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Properties

Label 98.16.a.b.1.1
Level $98$
Weight $16$
Character 98.1
Self dual yes
Analytic conductor $139.840$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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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(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, 16); 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, 16, names="a")
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 98.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-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: yes
Analytic conductor: \(139.839634998\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
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)
 
\(f(q)\) \(=\) \(q-128.000 q^{2} -1350.00 q^{3} +16384.0 q^{4} +81060.0 q^{5} +172800. q^{6} -2.09715e6 q^{8} -1.25264e7 q^{9} -1.03757e7 q^{10} +7.01212e7 q^{11} -2.21184e7 q^{12} -1.51470e8 q^{13} -1.09431e8 q^{15} +2.68435e8 q^{16} +2.49757e8 q^{17} +1.60338e9 q^{18} +6.47686e9 q^{19} +1.32809e9 q^{20} -8.97551e9 q^{22} -2.11292e10 q^{23} +2.83116e9 q^{24} -2.39469e10 q^{25} +1.93881e10 q^{26} +3.62817e10 q^{27} +7.79483e9 q^{29} +1.40072e10 q^{30} +9.50321e10 q^{31} -3.43597e10 q^{32} -9.46636e10 q^{33} -3.19688e10 q^{34} -2.05233e11 q^{36} -8.70082e11 q^{37} -8.29038e11 q^{38} +2.04484e11 q^{39} -1.69995e11 q^{40} -1.00767e12 q^{41} +1.55008e11 q^{43} +1.14887e12 q^{44} -1.01539e12 q^{45} +2.70454e12 q^{46} +2.55197e12 q^{47} -3.62388e11 q^{48} +3.06520e12 q^{50} -3.37171e11 q^{51} -2.48168e12 q^{52} +4.04765e12 q^{53} -4.64405e12 q^{54} +5.68402e12 q^{55} -8.74376e12 q^{57} -9.97738e11 q^{58} +1.25992e13 q^{59} -1.79292e12 q^{60} +3.99250e13 q^{61} -1.21641e13 q^{62} +4.39805e12 q^{64} -1.22781e13 q^{65} +1.21169e13 q^{66} -4.84238e13 q^{67} +4.09201e12 q^{68} +2.85244e13 q^{69} +3.76931e13 q^{71} +2.62698e13 q^{72} -1.41416e14 q^{73} +1.11371e14 q^{74} +3.23283e13 q^{75} +1.06117e14 q^{76} -2.61739e13 q^{78} +2.47021e14 q^{79} +2.17594e13 q^{80} +1.30760e14 q^{81} +1.28981e14 q^{82} -2.78879e12 q^{83} +2.02453e13 q^{85} -1.98410e13 q^{86} -1.05230e13 q^{87} -1.47055e14 q^{88} +5.83963e12 q^{89} +1.29970e14 q^{90} -3.46181e14 q^{92} -1.28293e14 q^{93} -3.26652e14 q^{94} +5.25014e14 q^{95} +4.63856e13 q^{96} -2.78027e14 q^{97} -8.78366e14 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −128.000 −0.707107
\(3\) −1350.00 −0.356389 −0.178195 − 0.983995i \(-0.557026\pi\)
−0.178195 + 0.983995i \(0.557026\pi\)
\(4\) 16384.0 0.500000
\(5\) 81060.0 0.464015 0.232007 − 0.972714i \(-0.425471\pi\)
0.232007 + 0.972714i \(0.425471\pi\)
\(6\) 172800. 0.252005
\(7\) 0 0
\(8\) −2.09715e6 −0.353553
\(9\) −1.25264e7 −0.872987
\(10\) −1.03757e7 −0.328108
\(11\) 7.01212e7 1.08494 0.542468 − 0.840076i \(-0.317490\pi\)
0.542468 + 0.840076i \(0.317490\pi\)
\(12\) −2.21184e7 −0.178195
\(13\) −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
\(16\) 2.68435e8 0.250000
\(17\) 2.49757e8 0.147622 0.0738108 − 0.997272i \(-0.476484\pi\)
0.0738108 + 0.997272i \(0.476484\pi\)
\(18\) 1.60338e9 0.617295
\(19\) 6.47686e9 1.66231 0.831155 − 0.556040i \(-0.187680\pi\)
0.831155 + 0.556040i \(0.187680\pi\)
\(20\) 1.32809e9 0.232007
\(21\) 0 0
\(22\) −8.97551e9 −0.767166
\(23\) −2.11292e10 −1.29397 −0.646985 − 0.762503i \(-0.723970\pi\)
−0.646985 + 0.762503i \(0.723970\pi\)
\(24\) 2.83116e9 0.126003
\(25\) −2.39469e10 −0.784691
\(26\) 1.93881e10 0.473408
\(27\) 3.62817e10 0.667512
\(28\) 0 0
\(29\) 7.79483e9 0.0839115 0.0419557 − 0.999119i \(-0.486641\pi\)
0.0419557 + 0.999119i \(0.486641\pi\)
\(30\) 1.40072e10 0.116934
\(31\) 9.50321e10 0.620379 0.310190 − 0.950675i \(-0.399607\pi\)
0.310190 + 0.950675i \(0.399607\pi\)
\(32\) −3.43597e10 −0.176777
\(33\) −9.46636e10 −0.386659
\(34\) −3.19688e10 −0.104384
\(35\) 0 0
\(36\) −2.05233e11 −0.436493
\(37\) −8.70082e11 −1.50677 −0.753386 − 0.657579i \(-0.771581\pi\)
−0.753386 + 0.657579i \(0.771581\pi\)
\(38\) −8.29038e11 −1.17543
\(39\) 2.04484e11 0.238602
\(40\) −1.69995e11 −0.164054
\(41\) −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 \(0.486155\pi\)
\(44\) 1.14887e12 0.542468
\(45\) −1.01539e12 −0.405079
\(46\) 2.70454e12 0.914975
\(47\) 2.55197e12 0.734753 0.367377 − 0.930072i \(-0.380256\pi\)
0.367377 + 0.930072i \(0.380256\pi\)
\(48\) −3.62388e11 −0.0890973
\(49\) 0 0
\(50\) 3.06520e12 0.554860
\(51\) −3.37171e11 −0.0526107
\(52\) −2.48168e12 −0.334750
\(53\) 4.04765e12 0.473297 0.236648 − 0.971595i \(-0.423951\pi\)
0.236648 + 0.971595i \(0.423951\pi\)
\(54\) −4.64405e12 −0.472002
\(55\) 5.68402e12 0.503426
\(56\) 0 0
\(57\) −8.74376e12 −0.592429
\(58\) −9.97738e11 −0.0593344
\(59\) 1.25992e13 0.659105 0.329552 − 0.944137i \(-0.393102\pi\)
0.329552 + 0.944137i \(0.393102\pi\)
\(60\) −1.79292e12 −0.0826848
\(61\) 3.99250e13 1.62657 0.813283 − 0.581869i \(-0.197678\pi\)
0.813283 + 0.581869i \(0.197678\pi\)
\(62\) −1.21641e13 −0.438675
\(63\) 0 0
\(64\) 4.39805e12 0.125000
\(65\) −1.22781e13 −0.310657
\(66\) 1.21169e13 0.273409
\(67\) −4.84238e13 −0.976108 −0.488054 − 0.872814i \(-0.662293\pi\)
−0.488054 + 0.872814i \(0.662293\pi\)
\(68\) 4.09201e12 0.0738108
\(69\) 2.85244e13 0.461157
\(70\) 0 0
\(71\) 3.76931e13 0.491840 0.245920 − 0.969290i \(-0.420910\pi\)
0.245920 + 0.969290i \(0.420910\pi\)
\(72\) 2.62698e13 0.308647
\(73\) −1.41416e14 −1.49823 −0.749114 − 0.662442i \(-0.769520\pi\)
−0.749114 + 0.662442i \(0.769520\pi\)
\(74\) 1.11371e14 1.06545
\(75\) 3.23283e13 0.279655
\(76\) 1.06117e14 0.831155
\(77\) 0 0
\(78\) −2.61739e13 −0.168717
\(79\) 2.47021e14 1.44720 0.723602 − 0.690217i \(-0.242485\pi\)
0.723602 + 0.690217i \(0.242485\pi\)
\(80\) 2.17594e13 0.116004
\(81\) 1.30760e14 0.635093
\(82\) 1.28981e14 0.571377
\(83\) −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
\(86\) −1.98410e13 −0.0614928
\(87\) −1.05230e13 −0.0299051
\(88\) −1.47055e14 −0.383583
\(89\) 5.83963e12 0.0139946 0.00699730 − 0.999976i \(-0.497773\pi\)
0.00699730 + 0.999976i \(0.497773\pi\)
\(90\) 1.29970e14 0.286434
\(91\) 0 0
\(92\) −3.46181e14 −0.646985
\(93\) −1.28293e14 −0.221096
\(94\) −3.26652e14 −0.519549
\(95\) 5.25014e14 0.771336
\(96\) 4.63856e13 0.0630013
\(97\) −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.a.b.1.1 1
7.2 even 3 98.16.c.c.67.1 2
7.3 odd 6 98.16.c.b.79.1 2
7.4 even 3 98.16.c.c.79.1 2
7.5 odd 6 98.16.c.b.67.1 2
7.6 odd 2 14.16.a.a.1.1 ✓ 1
21.20 even 2 126.16.a.e.1.1 1
28.27 even 2 112.16.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.16.a.a.1.1 ✓ 1 7.6 odd 2
98.16.a.b.1.1 1 1.1 even 1 trivial
98.16.c.b.67.1 2 7.5 odd 6
98.16.c.b.79.1 2 7.3 odd 6
98.16.c.c.67.1 2 7.2 even 3
98.16.c.c.79.1 2 7.4 even 3
112.16.a.a.1.1 1 28.27 even 2
126.16.a.e.1.1 1 21.20 even 2