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Properties

Label 1.100.a.a.1.5
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 1.5
Root \(4.33987e12\) 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)
 
\(f(q)\) \(=\) \(q+2.86466e14 q^{2} +2.01536e23 q^{3} -5.51763e29 q^{4} -4.79915e34 q^{5} +5.77333e37 q^{6} -6.92478e41 q^{7} -3.39630e44 q^{8} -1.31176e47 q^{9} -1.37479e49 q^{10} +1.55674e51 q^{11} -1.11200e53 q^{12} -2.14548e55 q^{13} -1.98371e56 q^{14} -9.67203e57 q^{15} +2.52429e59 q^{16} -1.39488e61 q^{17} -3.75773e61 q^{18} +1.51628e63 q^{19} +2.64799e64 q^{20} -1.39560e65 q^{21} +4.45952e65 q^{22} +2.55387e67 q^{23} -6.84479e67 q^{24} +7.25462e68 q^{25} -6.14606e69 q^{26} -6.10591e70 q^{27} +3.82084e71 q^{28} +1.39180e72 q^{29} -2.77071e72 q^{30} -1.44293e73 q^{31} +2.87578e74 q^{32} +3.13740e74 q^{33} -3.99585e75 q^{34} +3.32331e76 q^{35} +7.23778e76 q^{36} -5.19313e77 q^{37} +4.34363e77 q^{38} -4.32392e78 q^{39} +1.62994e79 q^{40} -3.58403e79 q^{41} -3.99790e79 q^{42} +2.84099e80 q^{43} -8.58951e80 q^{44} +6.29531e81 q^{45} +7.31597e81 q^{46} -7.73779e82 q^{47} +5.08736e82 q^{48} +1.74580e82 q^{49} +2.07820e83 q^{50} -2.81119e84 q^{51} +1.18380e85 q^{52} +1.89817e84 q^{53} -1.74913e85 q^{54} -7.47103e85 q^{55} +2.35187e86 q^{56} +3.05587e86 q^{57} +3.98704e86 q^{58} -2.13540e87 q^{59} +5.33667e87 q^{60} +4.67707e88 q^{61} -4.13350e87 q^{62} +9.08362e88 q^{63} -7.76144e88 q^{64} +1.02965e90 q^{65} +8.98757e88 q^{66} +3.33536e90 q^{67} +7.69642e90 q^{68} +5.14698e90 q^{69} +9.52013e90 q^{70} +2.25284e91 q^{71} +4.45512e91 q^{72} +2.47879e90 q^{73} -1.48765e92 q^{74} +1.46207e92 q^{75} -8.36629e92 q^{76} -1.07801e93 q^{77} -1.23866e93 q^{78} -1.29965e94 q^{79} -1.21144e94 q^{80} +1.02293e94 q^{81} -1.02670e94 q^{82} -1.54205e95 q^{83} +7.70038e94 q^{84} +6.69423e95 q^{85} +8.13847e94 q^{86} +2.80499e95 q^{87} -5.28716e95 q^{88} +2.29359e96 q^{89} +1.80339e96 q^{90} +1.48570e97 q^{91} -1.40913e97 q^{92} -2.90803e96 q^{93} -2.21661e97 q^{94} -7.27688e97 q^{95} +5.79575e97 q^{96} -3.70442e98 q^{97} +5.00113e96 q^{98} -2.04206e98 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 208040616902520 q^{2} - 28\!\cdots\!20 q^{3} + 28\!\cdots\!24 q^{4} - 48\!\cdots\!60 q^{5} - 77\!\cdots\!44 q^{6} - 56\!\cdots\!00 q^{7} + 59\!\cdots\!60 q^{8} + 15\!\cdots\!76 q^{9} - 20\!\cdots\!60 q^{10}+ \cdots - 13\!\cdots\!08 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.86466e14 0.359822 0.179911 − 0.983683i \(-0.442419\pi\)
0.179911 + 0.983683i \(0.442419\pi\)
\(3\) 2.01536e23 0.486241 0.243120 − 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(4\) −5.51763e29 −0.870528
\(5\) −4.79915e34 −1.20823 −0.604114 − 0.796898i \(-0.706473\pi\)
−0.604114 + 0.796898i \(0.706473\pi\)
\(6\) 5.77333e37 0.174960
\(7\) −6.92478e41 −1.01872 −0.509358 − 0.860555i \(-0.670117\pi\)
−0.509358 + 0.860555i \(0.670117\pi\)
\(8\) −3.39630e44 −0.673057
\(9\) −1.31176e47 −0.763570
\(10\) −1.37479e49 −0.434747
\(11\) 1.55674e51 0.439823 0.219912 − 0.975520i \(-0.429423\pi\)
0.219912 + 0.975520i \(0.429423\pi\)
\(12\) −1.11200e53 −0.423286
\(13\) −2.14548e55 −1.55356 −0.776781 − 0.629771i \(-0.783149\pi\)
−0.776781 + 0.629771i \(0.783149\pi\)
\(14\) −1.98371e56 −0.366557
\(15\) −9.67203e57 −0.587490
\(16\) 2.52429e59 0.628347
\(17\) −1.39488e61 −1.72709 −0.863545 − 0.504272i \(-0.831761\pi\)
−0.863545 + 0.504272i \(0.831761\pi\)
\(18\) −3.75773e61 −0.274749
\(19\) 1.51628e63 0.762917 0.381459 − 0.924386i \(-0.375422\pi\)
0.381459 + 0.924386i \(0.375422\pi\)
\(20\) 2.64799e64 1.05180
\(21\) −1.39560e65 −0.495341
\(22\) 4.45952e65 0.158258
\(23\) 2.55387e67 1.00386 0.501928 − 0.864910i \(-0.332624\pi\)
0.501928 + 0.864910i \(0.332624\pi\)
\(24\) −6.84479e67 −0.327268
\(25\) 7.25462e68 0.459816
\(26\) −6.14606e69 −0.559006
\(27\) −6.10591e70 −0.857520
\(28\) 3.82084e71 0.886821
\(29\) 1.39180e72 0.568692 0.284346 − 0.958722i \(-0.408224\pi\)
0.284346 + 0.958722i \(0.408224\pi\)
\(30\) −2.77071e72 −0.211392
\(31\) −1.44293e73 −0.217191 −0.108595 − 0.994086i \(-0.534635\pi\)
−0.108595 + 0.994086i \(0.534635\pi\)
\(32\) 2.87578e74 0.899150
\(33\) 3.13740e74 0.213860
\(34\) −3.99585e75 −0.621445
\(35\) 3.32331e76 1.23084
\(36\) 7.23778e76 0.664709
\(37\) −5.19313e77 −1.22869 −0.614347 − 0.789036i \(-0.710580\pi\)
−0.614347 + 0.789036i \(0.710580\pi\)
\(38\) 4.34363e77 0.274514
\(39\) −4.32392e78 −0.755405
\(40\) 1.62994e79 0.813207
\(41\) −3.58403e79 −0.526709 −0.263354 − 0.964699i \(-0.584829\pi\)
−0.263354 + 0.964699i \(0.584829\pi\)
\(42\) −3.99790e79 −0.178235
\(43\) 2.84099e80 0.395168 0.197584 − 0.980286i \(-0.436690\pi\)
0.197584 + 0.980286i \(0.436690\pi\)
\(44\) −8.58951e80 −0.382878
\(45\) 6.29531e81 0.922567
\(46\) 7.31597e81 0.361209
\(47\) −7.73779e82 −1.31757 −0.658784 − 0.752332i \(-0.728929\pi\)
−0.658784 + 0.752332i \(0.728929\pi\)
\(48\) 5.08736e82 0.305528
\(49\) 1.74580e82 0.0377824
\(50\) 2.07820e83 0.165452
\(51\) −2.81119e84 −0.839782
\(52\) 1.18380e85 1.35242
\(53\) 1.89817e84 0.0844655 0.0422328 − 0.999108i \(-0.486553\pi\)
0.0422328 + 0.999108i \(0.486553\pi\)
\(54\) −1.74913e85 −0.308555
\(55\) −7.47103e85 −0.531407
\(56\) 2.35187e86 0.685654
\(57\) 3.05587e86 0.370962
\(58\) 3.98704e86 0.204628
\(59\) −2.13540e87 −0.470224 −0.235112 − 0.971968i \(-0.575546\pi\)
−0.235112 + 0.971968i \(0.575546\pi\)
\(60\) 5.33667e87 0.511427
\(61\) 4.67707e88 1.97764 0.988819 − 0.149121i \(-0.0476444\pi\)
0.988819 + 0.149121i \(0.0476444\pi\)
\(62\) −4.13350e87 −0.0781501
\(63\) 9.08362e88 0.777861
\(64\) −7.76144e88 −0.304813
\(65\) 1.02965e90 1.87706
\(66\) 8.98757e88 0.0769515
\(67\) 3.33536e90 1.35656 0.678282 − 0.734802i \(-0.262725\pi\)
0.678282 + 0.734802i \(0.262725\pi\)
\(68\) 7.69642e90 1.50348
\(69\) 5.14698e90 0.488116
\(70\) 9.52013e90 0.442884
\(71\) 2.25284e91 0.519328 0.259664 − 0.965699i \(-0.416388\pi\)
0.259664 + 0.965699i \(0.416388\pi\)
\(72\) 4.45512e91 0.513926
\(73\) 2.47879e90 0.0144464 0.00722319 − 0.999974i \(-0.497701\pi\)
0.00722319 + 0.999974i \(0.497701\pi\)
\(74\) −1.48765e92 −0.442111
\(75\) 1.46207e92 0.223581
\(76\) −8.36629e92 −0.664141
\(77\) −1.07801e93 −0.448055
\(78\) −1.23866e93 −0.271812
\(79\) −1.29965e94 −1.51805 −0.759025 − 0.651062i \(-0.774324\pi\)
−0.759025 + 0.651062i \(0.774324\pi\)
\(80\) −1.21144e94 −0.759187
\(81\) 1.02293e94 0.346609
\(82\) −1.02670e94 −0.189521
\(83\) −1.54205e95 −1.56219 −0.781094 − 0.624414i \(-0.785338\pi\)
−0.781094 + 0.624414i \(0.785338\pi\)
\(84\) 7.70038e94 0.431209
\(85\) 6.69423e95 2.08672
\(86\) 8.13847e94 0.142190
\(87\) 2.80499e95 0.276521
\(88\) −5.28716e95 −0.296026
\(89\) 2.29359e96 0.734025 0.367013 − 0.930216i \(-0.380381\pi\)
0.367013 + 0.930216i \(0.380381\pi\)
\(90\) 1.80339e96 0.331960
\(91\) 1.48570e97 1.58264
\(92\) −1.40913e97 −0.873884
\(93\) −2.90803e96 −0.105607
\(94\) −2.21661e97 −0.474090
\(95\) −7.27688e97 −0.921778
\(96\) 5.79575e97 0.437204
\(97\) −3.70442e98 −1.67309 −0.836545 − 0.547898i \(-0.815428\pi\)
−0.836545 + 0.547898i \(0.815428\pi\)
\(98\) 5.00113e96 0.0135949
\(99\) −2.04206e98 −0.335836
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.1.5 ✓ 8
3.2 odd 2 9.100.a.d.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.100.a.a.1.5 ✓ 8 1.1 even 1 trivial
9.100.a.d.1.4 8 3.2 odd 2