↑

Properties

Label 1.100.a.a.1.7
Level $1$
Weight $100$
Character 1.1
Self dual yes
Analytic conductor $62.068$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.7
Root \(1.56804e13\) 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)
 
\(f(q)\) \(=\) \(q+1.10299e15 q^{2} +5.08811e23 q^{3} +5.82752e29 q^{4} +3.48045e34 q^{5} +5.61211e38 q^{6} +4.01523e41 q^{7} -5.63335e43 q^{8} +8.70965e46 q^{9} +3.83889e49 q^{10} +4.15197e51 q^{11} +2.96511e53 q^{12} -4.59067e54 q^{13} +4.42874e56 q^{14} +1.77089e58 q^{15} -4.31498e59 q^{16} +1.61313e61 q^{17} +9.60661e61 q^{18} -2.54791e63 q^{19} +2.02824e64 q^{20} +2.04300e65 q^{21} +4.57956e66 q^{22} +2.64606e67 q^{23} -2.86631e67 q^{24} -3.66366e68 q^{25} -5.06345e69 q^{26} -4.30943e70 q^{27} +2.33988e71 q^{28} +1.04939e72 q^{29} +1.95327e73 q^{30} +1.04509e74 q^{31} -4.40230e74 q^{32} +2.11257e75 q^{33} +1.77925e76 q^{34} +1.39748e76 q^{35} +5.07556e76 q^{36} +1.95635e76 q^{37} -2.81031e78 q^{38} -2.33579e78 q^{39} -1.96066e78 q^{40} -7.32415e79 q^{41} +2.25339e80 q^{42} -3.18946e80 q^{43} +2.41957e81 q^{44} +3.03135e81 q^{45} +2.91856e82 q^{46} -8.27217e82 q^{47} -2.19551e83 q^{48} -3.00847e83 q^{49} -4.04096e83 q^{50} +8.20777e84 q^{51} -2.67522e84 q^{52} +8.37221e84 q^{53} -4.75324e85 q^{54} +1.44507e86 q^{55} -2.26192e85 q^{56} -1.29641e87 q^{57} +1.15746e87 q^{58} +4.40637e87 q^{59} +1.03199e88 q^{60} +2.09373e87 q^{61} +1.15272e89 q^{62} +3.49712e88 q^{63} -2.12073e89 q^{64} -1.59776e89 q^{65} +2.33013e90 q^{66} +2.84981e90 q^{67} +9.40052e90 q^{68} +1.34634e91 q^{69} +1.54140e91 q^{70} -5.10722e91 q^{71} -4.90645e90 q^{72} +1.71803e92 q^{73} +2.15783e91 q^{74} -1.86411e92 q^{75} -1.48480e93 q^{76} +1.66711e93 q^{77} -2.57634e93 q^{78} -7.83106e93 q^{79} -1.50181e94 q^{80} -3.68894e94 q^{81} -8.07843e94 q^{82} +1.37202e95 q^{83} +1.19056e95 q^{84} +5.61441e95 q^{85} -3.51793e95 q^{86} +5.33943e95 q^{87} -2.33895e95 q^{88} +4.51907e95 q^{89} +3.34354e96 q^{90} -1.84326e96 q^{91} +1.54199e97 q^{92} +5.31752e97 q^{93} -9.12409e97 q^{94} -8.86788e97 q^{95} -2.23994e98 q^{96} +1.05705e98 q^{97} -3.31830e98 q^{98} +3.61622e98 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 208040616902520 q^{2} - 28\!\cdots\!20 q^{3} + 28\!\cdots\!24 q^{4} - 48\!\cdots\!60 q^{5} - 77\!\cdots\!44 q^{6} - 56\!\cdots\!00 q^{7} + 59\!\cdots\!60 q^{8} + 15\!\cdots\!76 q^{9} - 20\!\cdots\!60 q^{10}+ \cdots - 13\!\cdots\!08 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.10299e15 1.38543 0.692716 − 0.721211i \(-0.256414\pi\)
0.692716 + 0.721211i \(0.256414\pi\)
\(3\) 5.08811e23 1.22759 0.613797 − 0.789464i \(-0.289641\pi\)
0.613797 + 0.789464i \(0.289641\pi\)
\(4\) 5.82752e29 0.919420
\(5\) 3.48045e34 0.876235 0.438118 − 0.898918i \(-0.355645\pi\)
0.438118 + 0.898918i \(0.355645\pi\)
\(6\) 5.61211e38 1.70075
\(7\) 4.01523e41 0.590687 0.295344 − 0.955391i \(-0.404566\pi\)
0.295344 + 0.955391i \(0.404566\pi\)
\(8\) −5.63335e43 −0.111638
\(9\) 8.70965e46 0.506986
\(10\) 3.83889e49 1.21396
\(11\) 4.15197e51 1.17305 0.586524 − 0.809932i \(-0.300496\pi\)
0.586524 + 0.809932i \(0.300496\pi\)
\(12\) 2.96511e53 1.12867
\(13\) −4.59067e54 −0.332415 −0.166207 − 0.986091i \(-0.553152\pi\)
−0.166207 + 0.986091i \(0.553152\pi\)
\(14\) 4.42874e56 0.818357
\(15\) 1.77089e58 1.07566
\(16\) −4.31498e59 −1.07409
\(17\) 1.61313e61 1.99732 0.998658 − 0.0517994i \(-0.0164956\pi\)
0.998658 + 0.0517994i \(0.0164956\pi\)
\(18\) 9.60661e61 0.702395
\(19\) −2.54791e63 −1.28198 −0.640989 − 0.767550i \(-0.721476\pi\)
−0.640989 + 0.767550i \(0.721476\pi\)
\(20\) 2.02824e64 0.805628
\(21\) 2.04300e65 0.725124
\(22\) 4.57956e66 1.62518
\(23\) 2.64606e67 1.04009 0.520045 − 0.854139i \(-0.325915\pi\)
0.520045 + 0.854139i \(0.325915\pi\)
\(24\) −2.86631e67 −0.137046
\(25\) −3.66366e68 −0.232212
\(26\) −5.06345e69 −0.460538
\(27\) −4.30943e70 −0.605220
\(28\) 2.33988e71 0.543090
\(29\) 1.04939e72 0.428782 0.214391 − 0.976748i \(-0.431223\pi\)
0.214391 + 0.976748i \(0.431223\pi\)
\(30\) 1.95327e73 1.49025
\(31\) 1.04509e74 1.57307 0.786536 − 0.617545i \(-0.211873\pi\)
0.786536 + 0.617545i \(0.211873\pi\)
\(32\) −4.40230e74 −1.37644
\(33\) 2.11257e75 1.44003
\(34\) 1.77925e76 2.76714
\(35\) 1.39748e76 0.517581
\(36\) 5.07556e76 0.466133
\(37\) 1.95635e76 0.0462873 0.0231436 − 0.999732i \(-0.492632\pi\)
0.0231436 + 0.999732i \(0.492632\pi\)
\(38\) −2.81031e78 −1.77609
\(39\) −2.33579e78 −0.408070
\(40\) −1.96066e78 −0.0978213
\(41\) −7.32415e79 −1.07636 −0.538179 − 0.842831i \(-0.680887\pi\)
−0.538179 + 0.842831i \(0.680887\pi\)
\(42\) 2.25339e80 1.00461
\(43\) −3.18946e80 −0.443638 −0.221819 − 0.975088i \(-0.571199\pi\)
−0.221819 + 0.975088i \(0.571199\pi\)
\(44\) 2.41957e81 1.07852
\(45\) 3.03135e81 0.444239
\(46\) 2.91856e82 1.44097
\(47\) −8.27217e82 −1.40856 −0.704281 − 0.709922i \(-0.748730\pi\)
−0.704281 + 0.709922i \(0.748730\pi\)
\(48\) −2.19551e83 −1.31854
\(49\) −3.00847e83 −0.651089
\(50\) −4.04096e83 −0.321714
\(51\) 8.20777e84 2.45189
\(52\) −2.67522e84 −0.305629
\(53\) 8.37221e84 0.372550 0.186275 − 0.982498i \(-0.440358\pi\)
0.186275 + 0.982498i \(0.440358\pi\)
\(54\) −4.75324e85 −0.838491
\(55\) 1.44507e86 1.02787
\(56\) −2.26192e85 −0.0659432
\(57\) −1.29641e87 −1.57375
\(58\) 1.15746e87 0.594049
\(59\) 4.40637e87 0.970302 0.485151 − 0.874430i \(-0.338765\pi\)
0.485151 + 0.874430i \(0.338765\pi\)
\(60\) 1.03199e88 0.988984
\(61\) 2.09373e87 0.0885307 0.0442653 − 0.999020i \(-0.485905\pi\)
0.0442653 + 0.999020i \(0.485905\pi\)
\(62\) 1.15272e89 2.17938
\(63\) 3.49712e88 0.299470
\(64\) −2.12073e89 −0.832870
\(65\) −1.59776e89 −0.291274
\(66\) 2.33013e90 1.99506
\(67\) 2.84981e90 1.15908 0.579540 − 0.814944i \(-0.303232\pi\)
0.579540 + 0.814944i \(0.303232\pi\)
\(68\) 9.40052e90 1.83637
\(69\) 1.34634e91 1.27681
\(70\) 1.54140e91 0.717073
\(71\) −5.10722e91 −1.17733 −0.588663 − 0.808379i \(-0.700345\pi\)
−0.588663 + 0.808379i \(0.700345\pi\)
\(72\) −4.90645e90 −0.0565990
\(73\) 1.71803e92 1.00127 0.500634 − 0.865659i \(-0.333100\pi\)
0.500634 + 0.865659i \(0.333100\pi\)
\(74\) 2.15783e91 0.0641279
\(75\) −1.86411e92 −0.285062
\(76\) −1.48480e93 −1.17868
\(77\) 1.66711e93 0.692905
\(78\) −2.57634e93 −0.565354
\(79\) −7.83106e93 −0.914702 −0.457351 − 0.889286i \(-0.651202\pi\)
−0.457351 + 0.889286i \(0.651202\pi\)
\(80\) −1.50181e94 −0.941153
\(81\) −3.68894e94 −1.24995
\(82\) −8.07843e94 −1.49122
\(83\) 1.37202e95 1.38994 0.694968 − 0.719041i \(-0.255418\pi\)
0.694968 + 0.719041i \(0.255418\pi\)
\(84\) 1.19056e95 0.666693
\(85\) 5.61441e95 1.75012
\(86\) −3.51793e95 −0.614630
\(87\) 5.33943e95 0.526371
\(88\) −2.33895e95 −0.130957
\(89\) 4.51907e95 0.144625 0.0723125 − 0.997382i \(-0.476962\pi\)
0.0723125 + 0.997382i \(0.476962\pi\)
\(90\) 3.34354e96 0.615463
\(91\) −1.84326e96 −0.196353
\(92\) 1.54199e97 0.956280
\(93\) 5.31752e97 1.93109
\(94\) −9.12409e97 −1.95146
\(95\) −8.86788e97 −1.12331
\(96\) −2.23994e98 −1.68970
\(97\) 1.05705e98 0.477415 0.238708 − 0.971092i \(-0.423276\pi\)
0.238708 + 0.971092i \(0.423276\pi\)
\(98\) −3.31830e98 −0.902038
\(99\) 3.61622e98 0.594720
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.1.7 ✓ 8
3.2 odd 2 9.100.a.d.1.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.100.a.a.1.7 ✓ 8 1.1 even 1 trivial
9.100.a.d.1.2 8 3.2 odd 2