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Properties

Label 1.96.a.a.1.6
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 1.6
Root \(6.86315e12\) 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)
 
\(f(q)\) \(=\) \(q+1.63986e14 q^{2} +6.46739e21 q^{3} -1.27226e28 q^{4} +1.20020e33 q^{5} +1.06056e36 q^{6} -2.14893e40 q^{7} -8.58249e42 q^{8} -2.07907e45 q^{9} +1.96817e47 q^{10} +1.70307e49 q^{11} -8.22823e49 q^{12} +1.01181e53 q^{13} -3.52395e54 q^{14} +7.76219e54 q^{15} -9.03414e56 q^{16} +2.27059e58 q^{17} -3.40938e59 q^{18} +5.33207e60 q^{19} -1.52698e61 q^{20} -1.38980e62 q^{21} +2.79280e63 q^{22} +5.74222e62 q^{23} -5.55064e64 q^{24} -1.08387e66 q^{25} +1.65923e67 q^{26} -2.71628e67 q^{27} +2.73401e68 q^{28} +2.72768e69 q^{29} +1.27289e69 q^{30} +8.24705e70 q^{31} +1.91840e71 q^{32} +1.10144e71 q^{33} +3.72345e72 q^{34} -2.57916e73 q^{35} +2.64512e73 q^{36} -5.13786e74 q^{37} +8.74385e74 q^{38} +6.54379e74 q^{39} -1.03007e76 q^{40} +7.35670e76 q^{41} -2.27908e76 q^{42} -6.17805e76 q^{43} -2.16676e77 q^{44} -2.49530e78 q^{45} +9.41643e76 q^{46} +3.75012e78 q^{47} -5.84273e78 q^{48} +2.69343e80 q^{49} -1.77739e80 q^{50} +1.46848e80 q^{51} -1.28729e81 q^{52} +2.90261e81 q^{53} -4.45432e81 q^{54} +2.04403e82 q^{55} +1.84432e83 q^{56} +3.44846e82 q^{57} +4.47301e83 q^{58} +1.68978e84 q^{59} -9.87556e82 q^{60} +2.77082e84 q^{61} +1.35240e85 q^{62} +4.46778e85 q^{63} +6.72471e85 q^{64} +1.21438e86 q^{65} +1.80621e85 q^{66} +3.14536e85 q^{67} -2.88879e86 q^{68} +3.71372e84 q^{69} -4.22946e87 q^{70} -4.28767e87 q^{71} +1.78436e88 q^{72} -4.92275e88 q^{73} -8.42538e88 q^{74} -7.00979e87 q^{75} -6.78380e88 q^{76} -3.65979e89 q^{77} +1.07309e89 q^{78} +2.67794e90 q^{79} -1.08428e90 q^{80} +4.23381e90 q^{81} +1.20640e91 q^{82} -1.35258e91 q^{83} +1.76819e90 q^{84} +2.72517e91 q^{85} -1.01311e91 q^{86} +1.76409e91 q^{87} -1.46166e92 q^{88} +3.19257e91 q^{89} -4.09195e92 q^{90} -2.17432e93 q^{91} -7.30562e90 q^{92} +5.33369e92 q^{93} +6.14967e92 q^{94} +6.39957e93 q^{95} +1.24071e93 q^{96} +2.36851e94 q^{97} +4.41685e94 q^{98} -3.54080e94 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 5835659138280 q^{2} - 95\!\cdots\!80 q^{3} + 20\!\cdots\!84 q^{4} + 19\!\cdots\!60 q^{5} + 10\!\cdots\!76 q^{6} + 31\!\cdots\!00 q^{7} - 14\!\cdots\!60 q^{8} + 92\!\cdots\!36 q^{9} - 35\!\cdots\!40 q^{10}+ \cdots + 30\!\cdots\!72 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.63986e14 0.823915 0.411957 − 0.911203i \(-0.364845\pi\)
0.411957 + 0.911203i \(0.364845\pi\)
\(3\) 6.46739e21 0.140433 0.0702166 − 0.997532i \(-0.477631\pi\)
0.0702166 + 0.997532i \(0.477631\pi\)
\(4\) −1.27226e28 −0.321165
\(5\) 1.20020e33 0.755405 0.377702 − 0.925927i \(-0.376714\pi\)
0.377702 + 0.925927i \(0.376714\pi\)
\(6\) 1.06056e36 0.115705
\(7\) −2.14893e40 −1.54905 −0.774526 − 0.632542i \(-0.782012\pi\)
−0.774526 + 0.632542i \(0.782012\pi\)
\(8\) −8.58249e42 −1.08853
\(9\) −2.07907e45 −0.980279
\(10\) 1.96817e47 0.622389
\(11\) 1.70307e49 0.582211 0.291106 − 0.956691i \(-0.405977\pi\)
0.291106 + 0.956691i \(0.405977\pi\)
\(12\) −8.22823e49 −0.0451022
\(13\) 1.01181e53 1.23820 0.619101 − 0.785312i \(-0.287497\pi\)
0.619101 + 0.785312i \(0.287497\pi\)
\(14\) −3.52395e54 −1.27629
\(15\) 7.76219e54 0.106084
\(16\) −9.03414e56 −0.575688
\(17\) 2.27059e58 0.812484 0.406242 − 0.913765i \(-0.366839\pi\)
0.406242 + 0.913765i \(0.366839\pi\)
\(18\) −3.40938e59 −0.807666
\(19\) 5.33207e60 0.968500 0.484250 − 0.874930i \(-0.339093\pi\)
0.484250 + 0.874930i \(0.339093\pi\)
\(20\) −1.52698e61 −0.242609
\(21\) −1.38980e62 −0.217538
\(22\) 2.79280e63 0.479692
\(23\) 5.74222e62 0.0119401 0.00597004 − 0.999982i \(-0.498100\pi\)
0.00597004 + 0.999982i \(0.498100\pi\)
\(24\) −5.55064e64 −0.152865
\(25\) −1.08387e66 −0.429364
\(26\) 1.65923e67 1.02017
\(27\) −2.71628e67 −0.278097
\(28\) 2.73401e68 0.497501
\(29\) 2.72768e69 0.937320 0.468660 − 0.883379i \(-0.344737\pi\)
0.468660 + 0.883379i \(0.344737\pi\)
\(30\) 1.27289e69 0.0874040
\(31\) 8.24705e70 1.19294 0.596469 − 0.802636i \(-0.296570\pi\)
0.596469 + 0.802636i \(0.296570\pi\)
\(32\) 1.91840e71 0.614209
\(33\) 1.10144e71 0.0817618
\(34\) 3.72345e72 0.669418
\(35\) −2.57916e73 −1.17016
\(36\) 2.64512e73 0.314831
\(37\) −5.13786e74 −1.66418 −0.832091 − 0.554639i \(-0.812856\pi\)
−0.832091 + 0.554639i \(0.812856\pi\)
\(38\) 8.74385e74 0.797961
\(39\) 6.54379e74 0.173884
\(40\) −1.03007e76 −0.822278
\(41\) 7.35670e76 1.81740 0.908699 − 0.417452i \(-0.137077\pi\)
0.908699 + 0.417452i \(0.137077\pi\)
\(42\) −2.27908e76 −0.179233
\(43\) −6.17805e76 −0.158891 −0.0794456 − 0.996839i \(-0.525315\pi\)
−0.0794456 + 0.996839i \(0.525315\pi\)
\(44\) −2.16676e77 −0.186986
\(45\) −2.49530e78 −0.740507
\(46\) 9.41643e76 0.00983761
\(47\) 3.75012e78 0.141058 0.0705288 − 0.997510i \(-0.477531\pi\)
0.0705288 + 0.997510i \(0.477531\pi\)
\(48\) −5.84273e78 −0.0808457
\(49\) 2.69343e80 1.39956
\(50\) −1.77739e80 −0.353759
\(51\) 1.46848e80 0.114100
\(52\) −1.28729e81 −0.397667
\(53\) 2.90261e81 0.362814 0.181407 − 0.983408i \(-0.441935\pi\)
0.181407 + 0.983408i \(0.441935\pi\)
\(54\) −4.45432e81 −0.229128
\(55\) 2.04403e82 0.439805
\(56\) 1.84432e83 1.68618
\(57\) 3.44846e82 0.136009
\(58\) 4.47301e83 0.772272
\(59\) 1.68978e84 1.29527 0.647635 − 0.761950i \(-0.275758\pi\)
0.647635 + 0.761950i \(0.275758\pi\)
\(60\) −9.87556e82 −0.0340704
\(61\) 2.77082e84 0.435954 0.217977 − 0.975954i \(-0.430054\pi\)
0.217977 + 0.975954i \(0.430054\pi\)
\(62\) 1.35240e85 0.982879
\(63\) 4.46778e85 1.51850
\(64\) 6.72471e85 1.08174
\(65\) 1.21438e86 0.935343
\(66\) 1.80621e85 0.0673647
\(67\) 3.14536e85 0.0574271 0.0287135 − 0.999588i \(-0.490859\pi\)
0.0287135 + 0.999588i \(0.490859\pi\)
\(68\) −2.88879e86 −0.260941
\(69\) 3.71372e84 0.00167678
\(70\) −4.22946e87 −0.964113
\(71\) −4.28767e87 −0.498252 −0.249126 − 0.968471i \(-0.580143\pi\)
−0.249126 + 0.968471i \(0.580143\pi\)
\(72\) 1.78436e88 1.06706
\(73\) −4.92275e88 −1.52888 −0.764438 − 0.644697i \(-0.776983\pi\)
−0.764438 + 0.644697i \(0.776983\pi\)
\(74\) −8.42538e88 −1.37114
\(75\) −7.00979e87 −0.0602969
\(76\) −6.78380e88 −0.311048
\(77\) −3.65979e89 −0.901876
\(78\) 1.07309e89 0.143266
\(79\) 2.67794e90 1.95216 0.976078 − 0.217419i \(-0.0697637\pi\)
0.976078 + 0.217419i \(0.0697637\pi\)
\(80\) −1.08428e90 −0.434878
\(81\) 4.23381e90 0.941225
\(82\) 1.20640e91 1.49738
\(83\) −1.35258e91 −0.943960 −0.471980 − 0.881609i \(-0.656461\pi\)
−0.471980 + 0.881609i \(0.656461\pi\)
\(84\) 1.76819e90 0.0698656
\(85\) 2.72517e91 0.613755
\(86\) −1.01311e91 −0.130913
\(87\) 1.76409e91 0.131631
\(88\) −1.46166e92 −0.633753
\(89\) 3.19257e91 0.0809310 0.0404655 − 0.999181i \(-0.487116\pi\)
0.0404655 + 0.999181i \(0.487116\pi\)
\(90\) −4.09195e92 −0.610115
\(91\) −2.17432e93 −1.91804
\(92\) −7.30562e90 −0.00383473
\(93\) 5.33369e92 0.167528
\(94\) 6.14967e92 0.116219
\(95\) 6.39957e93 0.731609
\(96\) 1.24071e93 0.0862553
\(97\) 2.36851e94 1.00651 0.503254 − 0.864139i \(-0.332136\pi\)
0.503254 + 0.864139i \(0.332136\pi\)
\(98\) 4.41685e94 1.15312
\(99\) −3.54080e94 −0.570729
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.1.6 ✓ 8
3.2 odd 2 9.96.a.c.1.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.96.a.a.1.6 ✓ 8 1.1 even 1 trivial
9.96.a.c.1.3 8 3.2 odd 2