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Properties

Label 9.100.a.d.1.5
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 1.5
Root \(-8.46093e12\) 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)
 
\(f(q)\) \(=\) \(q+6.35192e14 q^{2} -2.30357e29 q^{4} -4.78100e33 q^{5} +7.17401e41 q^{7} -5.48921e44 q^{8} -3.03685e48 q^{10} +4.28029e51 q^{11} +1.83718e55 q^{13} +4.55687e56 q^{14} -2.02664e59 q^{16} -6.21642e59 q^{17} +2.46616e63 q^{19} +1.10133e63 q^{20} +2.71880e66 q^{22} -1.40807e67 q^{23} -1.55486e69 q^{25} +1.16696e70 q^{26} -1.65258e71 q^{28} -2.85339e72 q^{29} -7.00608e73 q^{31} +2.19189e74 q^{32} -3.94862e74 q^{34} -3.42989e75 q^{35} -5.39350e76 q^{37} +1.56649e78 q^{38} +2.62439e78 q^{40} -1.04979e80 q^{41} -1.06727e81 q^{43} -9.85993e80 q^{44} -8.94393e81 q^{46} -6.71187e82 q^{47} +5.25966e82 q^{49} -9.87637e83 q^{50} -4.23207e84 q^{52} +2.07277e85 q^{53} -2.04640e85 q^{55} -3.93797e86 q^{56} -1.81245e87 q^{58} +1.47079e87 q^{59} +2.22361e88 q^{61} -4.45021e88 q^{62} +2.67681e89 q^{64} -8.78357e88 q^{65} +4.24589e90 q^{67} +1.43199e89 q^{68} -2.17864e90 q^{70} -4.39297e91 q^{71} +1.41929e92 q^{73} -3.42591e91 q^{74} -5.68097e92 q^{76} +3.07068e93 q^{77} +1.10262e94 q^{79} +9.68938e92 q^{80} -6.66819e94 q^{82} -8.17771e94 q^{83} +2.97207e93 q^{85} -6.77920e95 q^{86} -2.34954e96 q^{88} -5.46832e96 q^{89} +1.31800e97 q^{91} +3.24358e96 q^{92} -4.26333e97 q^{94} -1.17907e97 q^{95} +1.47493e98 q^{97} +3.34089e97 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 208040616902520 q^{2} + 28\!\cdots\!24 q^{4} + 48\!\cdots\!60 q^{5} - 56\!\cdots\!00 q^{7} - 59\!\cdots\!60 q^{8} - 20\!\cdots\!60 q^{10} - 66\!\cdots\!36 q^{11} - 53\!\cdots\!40 q^{13} + 27\!\cdots\!92 q^{14}+ \cdots + 12\!\cdots\!60 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.35192e14 0.797848 0.398924 − 0.916984i \(-0.369384\pi\)
0.398924 + 0.916984i \(0.369384\pi\)
\(3\) 0 0
\(4\) −2.30357e29 −0.363439
\(5\) −4.78100e33 −0.120366 −0.0601829 − 0.998187i \(-0.519168\pi\)
−0.0601829 + 0.998187i \(0.519168\pi\)
\(6\) 0 0
\(7\) 7.17401e41 1.05538 0.527690 − 0.849437i \(-0.323058\pi\)
0.527690 + 0.849437i \(0.323058\pi\)
\(8\) −5.48921e44 −1.08782
\(9\) 0 0
\(10\) −3.03685e48 −0.0960336
\(11\) 4.28029e51 1.20930 0.604652 − 0.796490i \(-0.293312\pi\)
0.604652 + 0.796490i \(0.293312\pi\)
\(12\) 0 0
\(13\) 1.83718e55 1.33032 0.665161 − 0.746700i \(-0.268363\pi\)
0.665161 + 0.746700i \(0.268363\pi\)
\(14\) 4.55687e56 0.842033
\(15\) 0 0
\(16\) −2.02664e59 −0.504474
\(17\) −6.21642e59 −0.0769695 −0.0384847 − 0.999259i \(-0.512253\pi\)
−0.0384847 + 0.999259i \(0.512253\pi\)
\(18\) 0 0
\(19\) 2.46616e63 1.24085 0.620424 − 0.784267i \(-0.286961\pi\)
0.620424 + 0.784267i \(0.286961\pi\)
\(20\) 1.10133e63 0.0437456
\(21\) 0 0
\(22\) 2.71880e66 0.964840
\(23\) −1.40807e67 −0.553472 −0.276736 − 0.960946i \(-0.589253\pi\)
−0.276736 + 0.960946i \(0.589253\pi\)
\(24\) 0 0
\(25\) −1.55486e69 −0.985512
\(26\) 1.16696e70 1.06139
\(27\) 0 0
\(28\) −1.65258e71 −0.383566
\(29\) −2.85339e72 −1.16590 −0.582948 − 0.812509i \(-0.698101\pi\)
−0.582948 + 0.812509i \(0.698101\pi\)
\(30\) 0 0
\(31\) −7.00608e73 −1.05456 −0.527280 − 0.849692i \(-0.676788\pi\)
−0.527280 + 0.849692i \(0.676788\pi\)
\(32\) 2.19189e74 0.685324
\(33\) 0 0
\(34\) −3.94862e74 −0.0614099
\(35\) −3.42989e75 −0.127032
\(36\) 0 0
\(37\) −5.39350e76 −0.127610 −0.0638051 − 0.997962i \(-0.520324\pi\)
−0.0638051 + 0.997962i \(0.520324\pi\)
\(38\) 1.56649e78 0.990008
\(39\) 0 0
\(40\) 2.62439e78 0.130936
\(41\) −1.04979e80 −1.54277 −0.771387 − 0.636367i \(-0.780436\pi\)
−0.771387 + 0.636367i \(0.780436\pi\)
\(42\) 0 0
\(43\) −1.06727e81 −1.48452 −0.742258 − 0.670115i \(-0.766245\pi\)
−0.742258 + 0.670115i \(0.766245\pi\)
\(44\) −9.85993e80 −0.439508
\(45\) 0 0
\(46\) −8.94393e81 −0.441586
\(47\) −6.71187e82 −1.14288 −0.571439 − 0.820645i \(-0.693615\pi\)
−0.571439 + 0.820645i \(0.693615\pi\)
\(48\) 0 0
\(49\) 5.25966e82 0.113829
\(50\) −9.87637e83 −0.786289
\(51\) 0 0
\(52\) −4.23207e84 −0.483490
\(53\) 2.07277e85 0.922351 0.461176 − 0.887309i \(-0.347428\pi\)
0.461176 + 0.887309i \(0.347428\pi\)
\(54\) 0 0
\(55\) −2.04640e85 −0.145559
\(56\) −3.93797e86 −1.14806
\(57\) 0 0
\(58\) −1.81245e87 −0.930208
\(59\) 1.47079e87 0.323874 0.161937 − 0.986801i \(-0.448226\pi\)
0.161937 + 0.986801i \(0.448226\pi\)
\(60\) 0 0
\(61\) 2.22361e88 0.940227 0.470113 − 0.882606i \(-0.344213\pi\)
0.470113 + 0.882606i \(0.344213\pi\)
\(62\) −4.45021e88 −0.841379
\(63\) 0 0
\(64\) 2.67681e89 1.05126
\(65\) −8.78357e88 −0.160125
\(66\) 0 0
\(67\) 4.24589e90 1.72689 0.863447 − 0.504439i \(-0.168301\pi\)
0.863447 + 0.504439i \(0.168301\pi\)
\(68\) 1.43199e89 0.0279737
\(69\) 0 0
\(70\) −2.17864e90 −0.101352
\(71\) −4.39297e91 −1.01268 −0.506338 − 0.862335i \(-0.669001\pi\)
−0.506338 + 0.862335i \(0.669001\pi\)
\(72\) 0 0
\(73\) 1.41929e92 0.827158 0.413579 − 0.910468i \(-0.364279\pi\)
0.413579 + 0.910468i \(0.364279\pi\)
\(74\) −3.42591e91 −0.101814
\(75\) 0 0
\(76\) −5.68097e92 −0.450972
\(77\) 3.07068e93 1.27628
\(78\) 0 0
\(79\) 1.10262e94 1.28791 0.643954 − 0.765064i \(-0.277293\pi\)
0.643954 + 0.765064i \(0.277293\pi\)
\(80\) 9.68938e92 0.0607214
\(81\) 0 0
\(82\) −6.66819e94 −1.23090
\(83\) −8.17771e94 −0.828449 −0.414225 − 0.910175i \(-0.635947\pi\)
−0.414225 + 0.910175i \(0.635947\pi\)
\(84\) 0 0
\(85\) 2.97207e93 0.00926449
\(86\) −6.77920e95 −1.18442
\(87\) 0 0
\(88\) −2.34954e96 −1.31550
\(89\) −5.46832e96 −1.75004 −0.875021 − 0.484084i \(-0.839153\pi\)
−0.875021 + 0.484084i \(0.839153\pi\)
\(90\) 0 0
\(91\) 1.31800e97 1.40400
\(92\) 3.24358e96 0.201153
\(93\) 0 0
\(94\) −4.26333e97 −0.911842
\(95\) −1.17907e97 −0.149356
\(96\) 0 0
\(97\) 1.47493e98 0.666147 0.333074 − 0.942901i \(-0.391914\pi\)
0.333074 + 0.942901i \(0.391914\pi\)
\(98\) 3.34089e97 0.0908180
\(99\) 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.1.5 8
3.2 odd 2 1.100.a.a.1.4 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.100.a.a.1.4 ✓ 8 3.2 odd 2
9.100.a.d.1.5 8 1.1 even 1 trivial