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Properties

Label 1.96.a.a.1.7
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 1.7
Root \(9.80521e12\) 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)
 
\(f(q)\) \(=\) \(q+2.34596e14 q^{2} -5.48368e22 q^{3} +1.54210e28 q^{4} -2.52565e33 q^{5} -1.28645e37 q^{6} +4.80687e39 q^{7} -5.67559e42 q^{8} +8.86182e44 q^{9} -5.92506e47 q^{10} -3.16113e49 q^{11} -8.45638e50 q^{12} -9.97774e52 q^{13} +1.12767e54 q^{14} +1.38499e56 q^{15} -1.94236e57 q^{16} -3.48353e57 q^{17} +2.07894e59 q^{18} -1.04863e61 q^{19} -3.89480e61 q^{20} -2.63593e62 q^{21} -7.41586e63 q^{22} +1.70644e64 q^{23} +3.11231e65 q^{24} +3.85455e66 q^{25} -2.34073e67 q^{26} +6.77078e67 q^{27} +7.41266e67 q^{28} +1.55539e69 q^{29} +3.24912e70 q^{30} +1.02679e71 q^{31} -2.30835e71 q^{32} +1.73346e72 q^{33} -8.17221e71 q^{34} -1.21405e73 q^{35} +1.36658e73 q^{36} -2.81257e74 q^{37} -2.46005e75 q^{38} +5.47148e75 q^{39} +1.43346e76 q^{40} +3.26782e76 q^{41} -6.18378e76 q^{42} +7.50961e75 q^{43} -4.87477e77 q^{44} -2.23818e78 q^{45} +4.00322e78 q^{46} -2.42784e79 q^{47} +1.06513e80 q^{48} -1.69342e80 q^{49} +9.04260e80 q^{50} +1.91026e80 q^{51} -1.53867e81 q^{52} -1.40525e82 q^{53} +1.58839e82 q^{54} +7.98389e82 q^{55} -2.72818e82 q^{56} +5.75037e83 q^{57} +3.64887e83 q^{58} -1.17208e84 q^{59} +2.13578e84 q^{60} +7.58248e84 q^{61} +2.40880e85 q^{62} +4.25976e84 q^{63} +2.27918e85 q^{64} +2.52003e86 q^{65} +4.06662e86 q^{66} -8.84858e86 q^{67} -5.37195e85 q^{68} -9.35755e86 q^{69} -2.84810e87 q^{70} +6.46617e86 q^{71} -5.02961e87 q^{72} +1.01177e88 q^{73} -6.59817e88 q^{74} -2.11371e89 q^{75} -1.61710e89 q^{76} -1.51951e89 q^{77} +1.28358e90 q^{78} +1.04527e89 q^{79} +4.90571e90 q^{80} -5.59238e90 q^{81} +7.66617e90 q^{82} +1.05596e91 q^{83} -4.06487e90 q^{84} +8.79818e90 q^{85} +1.76172e90 q^{86} -8.52926e91 q^{87} +1.79413e92 q^{88} -3.90606e92 q^{89} -5.25068e92 q^{90} -4.79617e92 q^{91} +2.63149e92 q^{92} -5.63058e93 q^{93} -5.69559e93 q^{94} +2.64848e94 q^{95} +1.26583e94 q^{96} +3.16941e94 q^{97} -3.97269e94 q^{98} -2.80133e94 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 5835659138280 q^{2} - 95\!\cdots\!80 q^{3} + 20\!\cdots\!84 q^{4} + 19\!\cdots\!60 q^{5} + 10\!\cdots\!76 q^{6} + 31\!\cdots\!00 q^{7} - 14\!\cdots\!60 q^{8} + 92\!\cdots\!36 q^{9} - 35\!\cdots\!40 q^{10}+ \cdots + 30\!\cdots\!72 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.34596e14 1.17868 0.589339 − 0.807886i \(-0.299388\pi\)
0.589339 + 0.807886i \(0.299388\pi\)
\(3\) −5.48368e22 −1.19073 −0.595364 − 0.803456i \(-0.702992\pi\)
−0.595364 + 0.803456i \(0.702992\pi\)
\(4\) 1.54210e28 0.389280
\(5\) −2.52565e33 −1.58964 −0.794818 − 0.606847i \(-0.792434\pi\)
−0.794818 + 0.606847i \(0.792434\pi\)
\(6\) −1.28645e37 −1.40348
\(7\) 4.80687e39 0.346502 0.173251 − 0.984878i \(-0.444573\pi\)
0.173251 + 0.984878i \(0.444573\pi\)
\(8\) −5.67559e42 −0.719841
\(9\) 8.86182e44 0.417834
\(10\) −5.92506e47 −1.87367
\(11\) −3.16113e49 −1.08066 −0.540330 − 0.841453i \(-0.681701\pi\)
−0.540330 + 0.841453i \(0.681701\pi\)
\(12\) −8.45638e50 −0.463527
\(13\) −9.97774e52 −1.22102 −0.610510 − 0.792008i \(-0.709036\pi\)
−0.610510 + 0.792008i \(0.709036\pi\)
\(14\) 1.12767e54 0.408414
\(15\) 1.38499e56 1.89283
\(16\) −1.94236e57 −1.23774
\(17\) −3.48353e57 −0.124651 −0.0623256 − 0.998056i \(-0.519852\pi\)
−0.0623256 + 0.998056i \(0.519852\pi\)
\(18\) 2.07894e59 0.492491
\(19\) −1.04863e61 −1.90470 −0.952352 − 0.305002i \(-0.901343\pi\)
−0.952352 + 0.305002i \(0.901343\pi\)
\(20\) −3.89480e61 −0.618814
\(21\) −2.63593e62 −0.412589
\(22\) −7.41586e63 −1.27375
\(23\) 1.70644e64 0.354828 0.177414 − 0.984136i \(-0.443227\pi\)
0.177414 + 0.984136i \(0.443227\pi\)
\(24\) 3.11231e65 0.857135
\(25\) 3.85455e66 1.52694
\(26\) −2.34073e67 −1.43919
\(27\) 6.77078e67 0.693202
\(28\) 7.41266e67 0.134886
\(29\) 1.55539e69 0.534483 0.267242 − 0.963630i \(-0.413888\pi\)
0.267242 + 0.963630i \(0.413888\pi\)
\(30\) 3.24912e70 2.23103
\(31\) 1.02679e71 1.48525 0.742626 − 0.669706i \(-0.233580\pi\)
0.742626 + 0.669706i \(0.233580\pi\)
\(32\) −2.30835e71 −0.739056
\(33\) 1.73346e72 1.28677
\(34\) −8.17221e71 −0.146923
\(35\) −1.21405e73 −0.550812
\(36\) 1.36658e73 0.162655
\(37\) −2.81257e74 −0.911007 −0.455503 − 0.890234i \(-0.650541\pi\)
−0.455503 + 0.890234i \(0.650541\pi\)
\(38\) −2.46005e75 −2.24503
\(39\) 5.47148e75 1.45390
\(40\) 1.43346e76 1.14429
\(41\) 3.26782e76 0.807282 0.403641 − 0.914917i \(-0.367744\pi\)
0.403641 + 0.914917i \(0.367744\pi\)
\(42\) −6.18378e76 −0.486310
\(43\) 7.50961e75 0.0193137 0.00965686 − 0.999953i \(-0.496926\pi\)
0.00965686 + 0.999953i \(0.496926\pi\)
\(44\) −4.87477e77 −0.420680
\(45\) −2.23818e78 −0.664204
\(46\) 4.00322e78 0.418227
\(47\) −2.42784e79 −0.913211 −0.456605 − 0.889669i \(-0.650935\pi\)
−0.456605 + 0.889669i \(0.650935\pi\)
\(48\) 1.06513e80 1.47381
\(49\) −1.69342e80 −0.879937
\(50\) 9.04260e80 1.79978
\(51\) 1.91026e80 0.148426
\(52\) −1.53867e81 −0.475320
\(53\) −1.40525e82 −1.75650 −0.878252 − 0.478197i \(-0.841290\pi\)
−0.878252 + 0.478197i \(0.841290\pi\)
\(54\) 1.58839e82 0.817061
\(55\) 7.98389e82 1.71786
\(56\) −2.72818e82 −0.249426
\(57\) 5.75037e83 2.26798
\(58\) 3.64887e83 0.629983
\(59\) −1.17208e84 −0.898437 −0.449218 − 0.893422i \(-0.648297\pi\)
−0.449218 + 0.893422i \(0.648297\pi\)
\(60\) 2.13578e84 0.736840
\(61\) 7.58248e84 1.19301 0.596505 − 0.802609i \(-0.296555\pi\)
0.596505 + 0.802609i \(0.296555\pi\)
\(62\) 2.40880e85 1.75063
\(63\) 4.25976e84 0.144780
\(64\) 2.27918e85 0.366632
\(65\) 2.52003e86 1.94098
\(66\) 4.06662e86 1.51669
\(67\) −8.84858e86 −1.61555 −0.807775 − 0.589491i \(-0.799328\pi\)
−0.807775 + 0.589491i \(0.799328\pi\)
\(68\) −5.37195e85 −0.0485242
\(69\) −9.35755e86 −0.422503
\(70\) −2.84810e87 −0.649229
\(71\) 6.46617e86 0.0751407 0.0375704 − 0.999294i \(-0.488038\pi\)
0.0375704 + 0.999294i \(0.488038\pi\)
\(72\) −5.02961e87 −0.300774
\(73\) 1.01177e88 0.314229 0.157115 − 0.987580i \(-0.449781\pi\)
0.157115 + 0.987580i \(0.449781\pi\)
\(74\) −6.59817e88 −1.07378
\(75\) −2.11371e89 −1.81818
\(76\) −1.61710e89 −0.741464
\(77\) −1.51951e89 −0.374450
\(78\) 1.28358e90 1.71368
\(79\) 1.04527e89 0.0761980 0.0380990 − 0.999274i \(-0.487870\pi\)
0.0380990 + 0.999274i \(0.487870\pi\)
\(80\) 4.90571e90 1.96756
\(81\) −5.59238e90 −1.24325
\(82\) 7.66617e90 0.951525
\(83\) 1.05596e91 0.736952 0.368476 − 0.929637i \(-0.379880\pi\)
0.368476 + 0.929637i \(0.379880\pi\)
\(84\) −4.06487e90 −0.160613
\(85\) 8.79818e90 0.198150
\(86\) 1.76172e90 0.0227646
\(87\) −8.52926e91 −0.636424
\(88\) 1.79413e92 0.777904
\(89\) −3.90606e92 −0.990178 −0.495089 − 0.868842i \(-0.664865\pi\)
−0.495089 + 0.868842i \(0.664865\pi\)
\(90\) −5.25068e92 −0.782882
\(91\) −4.79617e92 −0.423086
\(92\) 2.63149e92 0.138128
\(93\) −5.63058e93 −1.76853
\(94\) −5.69559e93 −1.07638
\(95\) 2.64848e94 3.02779
\(96\) 1.26583e94 0.880015
\(97\) 3.16941e94 1.34685 0.673427 − 0.739253i \(-0.264821\pi\)
0.673427 + 0.739253i \(0.264821\pi\)
\(98\) −3.97269e94 −1.03716
\(99\) −2.80133e94 −0.451536
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.1.7 ✓ 8
3.2 odd 2 9.96.a.c.1.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.96.a.a.1.7 ✓ 8 1.1 even 1 trivial
9.96.a.c.1.2 8 3.2 odd 2