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Properties

Label 1.100.a.a.1.2
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 1.2
Root \(-1.08597e13\) 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)
 
\(f(q)\) \(=\) \(q-8.07907e14 q^{2} -6.25114e23 q^{3} +1.88877e28 q^{4} -5.34820e34 q^{5} +5.05034e38 q^{6} +1.04094e42 q^{7} +4.96812e44 q^{8} +2.18975e47 q^{9} +4.32085e49 q^{10} +2.09751e51 q^{11} -1.18070e52 q^{12} -2.27740e55 q^{13} -8.40981e56 q^{14} +3.34324e58 q^{15} -4.13349e59 q^{16} +7.56916e60 q^{17} -1.76912e62 q^{18} +5.81301e62 q^{19} -1.01016e63 q^{20} -6.50706e65 q^{21} -1.69459e66 q^{22} -2.12928e67 q^{23} -3.10564e68 q^{24} +1.28261e69 q^{25} +1.83993e70 q^{26} -2.94946e70 q^{27} +1.96610e70 q^{28} -1.52350e72 q^{29} -2.70102e73 q^{30} +2.72377e73 q^{31} +1.90555e73 q^{32} -1.31118e75 q^{33} -6.11517e75 q^{34} -5.56715e76 q^{35} +4.13595e75 q^{36} +4.28140e77 q^{37} -4.69637e77 q^{38} +1.42363e79 q^{39} -2.65705e79 q^{40} -5.02606e79 q^{41} +5.25709e80 q^{42} -1.05419e81 q^{43} +3.96172e79 q^{44} -1.17112e82 q^{45} +1.72026e82 q^{46} -3.56207e82 q^{47} +2.58391e83 q^{48} +6.21485e83 q^{49} -1.03623e84 q^{50} -4.73159e84 q^{51} -4.30149e83 q^{52} +1.04502e85 q^{53} +2.38289e85 q^{54} -1.12179e86 q^{55} +5.17151e86 q^{56} -3.63379e86 q^{57} +1.23085e87 q^{58} +3.60957e87 q^{59} +6.31462e86 q^{60} -9.18510e87 q^{61} -2.20055e88 q^{62} +2.27940e89 q^{63} +2.46596e89 q^{64} +1.21800e90 q^{65} +1.05931e90 q^{66} -1.79075e90 q^{67} +1.42964e89 q^{68} +1.33104e91 q^{69} +4.49774e91 q^{70} -3.40784e91 q^{71} +1.08790e92 q^{72} +1.68443e92 q^{73} -3.45898e92 q^{74} -8.01775e92 q^{75} +1.09795e91 q^{76} +2.18338e93 q^{77} -1.15016e94 q^{78} -1.45180e94 q^{79} +2.21068e94 q^{80} -1.91808e94 q^{81} +4.06058e94 q^{82} -9.55043e94 q^{83} -1.22904e94 q^{84} -4.04814e95 q^{85} +8.51687e95 q^{86} +9.52362e95 q^{87} +1.04207e96 q^{88} +4.74685e96 q^{89} +9.46159e96 q^{90} -2.37063e97 q^{91} -4.02173e95 q^{92} -1.70267e97 q^{93} +2.87782e97 q^{94} -3.10891e97 q^{95} -1.19119e97 q^{96} -1.76496e98 q^{97} -5.02102e98 q^{98} +4.59303e98 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 208040616902520 q^{2} - 28\!\cdots\!20 q^{3} + 28\!\cdots\!24 q^{4} - 48\!\cdots\!60 q^{5} - 77\!\cdots\!44 q^{6} - 56\!\cdots\!00 q^{7} + 59\!\cdots\!60 q^{8} + 15\!\cdots\!76 q^{9} - 20\!\cdots\!60 q^{10}+ \cdots - 13\!\cdots\!08 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.07907e14 −1.01479 −0.507395 − 0.861713i \(-0.669392\pi\)
−0.507395 + 0.861713i \(0.669392\pi\)
\(3\) −6.25114e23 −1.50819 −0.754097 − 0.656763i \(-0.771925\pi\)
−0.754097 + 0.656763i \(0.771925\pi\)
\(4\) 1.88877e28 0.0297996
\(5\) −5.34820e34 −1.34646 −0.673229 − 0.739434i \(-0.735093\pi\)
−0.673229 + 0.739434i \(0.735093\pi\)
\(6\) 5.05034e38 1.53050
\(7\) 1.04094e42 1.53134 0.765671 − 0.643232i \(-0.222407\pi\)
0.765671 + 0.643232i \(0.222407\pi\)
\(8\) 4.96812e44 0.984550
\(9\) 2.18975e47 1.27465
\(10\) 4.32085e49 1.36637
\(11\) 2.09751e51 0.592606 0.296303 − 0.955094i \(-0.404246\pi\)
0.296303 + 0.955094i \(0.404246\pi\)
\(12\) −1.18070e52 −0.0449436
\(13\) −2.27740e55 −1.64909 −0.824543 − 0.565799i \(-0.808568\pi\)
−0.824543 + 0.565799i \(0.808568\pi\)
\(14\) −8.40981e56 −1.55399
\(15\) 3.34324e58 2.03072
\(16\) −4.13349e59 −1.02891
\(17\) 7.56916e60 0.937187 0.468593 − 0.883414i \(-0.344761\pi\)
0.468593 + 0.883414i \(0.344761\pi\)
\(18\) −1.76912e62 −1.29350
\(19\) 5.81301e62 0.292481 0.146240 − 0.989249i \(-0.453283\pi\)
0.146240 + 0.989249i \(0.453283\pi\)
\(20\) −1.01016e63 −0.0401239
\(21\) −6.50706e65 −2.30956
\(22\) −1.69459e66 −0.601371
\(23\) −2.12928e67 −0.836961 −0.418481 − 0.908226i \(-0.637437\pi\)
−0.418481 + 0.908226i \(0.637437\pi\)
\(24\) −3.10564e68 −1.48489
\(25\) 1.28261e69 0.812948
\(26\) 1.83993e70 1.67348
\(27\) −2.94946e70 −0.414225
\(28\) 1.96610e70 0.0456334
\(29\) −1.52350e72 −0.622504 −0.311252 − 0.950327i \(-0.600748\pi\)
−0.311252 + 0.950327i \(0.600748\pi\)
\(30\) −2.70102e73 −2.06075
\(31\) 2.72377e73 0.409984 0.204992 − 0.978764i \(-0.434283\pi\)
0.204992 + 0.978764i \(0.434283\pi\)
\(32\) 1.90555e73 0.0595796
\(33\) −1.31118e75 −0.893765
\(34\) −6.11517e75 −0.951048
\(35\) −5.56715e76 −2.06189
\(36\) 4.13595e75 0.0379841
\(37\) 4.28140e77 1.01298 0.506490 − 0.862246i \(-0.330943\pi\)
0.506490 + 0.862246i \(0.330943\pi\)
\(38\) −4.69637e77 −0.296807
\(39\) 1.42363e79 2.48714
\(40\) −2.65705e79 −1.32566
\(41\) −5.02606e79 −0.738629 −0.369315 − 0.929304i \(-0.620407\pi\)
−0.369315 + 0.929304i \(0.620407\pi\)
\(42\) 5.25709e80 2.34372
\(43\) −1.05419e81 −1.46633 −0.733163 − 0.680053i \(-0.761957\pi\)
−0.733163 + 0.680053i \(0.761957\pi\)
\(44\) 3.96172e79 0.0176594
\(45\) −1.17112e82 −1.71626
\(46\) 1.72026e82 0.849340
\(47\) −3.56207e82 −0.606539 −0.303269 − 0.952905i \(-0.598078\pi\)
−0.303269 + 0.952905i \(0.598078\pi\)
\(48\) 2.58391e83 1.55180
\(49\) 6.21485e83 1.34501
\(50\) −1.03623e84 −0.824972
\(51\) −4.73159e84 −1.41346
\(52\) −4.30149e83 −0.0491421
\(53\) 1.04502e85 0.465019 0.232510 − 0.972594i \(-0.425306\pi\)
0.232510 + 0.972594i \(0.425306\pi\)
\(54\) 2.38289e85 0.420351
\(55\) −1.12179e86 −0.797919
\(56\) 5.17151e86 1.50768
\(57\) −3.63379e86 −0.441118
\(58\) 1.23085e87 0.631711
\(59\) 3.60957e87 0.794841 0.397421 − 0.917637i \(-0.369905\pi\)
0.397421 + 0.917637i \(0.369905\pi\)
\(60\) 6.31462e86 0.0605146
\(61\) −9.18510e87 −0.388380 −0.194190 − 0.980964i \(-0.562208\pi\)
−0.194190 + 0.980964i \(0.562208\pi\)
\(62\) −2.20055e88 −0.416048
\(63\) 2.27940e89 1.95192
\(64\) 2.46596e89 0.968451
\(65\) 1.21800e90 2.22042
\(66\) 1.05931e90 0.906984
\(67\) −1.79075e90 −0.728336 −0.364168 − 0.931333i \(-0.618647\pi\)
−0.364168 + 0.931333i \(0.618647\pi\)
\(68\) 1.42964e89 0.0279278
\(69\) 1.33104e91 1.26230
\(70\) 4.49774e91 2.09238
\(71\) −3.40784e91 −0.785581 −0.392791 − 0.919628i \(-0.628490\pi\)
−0.392791 + 0.919628i \(0.628490\pi\)
\(72\) 1.08790e92 1.25496
\(73\) 1.68443e92 0.981681 0.490841 − 0.871249i \(-0.336690\pi\)
0.490841 + 0.871249i \(0.336690\pi\)
\(74\) −3.45898e92 −1.02796
\(75\) −8.01775e92 −1.22608
\(76\) 1.09795e91 0.00871582
\(77\) 2.18338e93 0.907483
\(78\) −1.15016e94 −2.52393
\(79\) −1.45180e94 −1.69576 −0.847880 − 0.530188i \(-0.822121\pi\)
−0.847880 + 0.530188i \(0.822121\pi\)
\(80\) 2.21068e94 1.38539
\(81\) −1.91808e94 −0.649918
\(82\) 4.06058e94 0.749554
\(83\) −9.55043e94 −0.967515 −0.483757 − 0.875202i \(-0.660728\pi\)
−0.483757 + 0.875202i \(0.660728\pi\)
\(84\) −1.22904e94 −0.0688240
\(85\) −4.04814e95 −1.26188
\(86\) 8.51687e95 1.48801
\(87\) 9.52362e95 0.938856
\(88\) 1.04207e96 0.583451
\(89\) 4.74685e96 1.51915 0.759575 − 0.650420i \(-0.225407\pi\)
0.759575 + 0.650420i \(0.225407\pi\)
\(90\) 9.46159e96 1.74165
\(91\) −2.37063e97 −2.52531
\(92\) −4.02173e95 −0.0249411
\(93\) −1.70267e97 −0.618335
\(94\) 2.87782e97 0.615510
\(95\) −3.10891e97 −0.393813
\(96\) −1.19119e97 −0.0898575
\(97\) −1.76496e98 −0.797138 −0.398569 − 0.917138i \(-0.630493\pi\)
−0.398569 + 0.917138i \(0.630493\pi\)
\(98\) −5.02102e98 −1.36490
\(99\) 4.59303e98 0.755365
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.1.2 ✓ 8
3.2 odd 2 9.100.a.d.1.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.100.a.a.1.2 ✓ 8 1.1 even 1 trivial
9.100.a.d.1.7 8 3.2 odd 2