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(1,98)] 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("98.1"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(98, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 98.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,128,-266,4096,-7504,-8512,0,131072,123520] 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: yes
Analytic conductor: \(75.2976316948\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 101802x^{2} - 3237299x + 1820791991 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(128.231\) of defining polynomial
Character \(\chi\) \(=\) 98.1

$q$-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)
 
\(f(q)\) \(=\) \(q+32.0000 q^{2} +189.461 q^{3} +1024.00 q^{4} +6422.27 q^{5} +6062.75 q^{6} +32768.0 q^{8} -141252. q^{9} +205512. q^{10} -919916. q^{11} +194008. q^{12} +1.22771e6 q^{13} +1.21677e6 q^{15} +1.04858e6 q^{16} -6.62411e6 q^{17} -4.52005e6 q^{18} -1.86628e7 q^{19} +6.57640e6 q^{20} -2.94373e7 q^{22} -2.86779e7 q^{23} +6.20826e6 q^{24} -7.58264e6 q^{25} +3.92868e7 q^{26} -6.03241e7 q^{27} -7.73286e6 q^{29} +3.89366e7 q^{30} +1.13740e8 q^{31} +3.35544e7 q^{32} -1.74288e8 q^{33} -2.11972e8 q^{34} -1.44642e8 q^{36} -1.90061e8 q^{37} -5.97210e8 q^{38} +2.32604e8 q^{39} +2.10445e8 q^{40} -1.74512e8 q^{41} +1.83161e9 q^{43} -9.41994e8 q^{44} -9.07155e8 q^{45} -9.17694e8 q^{46} +1.15173e9 q^{47} +1.98664e8 q^{48} -2.42644e8 q^{50} -1.25501e9 q^{51} +1.25718e9 q^{52} +6.37041e8 q^{53} -1.93037e9 q^{54} -5.90795e9 q^{55} -3.53588e9 q^{57} -2.47452e8 q^{58} -2.79032e9 q^{59} +1.24597e9 q^{60} -5.20074e9 q^{61} +3.63969e9 q^{62} +1.07374e9 q^{64} +7.88469e9 q^{65} -5.57723e9 q^{66} -6.38304e9 q^{67} -6.78309e9 q^{68} -5.43335e9 q^{69} +2.41383e10 q^{71} -4.62853e9 q^{72} -1.63670e10 q^{73} -6.08194e9 q^{74} -1.43661e9 q^{75} -1.91107e10 q^{76} +7.44331e9 q^{78} -3.82870e9 q^{79} +6.73423e9 q^{80} +1.35932e10 q^{81} -5.58438e9 q^{82} -2.84759e9 q^{83} -4.25418e10 q^{85} +5.86114e10 q^{86} -1.46508e9 q^{87} -3.01438e10 q^{88} -2.70419e10 q^{89} -2.90289e10 q^{90} -2.93662e10 q^{92} +2.15494e10 q^{93} +3.68555e10 q^{94} -1.19858e11 q^{95} +6.35726e9 q^{96} -1.49558e11 q^{97} +1.29940e11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 128 q^{2} - 266 q^{3} + 4096 q^{4} - 7504 q^{5} - 8512 q^{6} + 131072 q^{8} + 123520 q^{9} - 240128 q^{10} - 213026 q^{11} - 272384 q^{12} - 1304856 q^{13} + 1137750 q^{15} + 4194304 q^{16} - 8854244 q^{17}+ \cdots + 196707902868 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 32.0000 0.707107
\(3\) 189.461 0.450146 0.225073 − 0.974342i \(-0.427738\pi\)
0.225073 + 0.974342i \(0.427738\pi\)
\(4\) 1024.00 0.500000
\(5\) 6422.27 0.919080 0.459540 − 0.888157i \(-0.348014\pi\)
0.459540 + 0.888157i \(0.348014\pi\)
\(6\) 6062.75 0.318301
\(7\) 0 0
\(8\) 32768.0 0.353553
\(9\) −141252. −0.797369
\(10\) 205512. 0.649888
\(11\) −919916. −1.72222 −0.861110 − 0.508419i \(-0.830230\pi\)
−0.861110 + 0.508419i \(0.830230\pi\)
\(12\) 194008. 0.225073
\(13\) 1.22771e6 0.917081 0.458541 − 0.888673i \(-0.348372\pi\)
0.458541 + 0.888673i \(0.348372\pi\)
\(14\) 0 0
\(15\) 1.21677e6 0.413720
\(16\) 1.04858e6 0.250000
\(17\) −6.62411e6 −1.13151 −0.565755 − 0.824573i \(-0.691415\pi\)
−0.565755 + 0.824573i \(0.691415\pi\)
\(18\) −4.52005e6 −0.563825
\(19\) −1.86628e7 −1.72915 −0.864574 − 0.502505i \(-0.832412\pi\)
−0.864574 + 0.502505i \(0.832412\pi\)
\(20\) 6.57640e6 0.459540
\(21\) 0 0
\(22\) −2.94373e7 −1.21779
\(23\) −2.86779e7 −0.929063 −0.464531 − 0.885557i \(-0.653777\pi\)
−0.464531 + 0.885557i \(0.653777\pi\)
\(24\) 6.20826e6 0.159151
\(25\) −7.58264e6 −0.155292
\(26\) 3.92868e7 0.648474
\(27\) −6.03241e7 −0.809078
\(28\) 0 0
\(29\) −7.73286e6 −0.0700086 −0.0350043 − 0.999387i \(-0.511144\pi\)
−0.0350043 + 0.999387i \(0.511144\pi\)
\(30\) 3.89366e7 0.292544
\(31\) 1.13740e8 0.713552 0.356776 − 0.934190i \(-0.383876\pi\)
0.356776 + 0.934190i \(0.383876\pi\)
\(32\) 3.35544e7 0.176777
\(33\) −1.74288e8 −0.775250
\(34\) −2.11972e8 −0.800099
\(35\) 0 0
\(36\) −1.44642e8 −0.398684
\(37\) −1.90061e8 −0.450591 −0.225296 − 0.974290i \(-0.572335\pi\)
−0.225296 + 0.974290i \(0.572335\pi\)
\(38\) −5.97210e8 −1.22269
\(39\) 2.32604e8 0.412820
\(40\) 2.10445e8 0.324944
\(41\) −1.74512e8 −0.235241 −0.117621 − 0.993059i \(-0.537527\pi\)
−0.117621 + 0.993059i \(0.537527\pi\)
\(42\) 0 0
\(43\) 1.83161e9 1.90001 0.950005 − 0.312235i \(-0.101078\pi\)
0.950005 + 0.312235i \(0.101078\pi\)
\(44\) −9.41994e8 −0.861110
\(45\) −9.07155e8 −0.732846
\(46\) −9.17694e8 −0.656946
\(47\) 1.15173e9 0.732511 0.366255 − 0.930514i \(-0.380640\pi\)
0.366255 + 0.930514i \(0.380640\pi\)
\(48\) 1.98664e8 0.112536
\(49\) 0 0
\(50\) −2.42644e8 −0.109808
\(51\) −1.25501e9 −0.509344
\(52\) 1.25718e9 0.458541
\(53\) 6.37041e8 0.209243 0.104621 − 0.994512i \(-0.466637\pi\)
0.104621 + 0.994512i \(0.466637\pi\)
\(54\) −1.93037e9 −0.572104
\(55\) −5.90795e9 −1.58286
\(56\) 0 0
\(57\) −3.53588e9 −0.778369
\(58\) −2.47452e8 −0.0495035
\(59\) −2.79032e9 −0.508123 −0.254061 − 0.967188i \(-0.581766\pi\)
−0.254061 + 0.967188i \(0.581766\pi\)
\(60\) 1.24597e9 0.206860
\(61\) −5.20074e9 −0.788408 −0.394204 − 0.919023i \(-0.628980\pi\)
−0.394204 + 0.919023i \(0.628980\pi\)
\(62\) 3.63969e9 0.504557
\(63\) 0 0
\(64\) 1.07374e9 0.125000
\(65\) 7.88469e9 0.842871
\(66\) −5.57723e9 −0.548184
\(67\) −6.38304e9 −0.577585 −0.288793 − 0.957392i \(-0.593254\pi\)
−0.288793 + 0.957392i \(0.593254\pi\)
\(68\) −6.78309e9 −0.565755
\(69\) −5.43335e9 −0.418214
\(70\) 0 0
\(71\) 2.41383e10 1.58776 0.793880 − 0.608074i \(-0.208058\pi\)
0.793880 + 0.608074i \(0.208058\pi\)
\(72\) −4.62853e9 −0.281912
\(73\) −1.63670e10 −0.924043 −0.462021 − 0.886869i \(-0.652876\pi\)
−0.462021 + 0.886869i \(0.652876\pi\)
\(74\) −6.08194e9 −0.318616
\(75\) −1.43661e9 −0.0699042
\(76\) −1.91107e10 −0.864574
\(77\) 0 0
\(78\) 7.44331e9 0.291908
\(79\) −3.82870e9 −0.139992 −0.0699958 − 0.997547i \(-0.522299\pi\)
−0.0699958 + 0.997547i \(0.522299\pi\)
\(80\) 6.73423e9 0.229770
\(81\) 1.35932e10 0.433166
\(82\) −5.58438e9 −0.166341
\(83\) −2.84759e9 −0.0793502 −0.0396751 − 0.999213i \(-0.512632\pi\)
−0.0396751 + 0.999213i \(0.512632\pi\)
\(84\) 0 0
\(85\) −4.25418e10 −1.03995
\(86\) 5.86114e10 1.34351
\(87\) −1.46508e9 −0.0315141
\(88\) −3.01438e10 −0.608896
\(89\) −2.70419e10 −0.513325 −0.256663 − 0.966501i \(-0.582623\pi\)
−0.256663 + 0.966501i \(0.582623\pi\)
\(90\) −2.90289e10 −0.518200
\(91\) 0 0
\(92\) −2.93662e10 −0.464531
\(93\) 2.15494e10 0.321202
\(94\) 3.68555e10 0.517963
\(95\) −1.19858e11 −1.58923
\(96\) 6.35726e9 0.0795753
\(97\) −1.49558e11 −1.76834 −0.884168 − 0.467169i \(-0.845274\pi\)
−0.884168 + 0.467169i \(0.845274\pi\)
\(98\) 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.a.j.1.3 4
7.2 even 3 14.12.c.a.11.2 yes 8
7.3 odd 6 98.12.c.l.79.3 8
7.4 even 3 14.12.c.a.9.2 ✓ 8
7.5 odd 6 98.12.c.l.67.3 8
7.6 odd 2 98.12.a.l.1.2 4
21.2 odd 6 126.12.g.e.109.3 8
21.11 odd 6 126.12.g.e.37.3 8
28.11 odd 6 112.12.i.a.65.3 8
28.23 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 7.4 even 3
14.12.c.a.11.2 yes 8 7.2 even 3
98.12.a.j.1.3 4 1.1 even 1 trivial
98.12.a.l.1.2 4 7.6 odd 2
98.12.c.l.67.3 8 7.5 odd 6
98.12.c.l.79.3 8 7.3 odd 6
112.12.i.a.65.3 8 28.11 odd 6
112.12.i.a.81.3 8 28.23 odd 6
126.12.g.e.37.3 8 21.11 odd 6
126.12.g.e.109.3 8 21.2 odd 6