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Properties

Label 1.96.a.a.1.8
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 1.8
Root \(1.62871e13\) 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)
 
\(f(q)\) \(=\) \(q+3.90162e14 q^{2} +2.16924e22 q^{3} +1.12612e29 q^{4} +1.39914e33 q^{5} +8.46353e36 q^{6} +1.75361e40 q^{7} +2.84812e43 q^{8} -1.65034e45 q^{9} +5.45891e47 q^{10} -5.41264e48 q^{11} +2.44283e51 q^{12} +1.04831e52 q^{13} +6.84194e54 q^{14} +3.03506e55 q^{15} +6.65124e57 q^{16} -2.56029e58 q^{17} -6.43899e59 q^{18} -4.34204e60 q^{19} +1.57560e62 q^{20} +3.80400e62 q^{21} -2.11181e63 q^{22} -7.48379e64 q^{23} +6.17824e65 q^{24} -5.66766e65 q^{25} +4.09009e66 q^{26} -8.18069e67 q^{27} +1.97479e69 q^{28} +2.03053e69 q^{29} +1.18417e70 q^{30} +3.72805e70 q^{31} +1.46681e72 q^{32} -1.17413e71 q^{33} -9.98929e72 q^{34} +2.45355e73 q^{35} -1.85848e74 q^{36} -1.43964e74 q^{37} -1.69410e75 q^{38} +2.27402e74 q^{39} +3.98491e76 q^{40} -4.16311e76 q^{41} +1.48418e77 q^{42} +4.94148e76 q^{43} -6.09530e77 q^{44} -2.30905e78 q^{45} -2.91989e79 q^{46} +4.71948e79 q^{47} +1.44281e80 q^{48} +1.15068e80 q^{49} -2.21131e80 q^{50} -5.55388e80 q^{51} +1.18052e81 q^{52} +9.21253e80 q^{53} -3.19180e82 q^{54} -7.57304e81 q^{55} +4.99450e83 q^{56} -9.41891e82 q^{57} +7.92238e83 q^{58} -1.44917e84 q^{59} +3.41785e84 q^{60} +2.68780e84 q^{61} +1.45454e85 q^{62} -2.89406e85 q^{63} +3.08809e86 q^{64} +1.46673e85 q^{65} -4.58101e85 q^{66} +3.24337e86 q^{67} -2.88321e87 q^{68} -1.62341e87 q^{69} +9.57282e87 q^{70} +2.78222e87 q^{71} -4.70035e88 q^{72} +2.94316e88 q^{73} -5.61691e88 q^{74} -1.22945e88 q^{75} -4.88967e89 q^{76} -9.49169e88 q^{77} +8.87238e88 q^{78} +9.20074e89 q^{79} +9.30600e90 q^{80} +1.72561e90 q^{81} -1.62429e91 q^{82} -2.26821e90 q^{83} +4.28378e91 q^{84} -3.58220e91 q^{85} +1.92798e91 q^{86} +4.40471e91 q^{87} -1.54158e92 q^{88} +3.95651e92 q^{89} -9.00904e92 q^{90} +1.83833e92 q^{91} -8.42767e93 q^{92} +8.08702e92 q^{93} +1.84136e94 q^{94} -6.07512e93 q^{95} +3.18185e94 q^{96} -3.21613e94 q^{97} +4.48953e94 q^{98} +8.93268e93 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 5835659138280 q^{2} - 95\!\cdots\!80 q^{3} + 20\!\cdots\!84 q^{4} + 19\!\cdots\!60 q^{5} + 10\!\cdots\!76 q^{6} + 31\!\cdots\!00 q^{7} - 14\!\cdots\!60 q^{8} + 92\!\cdots\!36 q^{9} - 35\!\cdots\!40 q^{10}+ \cdots + 30\!\cdots\!72 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.90162e14 1.96029 0.980145 − 0.198283i \(-0.0635364\pi\)
0.980145 + 0.198283i \(0.0635364\pi\)
\(3\) 2.16924e22 0.471028 0.235514 − 0.971871i \(-0.424323\pi\)
0.235514 + 0.971871i \(0.424323\pi\)
\(4\) 1.12612e29 2.84274
\(5\) 1.39914e33 0.880614 0.440307 − 0.897847i \(-0.354870\pi\)
0.440307 + 0.897847i \(0.354870\pi\)
\(6\) 8.46353e36 0.923352
\(7\) 1.75361e40 1.26409 0.632044 − 0.774932i \(-0.282216\pi\)
0.632044 + 0.774932i \(0.282216\pi\)
\(8\) 2.84812e43 3.61230
\(9\) −1.65034e45 −0.778132
\(10\) 5.45891e47 1.72626
\(11\) −5.41264e48 −0.185036 −0.0925181 − 0.995711i \(-0.529492\pi\)
−0.0925181 + 0.995711i \(0.529492\pi\)
\(12\) 2.44283e51 1.33901
\(13\) 1.04831e52 0.128286 0.0641430 − 0.997941i \(-0.479569\pi\)
0.0641430 + 0.997941i \(0.479569\pi\)
\(14\) 6.84194e54 2.47798
\(15\) 3.03506e55 0.414794
\(16\) 6.65124e57 4.23841
\(17\) −2.56029e58 −0.916148 −0.458074 − 0.888914i \(-0.651461\pi\)
−0.458074 + 0.888914i \(0.651461\pi\)
\(18\) −6.43899e59 −1.52536
\(19\) −4.34204e60 −0.788674 −0.394337 − 0.918966i \(-0.629026\pi\)
−0.394337 + 0.918966i \(0.629026\pi\)
\(20\) 1.57560e62 2.50335
\(21\) 3.80400e62 0.595421
\(22\) −2.11181e63 −0.362725
\(23\) −7.48379e64 −1.55614 −0.778071 − 0.628177i \(-0.783802\pi\)
−0.778071 + 0.628177i \(0.783802\pi\)
\(24\) 6.17824e65 1.70149
\(25\) −5.66766e65 −0.224519
\(26\) 4.09009e66 0.251478
\(27\) −8.18069e67 −0.837551
\(28\) 1.97479e69 3.59347
\(29\) 2.03053e69 0.697759 0.348880 − 0.937168i \(-0.386562\pi\)
0.348880 + 0.937168i \(0.386562\pi\)
\(30\) 1.18417e70 0.813117
\(31\) 3.72805e70 0.539264 0.269632 − 0.962963i \(-0.413098\pi\)
0.269632 + 0.962963i \(0.413098\pi\)
\(32\) 1.46681e72 4.69622
\(33\) −1.17413e71 −0.0871573
\(34\) −9.98929e72 −1.79592
\(35\) 2.45355e73 1.11317
\(36\) −1.85848e74 −2.21202
\(37\) −1.43964e74 −0.466305 −0.233153 − 0.972440i \(-0.574904\pi\)
−0.233153 + 0.972440i \(0.574904\pi\)
\(38\) −1.69410e75 −1.54603
\(39\) 2.27402e74 0.0604263
\(40\) 3.98491e76 3.18104
\(41\) −4.16311e76 −1.02845 −0.514227 − 0.857654i \(-0.671921\pi\)
−0.514227 + 0.857654i \(0.671921\pi\)
\(42\) 1.48418e77 1.16720
\(43\) 4.94148e76 0.127088 0.0635441 − 0.997979i \(-0.479760\pi\)
0.0635441 + 0.997979i \(0.479760\pi\)
\(44\) −6.09530e77 −0.526009
\(45\) −2.30905e78 −0.685234
\(46\) −2.91989e79 −3.05049
\(47\) 4.71948e79 1.77519 0.887596 − 0.460622i \(-0.152374\pi\)
0.887596 + 0.460622i \(0.152374\pi\)
\(48\) 1.44281e80 1.99641
\(49\) 1.15068e80 0.597919
\(50\) −2.21131e80 −0.440122
\(51\) −5.55388e80 −0.431532
\(52\) 1.18052e81 0.364683
\(53\) 9.21253e80 0.115153 0.0575765 − 0.998341i \(-0.481663\pi\)
0.0575765 + 0.998341i \(0.481663\pi\)
\(54\) −3.19180e82 −1.64184
\(55\) −7.57304e81 −0.162945
\(56\) 4.99450e83 4.56626
\(57\) −9.41891e82 −0.371488
\(58\) 7.92238e83 1.36781
\(59\) −1.44917e84 −1.11083 −0.555417 − 0.831572i \(-0.687442\pi\)
−0.555417 + 0.831572i \(0.687442\pi\)
\(60\) 3.41785e84 1.17915
\(61\) 2.68780e84 0.422892 0.211446 − 0.977390i \(-0.432183\pi\)
0.211446 + 0.977390i \(0.432183\pi\)
\(62\) 1.45454e85 1.05711
\(63\) −2.89406e85 −0.983628
\(64\) 3.08809e86 4.96754
\(65\) 1.46673e85 0.112970
\(66\) −4.58101e85 −0.170854
\(67\) 3.24337e86 0.592165 0.296083 − 0.955162i \(-0.404320\pi\)
0.296083 + 0.955162i \(0.404320\pi\)
\(68\) −2.88321e87 −2.60437
\(69\) −1.62341e87 −0.732987
\(70\) 9.57282e87 2.18214
\(71\) 2.78222e87 0.323311 0.161656 − 0.986847i \(-0.448317\pi\)
0.161656 + 0.986847i \(0.448317\pi\)
\(72\) −4.70035e88 −2.81084
\(73\) 2.94316e88 0.914067 0.457033 − 0.889450i \(-0.348912\pi\)
0.457033 + 0.889450i \(0.348912\pi\)
\(74\) −5.61691e88 −0.914094
\(75\) −1.22945e88 −0.105755
\(76\) −4.88967e89 −2.24199
\(77\) −9.49169e88 −0.233902
\(78\) 8.87238e88 0.118453
\(79\) 9.20074e89 0.670712 0.335356 − 0.942092i \(-0.391143\pi\)
0.335356 + 0.942092i \(0.391143\pi\)
\(80\) 9.30600e90 3.73241
\(81\) 1.72561e90 0.383622
\(82\) −1.62429e91 −2.01607
\(83\) −2.26821e90 −0.158298 −0.0791488 − 0.996863i \(-0.525220\pi\)
−0.0791488 + 0.996863i \(0.525220\pi\)
\(84\) 4.28378e91 1.69263
\(85\) −3.58220e91 −0.806773
\(86\) 1.92798e91 0.249130
\(87\) 4.40471e91 0.328664
\(88\) −1.54158e92 −0.668406
\(89\) 3.95651e92 1.00297 0.501484 − 0.865167i \(-0.332788\pi\)
0.501484 + 0.865167i \(0.332788\pi\)
\(90\) −9.00904e92 −1.34326
\(91\) 1.83833e92 0.162165
\(92\) −8.42767e93 −4.42370
\(93\) 8.08702e92 0.254009
\(94\) 1.84136e94 3.47989
\(95\) −6.07512e93 −0.694518
\(96\) 3.18185e94 2.21205
\(97\) −3.21613e94 −1.36671 −0.683354 − 0.730087i \(-0.739479\pi\)
−0.683354 + 0.730087i \(0.739479\pi\)
\(98\) 4.48953e94 1.17209
\(99\) 8.93268e93 0.143983
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.1.8 ✓ 8
3.2 odd 2 9.96.a.c.1.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.96.a.a.1.8 ✓ 8 1.1 even 1 trivial
9.96.a.c.1.1 8 3.2 odd 2