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Properties

Label 1.100.a.a.1.3
Level $1$
Weight $100$
Character 1.1
Self dual yes
Analytic conductor $62.068$
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] = [1,100,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, 100); 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, 100, names="a")
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 100 \)
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: \(62.0676682981\)
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: \( x^{8} - x^{7} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{109}\cdot 3^{44}\cdot 5^{13}\cdot 7^{9}\cdot 11^{3}\cdot 13\cdot 17 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-9.79932e12\) 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)
 
\(f(q)\) \(=\) \(q-7.31556e14 q^{2} -2.11461e23 q^{3} -9.86512e28 q^{4} +5.46163e34 q^{5} +1.54696e38 q^{6} -9.42405e41 q^{7} +5.35848e44 q^{8} -1.27077e47 q^{9} -3.99548e49 q^{10} +6.80844e51 q^{11} +2.08609e52 q^{12} +1.27244e55 q^{13} +6.89422e56 q^{14} -1.15492e58 q^{15} -3.29475e59 q^{16} +1.40992e60 q^{17} +9.29638e61 q^{18} -1.83028e63 q^{19} -5.38796e63 q^{20} +1.99282e65 q^{21} -4.98075e66 q^{22} +7.80909e66 q^{23} -1.13311e68 q^{24} +1.40521e69 q^{25} -9.30860e69 q^{26} +6.31992e70 q^{27} +9.29694e70 q^{28} -4.49461e72 q^{29} +8.44889e72 q^{30} -6.78211e73 q^{31} -9.86044e73 q^{32} -1.43972e75 q^{33} -1.03144e75 q^{34} -5.14706e76 q^{35} +1.25363e76 q^{36} +1.72780e77 q^{37} +1.33895e78 q^{38} -2.69071e78 q^{39} +2.92660e79 q^{40} -3.13802e78 q^{41} -1.45786e80 q^{42} +1.67273e80 q^{43} -6.71660e80 q^{44} -6.94046e81 q^{45} -5.71278e81 q^{46} +5.53499e82 q^{47} +6.96711e82 q^{48} +4.26060e83 q^{49} -1.02799e84 q^{50} -2.98144e83 q^{51} -1.25527e84 q^{52} -9.83283e84 q^{53} -4.62337e85 q^{54} +3.71851e86 q^{55} -5.04986e86 q^{56} +3.87034e86 q^{57} +3.28806e87 q^{58} -2.85418e87 q^{59} +1.13934e87 q^{60} -7.36889e87 q^{61} +4.96150e88 q^{62} +1.19758e89 q^{63} +2.80964e89 q^{64} +6.94958e89 q^{65} +1.05324e90 q^{66} +2.22058e90 q^{67} -1.39091e89 q^{68} -1.65132e90 q^{69} +3.76537e91 q^{70} +7.03212e91 q^{71} -6.80938e91 q^{72} +1.65280e92 q^{73} -1.26398e92 q^{74} -2.97148e92 q^{75} +1.80560e92 q^{76} -6.41631e93 q^{77} +1.96841e93 q^{78} -2.18798e93 q^{79} -1.79947e94 q^{80} +8.46667e93 q^{81} +2.29564e93 q^{82} -6.95611e94 q^{83} -1.96594e94 q^{84} +7.70048e94 q^{85} -1.22369e95 q^{86} +9.50434e95 q^{87} +3.64828e96 q^{88} +8.71787e95 q^{89} +5.07733e96 q^{90} -1.19915e97 q^{91} -7.70375e95 q^{92} +1.43415e97 q^{93} -4.04915e97 q^{94} -9.99632e97 q^{95} +2.08510e97 q^{96} +1.85375e98 q^{97} -3.11686e98 q^{98} -8.65194e98 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 208040616902520 q^{2} - 28\!\cdots\!20 q^{3} + 28\!\cdots\!24 q^{4} - 48\!\cdots\!60 q^{5} - 77\!\cdots\!44 q^{6} - 56\!\cdots\!00 q^{7} + 59\!\cdots\!60 q^{8} + 15\!\cdots\!76 q^{9} - 20\!\cdots\!60 q^{10}+ \cdots - 13\!\cdots\!08 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.31556e14 −0.918888 −0.459444 − 0.888207i \(-0.651951\pi\)
−0.459444 + 0.888207i \(0.651951\pi\)
\(3\) −2.11461e23 −0.510186 −0.255093 − 0.966917i \(-0.582106\pi\)
−0.255093 + 0.966917i \(0.582106\pi\)
\(4\) −9.86512e28 −0.155644
\(5\) 5.46163e34 1.37501 0.687506 − 0.726178i \(-0.258705\pi\)
0.687506 + 0.726178i \(0.258705\pi\)
\(6\) 1.54696e38 0.468804
\(7\) −9.42405e41 −1.38639 −0.693194 − 0.720751i \(-0.743797\pi\)
−0.693194 + 0.720751i \(0.743797\pi\)
\(8\) 5.35848e44 1.06191
\(9\) −1.27077e47 −0.739711
\(10\) −3.99548e49 −1.26348
\(11\) 6.80844e51 1.92358 0.961788 − 0.273794i \(-0.0882787\pi\)
0.961788 + 0.273794i \(0.0882787\pi\)
\(12\) 2.08609e52 0.0794074
\(13\) 1.27244e55 0.921384 0.460692 − 0.887560i \(-0.347601\pi\)
0.460692 + 0.887560i \(0.347601\pi\)
\(14\) 6.89422e56 1.27394
\(15\) −1.15492e58 −0.701512
\(16\) −3.29475e59 −0.820131
\(17\) 1.40992e60 0.174572 0.0872860 − 0.996183i \(-0.472181\pi\)
0.0872860 + 0.996183i \(0.472181\pi\)
\(18\) 9.29638e61 0.679712
\(19\) −1.83028e63 −0.920906 −0.460453 − 0.887684i \(-0.652313\pi\)
−0.460453 + 0.887684i \(0.652313\pi\)
\(20\) −5.38796e63 −0.214013
\(21\) 1.99282e65 0.707315
\(22\) −4.98075e66 −1.76755
\(23\) 7.80909e66 0.306953 0.153477 − 0.988152i \(-0.450953\pi\)
0.153477 + 0.988152i \(0.450953\pi\)
\(24\) −1.13311e68 −0.541770
\(25\) 1.40521e69 0.890660
\(26\) −9.30860e69 −0.846649
\(27\) 6.31992e70 0.887575
\(28\) 9.29694e70 0.215783
\(29\) −4.49461e72 −1.83650 −0.918250 − 0.396001i \(-0.870398\pi\)
−0.918250 + 0.396001i \(0.870398\pi\)
\(30\) 8.44889e72 0.644611
\(31\) −6.78211e73 −1.02085 −0.510424 − 0.859923i \(-0.670512\pi\)
−0.510424 + 0.859923i \(0.670512\pi\)
\(32\) −9.86044e73 −0.308299
\(33\) −1.43972e75 −0.981381
\(34\) −1.03144e75 −0.160412
\(35\) −5.14706e76 −1.90630
\(36\) 1.25363e76 0.115132
\(37\) 1.72780e77 0.408797 0.204399 − 0.978888i \(-0.434476\pi\)
0.204399 + 0.978888i \(0.434476\pi\)
\(38\) 1.33895e78 0.846209
\(39\) −2.69071e78 −0.470077
\(40\) 2.92660e79 1.46014
\(41\) −3.13802e78 −0.0461164 −0.0230582 − 0.999734i \(-0.507340\pi\)
−0.0230582 + 0.999734i \(0.507340\pi\)
\(42\) −1.45786e80 −0.649944
\(43\) 1.67273e80 0.232668 0.116334 − 0.993210i \(-0.462886\pi\)
0.116334 + 0.993210i \(0.462886\pi\)
\(44\) −6.71660e80 −0.299393
\(45\) −6.94046e81 −1.01711
\(46\) −5.71278e81 −0.282056
\(47\) 5.53499e82 0.942481 0.471241 − 0.882005i \(-0.343806\pi\)
0.471241 + 0.882005i \(0.343806\pi\)
\(48\) 6.96711e82 0.418419
\(49\) 4.26060e83 0.922071
\(50\) −1.02799e84 −0.818417
\(51\) −2.98144e83 −0.0890641
\(52\) −1.25527e84 −0.143408
\(53\) −9.83283e84 −0.437545 −0.218773 − 0.975776i \(-0.570205\pi\)
−0.218773 + 0.975776i \(0.570205\pi\)
\(54\) −4.62337e85 −0.815583
\(55\) 3.71851e86 2.64494
\(56\) −5.04986e86 −1.47222
\(57\) 3.87034e86 0.469833
\(58\) 3.28806e87 1.68754
\(59\) −2.85418e87 −0.628503 −0.314251 − 0.949340i \(-0.601753\pi\)
−0.314251 + 0.949340i \(0.601753\pi\)
\(60\) 1.13934e87 0.109186
\(61\) −7.36889e87 −0.311584 −0.155792 − 0.987790i \(-0.549793\pi\)
−0.155792 + 0.987790i \(0.549793\pi\)
\(62\) 4.96150e88 0.938046
\(63\) 1.19758e89 1.02553
\(64\) 2.80964e89 1.10342
\(65\) 6.94958e89 1.26691
\(66\) 1.05324e90 0.901780
\(67\) 2.22058e90 0.903159 0.451580 − 0.892231i \(-0.350861\pi\)
0.451580 + 0.892231i \(0.350861\pi\)
\(68\) −1.39091e89 −0.0271711
\(69\) −1.65132e90 −0.156603
\(70\) 3.76537e91 1.75168
\(71\) 7.03212e91 1.62106 0.810528 − 0.585699i \(-0.199180\pi\)
0.810528 + 0.585699i \(0.199180\pi\)
\(72\) −6.80938e91 −0.785505
\(73\) 1.65280e92 0.963250 0.481625 − 0.876377i \(-0.340047\pi\)
0.481625 + 0.876377i \(0.340047\pi\)
\(74\) −1.26398e92 −0.375639
\(75\) −2.97148e92 −0.454402
\(76\) 1.80560e92 0.143334
\(77\) −6.41631e93 −2.66682
\(78\) 1.96841e93 0.431948
\(79\) −2.18798e93 −0.255566 −0.127783 − 0.991802i \(-0.540786\pi\)
−0.127783 + 0.991802i \(0.540786\pi\)
\(80\) −1.79947e94 −1.12769
\(81\) 8.46667e93 0.286882
\(82\) 2.29564e93 0.0423758
\(83\) −6.95611e94 −0.704695 −0.352347 − 0.935869i \(-0.614616\pi\)
−0.352347 + 0.935869i \(0.614616\pi\)
\(84\) −1.96594e94 −0.110089
\(85\) 7.70048e94 0.240039
\(86\) −1.22369e95 −0.213796
\(87\) 9.50434e95 0.936956
\(88\) 3.64828e96 2.04266
\(89\) 8.71787e95 0.279000 0.139500 − 0.990222i \(-0.455450\pi\)
0.139500 + 0.990222i \(0.455450\pi\)
\(90\) 5.07733e96 0.934612
\(91\) −1.19915e97 −1.27740
\(92\) −7.70375e95 −0.0477755
\(93\) 1.43415e97 0.520822
\(94\) −4.04915e97 −0.866035
\(95\) −9.99632e97 −1.26626
\(96\) 2.08510e97 0.157290
\(97\) 1.85375e98 0.837239 0.418620 − 0.908162i \(-0.362514\pi\)
0.418620 + 0.908162i \(0.362514\pi\)
\(98\) −3.11686e98 −0.847280
\(99\) −8.65194e98 −1.42289
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 1.100.a.a.1.3 ✓ 8
3.2 odd 2 9.100.a.d.1.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.100.a.a.1.3 ✓ 8 1.1 even 1 trivial
9.100.a.d.1.6 8 3.2 odd 2