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Properties

Label 1.100.a.a.1.6
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 1.6
Root \(8.50040e12\) 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)
 
\(f(q)\) \(=\) \(q+5.86024e14 q^{2} -4.40976e23 q^{3} -2.90401e29 q^{4} +2.74639e34 q^{5} -2.58422e38 q^{6} +3.47138e41 q^{7} -5.41619e44 q^{8} +2.26671e46 q^{9} +1.60945e49 q^{10} -4.95910e51 q^{11} +1.28060e53 q^{12} +1.21492e55 q^{13} +2.03431e56 q^{14} -1.21109e58 q^{15} -1.33338e59 q^{16} -3.97916e60 q^{17} +1.32834e61 q^{18} -4.47251e62 q^{19} -7.97554e63 q^{20} -1.53079e65 q^{21} -2.90615e66 q^{22} -4.53817e67 q^{23} +2.38841e68 q^{24} -8.23457e68 q^{25} +7.11971e69 q^{26} +6.57607e70 q^{27} -1.00809e71 q^{28} -5.62741e71 q^{29} -7.09728e72 q^{30} +7.11109e73 q^{31} +2.65152e74 q^{32} +2.18684e75 q^{33} -2.33189e75 q^{34} +9.53375e75 q^{35} -6.58255e75 q^{36} +3.44683e77 q^{37} -2.62100e77 q^{38} -5.35749e78 q^{39} -1.48750e79 q^{40} +2.52241e78 q^{41} -8.97081e79 q^{42} +8.67636e80 q^{43} +1.44013e81 q^{44} +6.22526e80 q^{45} -2.65948e82 q^{46} +3.44728e82 q^{47} +5.87988e82 q^{48} -3.41563e83 q^{49} -4.82566e83 q^{50} +1.75471e84 q^{51} -3.52813e84 q^{52} +2.95920e85 q^{53} +3.85373e85 q^{54} -1.36196e86 q^{55} -1.88016e86 q^{56} +1.97227e86 q^{57} -3.29780e86 q^{58} +7.60447e87 q^{59} +3.51702e87 q^{60} +8.96058e87 q^{61} +4.16727e88 q^{62} +7.86860e87 q^{63} +2.39899e89 q^{64} +3.33663e89 q^{65} +1.28154e90 q^{66} +2.62496e90 q^{67} +1.15555e90 q^{68} +2.00122e91 q^{69} +5.58700e90 q^{70} +6.35730e91 q^{71} -1.22769e91 q^{72} -3.15102e92 q^{73} +2.01993e92 q^{74} +3.63125e92 q^{75} +1.29882e92 q^{76} -1.72149e93 q^{77} -3.13962e93 q^{78} -1.44064e94 q^{79} -3.66198e93 q^{80} -3.28929e94 q^{81} +1.47819e93 q^{82} +6.97978e94 q^{83} +4.44544e94 q^{84} -1.09283e95 q^{85} +5.08455e95 q^{86} +2.48155e95 q^{87} +2.68594e96 q^{88} -1.17382e96 q^{89} +3.64815e95 q^{90} +4.21744e96 q^{91} +1.31789e97 q^{92} -3.13582e97 q^{93} +2.02019e97 q^{94} -1.22833e97 q^{95} -1.16926e98 q^{96} -3.50742e98 q^{97} -2.00164e98 q^{98} -1.12408e98 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 208040616902520 q^{2} - 28\!\cdots\!20 q^{3} + 28\!\cdots\!24 q^{4} - 48\!\cdots\!60 q^{5} - 77\!\cdots\!44 q^{6} - 56\!\cdots\!00 q^{7} + 59\!\cdots\!60 q^{8} + 15\!\cdots\!76 q^{9} - 20\!\cdots\!60 q^{10}+ \cdots - 13\!\cdots\!08 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.86024e14 0.736089 0.368045 − 0.929808i \(-0.380027\pi\)
0.368045 + 0.929808i \(0.380027\pi\)
\(3\) −4.40976e23 −1.06393 −0.531964 − 0.846767i \(-0.678546\pi\)
−0.531964 + 0.846767i \(0.678546\pi\)
\(4\) −2.90401e29 −0.458172
\(5\) 2.74639e34 0.691427 0.345714 − 0.938340i \(-0.387637\pi\)
0.345714 + 0.938340i \(0.387637\pi\)
\(6\) −2.58422e38 −0.783147
\(7\) 3.47138e41 0.510680 0.255340 − 0.966851i \(-0.417813\pi\)
0.255340 + 0.966851i \(0.417813\pi\)
\(8\) −5.41619e44 −1.07335
\(9\) 2.26671e46 0.131944
\(10\) 1.60945e49 0.508952
\(11\) −4.95910e51 −1.40109 −0.700543 − 0.713611i \(-0.747059\pi\)
−0.700543 + 0.713611i \(0.747059\pi\)
\(12\) 1.28060e53 0.487463
\(13\) 1.21492e55 0.879733 0.439866 − 0.898063i \(-0.355026\pi\)
0.439866 + 0.898063i \(0.355026\pi\)
\(14\) 2.03431e56 0.375906
\(15\) −1.21109e58 −0.735630
\(16\) −1.33338e59 −0.331906
\(17\) −3.97916e60 −0.492686 −0.246343 − 0.969183i \(-0.579229\pi\)
−0.246343 + 0.969183i \(0.579229\pi\)
\(18\) 1.32834e61 0.0971229
\(19\) −4.47251e62 −0.225034 −0.112517 − 0.993650i \(-0.535891\pi\)
−0.112517 + 0.993650i \(0.535891\pi\)
\(20\) −7.97554e63 −0.316793
\(21\) −1.53079e65 −0.543327
\(22\) −2.90615e66 −1.03132
\(23\) −4.53817e67 −1.78383 −0.891914 − 0.452205i \(-0.850637\pi\)
−0.891914 + 0.452205i \(0.850637\pi\)
\(24\) 2.38841e68 1.14196
\(25\) −8.23457e68 −0.521928
\(26\) 7.11971e69 0.647562
\(27\) 6.57607e70 0.923549
\(28\) −1.00809e71 −0.233979
\(29\) −5.62741e71 −0.229936 −0.114968 − 0.993369i \(-0.536677\pi\)
−0.114968 + 0.993369i \(0.536677\pi\)
\(30\) −7.09728e72 −0.541489
\(31\) 7.11109e73 1.07037 0.535183 − 0.844736i \(-0.320243\pi\)
0.535183 + 0.844736i \(0.320243\pi\)
\(32\) 2.65152e74 0.829033
\(33\) 2.18684e75 1.49065
\(34\) −2.33189e75 −0.362661
\(35\) 9.53375e75 0.353098
\(36\) −6.58255e75 −0.0604533
\(37\) 3.44683e77 0.815520 0.407760 − 0.913089i \(-0.366310\pi\)
0.407760 + 0.913089i \(0.366310\pi\)
\(38\) −2.62100e77 −0.165645
\(39\) −5.35749e78 −0.935973
\(40\) −1.48750e79 −0.742140
\(41\) 2.52241e78 0.0370693 0.0185346 − 0.999828i \(-0.494100\pi\)
0.0185346 + 0.999828i \(0.494100\pi\)
\(42\) −8.97081e79 −0.399937
\(43\) 8.67636e80 1.20684 0.603419 − 0.797425i \(-0.293805\pi\)
0.603419 + 0.797425i \(0.293805\pi\)
\(44\) 1.44013e81 0.641939
\(45\) 6.22526e80 0.0912300
\(46\) −2.65948e82 −1.31306
\(47\) 3.44728e82 0.586992 0.293496 − 0.955960i \(-0.405181\pi\)
0.293496 + 0.955960i \(0.405181\pi\)
\(48\) 5.87988e82 0.353124
\(49\) −3.41563e83 −0.739206
\(50\) −4.82566e83 −0.384186
\(51\) 1.75471e84 0.524183
\(52\) −3.52813e84 −0.403069
\(53\) 2.95920e85 1.31680 0.658399 − 0.752669i \(-0.271234\pi\)
0.658399 + 0.752669i \(0.271234\pi\)
\(54\) 3.85373e85 0.679815
\(55\) −1.36196e86 −0.968749
\(56\) −1.88016e86 −0.548136
\(57\) 1.97227e86 0.239420
\(58\) −3.29780e86 −0.169254
\(59\) 7.60447e87 1.67454 0.837268 − 0.546792i \(-0.184151\pi\)
0.837268 + 0.546792i \(0.184151\pi\)
\(60\) 3.51702e87 0.337045
\(61\) 8.96058e87 0.378887 0.189443 − 0.981892i \(-0.439332\pi\)
0.189443 + 0.981892i \(0.439332\pi\)
\(62\) 4.16727e88 0.787885
\(63\) 7.86860e87 0.0673814
\(64\) 2.39899e89 0.942148
\(65\) 3.33663e89 0.608271
\(66\) 1.28154e90 1.09726
\(67\) 2.62496e90 1.06763 0.533813 − 0.845602i \(-0.320759\pi\)
0.533813 + 0.845602i \(0.320759\pi\)
\(68\) 1.15555e90 0.225735
\(69\) 2.00122e91 1.89787
\(70\) 5.58700e90 0.259912
\(71\) 6.35730e91 1.46550 0.732748 − 0.680501i \(-0.238238\pi\)
0.732748 + 0.680501i \(0.238238\pi\)
\(72\) −1.22769e91 −0.141622
\(73\) −3.15102e92 −1.83641 −0.918205 − 0.396105i \(-0.870362\pi\)
−0.918205 + 0.396105i \(0.870362\pi\)
\(74\) 2.01993e92 0.600296
\(75\) 3.63125e92 0.555294
\(76\) 1.29882e92 0.103104
\(77\) −1.72149e93 −0.715506
\(78\) −3.13962e93 −0.688960
\(79\) −1.44064e94 −1.68273 −0.841365 − 0.540468i \(-0.818247\pi\)
−0.841365 + 0.540468i \(0.818247\pi\)
\(80\) −3.66198e93 −0.229489
\(81\) −3.28929e94 −1.11454
\(82\) 1.47819e93 0.0272863
\(83\) 6.97978e94 0.707093 0.353546 − 0.935417i \(-0.384976\pi\)
0.353546 + 0.935417i \(0.384976\pi\)
\(84\) 4.44544e94 0.248937
\(85\) −1.09283e95 −0.340657
\(86\) 5.08455e95 0.888340
\(87\) 2.48155e95 0.244636
\(88\) 2.68594e96 1.50385
\(89\) −1.17382e96 −0.375660 −0.187830 − 0.982202i \(-0.560145\pi\)
−0.187830 + 0.982202i \(0.560145\pi\)
\(90\) 3.64815e95 0.0671535
\(91\) 4.21744e96 0.449262
\(92\) 1.31789e97 0.817301
\(93\) −3.13582e97 −1.13879
\(94\) 2.02019e97 0.432079
\(95\) −1.22833e97 −0.155595
\(96\) −1.16926e98 −0.882032
\(97\) −3.50742e98 −1.58412 −0.792059 − 0.610445i \(-0.790991\pi\)
−0.792059 + 0.610445i \(0.790991\pi\)
\(98\) −2.00164e98 −0.544122
\(99\) −1.12408e98 −0.184865
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.1.6 ✓ 8
3.2 odd 2 9.100.a.d.1.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.100.a.a.1.6 ✓ 8 1.1 even 1 trivial
9.100.a.d.1.3 8 3.2 odd 2