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,-7504] 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: \( x^{8} - x^{7} + 101803 x^{6} + 6576400 x^{5} + 8539617914 x^{4} + 333205096780 x^{3} + \cdots + 33\!\cdots\!81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{3}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 37.3
Root \(64.1153 + 111.051i\) of defining polynomial
Character \(\chi\) \(=\) 126.37
Dual form 126.12.g.e.109.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(16.0000 - 27.7128i) q^{2} +(-512.000 - 886.810i) q^{4} +(3211.13 - 5561.84i) q^{5} +(-9575.58 - 43423.9i) q^{7} -32768.0 q^{8} +(-102756. - 177979. i) q^{10} +(-459958. - 796671. i) q^{11} +1.22771e6 q^{13} +(-1.35661e6 - 429416. i) q^{14} +(-524288. + 908093. i) q^{16} +(-3.31206e6 - 5.73665e6i) q^{17} +(9.33141e6 - 1.61625e7i) q^{19} -6.57640e6 q^{20} -2.94373e7 q^{22} +(-1.43390e7 + 2.48358e7i) q^{23} +(3.79132e6 + 6.56676e6i) q^{25} +(1.96434e7 - 3.40233e7i) q^{26} +(-3.36060e7 + 3.07248e7i) q^{28} +7.73286e6 q^{29} +(-5.68702e7 - 9.85021e7i) q^{31} +(1.67772e7 + 2.90590e7i) q^{32} -2.11972e8 q^{34} +(-2.72265e8 - 8.61820e7i) q^{35} +(9.50304e7 - 1.64597e8i) q^{37} +(-2.98605e8 - 5.17199e8i) q^{38} +(-1.05222e8 + 1.82251e8i) q^{40} +1.74512e8 q^{41} +1.83161e9 q^{43} +(-4.70997e8 + 8.15791e8i) q^{44} +(4.58847e8 + 7.94747e8i) q^{46} +(5.75867e8 - 9.97431e8i) q^{47} +(-1.79394e9 + 8.31618e8i) q^{49} +2.42644e8 q^{50} +(-6.28588e8 - 1.08875e9i) q^{52} +(3.18521e8 + 5.51694e8i) q^{53} -5.90795e9 q^{55} +(3.13773e8 + 1.42291e9i) q^{56} +(1.23726e8 - 2.14299e8i) q^{58} +(-1.39516e9 - 2.41649e9i) q^{59} +(2.60037e9 - 4.50397e9i) q^{61} -3.63969e9 q^{62} +1.07374e9 q^{64} +(3.94234e9 - 6.82834e9i) q^{65} +(3.19152e9 + 5.52788e9i) q^{67} +(-3.39154e9 + 5.87433e9i) q^{68} +(-6.74459e9 + 6.16633e9i) q^{70} -2.41383e10 q^{71} +(8.18348e9 + 1.41742e10i) q^{73} +(-3.04097e9 - 5.26712e9i) q^{74} -1.91107e10 q^{76} +(-3.01902e10 + 2.76018e10i) q^{77} +(1.91435e9 - 3.31575e9i) q^{79} +(3.36712e9 + 5.83202e9i) q^{80} +(2.79219e9 - 4.83621e9i) q^{82} +2.84759e9 q^{83} -4.25418e10 q^{85} +(2.93057e10 - 5.07590e10i) q^{86} +(1.50719e10 + 2.61053e10i) q^{88} +(-1.35210e10 + 2.34190e10i) q^{89} +(-1.17561e10 - 5.33120e10i) q^{91} +2.93662e10 q^{92} +(-1.84277e10 - 3.19178e10i) q^{94} +(-5.99288e10 - 1.03800e11i) q^{95} -1.49558e11 q^{97} +(-5.65661e9 + 6.30211e10i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 128 q^{2} - 4096 q^{4} - 7504 q^{5} - 42224 q^{7} - 262144 q^{8} + 240128 q^{10} - 213026 q^{11} - 2609712 q^{13} - 1257536 q^{14} - 4194304 q^{16} - 8854244 q^{17} + 7232806 q^{19} + 15368192 q^{20}+ \cdots - 210290620288 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) 3211.13 − 5561.84i 0.459540 − 0.795946i −0.539397 − 0.842052i \(-0.681348\pi\)
0.998937 + 0.0461054i \(0.0146810\pi\)
\(6\) 0 0
\(7\) −9575.58 − 43423.9i −0.215341 − 0.976539i
\(8\) −32768.0 −0.353553
\(9\) 0 0
\(10\) −102756. − 177979.i −0.324944 − 0.562819i
\(11\) −459958. − 796671.i −0.861110 − 1.49149i −0.870859 − 0.491533i \(-0.836437\pi\)
0.00974918 − 0.999952i \(-0.496897\pi\)
\(12\) 0 0
\(13\) 1.22771e6 0.917081 0.458541 − 0.888673i \(-0.348372\pi\)
0.458541 + 0.888673i \(0.348372\pi\)
\(14\) −1.35661e6 − 429416.i −0.674140 − 0.213390i
\(15\) 0 0
\(16\) −524288. + 908093.i −0.125000 + 0.216506i
\(17\) −3.31206e6 − 5.73665e6i −0.565755 − 0.979917i −0.996979 − 0.0776717i \(-0.975251\pi\)
0.431224 − 0.902245i \(-0.358082\pi\)
\(18\) 0 0
\(19\) 9.33141e6 − 1.61625e7i 0.864574 − 1.49749i −0.00289490 − 0.999996i \(-0.500921\pi\)
0.867469 − 0.497491i \(-0.165745\pi\)
\(20\) −6.57640e6 −0.459540
\(21\) 0 0
\(22\) −2.94373e7 −1.21779
\(23\) −1.43390e7 + 2.48358e7i −0.464531 + 0.804592i −0.999180 − 0.0404825i \(-0.987111\pi\)
0.534649 + 0.845074i \(0.320444\pi\)
\(24\) 0 0
\(25\) 3.79132e6 + 6.56676e6i 0.0776462 + 0.134487i
\(26\) 1.96434e7 − 3.40233e7i 0.324237 − 0.561595i
\(27\) 0 0
\(28\) −3.36060e7 + 3.07248e7i −0.369019 + 0.337380i
\(29\) 7.73286e6 0.0700086 0.0350043 − 0.999387i \(-0.488856\pi\)
0.0350043 + 0.999387i \(0.488856\pi\)
\(30\) 0 0
\(31\) −5.68702e7 − 9.85021e7i −0.356776 − 0.617954i 0.630644 − 0.776072i \(-0.282791\pi\)
−0.987420 + 0.158118i \(0.949457\pi\)
\(32\) 1.67772e7 + 2.90590e7i 0.0883883 + 0.153093i
\(33\) 0 0
\(34\) −2.11972e8 −0.800099
\(35\) −2.72265e8 − 8.61820e7i −0.876230 − 0.277359i
\(36\) 0 0
\(37\) 9.50304e7 − 1.64597e8i 0.225296 − 0.390224i −0.731112 − 0.682257i \(-0.760999\pi\)
0.956408 + 0.292033i \(0.0943318\pi\)
\(38\) −2.98605e8 − 5.17199e8i −0.611346 − 1.05888i
\(39\) 0 0
\(40\) −1.05222e8 + 1.82251e8i −0.162472 + 0.281410i
\(41\) 1.74512e8 0.235241 0.117621 − 0.993059i \(-0.462473\pi\)
0.117621 + 0.993059i \(0.462473\pi\)
\(42\) 0 0
\(43\) 1.83161e9 1.90001 0.950005 − 0.312235i \(-0.101078\pi\)
0.950005 + 0.312235i \(0.101078\pi\)
\(44\) −4.70997e8 + 8.15791e8i −0.430555 + 0.745743i
\(45\) 0 0
\(46\) 4.58847e8 + 7.94747e8i 0.328473 + 0.568932i
\(47\) 5.75867e8 − 9.97431e8i 0.366255 − 0.634373i −0.622721 − 0.782444i \(-0.713973\pi\)
0.988977 + 0.148071i \(0.0473063\pi\)
\(48\) 0 0
\(49\) −1.79394e9 + 8.31618e8i −0.907257 + 0.420577i
\(50\) 2.42644e8 0.109808
\(51\) 0 0
\(52\) −6.28588e8 − 1.08875e9i −0.229270 − 0.397108i
\(53\) 3.18521e8 + 5.51694e8i 0.104621 + 0.181209i 0.913583 − 0.406651i \(-0.133304\pi\)
−0.808962 + 0.587861i \(0.799970\pi\)
\(54\) 0 0
\(55\) −5.90795e9 −1.58286
\(56\) 3.13773e8 + 1.42291e9i 0.0761344 + 0.345259i
\(57\) 0 0
\(58\) 1.23726e8 − 2.14299e8i 0.0247518 − 0.0428713i
\(59\) −1.39516e9 − 2.41649e9i −0.254061 − 0.440047i 0.710579 − 0.703618i \(-0.248433\pi\)
−0.964640 + 0.263571i \(0.915100\pi\)
\(60\) 0 0
\(61\) 2.60037e9 − 4.50397e9i 0.394204 − 0.682781i −0.598795 − 0.800902i \(-0.704354\pi\)
0.992999 + 0.118121i \(0.0376870\pi\)
\(62\) −3.63969e9 −0.504557
\(63\) 0 0
\(64\) 1.07374e9 0.125000
\(65\) 3.94234e9 − 6.82834e9i 0.421435 − 0.729948i
\(66\) 0 0
\(67\) 3.19152e9 + 5.52788e9i 0.288793 + 0.500204i 0.973522 − 0.228594i \(-0.0734128\pi\)
−0.684729 + 0.728798i \(0.740080\pi\)
\(68\) −3.39154e9 + 5.87433e9i −0.282878 + 0.489958i
\(69\) 0 0
\(70\) −6.74459e9 + 6.16633e9i −0.479641 + 0.438518i
\(71\) −2.41383e10 −1.58776 −0.793880 − 0.608074i \(-0.791942\pi\)
−0.793880 + 0.608074i \(0.791942\pi\)
\(72\) 0 0
\(73\) 8.18348e9 + 1.41742e10i 0.462021 + 0.800245i 0.999062 − 0.0433121i \(-0.0137910\pi\)
−0.537040 + 0.843557i \(0.680458\pi\)
\(74\) −3.04097e9 − 5.26712e9i −0.159308 − 0.275930i
\(75\) 0 0
\(76\) −1.91107e10 −0.864574
\(77\) −3.01902e10 + 2.76018e10i −1.27106 + 1.16208i
\(78\) 0 0
\(79\) 1.91435e9 − 3.31575e9i 0.0699958 − 0.121236i −0.828903 − 0.559392i \(-0.811035\pi\)
0.898899 + 0.438155i \(0.144368\pi\)
\(80\) 3.36712e9 + 5.83202e9i 0.114885 + 0.198987i
\(81\) 0 0
\(82\) 2.79219e9 − 4.83621e9i 0.0831704 − 0.144055i
\(83\) 2.84759e9 0.0793502 0.0396751 − 0.999213i \(-0.487368\pi\)
0.0396751 + 0.999213i \(0.487368\pi\)
\(84\) 0 0
\(85\) −4.25418e10 −1.03995
\(86\) 2.93057e10 − 5.07590e10i 0.671755 − 1.16351i
\(87\) 0 0
\(88\) 1.50719e10 + 2.61053e10i 0.304448 + 0.527320i
\(89\) −1.35210e10 + 2.34190e10i −0.256663 + 0.444553i −0.965346 − 0.260974i \(-0.915956\pi\)
0.708683 + 0.705527i \(0.249290\pi\)
\(90\) 0 0
\(91\) −1.17561e10 − 5.33120e10i −0.197485 − 0.895566i
\(92\) 2.93662e10 0.464531
\(93\) 0 0
\(94\) −1.84277e10 − 3.19178e10i −0.258982 − 0.448569i
\(95\) −5.99288e10 − 1.03800e11i −0.794613 − 1.37631i
\(96\) 0 0
\(97\) −1.49558e11 −1.76834 −0.884168 − 0.467169i \(-0.845274\pi\)
−0.884168 + 0.467169i \(0.845274\pi\)
\(98\) −5.65661e9 + 6.30211e10i −0.0632139 + 0.704276i
\(99\) 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.e.37.3 8
3.2 odd 2 14.12.c.a.9.2 ✓ 8
7.4 even 3 inner 126.12.g.e.109.3 8
12.11 even 2 112.12.i.a.65.3 8
21.2 odd 6 98.12.a.j.1.3 4
21.5 even 6 98.12.a.l.1.2 4
21.11 odd 6 14.12.c.a.11.2 yes 8
21.17 even 6 98.12.c.l.67.3 8
21.20 even 2 98.12.c.l.79.3 8
84.11 even 6 112.12.i.a.81.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.12.c.a.9.2 ✓ 8 3.2 odd 2
14.12.c.a.11.2 yes 8 21.11 odd 6
98.12.a.j.1.3 4 21.2 odd 6
98.12.a.l.1.2 4 21.5 even 6
98.12.c.l.67.3 8 21.17 even 6
98.12.c.l.79.3 8 21.20 even 2
112.12.i.a.65.3 8 12.11 even 2
112.12.i.a.81.3 8 84.11 even 6
126.12.g.e.37.3 8 1.1 even 1 trivial
126.12.g.e.109.3 8 7.4 even 3 inner