i) q^{14} +278589. q^{15} +237479. q^{16} +(-143187. + 248007. i) q^{17} +(-62611.2 - 108446. i) q^{18} +(387797. - 671684. i) q^{19} +(306496. + 530866. i) q^{20} +(-762958. - 1.32148e6i) q^{21} +(-54038.0 - 93596.5i) q^{22} +317995. q^{23} +(457372. - 792192. i) q^{24} +(212169. - 367488. i) q^{25} +(306741. + 531290. i) q^{26} +2.56877e6 q^{27} +(1.67877e6 - 2.90772e6i) q^{28} -2.34123e6 q^{29} +1.12231e6 q^{30} +(-4.97138e6 - 1.31338e6i) q^{31} +3.03534e6 q^{32} +6.04464e6 q^{33} +(-576834. + 999107. i) q^{34} +8.37364e6 q^{35} +(7.70522e6 + 1.33458e7i) q^{36} +(2.71867e6 - 4.70888e6i) q^{37} +(1.56226e6 - 2.70591e6i) q^{38} -3.43117e7 q^{39} +(2.50988e6 + 4.34724e6i) q^{40} +(-3.79864e6 - 6.57944e6i) q^{41} +(-3.07361e6 - 5.32365e6i) q^{42} +(1.12825e7 - 1.95418e7i) q^{43} +(6.65015e6 + 1.15184e7i) q^{44} +(-1.92166e7 + 3.32842e7i) q^{45} +1.28106e6 q^{46} -4.92633e7 q^{47} +(-2.67538e7 + 4.63390e7i) q^{48} +(-2.75572e6 - 4.77304e6i) q^{49} +(854734. - 1.48044e6i) q^{50} +(-3.22621e7 - 5.58796e7i) q^{51} +(-3.77489e7 - 6.53830e7i) q^{52} +(5.50712e7 + 9.53862e7i) q^{53} +1.03484e7 q^{54} +(-1.65853e7 + 2.87266e7i) q^{55} +(1.37474e7 - 2.38112e7i) q^{56} +(8.73764e7 + 1.51340e8i) q^{57} -9.43176e6 q^{58} +(1.68650e7 - 2.92110e7i) q^{59} -1.38116e8 q^{60} +1.02896e8 q^{61} +(-2.00274e7 - 5.29103e6i) q^{62} +2.10511e8 q^{63} -1.09361e8 q^{64} +(9.41448e7 - 1.63064e8i) q^{65} +2.43511e7 q^{66} +(-4.96203e7 - 8.59450e7i) q^{67} +(7.09878e7 - 1.22955e8i) q^{68} +(-3.58245e7 + 6.20498e7i) q^{69} +3.37336e7 q^{70} +(-1.66564e8 - 2.88498e8i) q^{71} +(6.30978e7 + 1.09289e8i) q^{72} +(-7.81676e7 - 1.35390e8i) q^{73} +(1.09523e7 - 1.89699e7i) q^{74} +(4.78049e7 + 8.28005e7i) q^{75} +(-1.92258e8 + 3.33001e8i) q^{76} +1.81686e8 q^{77} -1.38226e8 q^{78} +(2.89840e8 - 5.02018e8i) q^{79} +(-1.46815e8 - 2.54290e8i) q^{80} +(1.65202e7 - 2.86139e7i) q^{81} +(-1.53030e7 - 2.65056e7i) q^{82} +(3.77391e8 + 6.53661e8i) q^{83} +(3.78252e8 + 6.55152e8i) q^{84} +3.54084e8 q^{85} +(4.54520e7 - 7.87252e7i) q^{86} +(2.63757e8 - 4.56841e8i) q^{87} +(5.44579e7 + 9.43238e7i) q^{88} +1.57289e8 q^{89} +(-7.74151e7 + 1.34087e8i) q^{90} -1.03132e9 q^{91} -1.57652e8 q^{92} +(8.16342e8 - 8.22096e8i) q^{93} -1.98460e8 q^{94} -9.58976e8 q^{95} +(-3.41954e8 + 5.92281e8i) q^{96} +2.53580e8 q^{97} +(-1.11015e7 - 1.92284e7i) q^{98} +(-4.16951e8 + 7.22180e8i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46 q - 4 q^{2} + 13 q^{3} + 10576 q^{4} - 797 q^{5} + 96 q^{6} + 6359 q^{7} + 7740 q^{8} - 157910 q^{9} + 53094 q^{10} + 55493 q^{11} + 53834 q^{12} + 145047 q^{13} + 318552 q^{14} - 384066 q^{15} + 1197768 q^{16}+ \cdots + 644680976 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.02855 0.178038 0.0890191 − 0.996030i \(-0.471627\pi\)
0.0890191 + 0.996030i \(0.471627\pi\)
\(3\) −112.657 + 195.128i −0.802998 + 1.39083i 0.114637 + 0.993407i \(0.463430\pi\)
−0.917635 + 0.397425i \(0.869904\pi\)
\(4\) −495.771 −0.968302
\(5\) −618.220 − 1070.79i −0.442363 − 0.766194i 0.555502 − 0.831515i \(-0.312526\pi\)
−0.997864 + 0.0653209i \(0.979193\pi\)
\(6\) −453.846 + 786.084i −0.142964 + 0.247621i
\(7\) −3386.19 + 5865.05i −0.533052 + 0.923273i 0.466203 + 0.884678i \(0.345622\pi\)
−0.999255 + 0.0385955i \(0.987712\pi\)
\(8\) −4059.85 −0.350433
\(9\) −15541.9 − 26919.4i −0.789610 − 1.36765i
\(10\) −2490.53 − 4313.72i −0.0787575 − 0.136412i
\(11\) −13413.8 − 23233.3i −0.276238 − 0.478459i 0.694209 − 0.719774i \(-0.255755\pi\)
−0.970447 + 0.241315i \(0.922421\pi\)
\(12\) 55852.3 − 96739.0i 0.777544 − 1.34675i
\(13\) 76141.8 + 131881.i 0.739397 + 1.28067i 0.952767 + 0.303702i \(0.0982228\pi\)
−0.213370 + 0.976972i \(0.568444\pi\)
\(14\) −13641.4 + 23627.6i −0.0949037 + 0.164378i
\(15\) 278589. 1.42086
\(16\) 237479. 0.905912
\(17\) −143187. + 248007.i −0.415798 + 0.720184i −0.995512 − 0.0946364i \(-0.969831\pi\)
0.579714 + 0.814820i \(0.303164\pi\)
\(18\) −62611.2 − 108446.i −0.140581 − 0.243493i
\(19\) 387797. − 671684.i 0.682673 − 1.18242i −0.291489 − 0.956574i \(-0.594151\pi\)
0.974162 − 0.225851i \(-0.0725161\pi\)
\(20\) 306496. + 530866.i 0.428341 + 0.741908i
\(21\) −762958. − 1.32148e6i −0.856079 − 1.48277i
\(22\) −54038.0 − 93596.5i −0.0491810 − 0.0851839i
\(23\) 317995. 0.236943 0.118472 − 0.992957i \(-0.462201\pi\)
0.118472 + 0.992957i \(0.462201\pi\)
\(24\) 457372. − 792192.i 0.281397 − 0.487394i
\(25\) 212169. − 367488.i 0.108631 − 0.188154i
\(26\) 306741. + 531290.i 0.131641 + 0.228009i
\(27\) 2.56877e6 0.930225
\(28\) 1.67877e6 − 2.90772e6i 0.516156 − 0.894008i
\(29\) −2.34123e6 −0.614686 −0.307343 − 0.951599i \(-0.599440\pi\)
−0.307343 + 0.951599i \(0.599440\pi\)
\(30\) 1.12231e6 0.252968
\(31\) −4.97138e6 − 1.31338e6i −0.966829 − 0.255425i
\(32\) 3.03534e6 0.511720
\(33\) 6.04464e6 0.887274
\(34\) −576834. + 999107.i −0.0740280 + 0.128220i
\(35\) 8.37364e6 0.943209
\(36\) 7.70522e6 + 1.33458e7i 0.764581 + 1.32429i
\(37\) 2.71867e6 − 4.70888e6i 0.238478 − 0.413057i −0.721799 − 0.692102i \(-0.756685\pi\)
0.960278 + 0.279045i \(0.0900180\pi\)
\(38\) 1.56226e6 − 2.70591e6i 0.121542 − 0.210517i
\(39\) −3.43117e7 −2.37494
\(40\) 2.50988e6 + 4.34724e6i 0.155018 + 0.268500i
\(41\) −3.79864e6 − 6.57944e6i −0.209943 − 0.363632i 0.741753 − 0.670673i \(-0.233994\pi\)
−0.951696 + 0.307041i \(0.900661\pi\)
\(42\) −3.07361e6 − 5.32365e6i −0.152415 − 0.263990i
\(43\) 1.12825e7 − 1.95418e7i 0.503265 − 0.871680i −0.496728 − 0.867906i \(-0.665465\pi\)
0.999993 − 0.00377400i \(-0.00120131\pi\)
\(44\) 6.65015e6 + 1.15184e7i 0.267482 + 0.463293i
\(45\) −1.92166e7 + 3.32842e7i −0.698588 + 1.20999i
\(46\) 1.28106e6 0.0421850
\(47\) −4.92633e7 −1.47260 −0.736298 − 0.676658i \(-0.763428\pi\)
−0.736298 + 0.676658i \(0.763428\pi\)
\(48\) −2.67538e7 + 4.63390e7i −0.727445 + 1.25997i
\(49\) −2.75572e6 − 4.77304e6i −0.0682893 − 0.118280i
\(50\) 854734. − 1.48044e6i 0.0193404 − 0.0334986i
\(51\) −3.22621e7 − 5.58796e7i −0.667770 − 1.15661i
\(52\) −3.77489e7 − 6.53830e7i −0.715960 − 1.24008i
\(53\) 5.50712e7 + 9.53862e7i 0.958702 + 1.66052i 0.725660 + 0.688053i \(0.241534\pi\)
0.233041 + 0.972467i \(0.425132\pi\)
\(54\) 1.03484e7 0.165616
\(55\) −1.65853e7 + 2.87266e7i −0.244395 + 0.423304i
\(56\) 1.37474e7 − 2.38112e7i 0.186799 − 0.323546i
\(57\) 8.73764e7 + 1.51340e8i 1.09637 + 1.89897i
\(58\) −9.43176e6 −0.109438
\(59\) 1.68650e7 − 2.92110e7i 0.181197 − 0.313843i −0.761091 − 0.648645i \(-0.775336\pi\)
0.942289 + 0.334802i \(0.108669\pi\)
\(60\) −1.38116e8 −1.37583
\(61\) 1.02896e8 0.951511 0.475755 − 0.879578i \(-0.342175\pi\)
0.475755 + 0.879578i \(0.342175\pi\)
\(62\) −2.00274e7 − 5.29103e6i −0.172132 − 0.0454755i
\(63\) 2.10511e8 1.68361
\(64\) −1.09361e8 −0.814806
\(65\) 9.41448e7 − 1.63064e8i 0.654163 − 1.13304i
\(66\) 2.43511e7 0.157969
\(67\) −4.96203e7 − 8.59450e7i −0.300831 − 0.521055i 0.675493 − 0.737366i \(-0.263931\pi\)
−0.976324 + 0.216311i \(0.930597\pi\)
\(68\) 7.09878e7 − 1.22955e8i 0.402619 − 0.697356i
\(69\) −3.58245e7 + 6.20498e7i −0.190265 + 0.329549i
\(70\) 3.37336e7 0.167927
\(71\) −1.66564e8 − 2.88498e8i −0.777893 − 1.34735i −0.933154 − 0.359476i \(-0.882955\pi\)
0.155262 − 0.987873i \(-0.450378\pi\)
\(72\) 6.30978e7 + 1.09289e8i 0.276706 + 0.479268i
\(73\) −7.81676e7 − 1.35390e8i −0.322162 − 0.558000i 0.658772 − 0.752342i \(-0.271076\pi\)
−0.980934 + 0.194342i \(0.937743\pi\)
\(74\) 1.09523e7 − 1.89699e7i 0.0424583 − 0.0735399i
\(75\) 4.78049e7 + 8.28005e7i 0.174460 + 0.302174i
\(76\) −1.92258e8 + 3.33001e8i −0.661034 + 1.14494i
\(77\) 1.81686e8 0.588997
\(78\) −1.38226e8 −0.422830
\(79\) 2.89840e8 − 5.02018e8i 0.837214 − 1.45010i −0.0550011 − 0.998486i \(-0.517516\pi\)
0.892215 − 0.451611i \(-0.149150\pi\)
\(80\) −1.46815e8 − 2.54290e8i −0.400742 − 0.694105i
\(81\) 1.65202e7 − 2.86139e7i 0.0426416 − 0.0738574i
\(82\) −1.53030e7 − 2.65056e7i −0.0373778 − 0.0647403i
\(83\) 3.77391e8 + 6.53661e8i 0.872852 + 1.51182i 0.859034 + 0.511919i \(0.171065\pi\)
0.0138178 + 0.999905i \(0.495602\pi\)
\(84\) 3.78252e8 + 6.55152e8i 0.828944 + 1.43577i
\(85\) 3.54084e8 0.735735
\(86\) 4.54520e7 − 7.87252e7i 0.0896004 − 0.155192i
\(87\) 2.63757e8 − 4.56841e8i 0.493592 − 0.854926i
\(88\) 5.44579e7 + 9.43238e7i 0.0968030 + 0.167668i
\(89\) 1.57289e8 0.265732 0.132866 − 0.991134i \(-0.457582\pi\)
0.132866 + 0.991134i \(0.457582\pi\)
\(90\) −7.74151e7 + 1.34087e8i −0.124375 + 0.215424i
\(91\) −1.03132e9 −1.57655
\(92\) −1.57652e8 −0.229433
\(93\) 8.16342e8 − 8.22096e8i 1.13162 − 1.13959i
\(94\) −1.98460e8 −0.262178
\(95\) −9.58976e8 −1.20796
\(96\) −3.41954e8 + 5.92281e8i −0.410910 + 0.711717i
\(97\) 2.53580e8 0.290832 0.145416 − 0.989371i \(-0.453548\pi\)
0.145416 + 0.989371i \(0.453548\pi\)
\(98\) −1.11015e7 − 1.92284e7i −0.0121581 − 0.0210584i
\(99\) −4.16951e8 + 7.22180e8i −0.436241 + 0.755591i
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 31.10.c.a.5.13 ✓ 46
31.25 even 3 inner 31.10.c.a.25.13 yes 46
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
31.10.c.a.5.13 ✓ 46 1.1 even 1 trivial
31.10.c.a.25.13 yes 46 31.25 even 3 inner