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Properties

Label 1.96.a.a.1.5
Level $1$
Weight $96$
Character 1.1
Self dual yes
Analytic conductor $57.154$
Analytic rank $0$
Dimension $8$
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] = [1,96,Mod(1,1)] 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("1.1"); S:= CuspForms(chi, 96); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1, base_ring=CyclotomicField(1)) chi = DirichletCharacter(H, H._module([])) N = Newforms(chi, 96, names="a")
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 96 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(57.1535908815\)
Analytic rank: \(0\)
Dimension: \(8\)
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: \( x^{8} - x^{7} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{104}\cdot 3^{38}\cdot 5^{12}\cdot 7^{7}\cdot 11\cdot 13\cdot 19^{3} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(4.79736e12\) of defining polynomial
Character \(\chi\) \(=\) 1.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.14407e14 q^{2} +8.75958e22 q^{3} -2.65251e28 q^{4} -1.07741e33 q^{5} +1.00216e37 q^{6} +1.86215e40 q^{7} -7.56680e42 q^{8} +5.55214e45 q^{9} -1.23264e47 q^{10} +1.86898e49 q^{11} -2.32348e51 q^{12} -5.82728e52 q^{13} +2.13044e54 q^{14} -9.43769e55 q^{15} +1.85069e56 q^{16} -1.12709e58 q^{17} +6.35205e59 q^{18} +8.30862e60 q^{19} +2.85785e61 q^{20} +1.63117e63 q^{21} +2.13824e63 q^{22} +1.09549e64 q^{23} -6.62820e65 q^{24} -1.36353e66 q^{25} -6.66683e66 q^{26} +3.00562e68 q^{27} -4.93937e68 q^{28} +2.28760e69 q^{29} -1.07974e70 q^{30} +3.05026e70 q^{31} +3.20925e71 q^{32} +1.63715e72 q^{33} -1.28947e72 q^{34} -2.00631e73 q^{35} -1.47271e74 q^{36} +3.99750e74 q^{37} +9.50566e74 q^{38} -5.10446e75 q^{39} +8.15257e75 q^{40} -3.81589e76 q^{41} +1.86617e77 q^{42} +2.50487e77 q^{43} -4.95747e77 q^{44} -5.98195e78 q^{45} +1.25332e78 q^{46} +2.28370e79 q^{47} +1.62113e79 q^{48} +1.54313e80 q^{49} -1.55998e80 q^{50} -9.87281e80 q^{51} +1.54569e81 q^{52} -5.87919e80 q^{53} +3.43865e82 q^{54} -2.01366e82 q^{55} -1.40905e83 q^{56} +7.27800e83 q^{57} +2.61719e83 q^{58} +1.48881e84 q^{59} +2.50335e84 q^{60} -5.46013e84 q^{61} +3.48972e84 q^{62} +1.03389e86 q^{63} +2.93848e85 q^{64} +6.27839e85 q^{65} +1.87301e86 q^{66} -6.62877e86 q^{67} +2.98960e86 q^{68} +9.59605e86 q^{69} -2.29536e87 q^{70} +5.25075e87 q^{71} -4.20119e88 q^{72} -2.09882e88 q^{73} +4.57343e88 q^{74} -1.19440e89 q^{75} -2.20387e89 q^{76} +3.48032e89 q^{77} -5.83987e89 q^{78} +6.42811e89 q^{79} -1.99396e89 q^{80} +1.45525e91 q^{81} -4.36566e90 q^{82} +1.28783e91 q^{83} -4.32668e91 q^{84} +1.21434e91 q^{85} +2.86575e91 q^{86} +2.00385e92 q^{87} -1.41422e92 q^{88} -5.79584e92 q^{89} -6.84378e92 q^{90} -1.08513e93 q^{91} -2.90580e92 q^{92} +2.67190e93 q^{93} +2.61272e93 q^{94} -8.95182e93 q^{95} +2.81117e94 q^{96} -1.99071e94 q^{97} +1.76545e94 q^{98} +1.03768e95 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 5835659138280 q^{2} - 95\!\cdots\!80 q^{3} + 20\!\cdots\!84 q^{4} + 19\!\cdots\!60 q^{5} + 10\!\cdots\!76 q^{6} + 31\!\cdots\!00 q^{7} - 14\!\cdots\!60 q^{8} + 92\!\cdots\!36 q^{9} - 35\!\cdots\!40 q^{10}+ \cdots + 30\!\cdots\!72 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.14407e14 0.574816 0.287408 − 0.957808i \(-0.407206\pi\)
0.287408 + 0.957808i \(0.407206\pi\)
\(3\) 8.75958e22 1.90206 0.951029 − 0.309101i \(-0.100028\pi\)
0.951029 + 0.309101i \(0.100028\pi\)
\(4\) −2.65251e28 −0.669587
\(5\) −1.07741e33 −0.678121 −0.339061 − 0.940765i \(-0.610109\pi\)
−0.339061 + 0.940765i \(0.610109\pi\)
\(6\) 1.00216e37 1.09333
\(7\) 1.86215e40 1.34233 0.671163 − 0.741309i \(-0.265795\pi\)
0.671163 + 0.741309i \(0.265795\pi\)
\(8\) −7.56680e42 −0.959705
\(9\) 5.55214e45 2.61783
\(10\) −1.23264e47 −0.389795
\(11\) 1.86898e49 0.638927 0.319464 − 0.947599i \(-0.396497\pi\)
0.319464 + 0.947599i \(0.396497\pi\)
\(12\) −2.32348e51 −1.27359
\(13\) −5.82728e52 −0.713110 −0.356555 − 0.934274i \(-0.616049\pi\)
−0.356555 + 0.934274i \(0.616049\pi\)
\(14\) 2.13044e54 0.771591
\(15\) −9.43769e55 −1.28983
\(16\) 1.85069e56 0.117933
\(17\) −1.12709e58 −0.403305 −0.201652 − 0.979457i \(-0.564631\pi\)
−0.201652 + 0.979457i \(0.564631\pi\)
\(18\) 6.35205e59 1.50477
\(19\) 8.30862e60 1.50915 0.754576 − 0.656213i \(-0.227843\pi\)
0.754576 + 0.656213i \(0.227843\pi\)
\(20\) 2.85785e61 0.454061
\(21\) 1.63117e63 2.55318
\(22\) 2.13824e63 0.367265
\(23\) 1.09549e64 0.227791 0.113896 − 0.993493i \(-0.463667\pi\)
0.113896 + 0.993493i \(0.463667\pi\)
\(24\) −6.62820e65 −1.82542
\(25\) −1.36353e66 −0.540152
\(26\) −6.66683e66 −0.409907
\(27\) 3.00562e68 3.07720
\(28\) −4.93937e68 −0.898804
\(29\) 2.28760e69 0.786097 0.393048 − 0.919518i \(-0.371421\pi\)
0.393048 + 0.919518i \(0.371421\pi\)
\(30\) −1.07974e70 −0.741413
\(31\) 3.05026e70 0.441221 0.220610 − 0.975362i \(-0.429195\pi\)
0.220610 + 0.975362i \(0.429195\pi\)
\(32\) 3.20925e71 1.02749
\(33\) 1.63715e72 1.21528
\(34\) −1.28947e72 −0.231826
\(35\) −2.00631e73 −0.910260
\(36\) −1.47271e74 −1.75286
\(37\) 3.99750e74 1.29481 0.647406 − 0.762145i \(-0.275854\pi\)
0.647406 + 0.762145i \(0.275854\pi\)
\(38\) 9.50566e74 0.867484
\(39\) −5.10446e75 −1.35638
\(40\) 8.15257e75 0.650796
\(41\) −3.81589e76 −0.942677 −0.471338 − 0.881952i \(-0.656229\pi\)
−0.471338 + 0.881952i \(0.656229\pi\)
\(42\) 1.86617e77 1.46761
\(43\) 2.50487e77 0.644219 0.322110 − 0.946702i \(-0.395608\pi\)
0.322110 + 0.946702i \(0.395608\pi\)
\(44\) −4.95747e77 −0.427817
\(45\) −5.98195e78 −1.77520
\(46\) 1.25332e78 0.130938
\(47\) 2.28370e79 0.858994 0.429497 − 0.903068i \(-0.358691\pi\)
0.429497 + 0.903068i \(0.358691\pi\)
\(48\) 1.62113e79 0.224315
\(49\) 1.54313e80 0.801841
\(50\) −1.55998e80 −0.310488
\(51\) −9.87281e80 −0.767109
\(52\) 1.54569e81 0.477489
\(53\) −5.87919e80 −0.0734875 −0.0367438 − 0.999325i \(-0.511699\pi\)
−0.0367438 + 0.999325i \(0.511699\pi\)
\(54\) 3.43865e82 1.76882
\(55\) −2.01366e82 −0.433270
\(56\) −1.40905e83 −1.28824
\(57\) 7.27800e83 2.87049
\(58\) 2.61719e83 0.451861
\(59\) 1.48881e84 1.14122 0.570610 − 0.821221i \(-0.306707\pi\)
0.570610 + 0.821221i \(0.306707\pi\)
\(60\) 2.50335e84 0.863650
\(61\) −5.46013e84 −0.859084 −0.429542 − 0.903047i \(-0.641325\pi\)
−0.429542 + 0.903047i \(0.641325\pi\)
\(62\) 3.48972e84 0.253621
\(63\) 1.03389e86 3.51398
\(64\) 2.93848e85 0.472688
\(65\) 6.27839e85 0.483575
\(66\) 1.87301e86 0.698560
\(67\) −6.62877e86 −1.21026 −0.605132 − 0.796125i \(-0.706880\pi\)
−0.605132 + 0.796125i \(0.706880\pi\)
\(68\) 2.98960e86 0.270047
\(69\) 9.59605e86 0.433272
\(70\) −2.29536e87 −0.523232
\(71\) 5.25075e87 0.610168 0.305084 − 0.952325i \(-0.401315\pi\)
0.305084 + 0.952325i \(0.401315\pi\)
\(72\) −4.20119e88 −2.51234
\(73\) −2.09882e88 −0.651838 −0.325919 − 0.945398i \(-0.605674\pi\)
−0.325919 + 0.945398i \(0.605674\pi\)
\(74\) 4.57343e88 0.744279
\(75\) −1.19440e89 −1.02740
\(76\) −2.20387e89 −1.01051
\(77\) 3.48032e89 0.857649
\(78\) −5.83987e89 −0.779668
\(79\) 6.42811e89 0.468594 0.234297 − 0.972165i \(-0.424721\pi\)
0.234297 + 0.972165i \(0.424721\pi\)
\(80\) −1.99396e89 −0.0799727
\(81\) 1.45525e91 3.23519
\(82\) −4.36566e90 −0.541866
\(83\) 1.28783e91 0.898770 0.449385 − 0.893338i \(-0.351643\pi\)
0.449385 + 0.893338i \(0.351643\pi\)
\(84\) −4.32668e91 −1.70958
\(85\) 1.21434e91 0.273489
\(86\) 2.86575e91 0.370307
\(87\) 2.00385e92 1.49520
\(88\) −1.41422e92 −0.613182
\(89\) −5.79584e92 −1.46923 −0.734617 − 0.678482i \(-0.762638\pi\)
−0.734617 + 0.678482i \(0.762638\pi\)
\(90\) −6.84378e92 −1.02042
\(91\) −1.08513e93 −0.957227
\(92\) −2.90580e92 −0.152526
\(93\) 2.67190e93 0.839228
\(94\) 2.61272e93 0.493764
\(95\) −8.95182e93 −1.02339
\(96\) 2.81117e94 1.95436
\(97\) −1.99071e94 −0.845960 −0.422980 − 0.906139i \(-0.639016\pi\)
−0.422980 + 0.906139i \(0.639016\pi\)
\(98\) 1.76545e94 0.460911
\(99\) 1.03768e95 1.67260
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 1.96.a.a.1.5 ✓ 8
3.2 odd 2 9.96.a.c.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.96.a.a.1.5 ✓ 8 1.1 even 1 trivial
9.96.a.c.1.4 8 3.2 odd 2