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Properties

Label 1.96.a.a.1.3
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 1.3
Root \(-6.56924e12\) 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)
 
\(f(q)\) \(=\) \(q-1.58391e14 q^{2} +9.80327e21 q^{3} -1.45263e28 q^{4} -7.31691e32 q^{5} -1.55275e36 q^{6} -1.37055e39 q^{7} +8.57536e42 q^{8} -2.02479e45 q^{9} +1.15893e47 q^{10} -1.77564e49 q^{11} -1.42405e50 q^{12} -4.53188e52 q^{13} +2.17083e53 q^{14} -7.17297e54 q^{15} -7.82816e56 q^{16} -3.88116e57 q^{17} +3.20709e59 q^{18} -8.75641e59 q^{19} +1.06288e61 q^{20} -1.34359e61 q^{21} +2.81246e63 q^{22} -1.90280e64 q^{23} +8.40666e64 q^{24} -1.98898e66 q^{25} +7.17809e66 q^{26} -4.06413e67 q^{27} +1.99090e67 q^{28} -4.05377e69 q^{29} +1.13613e69 q^{30} -7.95372e70 q^{31} -2.15714e71 q^{32} -1.74071e71 q^{33} +6.14742e71 q^{34} +1.00282e72 q^{35} +2.94127e73 q^{36} +2.93459e74 q^{37} +1.38694e74 q^{38} -4.44272e74 q^{39} -6.27451e75 q^{40} -2.78348e76 q^{41} +2.12812e75 q^{42} +4.90861e76 q^{43} +2.57935e77 q^{44} +1.48152e78 q^{45} +3.01387e78 q^{46} -1.43188e78 q^{47} -7.67415e78 q^{48} -1.90570e80 q^{49} +3.15037e80 q^{50} -3.80481e79 q^{51} +6.58314e80 q^{52} +9.09844e81 q^{53} +6.43722e81 q^{54} +1.29922e82 q^{55} -1.17530e82 q^{56} -8.58415e81 q^{57} +6.42081e83 q^{58} +2.06207e84 q^{59} +1.04197e83 q^{60} +1.22605e85 q^{61} +1.25980e85 q^{62} +2.77508e84 q^{63} +6.51777e85 q^{64} +3.31593e85 q^{65} +2.75713e85 q^{66} +6.33834e86 q^{67} +5.63790e85 q^{68} -1.86537e86 q^{69} -1.58838e86 q^{70} -1.27499e88 q^{71} -1.73633e88 q^{72} -1.33744e88 q^{73} -4.64812e88 q^{74} -1.94985e88 q^{75} +1.27198e88 q^{76} +2.43360e88 q^{77} +7.03688e88 q^{78} -3.85669e89 q^{79} +5.72779e89 q^{80} +3.89595e90 q^{81} +4.40879e90 q^{82} +2.10785e91 q^{83} +1.95174e89 q^{84} +2.83981e90 q^{85} -7.77481e90 q^{86} -3.97402e91 q^{87} -1.52267e92 q^{88} -5.00386e92 q^{89} -2.34660e92 q^{90} +6.21116e91 q^{91} +2.76407e92 q^{92} -7.79725e92 q^{93} +2.26797e92 q^{94} +6.40699e92 q^{95} -2.11470e93 q^{96} +1.70309e94 q^{97} +3.01846e94 q^{98} +3.59530e94 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 5835659138280 q^{2} - 95\!\cdots\!80 q^{3} + 20\!\cdots\!84 q^{4} + 19\!\cdots\!60 q^{5} + 10\!\cdots\!76 q^{6} + 31\!\cdots\!00 q^{7} - 14\!\cdots\!60 q^{8} + 92\!\cdots\!36 q^{9} - 35\!\cdots\!40 q^{10}+ \cdots + 30\!\cdots\!72 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.58391e14 −0.795804 −0.397902 − 0.917428i \(-0.630262\pi\)
−0.397902 + 0.917428i \(0.630262\pi\)
\(3\) 9.80327e21 0.212869 0.106434 − 0.994320i \(-0.466057\pi\)
0.106434 + 0.994320i \(0.466057\pi\)
\(4\) −1.45263e28 −0.366696
\(5\) −7.31691e32 −0.460524 −0.230262 − 0.973129i \(-0.573958\pi\)
−0.230262 + 0.973129i \(0.573958\pi\)
\(6\) −1.55275e36 −0.169402
\(7\) −1.37055e39 −0.0987957 −0.0493978 − 0.998779i \(-0.515730\pi\)
−0.0493978 + 0.998779i \(0.515730\pi\)
\(8\) 8.57536e42 1.08762
\(9\) −2.02479e45 −0.954687
\(10\) 1.15893e47 0.366487
\(11\) −1.77564e49 −0.607019 −0.303509 − 0.952828i \(-0.598158\pi\)
−0.303509 + 0.952828i \(0.598158\pi\)
\(12\) −1.42405e50 −0.0780579
\(13\) −4.53188e52 −0.554586 −0.277293 − 0.960785i \(-0.589437\pi\)
−0.277293 + 0.960785i \(0.589437\pi\)
\(14\) 2.17083e53 0.0786220
\(15\) −7.17297e54 −0.0980311
\(16\) −7.82816e56 −0.498839
\(17\) −3.88116e57 −0.138880 −0.0694398 − 0.997586i \(-0.522121\pi\)
−0.0694398 + 0.997586i \(0.522121\pi\)
\(18\) 3.20709e59 0.759744
\(19\) −8.75641e59 −0.159049 −0.0795243 − 0.996833i \(-0.525340\pi\)
−0.0795243 + 0.996833i \(0.525340\pi\)
\(20\) 1.06288e61 0.168872
\(21\) −1.34359e61 −0.0210305
\(22\) 2.81246e63 0.483068
\(23\) −1.90280e64 −0.395659 −0.197829 − 0.980236i \(-0.563389\pi\)
−0.197829 + 0.980236i \(0.563389\pi\)
\(24\) 8.40666e64 0.231521
\(25\) −1.98898e66 −0.787917
\(26\) 7.17809e66 0.441342
\(27\) −4.06413e67 −0.416091
\(28\) 1.99090e67 0.0362279
\(29\) −4.05377e69 −1.39301 −0.696505 − 0.717552i \(-0.745263\pi\)
−0.696505 + 0.717552i \(0.745263\pi\)
\(30\) 1.13613e69 0.0780136
\(31\) −7.95372e70 −1.15051 −0.575255 − 0.817974i \(-0.695097\pi\)
−0.575255 + 0.817974i \(0.695097\pi\)
\(32\) −2.15714e71 −0.690644
\(33\) −1.74071e71 −0.129215
\(34\) 6.14742e71 0.110521
\(35\) 1.00282e72 0.0454978
\(36\) 2.94127e73 0.350079
\(37\) 2.93459e74 0.950528 0.475264 − 0.879843i \(-0.342353\pi\)
0.475264 + 0.879843i \(0.342353\pi\)
\(38\) 1.38694e74 0.126572
\(39\) −4.44272e74 −0.118054
\(40\) −6.27451e75 −0.500876
\(41\) −2.78348e76 −0.687631 −0.343815 − 0.939037i \(-0.611719\pi\)
−0.343815 + 0.939037i \(0.611719\pi\)
\(42\) 2.12812e75 0.0167362
\(43\) 4.90861e76 0.126243 0.0631215 − 0.998006i \(-0.479894\pi\)
0.0631215 + 0.998006i \(0.479894\pi\)
\(44\) 2.57935e77 0.222591
\(45\) 1.48152e78 0.439656
\(46\) 3.01387e78 0.314867
\(47\) −1.43188e78 −0.0538589 −0.0269295 − 0.999637i \(-0.508573\pi\)
−0.0269295 + 0.999637i \(0.508573\pi\)
\(48\) −7.67415e78 −0.106187
\(49\) −1.90570e80 −0.990239
\(50\) 3.15037e80 0.627028
\(51\) −3.80481e79 −0.0295631
\(52\) 6.58314e80 0.203364
\(53\) 9.09844e81 1.13727 0.568634 − 0.822590i \(-0.307472\pi\)
0.568634 + 0.822590i \(0.307472\pi\)
\(54\) 6.43722e81 0.331127
\(55\) 1.29922e82 0.279547
\(56\) −1.17530e82 −0.107452
\(57\) −8.58415e81 −0.0338565
\(58\) 6.42081e83 1.10856
\(59\) 2.06207e84 1.58064 0.790318 − 0.612697i \(-0.209915\pi\)
0.790318 + 0.612697i \(0.209915\pi\)
\(60\) 1.04197e83 0.0359476
\(61\) 1.22605e85 1.92904 0.964521 − 0.264005i \(-0.0850437\pi\)
0.964521 + 0.264005i \(0.0850437\pi\)
\(62\) 1.25980e85 0.915580
\(63\) 2.77508e84 0.0943189
\(64\) 6.51777e85 1.04846
\(65\) 3.31593e85 0.255400
\(66\) 2.75713e85 0.102830
\(67\) 6.33834e86 1.15724 0.578618 − 0.815598i \(-0.303592\pi\)
0.578618 + 0.815598i \(0.303592\pi\)
\(68\) 5.63790e85 0.0509265
\(69\) −1.86537e86 −0.0842233
\(70\) −1.58838e86 −0.0362073
\(71\) −1.27499e88 −1.48162 −0.740809 − 0.671715i \(-0.765558\pi\)
−0.740809 + 0.671715i \(0.765558\pi\)
\(72\) −1.73633e88 −1.03834
\(73\) −1.33744e88 −0.415374 −0.207687 − 0.978195i \(-0.566594\pi\)
−0.207687 + 0.978195i \(0.566594\pi\)
\(74\) −4.64812e88 −0.756434
\(75\) −1.94985e88 −0.167723
\(76\) 1.27198e88 0.0583224
\(77\) 2.43360e88 0.0599708
\(78\) 7.03688e88 0.0939478
\(79\) −3.85669e89 −0.281144 −0.140572 − 0.990070i \(-0.544894\pi\)
−0.140572 + 0.990070i \(0.544894\pi\)
\(80\) 5.72779e89 0.229727
\(81\) 3.89595e90 0.866114
\(82\) 4.40879e90 0.547219
\(83\) 2.10785e91 1.47106 0.735531 − 0.677491i \(-0.236933\pi\)
0.735531 + 0.677491i \(0.236933\pi\)
\(84\) 1.95174e89 0.00771179
\(85\) 2.83981e90 0.0639574
\(86\) −7.77481e90 −0.100465
\(87\) −3.97402e91 −0.296528
\(88\) −1.52267e92 −0.660207
\(89\) −5.00386e92 −1.26847 −0.634235 − 0.773140i \(-0.718685\pi\)
−0.634235 + 0.773140i \(0.718685\pi\)
\(90\) −2.34660e92 −0.349880
\(91\) 6.21116e91 0.0547907
\(92\) 2.76407e92 0.145086
\(93\) −7.79725e92 −0.244907
\(94\) 2.26797e92 0.0428612
\(95\) 6.40699e92 0.0732458
\(96\) −2.11470e93 −0.147016
\(97\) 1.70309e94 0.723734 0.361867 − 0.932230i \(-0.382139\pi\)
0.361867 + 0.932230i \(0.382139\pi\)
\(98\) 3.01846e94 0.788037
\(99\) 3.59530e94 0.579513
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.1.3 ✓ 8
3.2 odd 2 9.96.a.c.1.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.96.a.a.1.3 ✓ 8 1.1 even 1 trivial
9.96.a.c.1.6 8 3.2 odd 2