Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [98,12,Mod(67,98)] 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("98.67"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(98, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 98.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-128,-266,-4096,-7504,17024,0,262144,-123520] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(75.2976316948\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 79.3
Root \(64.1153 + 111.051i\) of defining polynomial
Character \(\chi\) \(=\) 98.79
Dual form 98.12.c.l.67.3

$q$-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} +(94.7305 + 164.078i) q^{3} +(-512.000 - 886.810i) q^{4} +(3211.13 - 5561.84i) q^{5} -6062.75 q^{6} +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} +1.21677e6 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} -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} -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} -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} +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} -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} -3.53588e9 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} +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} +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} +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} -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} -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} +1.29940e11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 128 q^{2} - 266 q^{3} - 4096 q^{4} - 7504 q^{5} + 17024 q^{6} + 262144 q^{8} - 123520 q^{9} - 240128 q^{10} + 213026 q^{11} - 272384 q^{12} + 2609712 q^{13} + 2275500 q^{15} - 4194304 q^{16} - 8854244 q^{17}+ \cdots + 393415805736 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).

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

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 0.956341 − 0.292252i \(-0.0944047\pi\)
−0.731268 + 0.682090i \(0.761071\pi\)
\(4\) −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 \(0.0146810\pi\)
\(6\) −6062.75 −0.318301
\(7\) 0 0
\(8\) 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 \(-0.651628\pi\)
−0.458541 + 0.888673i \(0.651628\pi\)
\(14\) 0 0
\(15\) 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 \(-0.975251\pi\)
0.431224 − 0.902245i \(-0.358082\pi\)
\(18\) 2.26002e6 + 3.91448e6i 0.281912 + 0.488287i
\(19\) −9.33141e6 + 1.61625e7i −0.864574 + 1.49749i 0.00289490 + 0.999996i \(0.499079\pi\)
−0.867469 + 0.497491i \(0.834255\pi\)
\(20\) −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 \(-0.679556\pi\)
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
\(28\) 0 0
\(29\) −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 \(-0.0505427\pi\)
−0.630644 + 0.776072i \(0.717209\pi\)
\(32\) −1.67772e7 − 2.90590e7i −0.0883883 − 0.153093i
\(33\) −8.71441e7 + 1.50938e8i −0.387625 + 0.671386i
\(34\) 2.11972e8 0.800099
\(35\) 0 0
\(36\) −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 \(-0.462473\pi\)
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
\(45\) −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 \(-0.713973\pi\)
0.988977 + 0.148071i \(0.0473063\pi\)
\(48\) −1.98664e8 −0.112536
\(49\) 0 0
\(50\) −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
\(56\) 0 0
\(57\) −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 \(-0.248433\pi\)
−0.964640 + 0.263571i \(0.915100\pi\)
\(60\) −6.22986e8 − 1.07904e9i −0.103430 − 0.179146i
\(61\) −2.60037e9 + 4.50397e9i −0.394204 + 0.682781i −0.992999 − 0.118121i \(-0.962313\pi\)
0.598795 + 0.800902i \(0.295646\pi\)
\(62\) −3.63969e9 −0.504557
\(63\) 0 0
\(64\) 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
\(70\) 0 0
\(71\) 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 \(-0.319542\pi\)
−0.999062 + 0.0433121i \(0.986209\pi\)
\(74\) 3.04097e9 + 5.26712e9i 0.159308 + 0.275930i
\(75\) −7.18307e8 + 1.24414e9i −0.0349521 + 0.0605388i
\(76\) 1.91107e10 0.864574
\(77\) 0 0
\(78\) 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 \(-0.487368\pi\)
0.0396751 + 0.999213i \(0.487368\pi\)
\(84\) 0 0
\(85\) −4.25418e10 −1.03995
\(86\) −2.93057e10 + 5.07590e10i −0.671755 + 1.16351i
\(87\) −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 \(-0.915956\pi\)
0.708683 + 0.705527i \(0.249290\pi\)
\(90\) 2.90289e10 0.518200
\(91\) 0 0
\(92\) −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 \(-0.154726\pi\)
0.884168 + 0.467169i \(0.154726\pi\)
\(98\) 0 0
\(99\) 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 98.12.c.l.79.3 8
7.2 even 3 98.12.a.l.1.2 4
7.3 odd 6 14.12.c.a.11.2 yes 8
7.4 even 3 inner 98.12.c.l.67.3 8
7.5 odd 6 98.12.a.j.1.3 4
7.6 odd 2 14.12.c.a.9.2 ✓ 8
21.17 even 6 126.12.g.e.109.3 8
21.20 even 2 126.12.g.e.37.3 8
28.3 even 6 112.12.i.a.81.3 8
28.27 even 2 112.12.i.a.65.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.12.c.a.9.2 ✓ 8 7.6 odd 2
14.12.c.a.11.2 yes 8 7.3 odd 6
98.12.a.j.1.3 4 7.5 odd 6
98.12.a.l.1.2 4 7.2 even 3
98.12.c.l.67.3 8 7.4 even 3 inner
98.12.c.l.79.3 8 1.1 even 1 trivial
112.12.i.a.65.3 8 28.27 even 2
112.12.i.a.81.3 8 28.3 even 6
126.12.g.e.37.3 8 21.20 even 2
126.12.g.e.109.3 8 21.17 even 6