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Properties

Label 1.100.a.a.1.8
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 1.8
Root \(2.16174e13\) 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)
 
\(f(q)\) \(=\) \(q+1.53045e15 q^{2} -3.78025e23 q^{3} +1.70845e30 q^{4} -5.47796e34 q^{5} -5.78549e38 q^{6} -5.03791e41 q^{7} +1.64466e45 q^{8} -2.88893e46 q^{9} -8.38375e49 q^{10} +3.39044e50 q^{11} -6.45838e53 q^{12} +1.35923e55 q^{13} -7.71028e56 q^{14} +2.07081e58 q^{15} +1.43421e60 q^{16} +1.35236e61 q^{17} -4.42136e61 q^{18} +1.25642e63 q^{19} -9.35884e64 q^{20} +1.90446e65 q^{21} +5.18889e65 q^{22} +5.45698e66 q^{23} -6.21724e68 q^{24} +1.42309e69 q^{25} +2.08023e70 q^{26} +7.58628e70 q^{27} -8.60704e71 q^{28} +1.00507e72 q^{29} +3.16927e73 q^{30} -5.43714e73 q^{31} +1.15256e75 q^{32} -1.28167e74 q^{33} +2.06972e76 q^{34} +2.75975e76 q^{35} -4.93560e76 q^{36} -2.04705e77 q^{37} +1.92288e78 q^{38} -5.13823e78 q^{39} -9.00940e79 q^{40} +5.05152e79 q^{41} +2.91468e80 q^{42} +9.21616e80 q^{43} +5.79240e80 q^{44} +1.58254e81 q^{45} +8.35164e81 q^{46} +4.56701e82 q^{47} -5.42169e83 q^{48} -2.08262e83 q^{49} +2.17796e84 q^{50} -5.11227e84 q^{51} +2.32218e85 q^{52} -2.37631e85 q^{53} +1.16104e86 q^{54} -1.85727e85 q^{55} -8.28567e86 q^{56} -4.74957e86 q^{57} +1.53821e87 q^{58} +1.52955e87 q^{59} +3.53788e88 q^{60} +1.15621e88 q^{61} -8.32127e88 q^{62} +1.45542e88 q^{63} +8.54901e89 q^{64} -7.44581e89 q^{65} -1.96153e89 q^{66} +1.58442e90 q^{67} +2.31045e91 q^{68} -2.06288e90 q^{69} +4.22366e91 q^{70} -7.68861e91 q^{71} -4.75131e91 q^{72} +2.31521e92 q^{73} -3.13290e92 q^{74} -5.37963e92 q^{75} +2.14653e93 q^{76} -1.70807e92 q^{77} -7.86381e93 q^{78} +1.21213e94 q^{79} -7.85657e94 q^{80} -2.37151e94 q^{81} +7.73109e94 q^{82} +6.01316e94 q^{83} +3.25368e95 q^{84} -7.40819e95 q^{85} +1.41049e96 q^{86} -3.79942e95 q^{87} +5.57612e95 q^{88} +1.16422e96 q^{89} +2.42201e96 q^{90} -6.84768e96 q^{91} +9.32300e96 q^{92} +2.05538e97 q^{93} +6.98958e97 q^{94} -6.88260e97 q^{95} -4.35698e98 q^{96} -1.98012e98 q^{97} -3.18735e98 q^{98} -9.79473e96 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 208040616902520 q^{2} - 28\!\cdots\!20 q^{3} + 28\!\cdots\!24 q^{4} - 48\!\cdots\!60 q^{5} - 77\!\cdots\!44 q^{6} - 56\!\cdots\!00 q^{7} + 59\!\cdots\!60 q^{8} + 15\!\cdots\!76 q^{9} - 20\!\cdots\!60 q^{10}+ \cdots - 13\!\cdots\!08 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.53045e15 1.92236 0.961179 − 0.275925i \(-0.0889841\pi\)
0.961179 + 0.275925i \(0.0889841\pi\)
\(3\) −3.78025e23 −0.912051 −0.456025 − 0.889967i \(-0.650727\pi\)
−0.456025 + 0.889967i \(0.650727\pi\)
\(4\) 1.70845e30 2.69546
\(5\) −5.47796e34 −1.37913 −0.689563 − 0.724226i \(-0.742197\pi\)
−0.689563 + 0.724226i \(0.742197\pi\)
\(6\) −5.78549e38 −1.75329
\(7\) −5.03791e41 −0.741136 −0.370568 − 0.928805i \(-0.620837\pi\)
−0.370568 + 0.928805i \(0.620837\pi\)
\(8\) 1.64466e45 3.25929
\(9\) −2.88893e46 −0.168164
\(10\) −8.38375e49 −2.65117
\(11\) 3.39044e50 0.0957895 0.0478947 − 0.998852i \(-0.484749\pi\)
0.0478947 + 0.998852i \(0.484749\pi\)
\(12\) −6.45838e53 −2.45840
\(13\) 1.35923e55 0.984231 0.492115 − 0.870530i \(-0.336224\pi\)
0.492115 + 0.870530i \(0.336224\pi\)
\(14\) −7.71028e56 −1.42473
\(15\) 2.07081e58 1.25783
\(16\) 1.43421e60 3.57005
\(17\) 1.35236e61 1.67445 0.837224 − 0.546860i \(-0.184177\pi\)
0.837224 + 0.546860i \(0.184177\pi\)
\(18\) −4.42136e61 −0.323271
\(19\) 1.25642e63 0.632165 0.316082 − 0.948732i \(-0.397632\pi\)
0.316082 + 0.948732i \(0.397632\pi\)
\(20\) −9.35884e64 −3.71738
\(21\) 1.90446e65 0.675953
\(22\) 5.18889e65 0.184142
\(23\) 5.45698e66 0.214499 0.107249 − 0.994232i \(-0.465796\pi\)
0.107249 + 0.994232i \(0.465796\pi\)
\(24\) −6.21724e68 −2.97263
\(25\) 1.42309e69 0.901988
\(26\) 2.08023e70 1.89204
\(27\) 7.58628e70 1.06542
\(28\) −8.60704e71 −1.99770
\(29\) 1.00507e72 0.410672 0.205336 − 0.978692i \(-0.434171\pi\)
0.205336 + 0.978692i \(0.434171\pi\)
\(30\) 3.16927e73 2.41800
\(31\) −5.43714e73 −0.818402 −0.409201 − 0.912444i \(-0.634193\pi\)
−0.409201 + 0.912444i \(0.634193\pi\)
\(32\) 1.15256e75 3.60364
\(33\) −1.28167e74 −0.0873648
\(34\) 2.06972e76 3.21889
\(35\) 2.75975e76 1.02212
\(36\) −4.93560e76 −0.453279
\(37\) −2.04705e77 −0.484331 −0.242166 − 0.970235i \(-0.577858\pi\)
−0.242166 + 0.970235i \(0.577858\pi\)
\(38\) 1.92288e78 1.21525
\(39\) −5.13823e78 −0.897668
\(40\) −9.00940e79 −4.49497
\(41\) 5.05152e79 0.742371 0.371185 − 0.928559i \(-0.378951\pi\)
0.371185 + 0.928559i \(0.378951\pi\)
\(42\) 2.91468e80 1.29942
\(43\) 9.21616e80 1.28192 0.640961 − 0.767573i \(-0.278536\pi\)
0.640961 + 0.767573i \(0.278536\pi\)
\(44\) 5.79240e80 0.258197
\(45\) 1.58254e81 0.231919
\(46\) 8.35164e81 0.412344
\(47\) 4.56701e82 0.777656 0.388828 − 0.921310i \(-0.372880\pi\)
0.388828 + 0.921310i \(0.372880\pi\)
\(48\) −5.42169e83 −3.25607
\(49\) −2.08262e83 −0.450718
\(50\) 2.17796e84 1.73394
\(51\) −5.11227e84 −1.52718
\(52\) 2.32218e85 2.65296
\(53\) −2.37631e85 −1.05742 −0.528709 − 0.848803i \(-0.677324\pi\)
−0.528709 + 0.848803i \(0.677324\pi\)
\(54\) 1.16104e86 2.04813
\(55\) −1.85727e85 −0.132106
\(56\) −8.28567e86 −2.41557
\(57\) −4.74957e86 −0.576566
\(58\) 1.53821e87 0.789459
\(59\) 1.52955e87 0.336813 0.168407 − 0.985718i \(-0.446138\pi\)
0.168407 + 0.985718i \(0.446138\pi\)
\(60\) 3.53788e88 3.39044
\(61\) 1.15621e88 0.488889 0.244444 − 0.969663i \(-0.421394\pi\)
0.244444 + 0.969663i \(0.421394\pi\)
\(62\) −8.32127e88 −1.57326
\(63\) 1.45542e88 0.124632
\(64\) 8.54901e89 3.35743
\(65\) −7.44581e89 −1.35738
\(66\) −1.96153e89 −0.167946
\(67\) 1.58442e90 0.644419 0.322209 − 0.946668i \(-0.395574\pi\)
0.322209 + 0.946668i \(0.395574\pi\)
\(68\) 2.31045e91 4.51341
\(69\) −2.06288e90 −0.195634
\(70\) 4.22366e91 1.96488
\(71\) −7.68861e91 −1.77239 −0.886195 − 0.463312i \(-0.846661\pi\)
−0.886195 + 0.463312i \(0.846661\pi\)
\(72\) −4.75131e91 −0.548094
\(73\) 2.31521e92 1.34930 0.674651 − 0.738137i \(-0.264294\pi\)
0.674651 + 0.738137i \(0.264294\pi\)
\(74\) −3.13290e92 −0.931058
\(75\) −5.37963e92 −0.822658
\(76\) 2.14653e93 1.70398
\(77\) −1.70807e92 −0.0709930
\(78\) −7.86381e93 −1.72564
\(79\) 1.21213e94 1.41582 0.707912 − 0.706300i \(-0.249637\pi\)
0.707912 + 0.706300i \(0.249637\pi\)
\(80\) −7.85657e94 −4.92355
\(81\) −2.37151e94 −0.803557
\(82\) 7.73109e94 1.42710
\(83\) 6.01316e94 0.609169 0.304584 − 0.952485i \(-0.401482\pi\)
0.304584 + 0.952485i \(0.401482\pi\)
\(84\) 3.25368e95 1.82201
\(85\) −7.40819e95 −2.30927
\(86\) 1.41049e96 2.46431
\(87\) −3.79942e95 −0.374554
\(88\) 5.57612e95 0.312205
\(89\) 1.16422e96 0.372587 0.186294 − 0.982494i \(-0.440352\pi\)
0.186294 + 0.982494i \(0.440352\pi\)
\(90\) 2.42201e96 0.445832
\(91\) −6.84768e96 −0.729449
\(92\) 9.32300e96 0.578173
\(93\) 2.05538e97 0.746424
\(94\) 6.98958e97 1.49493
\(95\) −6.88260e97 −0.871834
\(96\) −4.35698e98 −3.28670
\(97\) −1.98012e98 −0.894315 −0.447157 − 0.894455i \(-0.647564\pi\)
−0.447157 + 0.894455i \(0.647564\pi\)
\(98\) −3.18735e98 −0.866441
\(99\) −9.79473e96 −0.0161083
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.1.8 ✓ 8
3.2 odd 2 9.100.a.d.1.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.100.a.a.1.8 ✓ 8 1.1 even 1 trivial
9.100.a.d.1.1 8 3.2 odd 2