Properties

Label 98.8.a.c.1.1
Level $98$
Weight $8$
Character 98.1
Self dual yes
Analytic conductor $30.614$
Analytic rank $0$
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] = [98,8,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, 8); 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, 8, names="a")
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 98.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,8,66,64,400,528,0,512,2169] 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: \(30.6137324974\)
Analytic rank: \(0\)
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)
 
\(f(q)\) \(=\) \(q+8.00000 q^{2} +66.0000 q^{3} +64.0000 q^{4} +400.000 q^{5} +528.000 q^{6} +512.000 q^{8} +2169.00 q^{9} +3200.00 q^{10} +40.0000 q^{11} +4224.00 q^{12} +4452.00 q^{13} +26400.0 q^{15} +4096.00 q^{16} -36502.0 q^{17} +17352.0 q^{18} +46222.0 q^{19} +25600.0 q^{20} +320.000 q^{22} -105200. q^{23} +33792.0 q^{24} +81875.0 q^{25} +35616.0 q^{26} -1188.00 q^{27} -126334. q^{29} +211200. q^{30} +170964. q^{31} +32768.0 q^{32} +2640.00 q^{33} -292016. q^{34} +138816. q^{36} +20954.0 q^{37} +369776. q^{38} +293832. q^{39} +204800. q^{40} -318486. q^{41} +77744.0 q^{43} +2560.00 q^{44} +867600. q^{45} -841600. q^{46} -703716. q^{47} +270336. q^{48} +655000. q^{50} -2.40913e6 q^{51} +284928. q^{52} +1.60328e6 q^{53} -9504.00 q^{54} +16000.0 q^{55} +3.05065e6 q^{57} -1.01067e6 q^{58} +1.17189e6 q^{59} +1.68960e6 q^{60} +2.06887e6 q^{61} +1.36771e6 q^{62} +262144. q^{64} +1.78080e6 q^{65} +21120.0 q^{66} -994268. q^{67} -2.33613e6 q^{68} -6.94320e6 q^{69} +33280.0 q^{71} +1.11053e6 q^{72} +2.97145e6 q^{73} +167632. q^{74} +5.40375e6 q^{75} +2.95821e6 q^{76} +2.35066e6 q^{78} -2.37617e6 q^{79} +1.63840e6 q^{80} -4.82201e6 q^{81} -2.54789e6 q^{82} +2.12236e6 q^{83} -1.46008e7 q^{85} +621952. q^{86} -8.33804e6 q^{87} +20480.0 q^{88} -6.92035e6 q^{89} +6.94080e6 q^{90} -6.73280e6 q^{92} +1.12836e7 q^{93} -5.62973e6 q^{94} +1.84888e7 q^{95} +2.16269e6 q^{96} -4.95271e6 q^{97} +86760.0 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000 0.707107
\(3\) 66.0000 1.41130 0.705650 − 0.708560i \(-0.250655\pi\)
0.705650 + 0.708560i \(0.250655\pi\)
\(4\) 64.0000 0.500000
\(5\) 400.000 1.43108 0.715542 − 0.698570i \(-0.246180\pi\)
0.715542 + 0.698570i \(0.246180\pi\)
\(6\) 528.000 0.997940
\(7\) 0 0
\(8\) 512.000 0.353553
\(9\) 2169.00 0.991770
\(10\) 3200.00 1.01193
\(11\) 40.0000 0.00906120 0.00453060 − 0.999990i \(-0.498558\pi\)
0.00453060 + 0.999990i \(0.498558\pi\)
\(12\) 4224.00 0.705650
\(13\) 4452.00 0.562022 0.281011 − 0.959705i \(-0.409330\pi\)
0.281011 + 0.959705i \(0.409330\pi\)
\(14\) 0 0
\(15\) 26400.0 2.01969
\(16\) 4096.00 0.250000
\(17\) −36502.0 −1.80196 −0.900981 − 0.433859i \(-0.857151\pi\)
−0.900981 + 0.433859i \(0.857151\pi\)
\(18\) 17352.0 0.701287
\(19\) 46222.0 1.54601 0.773003 − 0.634402i \(-0.218754\pi\)
0.773003 + 0.634402i \(0.218754\pi\)
\(20\) 25600.0 0.715542
\(21\) 0 0
\(22\) 320.000 0.00640723
\(23\) −105200. −1.80289 −0.901443 − 0.432898i \(-0.857491\pi\)
−0.901443 + 0.432898i \(0.857491\pi\)
\(24\) 33792.0 0.498970
\(25\) 81875.0 1.04800
\(26\) 35616.0 0.397410
\(27\) −1188.00 −0.0116156
\(28\) 0 0
\(29\) −126334. −0.961894 −0.480947 − 0.876750i \(-0.659707\pi\)
−0.480947 + 0.876750i \(0.659707\pi\)
\(30\) 211200. 1.42814
\(31\) 170964. 1.03072 0.515358 − 0.856975i \(-0.327659\pi\)
0.515358 + 0.856975i \(0.327659\pi\)
\(32\) 32768.0 0.176777
\(33\) 2640.00 0.0127881
\(34\) −292016. −1.27418
\(35\) 0 0
\(36\) 138816. 0.495885
\(37\) 20954.0 0.0680081 0.0340041 − 0.999422i \(-0.489174\pi\)
0.0340041 + 0.999422i \(0.489174\pi\)
\(38\) 369776. 1.09319
\(39\) 293832. 0.793182
\(40\) 204800. 0.505964
\(41\) −318486. −0.721684 −0.360842 − 0.932627i \(-0.617511\pi\)
−0.360842 + 0.932627i \(0.617511\pi\)
\(42\) 0 0
\(43\) 77744.0 0.149117 0.0745585 − 0.997217i \(-0.476245\pi\)
0.0745585 + 0.997217i \(0.476245\pi\)
\(44\) 2560.00 0.00453060
\(45\) 867600. 1.41931
\(46\) −841600. −1.27483
\(47\) −703716. −0.988678 −0.494339 − 0.869269i \(-0.664590\pi\)
−0.494339 + 0.869269i \(0.664590\pi\)
\(48\) 270336. 0.352825
\(49\) 0 0
\(50\) 655000. 0.741048
\(51\) −2.40913e6 −2.54311
\(52\) 284928. 0.281011
\(53\) 1.60328e6 1.47926 0.739628 − 0.673016i \(-0.235002\pi\)
0.739628 + 0.673016i \(0.235002\pi\)
\(54\) −9504.00 −0.00821350
\(55\) 16000.0 0.0129673
\(56\) 0 0
\(57\) 3.05065e6 2.18188
\(58\) −1.01067e6 −0.680162
\(59\) 1.17189e6 0.742859 0.371429 − 0.928461i \(-0.378868\pi\)
0.371429 + 0.928461i \(0.378868\pi\)
\(60\) 1.68960e6 1.00984
\(61\) 2.06887e6 1.16702 0.583511 − 0.812105i \(-0.301678\pi\)
0.583511 + 0.812105i \(0.301678\pi\)
\(62\) 1.36771e6 0.728826
\(63\) 0 0
\(64\) 262144. 0.125000
\(65\) 1.78080e6 0.804301
\(66\) 21120.0 0.00904253
\(67\) −994268. −0.403870 −0.201935 − 0.979399i \(-0.564723\pi\)
−0.201935 + 0.979399i \(0.564723\pi\)
\(68\) −2.33613e6 −0.900981
\(69\) −6.94320e6 −2.54441
\(70\) 0 0
\(71\) 33280.0 0.0110352 0.00551759 − 0.999985i \(-0.498244\pi\)
0.00551759 + 0.999985i \(0.498244\pi\)
\(72\) 1.11053e6 0.350643
\(73\) 2.97145e6 0.894003 0.447002 − 0.894533i \(-0.352492\pi\)
0.447002 + 0.894533i \(0.352492\pi\)
\(74\) 167632. 0.0480890
\(75\) 5.40375e6 1.47904
\(76\) 2.95821e6 0.773003
\(77\) 0 0
\(78\) 2.35066e6 0.560865
\(79\) −2.37617e6 −0.542228 −0.271114 − 0.962547i \(-0.587392\pi\)
−0.271114 + 0.962547i \(0.587392\pi\)
\(80\) 1.63840e6 0.357771
\(81\) −4.82201e6 −1.00816
\(82\) −2.54789e6 −0.510307
\(83\) 2.12236e6 0.407423 0.203711 − 0.979031i \(-0.434700\pi\)
0.203711 + 0.979031i \(0.434700\pi\)
\(84\) 0 0
\(85\) −1.46008e7 −2.57876
\(86\) 621952. 0.105442
\(87\) −8.33804e6 −1.35752
\(88\) 20480.0 0.00320362
\(89\) −6.92035e6 −1.04055 −0.520275 − 0.853999i \(-0.674170\pi\)
−0.520275 + 0.853999i \(0.674170\pi\)
\(90\) 6.94080e6 1.00360
\(91\) 0 0
\(92\) −6.73280e6 −0.901443
\(93\) 1.12836e7 1.45465
\(94\) −5.62973e6 −0.699101
\(95\) 1.84888e7 2.21246
\(96\) 2.16269e6 0.249485
\(97\) −4.95271e6 −0.550988 −0.275494 − 0.961303i \(-0.588841\pi\)
−0.275494 + 0.961303i \(0.588841\pi\)
\(98\) 0 0
\(99\) 86760.0 0.00898662
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 98.8.a.c.1.1 1
7.2 even 3 98.8.c.a.67.1 2
7.3 odd 6 98.8.c.b.79.1 2
7.4 even 3 98.8.c.a.79.1 2
7.5 odd 6 98.8.c.b.67.1 2
7.6 odd 2 14.8.a.b.1.1 ✓ 1
21.20 even 2 126.8.a.c.1.1 1
28.27 even 2 112.8.a.d.1.1 1
35.13 even 4 350.8.c.b.99.1 2
35.27 even 4 350.8.c.b.99.2 2
35.34 odd 2 350.8.a.d.1.1 1
56.13 odd 2 448.8.a.i.1.1 1
56.27 even 2 448.8.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.8.a.b.1.1 ✓ 1 7.6 odd 2
98.8.a.c.1.1 1 1.1 even 1 trivial
98.8.c.a.67.1 2 7.2 even 3
98.8.c.a.79.1 2 7.4 even 3
98.8.c.b.67.1 2 7.5 odd 6
98.8.c.b.79.1 2 7.3 odd 6
112.8.a.d.1.1 1 28.27 even 2
126.8.a.c.1.1 1 21.20 even 2
350.8.a.d.1.1 1 35.34 odd 2
350.8.c.b.99.1 2 35.13 even 4
350.8.c.b.99.2 2 35.27 even 4
448.8.a.b.1.1 1 56.27 even 2
448.8.a.i.1.1 1 56.13 odd 2