Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [14,12,Mod(9,14)] 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("14.9"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(14, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 14.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-128] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.7568045278\)
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: \( 2^{12}\cdot 3\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 11.2
Root \(64.1153 + 111.051i\) of defining polynomial
Character \(\chi\) \(=\) 14.11
Dual form 14.12.c.a.9.2

$q$-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} +(-94.7305 + 164.078i) q^{3} +(-512.000 + 886.810i) q^{4} +(-3211.13 - 5561.84i) q^{5} +6062.75 q^{6} +(-9575.58 + 43423.9i) q^{7} +32768.0 q^{8} +(70625.8 + 122327. i) q^{9} +(-102756. + 177979. i) q^{10} +(459958. - 796671. i) q^{11} +(-97004.1 - 168016. i) q^{12} +1.22771e6 q^{13} +(1.35661e6 - 429416. i) q^{14} +1.21677e6 q^{15} +(-524288. - 908093. i) q^{16} +(3.31206e6 - 5.73665e6i) q^{17} +(2.26002e6 - 3.91448e6i) q^{18} +(9.33141e6 + 1.61625e7i) q^{19} +6.57640e6 q^{20} +(-6.21781e6 - 5.68471e6i) q^{21} -2.94373e7 q^{22} +(1.43390e7 + 2.48358e7i) q^{23} +(-3.10413e6 + 5.37651e6i) q^{24} +(3.79132e6 - 6.56676e6i) q^{25} +(-1.96434e7 - 3.40233e7i) q^{26} -6.03241e7 q^{27} +(-3.36060e7 - 3.07248e7i) q^{28} -7.73286e6 q^{29} +(-1.94683e7 - 3.37201e7i) q^{30} +(-5.68702e7 + 9.85021e7i) q^{31} +(-1.67772e7 + 2.90590e7i) q^{32} +(8.71441e7 + 1.50938e8i) q^{33} -2.11972e8 q^{34} +(2.72265e8 - 8.61820e7i) q^{35} -1.44642e8 q^{36} +(9.50304e7 + 1.64597e8i) q^{37} +(2.98605e8 - 5.17199e8i) q^{38} +(-1.16302e8 + 2.01441e8i) q^{39} +(-1.05222e8 - 1.82251e8i) q^{40} -1.74512e8 q^{41} +(-5.80544e7 + 2.63268e8i) q^{42} +1.83161e9 q^{43} +(4.70997e8 + 8.15791e8i) q^{44} +(4.53577e8 - 7.85619e8i) q^{45} +(4.58847e8 - 7.94747e8i) q^{46} +(-5.75867e8 - 9.97431e8i) q^{47} +1.98664e8 q^{48} +(-1.79394e9 - 8.31618e8i) q^{49} -2.42644e8 q^{50} +(6.27506e8 + 1.08687e9i) q^{51} +(-6.28588e8 + 1.08875e9i) q^{52} +(-3.18521e8 + 5.51694e8i) q^{53} +(9.65186e8 + 1.67175e9i) q^{54} -5.90795e9 q^{55} +(-3.13773e8 + 1.42291e9i) q^{56} -3.53588e9 q^{57} +(1.23726e8 + 2.14299e8i) q^{58} +(1.39516e9 - 2.41649e9i) q^{59} +(-6.22986e8 + 1.07904e9i) q^{60} +(2.60037e9 + 4.50397e9i) q^{61} +3.63969e9 q^{62} +(-5.98821e9 + 1.89549e9i) q^{63} +1.07374e9 q^{64} +(-3.94234e9 - 6.82834e9i) q^{65} +(2.78861e9 - 4.83002e9i) q^{66} +(3.19152e9 - 5.52788e9i) q^{67} +(3.39154e9 + 5.87433e9i) q^{68} -5.43335e9 q^{69} +(-6.74459e9 - 6.16633e9i) q^{70} +2.41383e10 q^{71} +(2.31426e9 + 4.00842e9i) q^{72} +(8.18348e9 - 1.41742e10i) q^{73} +(3.04097e9 - 5.26712e9i) q^{74} +(7.18307e8 + 1.24414e9i) q^{75} -1.91107e10 q^{76} +(3.01902e10 + 2.76018e10i) q^{77} +7.44331e9 q^{78} +(1.91435e9 + 3.31575e9i) q^{79} +(-3.36712e9 + 5.83202e9i) q^{80} +(-6.79660e9 + 1.17721e10i) q^{81} +(2.79219e9 + 4.83621e9i) q^{82} -2.84759e9 q^{83} +(8.22478e9 - 2.60344e9i) q^{84} -4.25418e10 q^{85} +(-2.93057e10 - 5.07590e10i) q^{86} +(7.32538e8 - 1.26879e9i) q^{87} +(1.50719e10 - 2.61053e10i) q^{88} +(1.35210e10 + 2.34190e10i) q^{89} -2.90289e10 q^{90} +(-1.17561e10 + 5.33120e10i) q^{91} -2.93662e10 q^{92} +(-1.07747e10 - 1.86623e10i) q^{93} +(-1.84277e10 + 3.19178e10i) q^{94} +(5.99288e10 - 1.03800e11i) q^{95} +(-3.17863e9 - 5.50555e9i) q^{96} -1.49558e11 q^{97} +(5.65661e9 + 6.30211e10i) q^{98} +1.29940e11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 128 q^{2} + 266 q^{3} - 4096 q^{4} + 7504 q^{5} - 17024 q^{6} - 42224 q^{7} + 262144 q^{8} - 123520 q^{9} + 240128 q^{10} + 213026 q^{11} + 272384 q^{12} - 2609712 q^{13} + 1257536 q^{14} + 2275500 q^{15}+ \cdots + 393415805736 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/14\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −16.0000 − 27.7128i −0.353553 − 0.612372i
\(3\) −94.7305 + 164.078i −0.225073 + 0.389838i −0.956341 − 0.292252i \(-0.905595\pi\)
0.731268 + 0.682090i \(0.238929\pi\)
\(4\) −512.000 + 886.810i −0.250000 + 0.433013i
\(5\) −3211.13 − 5561.84i −0.459540 − 0.795946i 0.539397 − 0.842052i \(-0.318652\pi\)
−0.998937 + 0.0461054i \(0.985319\pi\)
\(6\) 6062.75 0.318301
\(7\) −9575.58 + 43423.9i −0.215341 + 0.976539i
\(8\) 32768.0 0.353553
\(9\) 70625.8 + 122327.i 0.398684 + 0.690542i
\(10\) −102756. + 177979.i −0.324944 + 0.562819i
\(11\) 459958. − 796671.i 0.861110 − 1.49149i −0.00974918 − 0.999952i \(-0.503103\pi\)
0.870859 − 0.491533i \(-0.163563\pi\)
\(12\) −97004.1 − 168016.i −0.112536 − 0.194919i
\(13\) 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
\(15\) 1.21677e6 0.413720
\(16\) −524288. − 908093.i −0.125000 − 0.216506i
\(17\) 3.31206e6 − 5.73665e6i 0.565755 − 0.979917i −0.431224 − 0.902245i \(-0.641918\pi\)
0.996979 − 0.0776717i \(-0.0247486\pi\)
\(18\) 2.26002e6 − 3.91448e6i 0.281912 − 0.488287i
\(19\) 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
\(21\) −6.21781e6 − 5.68471e6i −0.332224 − 0.303740i
\(22\) −2.94373e7 −1.21779
\(23\) 1.43390e7 + 2.48358e7i 0.464531 + 0.804592i 0.999180 − 0.0404825i \(-0.0128895\pi\)
−0.534649 + 0.845074i \(0.679556\pi\)
\(24\) −3.10413e6 + 5.37651e6i −0.0795753 + 0.137828i
\(25\) 3.79132e6 − 6.56676e6i 0.0776462 − 0.134487i
\(26\) −1.96434e7 − 3.40233e7i −0.324237 − 0.561595i
\(27\) −6.03241e7 −0.809078
\(28\) −3.36060e7 − 3.07248e7i −0.369019 − 0.337380i
\(29\) −7.73286e6 −0.0700086 −0.0350043 − 0.999387i \(-0.511144\pi\)
−0.0350043 + 0.999387i \(0.511144\pi\)
\(30\) −1.94683e7 − 3.37201e7i −0.146272 − 0.253351i
\(31\) −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
\(33\) 8.71441e7 + 1.50938e8i 0.387625 + 0.671386i
\(34\) −2.11972e8 −0.800099
\(35\) 2.72265e8 − 8.61820e7i 0.876230 − 0.277359i
\(36\) −1.44642e8 −0.398684
\(37\) 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
\(39\) −1.16302e8 + 2.01441e8i −0.206410 + 0.357513i
\(40\) −1.05222e8 − 1.82251e8i −0.162472 − 0.281410i
\(41\) −1.74512e8 −0.235241 −0.117621 − 0.993059i \(-0.537527\pi\)
−0.117621 + 0.993059i \(0.537527\pi\)
\(42\) −5.80544e7 + 2.63268e8i −0.0685432 + 0.310833i
\(43\) 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
\(45\) 4.53577e8 − 7.85619e8i 0.366423 − 0.634663i
\(46\) 4.58847e8 − 7.94747e8i 0.328473 − 0.568932i
\(47\) −5.75867e8 − 9.97431e8i −0.366255 − 0.634373i 0.622721 − 0.782444i \(-0.286027\pi\)
−0.988977 + 0.148071i \(0.952694\pi\)
\(48\) 1.98664e8 0.112536
\(49\) −1.79394e9 − 8.31618e8i −0.907257 − 0.420577i
\(50\) −2.42644e8 −0.109808
\(51\) 6.27506e8 + 1.08687e9i 0.254672 + 0.441105i
\(52\) −6.28588e8 + 1.08875e9i −0.229270 + 0.397108i
\(53\) −3.18521e8 + 5.51694e8i −0.104621 + 0.181209i −0.913583 − 0.406651i \(-0.866696\pi\)
0.808962 + 0.587861i \(0.200030\pi\)
\(54\) 9.65186e8 + 1.67175e9i 0.286052 + 0.495457i
\(55\) −5.90795e9 −1.58286
\(56\) −3.13773e8 + 1.42291e9i −0.0761344 + 0.345259i
\(57\) −3.53588e9 −0.778369
\(58\) 1.23726e8 + 2.14299e8i 0.0247518 + 0.0428713i
\(59\) 1.39516e9 − 2.41649e9i 0.254061 − 0.440047i −0.710579 − 0.703618i \(-0.751567\pi\)
0.964640 + 0.263571i \(0.0849002\pi\)
\(60\) −6.22986e8 + 1.07904e9i −0.103430 + 0.179146i
\(61\) 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
\(63\) −5.98821e9 + 1.89549e9i −0.760194 + 0.240629i
\(64\) 1.07374e9 0.125000
\(65\) −3.94234e9 − 6.82834e9i −0.421435 − 0.729948i
\(66\) 2.78861e9 − 4.83002e9i 0.274092 − 0.474741i
\(67\) 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
\(69\) −5.43335e9 −0.418214
\(70\) −6.74459e9 − 6.16633e9i −0.479641 − 0.438518i
\(71\) 2.41383e10 1.58776 0.793880 − 0.608074i \(-0.208058\pi\)
0.793880 + 0.608074i \(0.208058\pi\)
\(72\) 2.31426e9 + 4.00842e9i 0.140956 + 0.244143i
\(73\) 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
\(75\) 7.18307e8 + 1.24414e9i 0.0349521 + 0.0605388i
\(76\) −1.91107e10 −0.864574
\(77\) 3.01902e10 + 2.76018e10i 1.27106 + 1.16208i
\(78\) 7.44331e9 0.291908
\(79\) 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
\(81\) −6.79660e9 + 1.17721e10i −0.216583 + 0.375133i
\(82\) 2.79219e9 + 4.83621e9i 0.0831704 + 0.144055i
\(83\) −2.84759e9 −0.0793502 −0.0396751 − 0.999213i \(-0.512632\pi\)
−0.0396751 + 0.999213i \(0.512632\pi\)
\(84\) 8.22478e9 − 2.60344e9i 0.214579 − 0.0679223i
\(85\) −4.25418e10 −1.03995
\(86\) −2.93057e10 − 5.07590e10i −0.671755 − 1.16351i
\(87\) 7.32538e8 − 1.26879e9i 0.0157570 − 0.0272920i
\(88\) 1.50719e10 − 2.61053e10i 0.304448 − 0.527320i
\(89\) 1.35210e10 + 2.34190e10i 0.256663 + 0.444553i 0.965346 − 0.260974i \(-0.0840438\pi\)
−0.708683 + 0.705527i \(0.750710\pi\)
\(90\) −2.90289e10 −0.518200
\(91\) −1.17561e10 + 5.33120e10i −0.197485 + 0.895566i
\(92\) −2.93662e10 −0.464531
\(93\) −1.07747e10 − 1.86623e10i −0.160601 − 0.278169i
\(94\) −1.84277e10 + 3.19178e10i −0.258982 + 0.448569i
\(95\) 5.99288e10 − 1.03800e11i 0.794613 − 1.37631i
\(96\) −3.17863e9 − 5.50555e9i −0.0397876 − 0.0689142i
\(97\) −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
\(99\) 1.29940e11 1.37324
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 14.12.c.a.11.2 yes 8
3.2 odd 2 126.12.g.e.109.3 8
4.3 odd 2 112.12.i.a.81.3 8
7.2 even 3 inner 14.12.c.a.9.2 ✓ 8
7.3 odd 6 98.12.a.l.1.2 4
7.4 even 3 98.12.a.j.1.3 4
7.5 odd 6 98.12.c.l.79.3 8
7.6 odd 2 98.12.c.l.67.3 8
21.2 odd 6 126.12.g.e.37.3 8
28.23 odd 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 7.2 even 3 inner
14.12.c.a.11.2 yes 8 1.1 even 1 trivial
98.12.a.j.1.3 4 7.4 even 3
98.12.a.l.1.2 4 7.3 odd 6
98.12.c.l.67.3 8 7.6 odd 2
98.12.c.l.79.3 8 7.5 odd 6
112.12.i.a.65.3 8 28.23 odd 6
112.12.i.a.81.3 8 4.3 odd 2
126.12.g.e.37.3 8 21.2 odd 6
126.12.g.e.109.3 8 3.2 odd 2