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Properties

Label 1.100.a.a.1.4
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 1.4
Root \(-8.46093e12\) 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)
 
\(f(q)\) \(=\) \(q-6.35192e14 q^{2} +5.52801e23 q^{3} -2.30357e29 q^{4} +4.78100e33 q^{5} -3.51135e38 q^{6} +7.17401e41 q^{7} +5.48921e44 q^{8} +1.33796e47 q^{9} -3.03685e48 q^{10} -4.28029e51 q^{11} -1.27341e53 q^{12} +1.83718e55 q^{13} -4.55687e56 q^{14} +2.64294e57 q^{15} -2.02664e59 q^{16} +6.21642e59 q^{17} -8.49864e61 q^{18} +2.46616e63 q^{19} -1.10133e63 q^{20} +3.96580e65 q^{21} +2.71880e66 q^{22} +1.40807e67 q^{23} +3.03444e68 q^{24} -1.55486e69 q^{25} -1.16696e70 q^{26} -2.10043e70 q^{27} -1.65258e71 q^{28} +2.85339e72 q^{29} -1.67877e72 q^{30} -7.00608e73 q^{31} -2.19189e74 q^{32} -2.36615e75 q^{33} -3.94862e74 q^{34} +3.42989e75 q^{35} -3.08209e76 q^{36} -5.39350e76 q^{37} -1.56649e78 q^{38} +1.01560e79 q^{39} +2.62439e78 q^{40} +1.04979e80 q^{41} -2.51904e80 q^{42} -1.06727e81 q^{43} +9.85993e80 q^{44} +6.39681e80 q^{45} -8.94393e81 q^{46} +6.71187e82 q^{47} -1.12033e83 q^{48} +5.25966e82 q^{49} +9.87637e83 q^{50} +3.43644e83 q^{51} -4.23207e84 q^{52} -2.07277e85 q^{53} +1.33417e85 q^{54} -2.04640e85 q^{55} +3.93797e86 q^{56} +1.36330e87 q^{57} -1.81245e87 q^{58} -1.47079e87 q^{59} -6.08819e86 q^{60} +2.22361e88 q^{61} +4.45021e88 q^{62} +9.59858e88 q^{63} +2.67681e89 q^{64} +8.78357e88 q^{65} +1.50296e90 q^{66} +4.24589e90 q^{67} -1.43199e89 q^{68} +7.78381e90 q^{69} -2.17864e90 q^{70} +4.39297e91 q^{71} +7.34437e91 q^{72} +1.41929e92 q^{73} +3.42591e91 q^{74} -8.59530e92 q^{75} -5.68097e92 q^{76} -3.07068e93 q^{77} -6.45099e93 q^{78} +1.10262e94 q^{79} -9.68938e92 q^{80} -3.45964e94 q^{81} -6.66819e94 q^{82} +8.17771e94 q^{83} -9.13549e94 q^{84} +2.97207e93 q^{85} +6.77920e95 q^{86} +1.57736e96 q^{87} -2.34954e96 q^{88} +5.46832e96 q^{89} -4.06320e95 q^{90} +1.31800e97 q^{91} -3.24358e96 q^{92} -3.87297e97 q^{93} -4.26333e97 q^{94} +1.17907e97 q^{95} -1.21168e98 q^{96} +1.47493e98 q^{97} -3.34089e97 q^{98} -5.72687e98 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 208040616902520 q^{2} - 28\!\cdots\!20 q^{3} + 28\!\cdots\!24 q^{4} - 48\!\cdots\!60 q^{5} - 77\!\cdots\!44 q^{6} - 56\!\cdots\!00 q^{7} + 59\!\cdots\!60 q^{8} + 15\!\cdots\!76 q^{9} - 20\!\cdots\!60 q^{10}+ \cdots - 13\!\cdots\!08 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −6.35192e14 −0.797848 −0.398924 − 0.916984i \(-0.630616\pi\)
−0.398924 + 0.916984i \(0.630616\pi\)
\(3\) 5.52801e23 1.33373 0.666863 − 0.745180i \(-0.267637\pi\)
0.666863 + 0.745180i \(0.267637\pi\)
\(4\) −2.30357e29 −0.363439
\(5\) 4.78100e33 0.120366 0.0601829 − 0.998187i \(-0.480832\pi\)
0.0601829 + 0.998187i \(0.480832\pi\)
\(6\) −3.51135e38 −1.06411
\(7\) 7.17401e41 1.05538 0.527690 − 0.849437i \(-0.323058\pi\)
0.527690 + 0.849437i \(0.323058\pi\)
\(8\) 5.48921e44 1.08782
\(9\) 1.33796e47 0.778826
\(10\) −3.03685e48 −0.0960336
\(11\) −4.28029e51 −1.20930 −0.604652 − 0.796490i \(-0.706688\pi\)
−0.604652 + 0.796490i \(0.706688\pi\)
\(12\) −1.27341e53 −0.484728
\(13\) 1.83718e55 1.33032 0.665161 − 0.746700i \(-0.268363\pi\)
0.665161 + 0.746700i \(0.268363\pi\)
\(14\) −4.55687e56 −0.842033
\(15\) 2.64294e57 0.160535
\(16\) −2.02664e59 −0.504474
\(17\) 6.21642e59 0.0769695 0.0384847 − 0.999259i \(-0.487747\pi\)
0.0384847 + 0.999259i \(0.487747\pi\)
\(18\) −8.49864e61 −0.621385
\(19\) 2.46616e63 1.24085 0.620424 − 0.784267i \(-0.286961\pi\)
0.620424 + 0.784267i \(0.286961\pi\)
\(20\) −1.10133e63 −0.0437456
\(21\) 3.96580e65 1.40759
\(22\) 2.71880e66 0.964840
\(23\) 1.40807e67 0.553472 0.276736 − 0.960946i \(-0.410747\pi\)
0.276736 + 0.960946i \(0.410747\pi\)
\(24\) 3.03444e68 1.45085
\(25\) −1.55486e69 −0.985512
\(26\) −1.16696e70 −1.06139
\(27\) −2.10043e70 −0.294986
\(28\) −1.65258e71 −0.383566
\(29\) 2.85339e72 1.16590 0.582948 − 0.812509i \(-0.301899\pi\)
0.582948 + 0.812509i \(0.301899\pi\)
\(30\) −1.67877e72 −0.128083
\(31\) −7.00608e73 −1.05456 −0.527280 − 0.849692i \(-0.676788\pi\)
−0.527280 + 0.849692i \(0.676788\pi\)
\(32\) −2.19189e74 −0.685324
\(33\) −2.36615e75 −1.61288
\(34\) −3.94862e74 −0.0614099
\(35\) 3.42989e75 0.127032
\(36\) −3.08209e76 −0.283056
\(37\) −5.39350e76 −0.127610 −0.0638051 − 0.997962i \(-0.520324\pi\)
−0.0638051 + 0.997962i \(0.520324\pi\)
\(38\) −1.56649e78 −0.990008
\(39\) 1.01560e79 1.77428
\(40\) 2.62439e78 0.130936
\(41\) 1.04979e80 1.54277 0.771387 − 0.636367i \(-0.219564\pi\)
0.771387 + 0.636367i \(0.219564\pi\)
\(42\) −2.51904e80 −1.12304
\(43\) −1.06727e81 −1.48452 −0.742258 − 0.670115i \(-0.766245\pi\)
−0.742258 + 0.670115i \(0.766245\pi\)
\(44\) 9.85993e80 0.439508
\(45\) 6.39681e80 0.0937440
\(46\) −8.94393e81 −0.441586
\(47\) 6.71187e82 1.14288 0.571439 − 0.820645i \(-0.306385\pi\)
0.571439 + 0.820645i \(0.306385\pi\)
\(48\) −1.12033e83 −0.672830
\(49\) 5.25966e82 0.113829
\(50\) 9.87637e83 0.786289
\(51\) 3.43644e83 0.102656
\(52\) −4.23207e84 −0.483490
\(53\) −2.07277e85 −0.922351 −0.461176 − 0.887309i \(-0.652572\pi\)
−0.461176 + 0.887309i \(0.652572\pi\)
\(54\) 1.33417e85 0.235354
\(55\) −2.04640e85 −0.145559
\(56\) 3.93797e86 1.14806
\(57\) 1.36330e87 1.65495
\(58\) −1.81245e87 −0.930208
\(59\) −1.47079e87 −0.323874 −0.161937 − 0.986801i \(-0.551774\pi\)
−0.161937 + 0.986801i \(0.551774\pi\)
\(60\) −6.08819e86 −0.0583447
\(61\) 2.22361e88 0.940227 0.470113 − 0.882606i \(-0.344213\pi\)
0.470113 + 0.882606i \(0.344213\pi\)
\(62\) 4.45021e88 0.841379
\(63\) 9.59858e88 0.821958
\(64\) 2.67681e89 1.05126
\(65\) 8.78357e88 0.160125
\(66\) 1.50296e90 1.28683
\(67\) 4.24589e90 1.72689 0.863447 − 0.504439i \(-0.168301\pi\)
0.863447 + 0.504439i \(0.168301\pi\)
\(68\) −1.43199e89 −0.0279737
\(69\) 7.78381e90 0.738180
\(70\) −2.17864e90 −0.101352
\(71\) 4.39297e91 1.01268 0.506338 − 0.862335i \(-0.330999\pi\)
0.506338 + 0.862335i \(0.330999\pi\)
\(72\) 7.34437e91 0.847220
\(73\) 1.41929e92 0.827158 0.413579 − 0.910468i \(-0.364279\pi\)
0.413579 + 0.910468i \(0.364279\pi\)
\(74\) 3.42591e91 0.101814
\(75\) −8.59530e92 −1.31440
\(76\) −5.68097e92 −0.450972
\(77\) −3.07068e93 −1.27628
\(78\) −6.45099e93 −1.41561
\(79\) 1.10262e94 1.28791 0.643954 − 0.765064i \(-0.277293\pi\)
0.643954 + 0.765064i \(0.277293\pi\)
\(80\) −9.68938e92 −0.0607214
\(81\) −3.45964e94 −1.17226
\(82\) −6.66819e94 −1.23090
\(83\) 8.17771e94 0.828449 0.414225 − 0.910175i \(-0.364053\pi\)
0.414225 + 0.910175i \(0.364053\pi\)
\(84\) −9.13549e94 −0.511572
\(85\) 2.97207e93 0.00926449
\(86\) 6.77920e95 1.18442
\(87\) 1.57736e96 1.55499
\(88\) −2.34954e96 −1.31550
\(89\) 5.46832e96 1.75004 0.875021 − 0.484084i \(-0.160847\pi\)
0.875021 + 0.484084i \(0.160847\pi\)
\(90\) −4.06320e95 −0.0747935
\(91\) 1.31800e97 1.40400
\(92\) −3.24358e96 −0.201153
\(93\) −3.87297e97 −1.40649
\(94\) −4.26333e97 −0.911842
\(95\) 1.17907e97 0.149356
\(96\) −1.21168e98 −0.914034
\(97\) 1.47493e98 0.666147 0.333074 − 0.942901i \(-0.391914\pi\)
0.333074 + 0.942901i \(0.391914\pi\)
\(98\) −3.34089e97 −0.0908180
\(99\) −5.72687e98 −0.941837
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.1.4 ✓ 8
3.2 odd 2 9.100.a.d.1.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.100.a.a.1.4 ✓ 8 1.1 even 1 trivial
9.100.a.d.1.5 8 3.2 odd 2