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Properties

Label 9.100.a.d.1.8
Level $9$
Weight $100$
Character 9.1
Self dual yes
Analytic conductor $558.609$
Analytic rank $1$
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] = [9,100,Mod(1,9)] 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("9.1"); S:= CuspForms(chi, 100); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 100, names="a")
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 100 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,208040616902520] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(558.609014683\)
Analytic rank: \(1\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: multiple of \( 2^{109}\cdot 3^{62}\cdot 5^{13}\cdot 7^{9}\cdot 11^{3}\cdot 13\cdot 17 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-2.10181e13\) of defining polynomial
Character \(\chi\) \(=\) 9.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.53931e15 q^{2} +1.73565e30 q^{4} +1.42424e34 q^{5} -4.25157e41 q^{7} +1.69606e45 q^{8} +2.19235e49 q^{10} +5.04571e51 q^{11} -1.33313e55 q^{13} -6.54449e56 q^{14} +1.51066e60 q^{16} -8.54293e60 q^{17} -3.57926e63 q^{19} +2.47199e64 q^{20} +7.76692e66 q^{22} +1.32632e67 q^{23} -1.37488e69 q^{25} -2.05211e70 q^{26} -7.37926e71 q^{28} +1.57811e71 q^{29} -3.89706e73 q^{31} +1.25037e75 q^{32} -1.31502e76 q^{34} -6.05525e75 q^{35} -1.80202e77 q^{37} -5.50960e78 q^{38} +2.41559e79 q^{40} -2.13003e79 q^{41} +8.43003e79 q^{43} +8.75761e81 q^{44} +2.04162e82 q^{46} -2.19872e82 q^{47} -2.81310e83 q^{49} -2.11636e84 q^{50} -2.31386e85 q^{52} +2.60808e85 q^{53} +7.18630e85 q^{55} -7.21091e86 q^{56} +2.42920e86 q^{58} -3.91766e87 q^{59} -3.67456e88 q^{61} -5.99879e88 q^{62} +9.67216e89 q^{64} -1.89870e89 q^{65} -3.53225e90 q^{67} -1.48276e91 q^{68} -9.32092e90 q^{70} -1.43616e91 q^{71} -5.96639e91 q^{73} -2.77387e92 q^{74} -6.21236e93 q^{76} -2.14522e93 q^{77} -1.81922e93 q^{79} +2.15154e94 q^{80} -3.27878e94 q^{82} +6.69942e93 q^{83} -1.21672e95 q^{85} +1.29764e95 q^{86} +8.55782e96 q^{88} +5.60112e95 q^{89} +5.66791e96 q^{91} +2.30203e97 q^{92} -3.38451e97 q^{94} -5.09772e97 q^{95} -2.13029e98 q^{97} -4.33023e98 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 208040616902520 q^{2} + 28\!\cdots\!24 q^{4} + 48\!\cdots\!60 q^{5} - 56\!\cdots\!00 q^{7} - 59\!\cdots\!60 q^{8} - 20\!\cdots\!60 q^{10} - 66\!\cdots\!36 q^{11} - 53\!\cdots\!40 q^{13} + 27\!\cdots\!92 q^{14}+ \cdots + 12\!\cdots\!60 q^{98}+O(q^{100}) \) 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 \(-0.0823205\pi\)
0.966744 + 0.255744i \(0.0823205\pi\)
\(3\) 0 0
\(4\) 1.73565e30 2.73838
\(5\) 1.42424e34 0.358565 0.179282 − 0.983798i \(-0.442622\pi\)
0.179282 + 0.983798i \(0.442622\pi\)
\(6\) 0 0
\(7\) −4.25157e41 −0.625455 −0.312728 − 0.949843i \(-0.601243\pi\)
−0.312728 + 0.949843i \(0.601243\pi\)
\(8\) 1.69606e45 3.36114
\(9\) 0 0
\(10\) 2.19235e49 0.693281
\(11\) 5.04571e51 1.42556 0.712778 − 0.701390i \(-0.247437\pi\)
0.712778 + 0.701390i \(0.247437\pi\)
\(12\) 0 0
\(13\) −1.33313e55 −0.965334 −0.482667 − 0.875804i \(-0.660332\pi\)
−0.482667 + 0.875804i \(0.660332\pi\)
\(14\) −6.54449e56 −1.20931
\(15\) 0 0
\(16\) 1.51066e60 3.76034
\(17\) −8.54293e60 −1.05776 −0.528878 − 0.848698i \(-0.677387\pi\)
−0.528878 + 0.848698i \(0.677387\pi\)
\(18\) 0 0
\(19\) −3.57926e63 −1.80090 −0.900451 − 0.434957i \(-0.856763\pi\)
−0.900451 + 0.434957i \(0.856763\pi\)
\(20\) 2.47199e64 0.981887
\(21\) 0 0
\(22\) 7.76692e66 2.75630
\(23\) 1.32632e67 0.521340 0.260670 − 0.965428i \(-0.416057\pi\)
0.260670 + 0.965428i \(0.416057\pi\)
\(24\) 0 0
\(25\) −1.37488e69 −0.871431
\(26\) −2.05211e70 −1.86646
\(27\) 0 0
\(28\) −7.37926e71 −1.71273
\(29\) 1.57811e71 0.0644817 0.0322409 − 0.999480i \(-0.489736\pi\)
0.0322409 + 0.999480i \(0.489736\pi\)
\(30\) 0 0
\(31\) −3.89706e73 −0.586588 −0.293294 − 0.956022i \(-0.594752\pi\)
−0.293294 + 0.956022i \(0.594752\pi\)
\(32\) 1.25037e75 3.90944
\(33\) 0 0
\(34\) −1.31502e76 −2.04516
\(35\) −6.05525e75 −0.224266
\(36\) 0 0
\(37\) −1.80202e77 −0.426357 −0.213179 − 0.977013i \(-0.568382\pi\)
−0.213179 + 0.977013i \(0.568382\pi\)
\(38\) −5.50960e78 −3.48202
\(39\) 0 0
\(40\) 2.41559e79 1.20519
\(41\) −2.13003e79 −0.313029 −0.156514 − 0.987676i \(-0.550026\pi\)
−0.156514 + 0.987676i \(0.550026\pi\)
\(42\) 0 0
\(43\) 8.43003e79 0.117257 0.0586287 − 0.998280i \(-0.481327\pi\)
0.0586287 + 0.998280i \(0.481327\pi\)
\(44\) 8.75761e81 3.90371
\(45\) 0 0
\(46\) 2.04162e82 1.00800
\(47\) −2.19872e82 −0.374391 −0.187195 − 0.982323i \(-0.559940\pi\)
−0.187195 + 0.982323i \(0.559940\pi\)
\(48\) 0 0
\(49\) −2.81310e83 −0.608805
\(50\) −2.11636e84 −1.68490
\(51\) 0 0
\(52\) −2.31386e85 −2.64345
\(53\) 2.60808e85 1.16056 0.580278 − 0.814418i \(-0.302944\pi\)
0.580278 + 0.814418i \(0.302944\pi\)
\(54\) 0 0
\(55\) 7.18630e85 0.511154
\(56\) −7.21091e86 −2.10224
\(57\) 0 0
\(58\) 2.42920e86 0.124675
\(59\) −3.91766e87 −0.862684 −0.431342 − 0.902188i \(-0.641960\pi\)
−0.431342 + 0.902188i \(0.641960\pi\)
\(60\) 0 0
\(61\) −3.67456e88 −1.55374 −0.776869 − 0.629662i \(-0.783193\pi\)
−0.776869 + 0.629662i \(0.783193\pi\)
\(62\) −5.99879e88 −1.13416
\(63\) 0 0
\(64\) 9.67216e89 3.79852
\(65\) −1.89870e89 −0.346135
\(66\) 0 0
\(67\) −3.53225e90 −1.43664 −0.718322 − 0.695711i \(-0.755089\pi\)
−0.718322 + 0.695711i \(0.755089\pi\)
\(68\) −1.48276e91 −2.89654
\(69\) 0 0
\(70\) −9.32092e90 −0.433617
\(71\) −1.43616e91 −0.331065 −0.165533 − 0.986204i \(-0.552934\pi\)
−0.165533 + 0.986204i \(0.552934\pi\)
\(72\) 0 0
\(73\) −5.96639e91 −0.347720 −0.173860 − 0.984770i \(-0.555624\pi\)
−0.173860 + 0.984770i \(0.555624\pi\)
\(74\) −2.77387e92 −0.824357
\(75\) 0 0
\(76\) −6.21236e93 −4.93155
\(77\) −2.14522e93 −0.891622
\(78\) 0 0
\(79\) −1.81922e93 −0.212493 −0.106247 − 0.994340i \(-0.533883\pi\)
−0.106247 + 0.994340i \(0.533883\pi\)
\(80\) 2.15154e94 1.34833
\(81\) 0 0
\(82\) −3.27878e94 −0.605238
\(83\) 6.69942e93 0.0678690 0.0339345 − 0.999424i \(-0.489196\pi\)
0.0339345 + 0.999424i \(0.489196\pi\)
\(84\) 0 0
\(85\) −1.21672e95 −0.379274
\(86\) 1.29764e95 0.226716
\(87\) 0 0
\(88\) 8.55782e96 4.79149
\(89\) 5.60112e95 0.179254 0.0896271 − 0.995975i \(-0.471432\pi\)
0.0896271 + 0.995975i \(0.471432\pi\)
\(90\) 0 0
\(91\) 5.66791e96 0.603774
\(92\) 2.30203e97 1.42763
\(93\) 0 0
\(94\) −3.38451e97 −0.723881
\(95\) −5.09772e97 −0.645740
\(96\) 0 0
\(97\) −2.13029e98 −0.962139 −0.481070 − 0.876682i \(-0.659752\pi\)
−0.481070 + 0.876682i \(0.659752\pi\)
\(98\) −4.33023e98 −1.17712
\(99\) 0 0
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 9.100.a.d.1.8 8
3.2 odd 2 1.100.a.a.1.1 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.100.a.a.1.1 ✓ 8 3.2 odd 2
9.100.a.d.1.8 8 1.1 even 1 trivial