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Properties

Label 1.100.a.a.1.1
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 1.1
Root \(-2.10181e13\) 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)
 
\(f(q)\) \(=\) \(q-1.53931e15 q^{2} +1.09471e23 q^{3} +1.73565e30 q^{4} -1.42424e34 q^{5} -1.68510e38 q^{6} -4.25157e41 q^{7} -1.69606e45 q^{8} -1.59809e47 q^{9} +2.19235e49 q^{10} -5.04571e51 q^{11} +1.90004e53 q^{12} -1.33313e55 q^{13} +6.54449e56 q^{14} -1.55913e57 q^{15} +1.51066e60 q^{16} +8.54293e60 q^{17} +2.45995e62 q^{18} -3.57926e63 q^{19} -2.47199e64 q^{20} -4.65425e64 q^{21} +7.76692e66 q^{22} -1.32632e67 q^{23} -1.85670e68 q^{24} -1.37488e69 q^{25} +2.05211e70 q^{26} -3.63008e70 q^{27} -7.37926e71 q^{28} -1.57811e71 q^{29} +2.39999e72 q^{30} -3.89706e73 q^{31} -1.25037e75 q^{32} -5.52360e74 q^{33} -1.31502e76 q^{34} +6.05525e75 q^{35} -2.77372e77 q^{36} -1.80202e77 q^{37} +5.50960e78 q^{38} -1.45940e78 q^{39} +2.41559e79 q^{40} +2.13003e79 q^{41} +7.16433e79 q^{42} +8.43003e79 q^{43} -8.75761e81 q^{44} +2.27606e81 q^{45} +2.04162e82 q^{46} +2.19872e82 q^{47} +1.65374e83 q^{48} -2.81310e83 q^{49} +2.11636e84 q^{50} +9.35205e83 q^{51} -2.31386e85 q^{52} -2.60808e85 q^{53} +5.58782e85 q^{54} +7.18630e85 q^{55} +7.21091e86 q^{56} -3.91826e86 q^{57} +2.42920e86 q^{58} +3.91766e87 q^{59} -2.70611e87 q^{60} -3.67456e88 q^{61} +5.99879e88 q^{62} +6.79437e88 q^{63} +9.67216e89 q^{64} +1.89870e89 q^{65} +8.50254e89 q^{66} -3.53225e90 q^{67} +1.48276e91 q^{68} -1.45194e90 q^{69} -9.32092e90 q^{70} +1.43616e91 q^{71} +2.71045e92 q^{72} -5.96639e91 q^{73} +2.77387e92 q^{74} -1.50509e92 q^{75} -6.21236e93 q^{76} +2.14522e93 q^{77} +2.24647e93 q^{78} -1.81922e93 q^{79} -2.15154e94 q^{80} +2.34800e94 q^{81} -3.27878e94 q^{82} -6.69942e93 q^{83} -8.07816e94 q^{84} -1.21672e95 q^{85} -1.29764e95 q^{86} -1.72758e94 q^{87} +8.55782e96 q^{88} -5.60112e95 q^{89} -3.50356e96 q^{90} +5.66791e96 q^{91} -2.30203e97 q^{92} -4.26616e96 q^{93} -3.38451e97 q^{94} +5.09772e97 q^{95} -1.36880e98 q^{96} -2.13029e98 q^{97} +4.33023e98 q^{98} +8.06348e98 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 208040616902520 q^{2} - 28\!\cdots\!20 q^{3} + 28\!\cdots\!24 q^{4} - 48\!\cdots\!60 q^{5} - 77\!\cdots\!44 q^{6} - 56\!\cdots\!00 q^{7} + 59\!\cdots\!60 q^{8} + 15\!\cdots\!76 q^{9} - 20\!\cdots\!60 q^{10}+ \cdots - 13\!\cdots\!08 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.53931e15 −1.93349 −0.966744 − 0.255744i \(-0.917680\pi\)
−0.966744 + 0.255744i \(0.917680\pi\)
\(3\) 1.09471e23 0.264118 0.132059 − 0.991242i \(-0.457841\pi\)
0.132059 + 0.991242i \(0.457841\pi\)
\(4\) 1.73565e30 2.73838
\(5\) −1.42424e34 −0.358565 −0.179282 − 0.983798i \(-0.557378\pi\)
−0.179282 + 0.983798i \(0.557378\pi\)
\(6\) −1.68510e38 −0.510669
\(7\) −4.25157e41 −0.625455 −0.312728 − 0.949843i \(-0.601243\pi\)
−0.312728 + 0.949843i \(0.601243\pi\)
\(8\) −1.69606e45 −3.36114
\(9\) −1.59809e47 −0.930242
\(10\) 2.19235e49 0.693281
\(11\) −5.04571e51 −1.42556 −0.712778 − 0.701390i \(-0.752563\pi\)
−0.712778 + 0.701390i \(0.752563\pi\)
\(12\) 1.90004e53 0.723255
\(13\) −1.33313e55 −0.965334 −0.482667 − 0.875804i \(-0.660332\pi\)
−0.482667 + 0.875804i \(0.660332\pi\)
\(14\) 6.54449e56 1.20931
\(15\) −1.55913e57 −0.0947034
\(16\) 1.51066e60 3.76034
\(17\) 8.54293e60 1.05776 0.528878 − 0.848698i \(-0.322613\pi\)
0.528878 + 0.848698i \(0.322613\pi\)
\(18\) 2.45995e62 1.79861
\(19\) −3.57926e63 −1.80090 −0.900451 − 0.434957i \(-0.856763\pi\)
−0.900451 + 0.434957i \(0.856763\pi\)
\(20\) −2.47199e64 −0.981887
\(21\) −4.65425e64 −0.165194
\(22\) 7.76692e66 2.75630
\(23\) −1.32632e67 −0.521340 −0.260670 − 0.965428i \(-0.583943\pi\)
−0.260670 + 0.965428i \(0.583943\pi\)
\(24\) −1.85670e68 −0.887737
\(25\) −1.37488e69 −0.871431
\(26\) 2.05211e70 1.86646
\(27\) −3.63008e70 −0.509811
\(28\) −7.37926e71 −1.71273
\(29\) −1.57811e71 −0.0644817 −0.0322409 − 0.999480i \(-0.510264\pi\)
−0.0322409 + 0.999480i \(0.510264\pi\)
\(30\) 2.39999e72 0.183108
\(31\) −3.89706e73 −0.586588 −0.293294 − 0.956022i \(-0.594752\pi\)
−0.293294 + 0.956022i \(0.594752\pi\)
\(32\) −1.25037e75 −3.90944
\(33\) −5.52360e74 −0.376515
\(34\) −1.31502e76 −2.04516
\(35\) 6.05525e75 0.224266
\(36\) −2.77372e77 −2.54735
\(37\) −1.80202e77 −0.426357 −0.213179 − 0.977013i \(-0.568382\pi\)
−0.213179 + 0.977013i \(0.568382\pi\)
\(38\) 5.50960e78 3.48202
\(39\) −1.45940e78 −0.254962
\(40\) 2.41559e79 1.20519
\(41\) 2.13003e79 0.313029 0.156514 − 0.987676i \(-0.449974\pi\)
0.156514 + 0.987676i \(0.449974\pi\)
\(42\) 7.16433e79 0.319401
\(43\) 8.43003e79 0.117257 0.0586287 − 0.998280i \(-0.481327\pi\)
0.0586287 + 0.998280i \(0.481327\pi\)
\(44\) −8.75761e81 −3.90371
\(45\) 2.27606e81 0.333552
\(46\) 2.04162e82 1.00800
\(47\) 2.19872e82 0.374391 0.187195 − 0.982323i \(-0.440060\pi\)
0.187195 + 0.982323i \(0.440060\pi\)
\(48\) 1.65374e83 0.993174
\(49\) −2.81310e83 −0.608805
\(50\) 2.11636e84 1.68490
\(51\) 9.35205e83 0.279372
\(52\) −2.31386e85 −2.64345
\(53\) −2.60808e85 −1.16056 −0.580278 − 0.814418i \(-0.697056\pi\)
−0.580278 + 0.814418i \(0.697056\pi\)
\(54\) 5.58782e85 0.985715
\(55\) 7.18630e85 0.511154
\(56\) 7.21091e86 2.10224
\(57\) −3.91826e86 −0.475651
\(58\) 2.42920e86 0.124675
\(59\) 3.91766e87 0.862684 0.431342 − 0.902188i \(-0.358040\pi\)
0.431342 + 0.902188i \(0.358040\pi\)
\(60\) −2.70611e87 −0.259334
\(61\) −3.67456e88 −1.55374 −0.776869 − 0.629662i \(-0.783193\pi\)
−0.776869 + 0.629662i \(0.783193\pi\)
\(62\) 5.99879e88 1.13416
\(63\) 6.79437e88 0.581825
\(64\) 9.67216e89 3.79852
\(65\) 1.89870e89 0.346135
\(66\) 8.50254e89 0.727987
\(67\) −3.53225e90 −1.43664 −0.718322 − 0.695711i \(-0.755089\pi\)
−0.718322 + 0.695711i \(0.755089\pi\)
\(68\) 1.48276e91 2.89654
\(69\) −1.45194e90 −0.137695
\(70\) −9.32092e90 −0.433617
\(71\) 1.43616e91 0.331065 0.165533 − 0.986204i \(-0.447066\pi\)
0.165533 + 0.986204i \(0.447066\pi\)
\(72\) 2.71045e92 3.12667
\(73\) −5.96639e91 −0.347720 −0.173860 − 0.984770i \(-0.555624\pi\)
−0.173860 + 0.984770i \(0.555624\pi\)
\(74\) 2.77387e92 0.824357
\(75\) −1.50509e92 −0.230161
\(76\) −6.21236e93 −4.93155
\(77\) 2.14522e93 0.891622
\(78\) 2.24647e93 0.492966
\(79\) −1.81922e93 −0.212493 −0.106247 − 0.994340i \(-0.533883\pi\)
−0.106247 + 0.994340i \(0.533883\pi\)
\(80\) −2.15154e94 −1.34833
\(81\) 2.34800e94 0.795591
\(82\) −3.27878e94 −0.605238
\(83\) −6.69942e93 −0.0678690 −0.0339345 − 0.999424i \(-0.510804\pi\)
−0.0339345 + 0.999424i \(0.510804\pi\)
\(84\) −8.07816e94 −0.452364
\(85\) −1.21672e95 −0.379274
\(86\) −1.29764e95 −0.226716
\(87\) −1.72758e94 −0.0170308
\(88\) 8.55782e96 4.79149
\(89\) −5.60112e95 −0.179254 −0.0896271 − 0.995975i \(-0.528568\pi\)
−0.0896271 + 0.995975i \(0.528568\pi\)
\(90\) −3.50356e96 −0.644919
\(91\) 5.66791e96 0.603774
\(92\) −2.30203e97 −1.42763
\(93\) −4.26616e96 −0.154929
\(94\) −3.38451e97 −0.723881
\(95\) 5.09772e97 0.645740
\(96\) −1.36880e98 −1.03255
\(97\) −2.13029e98 −0.962139 −0.481070 − 0.876682i \(-0.659752\pi\)
−0.481070 + 0.876682i \(0.659752\pi\)
\(98\) 4.33023e98 1.17712
\(99\) 8.06348e98 1.32611
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.1.1 ✓ 8
3.2 odd 2 9.100.a.d.1.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.100.a.a.1.1 ✓ 8 1.1 even 1 trivial
9.100.a.d.1.8 8 3.2 odd 2