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Properties

Label 98.16.c.c.67.1
Level $98$
Weight $16$
Character 98.67
Analytic conductor $139.840$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [98,16,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, 16); 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, 16, names="a")
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 98.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,128,1350] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(139.839634998\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 67.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 98.67
Dual form 98.16.c.c.79.1

$q$-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+(64.0000 + 110.851i) q^{2} +(675.000 - 1169.13i) q^{3} +(-8192.00 + 14189.0i) q^{4} +(-40530.0 - 70200.0i) q^{5} +172800. q^{6} -2.09715e6 q^{8} +(6.26320e6 + 1.08482e7i) q^{9} +(5.18784e6 - 8.98560e6i) q^{10} +(-3.50606e7 + 6.07267e7i) q^{11} +(1.10592e7 + 1.91551e7i) q^{12} -1.51470e8 q^{13} -1.09431e8 q^{15} +(-1.34218e8 - 2.32472e8i) q^{16} +(-1.24878e8 + 2.16296e8i) q^{17} +(-8.01690e8 + 1.38857e9i) q^{18} +(-3.23843e9 - 5.60912e9i) q^{19} +1.32809e9 q^{20} -8.97551e9 q^{22} +(1.05646e10 + 1.82984e10i) q^{23} +(-1.41558e9 + 2.45185e9i) q^{24} +(1.19734e10 - 2.07386e10i) q^{25} +(-9.69405e9 - 1.67906e10i) q^{26} +3.62817e10 q^{27} +7.79483e9 q^{29} +(-7.00358e9 - 1.21306e10i) q^{30} +(-4.75160e10 + 8.23002e10i) q^{31} +(1.71799e10 - 2.97564e10i) q^{32} +(4.73318e10 + 8.19811e10i) q^{33} -3.19688e10 q^{34} -2.05233e11 q^{36} +(4.35041e11 + 7.53513e11i) q^{37} +(4.14519e11 - 7.17968e11i) q^{38} +(-1.02242e11 + 1.77088e11i) q^{39} +(8.49976e10 + 1.47220e11i) q^{40} -1.00767e12 q^{41} +1.55008e11 q^{43} +(-5.74433e11 - 9.94947e11i) q^{44} +(5.07695e11 - 8.79354e11i) q^{45} +(-1.35227e12 + 2.34220e12i) q^{46} +(-1.27599e12 - 2.21007e12i) q^{47} -3.62388e11 q^{48} +3.06520e12 q^{50} +(1.68586e11 + 2.91999e11i) q^{51} +(1.24084e12 - 2.14920e12i) q^{52} +(-2.02382e12 + 3.50536e12i) q^{53} +(2.32203e12 + 4.02187e12i) q^{54} +5.68402e12 q^{55} -8.74376e12 q^{57} +(4.98869e11 + 8.64066e11i) q^{58} +(-6.29962e12 + 1.09113e13i) q^{59} +(8.96459e11 - 1.55271e12i) q^{60} +(-1.99625e13 - 3.45761e13i) q^{61} -1.21641e13 q^{62} +4.39805e12 q^{64} +(6.13906e12 + 1.06332e13i) q^{65} +(-6.05847e12 + 1.04936e13i) q^{66} +(2.42119e13 - 4.19362e13i) q^{67} +(-2.04601e12 - 3.54379e12i) q^{68} +2.85244e13 q^{69} +3.76931e13 q^{71} +(-1.31349e13 - 2.27503e13i) q^{72} +(7.07081e13 - 1.22470e14i) q^{73} +(-5.56853e13 + 9.64497e13i) q^{74} +(-1.61641e13 - 2.79971e13i) q^{75} +1.06117e14 q^{76} -2.61739e13 q^{78} +(-1.23510e14 - 2.13926e14i) q^{79} +(-1.08797e13 + 1.88442e13i) q^{80} +(-6.53800e13 + 1.13241e14i) q^{81} +(-6.44907e13 - 1.11701e14i) q^{82} -2.78879e12 q^{83} +2.02453e13 q^{85} +(9.92049e12 + 1.71828e13i) q^{86} +(5.26151e12 - 9.11320e12i) q^{87} +(7.35274e13 - 1.27353e14i) q^{88} +(-2.91982e12 - 5.05727e12i) q^{89} +1.29970e14 q^{90} -3.46181e14 q^{92} +(6.41466e13 + 1.11105e14i) q^{93} +(1.63326e14 - 2.82889e14i) q^{94} +(-2.62507e14 + 4.54675e14i) q^{95} +(-2.31928e13 - 4.01711e13i) q^{96} -2.78027e14 q^{97} -8.78366e14 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 128 q^{2} + 1350 q^{3} - 16384 q^{4} - 81060 q^{5} + 345600 q^{6} - 4194304 q^{8} + 12526407 q^{9} + 10375680 q^{10} - 70121184 q^{11} + 22118400 q^{12} - 302939104 q^{13} - 218862000 q^{15} - 268435456 q^{16}+ \cdots - 17\!\cdots\!76 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 64.0000 + 110.851i 0.353553 + 0.612372i
\(3\) 675.000 − 1169.13i 0.178195 − 0.308642i −0.763068 − 0.646319i \(-0.776308\pi\)
0.941262 + 0.337677i \(0.109641\pi\)
\(4\) −8192.00 + 14189.0i −0.250000 + 0.433013i
\(5\) −40530.0 − 70200.0i −0.232007 − 0.401848i 0.726391 − 0.687281i \(-0.241196\pi\)
−0.958399 + 0.285433i \(0.907863\pi\)
\(6\) 172800. 0.252005
\(7\) 0 0
\(8\) −2.09715e6 −0.353553
\(9\) 6.26320e6 + 1.08482e7i 0.436493 + 0.756029i
\(10\) 5.18784e6 − 8.98560e6i 0.164054 − 0.284150i
\(11\) −3.50606e7 + 6.07267e7i −0.542468 + 0.939582i 0.456294 + 0.889829i \(0.349177\pi\)
−0.998762 + 0.0497528i \(0.984157\pi\)
\(12\) 1.10592e7 + 1.91551e7i 0.0890973 + 0.154321i
\(13\) −1.51470e8 −0.669499 −0.334750 − 0.942307i \(-0.608652\pi\)
−0.334750 + 0.942307i \(0.608652\pi\)
\(14\) 0 0
\(15\) −1.09431e8 −0.165370
\(16\) −1.34218e8 − 2.32472e8i −0.125000 − 0.216506i
\(17\) −1.24878e8 + 2.16296e8i −0.0738108 + 0.127844i −0.900569 − 0.434714i \(-0.856849\pi\)
0.826758 + 0.562558i \(0.190183\pi\)
\(18\) −8.01690e8 + 1.38857e9i −0.308647 + 0.534593i
\(19\) −3.23843e9 − 5.60912e9i −0.831155 − 1.43960i −0.897123 − 0.441782i \(-0.854347\pi\)
0.0659671 − 0.997822i \(-0.478987\pi\)
\(20\) 1.32809e9 0.232007
\(21\) 0 0
\(22\) −8.97551e9 −0.767166
\(23\) 1.05646e10 + 1.82984e10i 0.646985 + 1.12061i 0.983839 + 0.179054i \(0.0573037\pi\)
−0.336854 + 0.941557i \(0.609363\pi\)
\(24\) −1.41558e9 + 2.45185e9i −0.0630013 + 0.109121i
\(25\) 1.19734e10 − 2.07386e10i 0.392345 − 0.679562i
\(26\) −9.69405e9 − 1.67906e10i −0.236704 − 0.409983i
\(27\) 3.62817e10 0.667512
\(28\) 0 0
\(29\) 7.79483e9 0.0839115 0.0419557 − 0.999119i \(-0.486641\pi\)
0.0419557 + 0.999119i \(0.486641\pi\)
\(30\) −7.00358e9 − 1.21306e10i −0.0584670 − 0.101268i
\(31\) −4.75160e10 + 8.23002e10i −0.310190 + 0.537264i −0.978403 − 0.206705i \(-0.933726\pi\)
0.668214 + 0.743970i \(0.267059\pi\)
\(32\) 1.71799e10 − 2.97564e10i 0.0883883 − 0.153093i
\(33\) 4.73318e10 + 8.19811e10i 0.193330 + 0.334857i
\(34\) −3.19688e10 −0.104384
\(35\) 0 0
\(36\) −2.05233e11 −0.436493
\(37\) 4.35041e11 + 7.53513e11i 0.753386 + 1.30490i 0.946173 + 0.323662i \(0.104914\pi\)
−0.192787 + 0.981241i \(0.561753\pi\)
\(38\) 4.14519e11 − 7.17968e11i 0.587716 − 1.01795i
\(39\) −1.02242e11 + 1.77088e11i −0.119301 + 0.206636i
\(40\) 8.49976e10 + 1.47220e11i 0.0820270 + 0.142075i
\(41\) −1.00767e12 −0.808049 −0.404025 − 0.914748i \(-0.632389\pi\)
−0.404025 + 0.914748i \(0.632389\pi\)
\(42\) 0 0
\(43\) 1.55008e11 0.0869640 0.0434820 − 0.999054i \(-0.486155\pi\)
0.0434820 + 0.999054i \(0.486155\pi\)
\(44\) −5.74433e11 − 9.94947e11i −0.271234 − 0.469791i
\(45\) 5.07695e11 − 8.79354e11i 0.202539 − 0.350808i
\(46\) −1.35227e12 + 2.34220e12i −0.457487 + 0.792392i
\(47\) −1.27599e12 − 2.21007e12i −0.367377 − 0.636315i 0.621778 − 0.783194i \(-0.286411\pi\)
−0.989155 + 0.146879i \(0.953077\pi\)
\(48\) −3.62388e11 −0.0890973
\(49\) 0 0
\(50\) 3.06520e12 0.554860
\(51\) 1.68586e11 + 2.91999e11i 0.0263054 + 0.0455622i
\(52\) 1.24084e12 − 2.14920e12i 0.167375 − 0.289902i
\(53\) −2.02382e12 + 3.50536e12i −0.236648 + 0.409887i −0.959750 − 0.280854i \(-0.909382\pi\)
0.723102 + 0.690741i \(0.242716\pi\)
\(54\) 2.32203e12 + 4.02187e12i 0.236001 + 0.408766i
\(55\) 5.68402e12 0.503426
\(56\) 0 0
\(57\) −8.74376e12 −0.592429
\(58\) 4.98869e11 + 8.64066e11i 0.0296672 + 0.0513851i
\(59\) −6.29962e12 + 1.09113e13i −0.329552 + 0.570802i −0.982423 − 0.186668i \(-0.940231\pi\)
0.652871 + 0.757469i \(0.273565\pi\)
\(60\) 8.96459e11 − 1.55271e12i 0.0413424 − 0.0716072i
\(61\) −1.99625e13 − 3.45761e13i −0.813283 − 1.40865i −0.910555 − 0.413389i \(-0.864345\pi\)
0.0972719 − 0.995258i \(-0.468988\pi\)
\(62\) −1.21641e13 −0.438675
\(63\) 0 0
\(64\) 4.39805e12 0.125000
\(65\) 6.13906e12 + 1.06332e13i 0.155329 + 0.269037i
\(66\) −6.05847e12 + 1.04936e13i −0.136705 + 0.236779i
\(67\) 2.42119e13 − 4.19362e13i 0.488054 − 0.845334i −0.511852 − 0.859074i \(-0.671040\pi\)
0.999906 + 0.0137398i \(0.00437366\pi\)
\(68\) −2.04601e12 − 3.54379e12i −0.0369054 − 0.0639220i
\(69\) 2.85244e13 0.461157
\(70\) 0 0
\(71\) 3.76931e13 0.491840 0.245920 − 0.969290i \(-0.420910\pi\)
0.245920 + 0.969290i \(0.420910\pi\)
\(72\) −1.31349e13 − 2.27503e13i −0.154324 − 0.267297i
\(73\) 7.07081e13 − 1.22470e14i 0.749114 − 1.29750i −0.199134 − 0.979972i \(-0.563813\pi\)
0.948248 − 0.317531i \(-0.102854\pi\)
\(74\) −5.56853e13 + 9.64497e13i −0.532724 + 0.922705i
\(75\) −1.61641e13 − 2.79971e13i −0.139828 − 0.242188i
\(76\) 1.06117e14 0.831155
\(77\) 0 0
\(78\) −2.61739e13 −0.168717
\(79\) −1.23510e14 − 2.13926e14i −0.723602 − 1.25332i −0.959547 − 0.281549i \(-0.909152\pi\)
0.235945 − 0.971766i \(-0.424182\pi\)
\(80\) −1.08797e13 + 1.88442e13i −0.0580018 + 0.100462i
\(81\) −6.53800e13 + 1.13241e14i −0.317546 + 0.550007i
\(82\) −6.44907e13 − 1.11701e14i −0.285689 − 0.494827i
\(83\) −2.78879e12 −0.0112805 −0.00564027 − 0.999984i \(-0.501795\pi\)
−0.00564027 + 0.999984i \(0.501795\pi\)
\(84\) 0 0
\(85\) 2.02453e13 0.0684986
\(86\) 9.92049e12 + 1.71828e13i 0.0307464 + 0.0532544i
\(87\) 5.26151e12 − 9.11320e12i 0.0149526 − 0.0258986i
\(88\) 7.35274e13 − 1.27353e14i 0.191791 − 0.332192i
\(89\) −2.91982e12 − 5.05727e12i −0.00699730 − 0.0121197i 0.862506 − 0.506048i \(-0.168894\pi\)
−0.869503 + 0.493928i \(0.835561\pi\)
\(90\) 1.29970e14 0.286434
\(91\) 0 0
\(92\) −3.46181e14 −0.646985
\(93\) 6.41466e13 + 1.11105e14i 0.110548 + 0.191475i
\(94\) 1.63326e14 − 2.82889e14i 0.259775 − 0.449943i
\(95\) −2.62507e14 + 4.54675e14i −0.385668 + 0.667997i
\(96\) −2.31928e13 − 4.01711e13i −0.0315006 − 0.0545607i
\(97\) −2.78027e14 −0.349381 −0.174690 − 0.984623i \(-0.555892\pi\)
−0.174690 + 0.984623i \(0.555892\pi\)
\(98\) 0 0
\(99\) −8.78366e14 −0.947135
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 98.16.c.c.67.1 2
7.2 even 3 inner 98.16.c.c.79.1 2
7.3 odd 6 14.16.a.a.1.1 ✓ 1
7.4 even 3 98.16.a.b.1.1 1
7.5 odd 6 98.16.c.b.79.1 2
7.6 odd 2 98.16.c.b.67.1 2
21.17 even 6 126.16.a.e.1.1 1
28.3 even 6 112.16.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.16.a.a.1.1 ✓ 1 7.3 odd 6
98.16.a.b.1.1 1 7.4 even 3
98.16.c.b.67.1 2 7.6 odd 2
98.16.c.b.79.1 2 7.5 odd 6
98.16.c.c.67.1 2 1.1 even 1 trivial
98.16.c.c.79.1 2 7.2 even 3 inner
112.16.a.a.1.1 1 28.3 even 6
126.16.a.e.1.1 1 21.17 even 6