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 109.3
Root \(64.1153 - 111.051i\) of defining polynomial
Character \(\chi\) \(=\) 126.109
Dual form 126.12.g.e.37.3

$q$-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 \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).

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

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
\(5\) 3211.13 + 5561.84i 0.459540 + 0.795946i 0.998937 − 0.0461054i \(-0.0146810\pi\)
−0.539397 + 0.842052i \(0.681348\pi\)
\(6\) 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 0.00974918 + 0.999952i \(0.496897\pi\)
−0.870859 + 0.491533i \(0.836437\pi\)
\(12\) 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 0.431224 + 0.902245i \(0.358082\pi\)
−0.996979 + 0.0776717i \(0.975251\pi\)
\(18\) 0 0
\(19\) 9.33141e6 + 1.61625e7i 0.864574 + 1.49749i 0.867469 + 0.497491i \(0.165745\pi\)
−0.00289490 + 0.999996i \(0.500921\pi\)
\(20\) −6.57640e6 −0.459540
\(21\) 0 0
\(22\) −2.94373e7 −1.21779
\(23\) −1.43390e7 − 2.48358e7i −0.464531 − 0.804592i 0.534649 − 0.845074i \(-0.320444\pi\)
−0.999180 + 0.0404825i \(0.987111\pi\)
\(24\) 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 −0.987420 − 0.158118i \(-0.949457\pi\)
0.630644 + 0.776072i \(0.282791\pi\)
\(32\) 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 0.956408 − 0.292033i \(-0.0943318\pi\)
−0.731112 + 0.682257i \(0.760999\pi\)
\(38\) −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 0.988977 − 0.148071i \(-0.0473063\pi\)
−0.622721 + 0.782444i \(0.713973\pi\)
\(48\) 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 −0.808962 − 0.587861i \(-0.799970\pi\)
0.913583 + 0.406651i \(0.133304\pi\)
\(54\) 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 −0.964640 − 0.263571i \(-0.915100\pi\)
0.710579 + 0.703618i \(0.248433\pi\)
\(60\) 0 0
\(61\) 2.60037e9 + 4.50397e9i 0.394204 + 0.682781i 0.992999 − 0.118121i \(-0.0376870\pi\)
−0.598795 + 0.800902i \(0.704354\pi\)
\(62\) −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 −0.684729 − 0.728798i \(-0.740080\pi\)
0.973522 + 0.228594i \(0.0734128\pi\)
\(68\) −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 −0.537040 − 0.843557i \(-0.680458\pi\)
0.999062 + 0.0433121i \(0.0137910\pi\)
\(74\) −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 0.898899 − 0.438155i \(-0.144368\pi\)
−0.828903 + 0.559392i \(0.811035\pi\)
\(80\) 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 0.708683 − 0.705527i \(-0.249290\pi\)
−0.965346 + 0.260974i \(0.915956\pi\)
\(90\) 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.109.3 8
3.2 odd 2 14.12.c.a.11.2 yes 8
7.2 even 3 inner 126.12.g.e.37.3 8
12.11 even 2 112.12.i.a.81.3 8
21.2 odd 6 14.12.c.a.9.2 ✓ 8
21.5 even 6 98.12.c.l.79.3 8
21.11 odd 6 98.12.a.j.1.3 4
21.17 even 6 98.12.a.l.1.2 4
21.20 even 2 98.12.c.l.67.3 8
84.23 even 6 112.12.i.a.65.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.12.c.a.9.2 ✓ 8 21.2 odd 6
14.12.c.a.11.2 yes 8 3.2 odd 2
98.12.a.j.1.3 4 21.11 odd 6
98.12.a.l.1.2 4 21.17 even 6
98.12.c.l.67.3 8 21.20 even 2
98.12.c.l.79.3 8 21.5 even 6
112.12.i.a.65.3 8 84.23 even 6
112.12.i.a.81.3 8 12.11 even 2
126.12.g.e.37.3 8 7.2 even 3 inner
126.12.g.e.109.3 8 1.1 even 1 trivial