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 9.2
Root \(64.1153 - 111.051i\) of defining polynomial
Character \(\chi\) \(=\) 14.9
Dual form 14.12.c.a.11.2

$q$-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 \(\left(\mathbb{Z}/14\mathbb{Z}\right)^\times\).

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

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
\(3\) −94.7305 − 164.078i −0.225073 − 0.389838i 0.731268 − 0.682090i \(-0.238929\pi\)
−0.956341 + 0.292252i \(0.905595\pi\)
\(4\) −512.000 − 886.810i −0.250000 − 0.433013i
\(5\) −3211.13 + 5561.84i −0.459540 + 0.795946i −0.998937 − 0.0461054i \(-0.985319\pi\)
0.539397 + 0.842052i \(0.318652\pi\)
\(6\) 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 0.870859 + 0.491533i \(0.163563\pi\)
−0.00974918 + 0.999952i \(0.503103\pi\)
\(12\) −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 0.996979 + 0.0776717i \(0.0247486\pi\)
−0.431224 + 0.902245i \(0.641918\pi\)
\(18\) 2.26002e6 + 3.91448e6i 0.281912 + 0.488287i
\(19\) 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
\(21\) −6.21781e6 + 5.68471e6i −0.332224 + 0.303740i
\(22\) −2.94373e7 −1.21779
\(23\) 1.43390e7 − 2.48358e7i 0.464531 − 0.804592i −0.534649 − 0.845074i \(-0.679556\pi\)
0.999180 + 0.0404825i \(0.0128895\pi\)
\(24\) −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 0.630644 − 0.776072i \(-0.282791\pi\)
−0.987420 + 0.158118i \(0.949457\pi\)
\(32\) −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 −0.731112 − 0.682257i \(-0.760999\pi\)
0.956408 + 0.292033i \(0.0943318\pi\)
\(38\) 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 −0.988977 − 0.148071i \(-0.952694\pi\)
0.622721 + 0.782444i \(0.286027\pi\)
\(48\) 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 0.808962 − 0.587861i \(-0.200030\pi\)
−0.913583 + 0.406651i \(0.866696\pi\)
\(54\) 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 0.964640 − 0.263571i \(-0.0849002\pi\)
−0.710579 + 0.703618i \(0.751567\pi\)
\(60\) −6.22986e8 − 1.07904e9i −0.103430 − 0.179146i
\(61\) 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
\(63\) −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 0.973522 − 0.228594i \(-0.0734128\pi\)
−0.684729 + 0.728798i \(0.740080\pi\)
\(68\) 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 0.999062 − 0.0433121i \(-0.0137910\pi\)
−0.537040 + 0.843557i \(0.680458\pi\)
\(74\) 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 −0.828903 − 0.559392i \(-0.811035\pi\)
0.898899 + 0.438155i \(0.144368\pi\)
\(80\) −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 −0.708683 − 0.705527i \(-0.750710\pi\)
0.965346 + 0.260974i \(0.0840438\pi\)
\(90\) −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.9.2 ✓ 8
3.2 odd 2 126.12.g.e.37.3 8
4.3 odd 2 112.12.i.a.65.3 8
7.2 even 3 98.12.a.j.1.3 4
7.3 odd 6 98.12.c.l.67.3 8
7.4 even 3 inner 14.12.c.a.11.2 yes 8
7.5 odd 6 98.12.a.l.1.2 4
7.6 odd 2 98.12.c.l.79.3 8
21.11 odd 6 126.12.g.e.109.3 8
28.11 odd 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 1.1 even 1 trivial
14.12.c.a.11.2 yes 8 7.4 even 3 inner
98.12.a.j.1.3 4 7.2 even 3
98.12.a.l.1.2 4 7.5 odd 6
98.12.c.l.67.3 8 7.3 odd 6
98.12.c.l.79.3 8 7.6 odd 2
112.12.i.a.65.3 8 4.3 odd 2
112.12.i.a.81.3 8 28.11 odd 6
126.12.g.e.37.3 8 3.2 odd 2
126.12.g.e.109.3 8 21.11 odd 6