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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 67.3
Root \(64.1153 - 111.051i\) of defining polynomial
Character \(\chi\) \(=\) 98.67
Dual form 98.12.c.l.79.3

$q$-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 \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).

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

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 −0.731268 − 0.682090i \(-0.761071\pi\)
0.956341 + 0.292252i \(0.0944047\pi\)
\(4\) −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 \(0.681348\pi\)
\(6\) −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 −0.00974918 − 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 \(-0.651628\pi\)
−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 0.431224 + 0.902245i \(0.358082\pi\)
−0.996979 + 0.0776717i \(0.975251\pi\)
\(18\) 2.26002e6 − 3.91448e6i 0.281912 − 0.488287i
\(19\) −9.33141e6 − 1.61625e7i −0.864574 − 1.49749i −0.867469 − 0.497491i \(-0.834255\pi\)
0.00289490 − 0.999996i \(-0.499079\pi\)
\(20\) −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 \(-0.0128895\pi\)
−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
\(28\) 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 −0.630644 − 0.776072i \(-0.717209\pi\)
0.987420 + 0.158118i \(0.0505427\pi\)
\(32\) −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 0.956408 − 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 \(-0.462473\pi\)
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 0.988977 − 0.148071i \(-0.0473063\pi\)
−0.622721 + 0.782444i \(0.713973\pi\)
\(48\) −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 −0.913583 − 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
\(56\) 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 −0.964640 − 0.263571i \(-0.915100\pi\)
0.710579 + 0.703618i \(0.248433\pi\)
\(60\) −6.22986e8 + 1.07904e9i −0.103430 + 0.179146i
\(61\) −2.60037e9 − 4.50397e9i −0.394204 − 0.682781i 0.598795 − 0.800902i \(-0.295646\pi\)
−0.992999 + 0.118121i \(0.962313\pi\)
\(62\) −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 −0.684729 − 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
\(70\) 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 −0.999062 − 0.0433121i \(-0.986209\pi\)
0.537040 + 0.843557i \(0.319542\pi\)
\(74\) 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 0.898899 − 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 \(-0.487368\pi\)
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 0.708683 − 0.705527i \(-0.249290\pi\)
−0.965346 + 0.260974i \(0.915956\pi\)
\(90\) 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.67.3 8
7.2 even 3 inner 98.12.c.l.79.3 8
7.3 odd 6 98.12.a.j.1.3 4
7.4 even 3 98.12.a.l.1.2 4
7.5 odd 6 14.12.c.a.9.2 ✓ 8
7.6 odd 2 14.12.c.a.11.2 yes 8
21.5 even 6 126.12.g.e.37.3 8
21.20 even 2 126.12.g.e.109.3 8
28.19 even 6 112.12.i.a.65.3 8
28.27 even 2 112.12.i.a.81.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.12.c.a.9.2 ✓ 8 7.5 odd 6
14.12.c.a.11.2 yes 8 7.6 odd 2
98.12.a.j.1.3 4 7.3 odd 6
98.12.a.l.1.2 4 7.4 even 3
98.12.c.l.67.3 8 1.1 even 1 trivial
98.12.c.l.79.3 8 7.2 even 3 inner
112.12.i.a.65.3 8 28.19 even 6
112.12.i.a.81.3 8 28.27 even 2
126.12.g.e.37.3 8 21.5 even 6
126.12.g.e.109.3 8 21.20 even 2