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Properties

Label 1.96.a.a.1.1
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 1.1
Root \(-1.48943e13\) 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)
 
\(f(q)\) \(=\) \(q-3.58194e14 q^{2} -7.12911e22 q^{3} +8.86885e28 q^{4} -1.64420e33 q^{5} +2.55360e37 q^{6} -1.35904e40 q^{7} -1.75781e43 q^{8} +2.96152e45 q^{9} +5.88943e47 q^{10} +4.57322e49 q^{11} -6.32270e51 q^{12} +7.91094e52 q^{13} +4.86801e54 q^{14} +1.17217e56 q^{15} +2.78306e57 q^{16} -3.60982e57 q^{17} -1.06080e60 q^{18} -1.15920e60 q^{19} -1.45822e62 q^{20} +9.68877e62 q^{21} -1.63810e64 q^{22} -4.43116e64 q^{23} +1.25316e66 q^{24} +1.79050e65 q^{25} -2.83365e67 q^{26} -5.99290e67 q^{27} -1.20532e69 q^{28} +3.11912e69 q^{29} -4.19864e70 q^{30} -6.79915e70 q^{31} -3.00533e71 q^{32} -3.26030e72 q^{33} +1.29301e72 q^{34} +2.23455e73 q^{35} +2.62653e74 q^{36} -2.19553e74 q^{37} +4.15219e74 q^{38} -5.63979e75 q^{39} +2.89021e76 q^{40} -7.21825e76 q^{41} -3.47046e77 q^{42} +9.95921e76 q^{43} +4.05592e78 q^{44} -4.86934e78 q^{45} +1.58721e79 q^{46} +1.61545e79 q^{47} -1.98408e80 q^{48} -7.74788e78 q^{49} -6.41346e79 q^{50} +2.57348e80 q^{51} +7.01609e81 q^{52} -2.46146e81 q^{53} +2.14662e82 q^{54} -7.51930e82 q^{55} +2.38895e83 q^{56} +8.26407e82 q^{57} -1.11725e84 q^{58} +1.31371e84 q^{59} +1.03958e85 q^{60} +8.03762e84 q^{61} +2.43541e85 q^{62} -4.02484e85 q^{63} -2.59939e84 q^{64} -1.30072e86 q^{65} +1.16782e87 q^{66} -4.90861e86 q^{67} -3.20149e86 q^{68} +3.15902e87 q^{69} -8.00400e87 q^{70} +4.39734e87 q^{71} -5.20580e88 q^{72} -3.36142e88 q^{73} +7.86425e88 q^{74} -1.27647e88 q^{75} -1.02808e89 q^{76} -6.21521e89 q^{77} +2.02014e90 q^{78} +1.11166e89 q^{79} -4.57592e90 q^{80} -2.00867e90 q^{81} +2.58553e91 q^{82} -1.87032e91 q^{83} +8.59283e91 q^{84} +5.93527e90 q^{85} -3.56733e91 q^{86} -2.22365e92 q^{87} -8.03887e92 q^{88} +3.70282e92 q^{89} +1.74417e93 q^{90} -1.07513e93 q^{91} -3.92993e93 q^{92} +4.84719e93 q^{93} -5.78643e93 q^{94} +1.90596e93 q^{95} +2.14253e94 q^{96} -2.62608e94 q^{97} +2.77524e93 q^{98} +1.35437e95 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 5835659138280 q^{2} - 95\!\cdots\!80 q^{3} + 20\!\cdots\!84 q^{4} + 19\!\cdots\!60 q^{5} + 10\!\cdots\!76 q^{6} + 31\!\cdots\!00 q^{7} - 14\!\cdots\!60 q^{8} + 92\!\cdots\!36 q^{9} - 35\!\cdots\!40 q^{10}+ \cdots + 30\!\cdots\!72 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.58194e14 −1.79967 −0.899835 − 0.436230i \(-0.856313\pi\)
−0.899835 + 0.436230i \(0.856313\pi\)
\(3\) −7.12911e22 −1.54802 −0.774008 − 0.633176i \(-0.781751\pi\)
−0.774008 + 0.633176i \(0.781751\pi\)
\(4\) 8.86885e28 2.23881
\(5\) −1.64420e33 −1.03486 −0.517429 − 0.855726i \(-0.673111\pi\)
−0.517429 + 0.855726i \(0.673111\pi\)
\(6\) 2.55360e37 2.78592
\(7\) −1.35904e40 −0.979663 −0.489832 − 0.871817i \(-0.662942\pi\)
−0.489832 + 0.871817i \(0.662942\pi\)
\(8\) −1.75781e43 −2.22946
\(9\) 2.96152e45 1.39635
\(10\) 5.88943e47 1.86240
\(11\) 4.57322e49 1.56340 0.781699 − 0.623656i \(-0.214354\pi\)
0.781699 + 0.623656i \(0.214354\pi\)
\(12\) −6.32270e51 −3.46572
\(13\) 7.91094e52 0.968096 0.484048 − 0.875041i \(-0.339166\pi\)
0.484048 + 0.875041i \(0.339166\pi\)
\(14\) 4.86801e54 1.76307
\(15\) 1.17217e56 1.60198
\(16\) 2.78306e57 1.77347
\(17\) −3.60982e57 −0.129170 −0.0645849 − 0.997912i \(-0.520572\pi\)
−0.0645849 + 0.997912i \(0.520572\pi\)
\(18\) −1.06080e60 −2.51298
\(19\) −1.15920e60 −0.210554 −0.105277 − 0.994443i \(-0.533573\pi\)
−0.105277 + 0.994443i \(0.533573\pi\)
\(20\) −1.45822e62 −2.31685
\(21\) 9.68877e62 1.51653
\(22\) −1.63810e64 −2.81360
\(23\) −4.43116e64 −0.921393 −0.460697 − 0.887558i \(-0.652400\pi\)
−0.460697 + 0.887558i \(0.652400\pi\)
\(24\) 1.25316e66 3.45123
\(25\) 1.79050e65 0.0709291
\(26\) −2.83365e67 −1.74225
\(27\) −5.99290e67 −0.613562
\(28\) −1.20532e69 −2.19328
\(29\) 3.11912e69 1.07183 0.535916 − 0.844271i \(-0.319966\pi\)
0.535916 + 0.844271i \(0.319966\pi\)
\(30\) −4.19864e70 −2.88303
\(31\) −6.79915e70 −0.983500 −0.491750 − 0.870737i \(-0.663643\pi\)
−0.491750 + 0.870737i \(0.663643\pi\)
\(32\) −3.00533e71 −0.962208
\(33\) −3.26030e72 −2.42016
\(34\) 1.29301e72 0.232463
\(35\) 2.23455e73 1.01381
\(36\) 2.62653e74 3.12617
\(37\) −2.19553e74 −0.711144 −0.355572 − 0.934649i \(-0.615714\pi\)
−0.355572 + 0.934649i \(0.615714\pi\)
\(38\) 4.15219e74 0.378927
\(39\) −5.63979e75 −1.49863
\(40\) 2.89021e76 2.30717
\(41\) −7.21825e76 −1.78320 −0.891598 − 0.452828i \(-0.850415\pi\)
−0.891598 + 0.452828i \(0.850415\pi\)
\(42\) −3.47046e77 −2.72926
\(43\) 9.95921e76 0.256138 0.128069 − 0.991765i \(-0.459122\pi\)
0.128069 + 0.991765i \(0.459122\pi\)
\(44\) 4.05592e78 3.50015
\(45\) −4.86934e78 −1.44503
\(46\) 1.58721e79 1.65820
\(47\) 1.61545e79 0.607638 0.303819 − 0.952730i \(-0.401738\pi\)
0.303819 + 0.952730i \(0.401738\pi\)
\(48\) −1.98408e80 −2.74536
\(49\) −7.74788e78 −0.0402596
\(50\) −6.41346e79 −0.127649
\(51\) 2.57348e80 0.199957
\(52\) 7.01609e81 2.16739
\(53\) −2.46146e81 −0.307673 −0.153836 − 0.988096i \(-0.549163\pi\)
−0.153836 + 0.988096i \(0.549163\pi\)
\(54\) 2.14662e82 1.10421
\(55\) −7.51930e82 −1.61789
\(56\) 2.38895e83 2.18412
\(57\) 8.26407e82 0.325941
\(58\) −1.11725e84 −1.92895
\(59\) 1.31371e84 1.00700 0.503499 − 0.863996i \(-0.332046\pi\)
0.503499 + 0.863996i \(0.332046\pi\)
\(60\) 1.03958e85 3.58652
\(61\) 8.03762e84 1.26462 0.632310 − 0.774715i \(-0.282107\pi\)
0.632310 + 0.774715i \(0.282107\pi\)
\(62\) 2.43541e85 1.76997
\(63\) −4.02484e85 −1.36796
\(64\) −2.59939e84 −0.0418141
\(65\) −1.30072e86 −1.00184
\(66\) 1.16782e87 4.35550
\(67\) −4.90861e86 −0.896201 −0.448101 − 0.893983i \(-0.647899\pi\)
−0.448101 + 0.893983i \(0.647899\pi\)
\(68\) −3.20149e86 −0.289187
\(69\) 3.15902e87 1.42633
\(70\) −8.00400e87 −1.82453
\(71\) 4.39734e87 0.510997 0.255498 − 0.966809i \(-0.417760\pi\)
0.255498 + 0.966809i \(0.417760\pi\)
\(72\) −5.20580e88 −3.11311
\(73\) −3.36142e88 −1.04397 −0.521983 − 0.852956i \(-0.674808\pi\)
−0.521983 + 0.852956i \(0.674808\pi\)
\(74\) 7.86425e88 1.27983
\(75\) −1.27647e88 −0.109799
\(76\) −1.02808e89 −0.471390
\(77\) −6.21521e89 −1.53160
\(78\) 2.02014e90 2.69704
\(79\) 1.11166e89 0.0810373 0.0405186 − 0.999179i \(-0.487099\pi\)
0.0405186 + 0.999179i \(0.487099\pi\)
\(80\) −4.57592e90 −1.83529
\(81\) −2.00867e90 −0.446550
\(82\) 2.58553e91 3.20916
\(83\) −1.87032e91 −1.30529 −0.652646 − 0.757663i \(-0.726341\pi\)
−0.652646 + 0.757663i \(0.726341\pi\)
\(84\) 8.59283e91 3.39524
\(85\) 5.93527e90 0.133672
\(86\) −3.56733e91 −0.460963
\(87\) −2.22365e92 −1.65921
\(88\) −8.03887e92 −3.48552
\(89\) 3.70282e92 0.938657 0.469329 − 0.883024i \(-0.344496\pi\)
0.469329 + 0.883024i \(0.344496\pi\)
\(90\) 1.74417e93 2.60057
\(91\) −1.07513e93 −0.948409
\(92\) −3.92993e93 −2.06283
\(93\) 4.84719e93 1.52247
\(94\) −5.78643e93 −1.09355
\(95\) 1.90596e93 0.217893
\(96\) 2.14253e94 1.48951
\(97\) −2.62608e94 −1.11597 −0.557983 − 0.829852i \(-0.688425\pi\)
−0.557983 + 0.829852i \(0.688425\pi\)
\(98\) 2.77524e93 0.0724540
\(99\) 1.35437e95 2.18306
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.1.1 ✓ 8
3.2 odd 2 9.96.a.c.1.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.96.a.a.1.1 ✓ 8 1.1 even 1 trivial
9.96.a.c.1.8 8 3.2 odd 2