Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [126,12,Mod(37,126)] 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("126.37"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(126, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 126.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,128,0,-4096,11080] 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: \(96.8112407505\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2 x^{7} + 27351029 x^{6} + 70092626926 x^{5} + 716243548908965 x^{4} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{3}\cdot 7^{3} \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 109.2
Root \(-2059.74 - 3567.58i\) of defining polynomial
Character \(\chi\) \(=\) 126.109
Dual form 126.12.g.f.37.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(16.0000 + 27.7128i) q^{2} +(-512.000 + 886.810i) q^{4} +(-731.917 - 1267.72i) q^{5} +(-39593.4 + 20240.8i) q^{7} -32768.0 q^{8} +(23421.4 - 40567.0i) q^{10} +(434862. - 753202. i) q^{11} +524814. q^{13} +(-1.19442e6 - 773392. i) q^{14} +(-524288. - 908093. i) q^{16} +(-399550. + 692041. i) q^{17} +(6.83529e6 + 1.18391e7i) q^{19} +1.49897e6 q^{20} +2.78311e7 q^{22} +(1.93468e7 + 3.35096e7i) q^{23} +(2.33427e7 - 4.04307e7i) q^{25} +(8.39703e6 + 1.45441e7i) q^{26} +(2.32210e6 - 4.54751e7i) q^{28} -7.81690e7 q^{29} +(-1.00177e8 + 1.73513e8i) q^{31} +(1.67772e7 - 2.90590e7i) q^{32} -2.55712e7 q^{34} +(5.46387e7 + 3.53787e7i) q^{35} +(-3.59693e8 - 6.23007e8i) q^{37} +(-2.18729e8 + 3.78850e8i) q^{38} +(2.39835e7 + 4.15406e7i) q^{40} +8.89034e8 q^{41} -8.31058e8 q^{43} +(4.45298e8 + 7.71279e8i) q^{44} +(-6.19097e8 + 1.07231e9i) q^{46} +(7.27046e8 + 1.25928e9i) q^{47} +(1.15795e9 - 1.60280e9i) q^{49} +1.49393e9 q^{50} +(-2.68705e8 + 4.65411e8i) q^{52} +(-1.76609e9 + 3.05896e9i) q^{53} -1.27313e9 q^{55} +(1.29740e9 - 6.63250e8i) q^{56} +(-1.25070e9 - 2.16628e9i) q^{58} +(-3.16128e9 + 5.47551e9i) q^{59} +(-1.48706e9 - 2.57566e9i) q^{61} -6.41136e9 q^{62} +1.07374e9 q^{64} +(-3.84121e8 - 6.65317e8i) q^{65} +(1.65479e9 - 2.86617e9i) q^{67} +(-4.09139e8 - 7.08650e8i) q^{68} +(-1.06224e8 + 2.08025e9i) q^{70} -1.63541e10 q^{71} +(-1.20806e10 + 2.09241e10i) q^{73} +(1.15102e10 - 1.99362e10i) q^{74} -1.39987e10 q^{76} +(-1.97225e9 + 3.86238e10i) q^{77} +(2.70516e8 + 4.68547e8i) q^{79} +(-7.67471e8 + 1.32930e9i) q^{80} +(1.42245e10 + 2.46376e10i) q^{82} +4.32612e8 q^{83} +1.16975e9 q^{85} +(-1.32969e10 - 2.30310e10i) q^{86} +(-1.42495e10 + 2.46809e10i) q^{88} +(-2.66015e10 - 4.60751e10i) q^{89} +(-2.07792e10 + 1.06226e10i) q^{91} -3.96222e10 q^{92} +(-2.32655e10 + 4.02970e10i) q^{94} +(1.00057e10 - 1.73304e10i) q^{95} +1.16010e11 q^{97} +(6.29453e10 + 6.44518e9i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 128 q^{2} - 4096 q^{4} + 11080 q^{5} + 31758 q^{7} - 262144 q^{8} - 354560 q^{10} + 601262 q^{11} + 1909820 q^{13} + 280512 q^{14} - 4194304 q^{16} + 5118224 q^{17} - 8583446 q^{19} - 22691840 q^{20}+ \cdots - 12286821184 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16.0000 + 27.7128i 0.353553 + 0.612372i
\(3\) 0 0
\(4\) −512.000 + 886.810i −0.250000 + 0.433013i
\(5\) −731.917 − 1267.72i −0.104743 − 0.181421i 0.808890 − 0.587960i \(-0.200069\pi\)
−0.913633 + 0.406539i \(0.866735\pi\)
\(6\) 0 0
\(7\) −39593.4 + 20240.8i −0.890397 + 0.455185i
\(8\) −32768.0 −0.353553
\(9\) 0 0
\(10\) 23421.4 − 40567.0i 0.0740648 − 0.128284i
\(11\) 434862. − 753202.i 0.814125 − 1.41011i −0.0958287 − 0.995398i \(-0.530550\pi\)
0.909954 − 0.414709i \(-0.136117\pi\)
\(12\) 0 0
\(13\) 524814. 0.392028 0.196014 − 0.980601i \(-0.437200\pi\)
0.196014 + 0.980601i \(0.437200\pi\)
\(14\) −1.19442e6 − 773392.i −0.593546 − 0.384322i
\(15\) 0 0
\(16\) −524288. − 908093.i −0.125000 − 0.216506i
\(17\) −399550. + 692041.i −0.0682499 + 0.118212i −0.898131 − 0.439728i \(-0.855075\pi\)
0.829881 + 0.557940i \(0.188408\pi\)
\(18\) 0 0
\(19\) 6.83529e6 + 1.18391e7i 0.633304 + 1.09691i 0.986872 + 0.161506i \(0.0516351\pi\)
−0.353568 + 0.935409i \(0.615032\pi\)
\(20\) 1.49897e6 0.104743
\(21\) 0 0
\(22\) 2.78311e7 1.15135
\(23\) 1.93468e7 + 3.35096e7i 0.626766 + 1.08559i 0.988196 + 0.153192i \(0.0489552\pi\)
−0.361430 + 0.932399i \(0.617711\pi\)
\(24\) 0 0
\(25\) 2.33427e7 − 4.04307e7i 0.478058 − 0.828020i
\(26\) 8.39703e6 + 1.45441e7i 0.138603 + 0.240067i
\(27\) 0 0
\(28\) 2.32210e6 − 4.54751e7i 0.0254983 − 0.499349i
\(29\) −7.81690e7 −0.707694 −0.353847 − 0.935303i \(-0.615127\pi\)
−0.353847 + 0.935303i \(0.615127\pi\)
\(30\) 0 0
\(31\) −1.00177e8 + 1.73513e8i −0.628465 + 1.08853i 0.359395 + 0.933185i \(0.382983\pi\)
−0.987860 + 0.155347i \(0.950350\pi\)
\(32\) 1.67772e7 − 2.90590e7i 0.0883883 − 0.153093i
\(33\) 0 0
\(34\) −2.55712e7 −0.0965199
\(35\) 5.46387e7 + 3.53787e7i 0.175843 + 0.113859i
\(36\) 0 0
\(37\) −3.59693e8 − 6.23007e8i −0.852752 − 1.47701i −0.878715 − 0.477347i \(-0.841598\pi\)
0.0259623 − 0.999663i \(-0.491735\pi\)
\(38\) −2.18729e8 + 3.78850e8i −0.447814 + 0.775636i
\(39\) 0 0
\(40\) 2.39835e7 + 4.15406e7i 0.0370324 + 0.0641420i
\(41\) 8.89034e8 1.19841 0.599207 − 0.800594i \(-0.295483\pi\)
0.599207 + 0.800594i \(0.295483\pi\)
\(42\) 0 0
\(43\) −8.31058e8 −0.862094 −0.431047 − 0.902329i \(-0.641856\pi\)
−0.431047 + 0.902329i \(0.641856\pi\)
\(44\) 4.45298e8 + 7.71279e8i 0.407063 + 0.705053i
\(45\) 0 0
\(46\) −6.19097e8 + 1.07231e9i −0.443191 + 0.767629i
\(47\) 7.27046e8 + 1.25928e9i 0.462406 + 0.800911i 0.999080 − 0.0428785i \(-0.0136528\pi\)
−0.536674 + 0.843790i \(0.680320\pi\)
\(48\) 0 0
\(49\) 1.15795e9 − 1.60280e9i 0.585613 − 0.810591i
\(50\) 1.49393e9 0.676076
\(51\) 0 0
\(52\) −2.68705e8 + 4.65411e8i −0.0980070 + 0.169753i
\(53\) −1.76609e9 + 3.05896e9i −0.580090 + 1.00475i 0.415378 + 0.909649i \(0.363649\pi\)
−0.995468 + 0.0950969i \(0.969684\pi\)
\(54\) 0 0
\(55\) −1.27313e9 −0.341097
\(56\) 1.29740e9 − 6.63250e8i 0.314803 − 0.160932i
\(57\) 0 0
\(58\) −1.25070e9 − 2.16628e9i −0.250208 − 0.433372i
\(59\) −3.16128e9 + 5.47551e9i −0.575675 + 0.997099i 0.420293 + 0.907389i \(0.361927\pi\)
−0.995968 + 0.0897102i \(0.971406\pi\)
\(60\) 0 0
\(61\) −1.48706e9 − 2.57566e9i −0.225431 − 0.390458i 0.731018 − 0.682359i \(-0.239046\pi\)
−0.956449 + 0.291901i \(0.905712\pi\)
\(62\) −6.41136e9 −0.888783
\(63\) 0 0
\(64\) 1.07374e9 0.125000
\(65\) −3.84121e8 − 6.65317e8i −0.0410624 − 0.0711221i
\(66\) 0 0
\(67\) 1.65479e9 − 2.86617e9i 0.149737 − 0.259353i −0.781393 − 0.624039i \(-0.785490\pi\)
0.931130 + 0.364686i \(0.118824\pi\)
\(68\) −4.09139e8 − 7.08650e8i −0.0341249 − 0.0591061i
\(69\) 0 0
\(70\) −1.06224e8 + 2.08025e9i −0.00755412 + 0.147937i
\(71\) −1.63541e10 −1.07574 −0.537869 − 0.843028i \(-0.680771\pi\)
−0.537869 + 0.843028i \(0.680771\pi\)
\(72\) 0 0
\(73\) −1.20806e10 + 2.09241e10i −0.682042 + 1.18133i 0.292314 + 0.956322i \(0.405575\pi\)
−0.974357 + 0.225010i \(0.927759\pi\)
\(74\) 1.15102e10 − 1.99362e10i 0.602987 − 1.04440i
\(75\) 0 0
\(76\) −1.39987e10 −0.633304
\(77\) −1.97225e9 + 3.86238e10i −0.0830354 + 1.62613i
\(78\) 0 0
\(79\) 2.70516e8 + 4.68547e8i 0.00989107 + 0.0171318i 0.870929 − 0.491410i \(-0.163518\pi\)
−0.861038 + 0.508541i \(0.830185\pi\)
\(80\) −7.67471e8 + 1.32930e9i −0.0261859 + 0.0453553i
\(81\) 0 0
\(82\) 1.42245e10 + 2.46376e10i 0.423703 + 0.733876i
\(83\) 4.32612e8 0.0120551 0.00602753 − 0.999982i \(-0.498081\pi\)
0.00602753 + 0.999982i \(0.498081\pi\)
\(84\) 0 0
\(85\) 1.16975e9 0.0285949
\(86\) −1.32969e10 − 2.30310e10i −0.304796 − 0.527923i
\(87\) 0 0
\(88\) −1.42495e10 + 2.46809e10i −0.287837 + 0.498548i
\(89\) −2.66015e10 − 4.60751e10i −0.504964 − 0.874624i −0.999984 − 0.00574172i \(-0.998172\pi\)
0.495019 − 0.868882i \(-0.335161\pi\)
\(90\) 0 0
\(91\) −2.07792e10 + 1.06226e10i −0.349061 + 0.178445i
\(92\) −3.96222e10 −0.626766
\(93\) 0 0
\(94\) −2.32655e10 + 4.02970e10i −0.326971 + 0.566330i
\(95\) 1.00057e10 − 1.73304e10i 0.132669 − 0.229789i
\(96\) 0 0
\(97\) 1.16010e11 1.37167 0.685837 − 0.727755i \(-0.259436\pi\)
0.685837 + 0.727755i \(0.259436\pi\)
\(98\) 6.29453e10 + 6.44518e9i 0.703429 + 0.0720264i
\(99\) 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 126.12.g.f.109.2 8
3.2 odd 2 42.12.e.c.25.3 ✓ 8
7.2 even 3 inner 126.12.g.f.37.2 8
21.2 odd 6 42.12.e.c.37.3 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.12.e.c.25.3 ✓ 8 3.2 odd 2
42.12.e.c.37.3 yes 8 21.2 odd 6
126.12.g.f.37.2 8 7.2 even 3 inner
126.12.g.f.109.2 8 1.1 even 1 trivial