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Properties

Label 9.100.a.d.1.3
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 1.3
Root \(8.50040e12\) 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)
 
\(f(q)\) \(=\) \(q-5.86024e14 q^{2} -2.90401e29 q^{4} -2.74639e34 q^{5} +3.47138e41 q^{7} +5.41619e44 q^{8} +1.60945e49 q^{10} +4.95910e51 q^{11} +1.21492e55 q^{13} -2.03431e56 q^{14} -1.33338e59 q^{16} +3.97916e60 q^{17} -4.47251e62 q^{19} +7.97554e63 q^{20} -2.90615e66 q^{22} +4.53817e67 q^{23} -8.23457e68 q^{25} -7.11971e69 q^{26} -1.00809e71 q^{28} +5.62741e71 q^{29} +7.11109e73 q^{31} -2.65152e74 q^{32} -2.33189e75 q^{34} -9.53375e75 q^{35} +3.44683e77 q^{37} +2.62100e77 q^{38} -1.48750e79 q^{40} -2.52241e78 q^{41} +8.67636e80 q^{43} -1.44013e81 q^{44} -2.65948e82 q^{46} -3.44728e82 q^{47} -3.41563e83 q^{49} +4.82566e83 q^{50} -3.52813e84 q^{52} -2.95920e85 q^{53} -1.36196e86 q^{55} +1.88016e86 q^{56} -3.29780e86 q^{58} -7.60447e87 q^{59} +8.96058e87 q^{61} -4.16727e88 q^{62} +2.39899e89 q^{64} -3.33663e89 q^{65} +2.62496e90 q^{67} -1.15555e90 q^{68} +5.58700e90 q^{70} -6.35730e91 q^{71} -3.15102e92 q^{73} -2.01993e92 q^{74} +1.29882e92 q^{76} +1.72149e93 q^{77} -1.44064e94 q^{79} +3.66198e93 q^{80} +1.47819e93 q^{82} -6.97978e94 q^{83} -1.09283e95 q^{85} -5.08455e95 q^{86} +2.68594e96 q^{88} +1.17382e96 q^{89} +4.21744e96 q^{91} -1.31789e97 q^{92} +2.02019e97 q^{94} +1.22833e97 q^{95} -3.50742e98 q^{97} +2.00164e98 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 208040616902520 q^{2} + 28\!\cdots\!24 q^{4} + 48\!\cdots\!60 q^{5} - 56\!\cdots\!00 q^{7} - 59\!\cdots\!60 q^{8} - 20\!\cdots\!60 q^{10} - 66\!\cdots\!36 q^{11} - 53\!\cdots\!40 q^{13} + 27\!\cdots\!92 q^{14}+ \cdots + 12\!\cdots\!60 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.86024e14 −0.736089 −0.368045 − 0.929808i \(-0.619973\pi\)
−0.368045 + 0.929808i \(0.619973\pi\)
\(3\) 0 0
\(4\) −2.90401e29 −0.458172
\(5\) −2.74639e34 −0.691427 −0.345714 − 0.938340i \(-0.612363\pi\)
−0.345714 + 0.938340i \(0.612363\pi\)
\(6\) 0 0
\(7\) 3.47138e41 0.510680 0.255340 − 0.966851i \(-0.417813\pi\)
0.255340 + 0.966851i \(0.417813\pi\)
\(8\) 5.41619e44 1.07335
\(9\) 0 0
\(10\) 1.60945e49 0.508952
\(11\) 4.95910e51 1.40109 0.700543 − 0.713611i \(-0.252941\pi\)
0.700543 + 0.713611i \(0.252941\pi\)
\(12\) 0 0
\(13\) 1.21492e55 0.879733 0.439866 − 0.898063i \(-0.355026\pi\)
0.439866 + 0.898063i \(0.355026\pi\)
\(14\) −2.03431e56 −0.375906
\(15\) 0 0
\(16\) −1.33338e59 −0.331906
\(17\) 3.97916e60 0.492686 0.246343 − 0.969183i \(-0.420771\pi\)
0.246343 + 0.969183i \(0.420771\pi\)
\(18\) 0 0
\(19\) −4.47251e62 −0.225034 −0.112517 − 0.993650i \(-0.535891\pi\)
−0.112517 + 0.993650i \(0.535891\pi\)
\(20\) 7.97554e63 0.316793
\(21\) 0 0
\(22\) −2.90615e66 −1.03132
\(23\) 4.53817e67 1.78383 0.891914 − 0.452205i \(-0.149363\pi\)
0.891914 + 0.452205i \(0.149363\pi\)
\(24\) 0 0
\(25\) −8.23457e68 −0.521928
\(26\) −7.11971e69 −0.647562
\(27\) 0 0
\(28\) −1.00809e71 −0.233979
\(29\) 5.62741e71 0.229936 0.114968 − 0.993369i \(-0.463323\pi\)
0.114968 + 0.993369i \(0.463323\pi\)
\(30\) 0 0
\(31\) 7.11109e73 1.07037 0.535183 − 0.844736i \(-0.320243\pi\)
0.535183 + 0.844736i \(0.320243\pi\)
\(32\) −2.65152e74 −0.829033
\(33\) 0 0
\(34\) −2.33189e75 −0.362661
\(35\) −9.53375e75 −0.353098
\(36\) 0 0
\(37\) 3.44683e77 0.815520 0.407760 − 0.913089i \(-0.366310\pi\)
0.407760 + 0.913089i \(0.366310\pi\)
\(38\) 2.62100e77 0.165645
\(39\) 0 0
\(40\) −1.48750e79 −0.742140
\(41\) −2.52241e78 −0.0370693 −0.0185346 − 0.999828i \(-0.505900\pi\)
−0.0185346 + 0.999828i \(0.505900\pi\)
\(42\) 0 0
\(43\) 8.67636e80 1.20684 0.603419 − 0.797425i \(-0.293805\pi\)
0.603419 + 0.797425i \(0.293805\pi\)
\(44\) −1.44013e81 −0.641939
\(45\) 0 0
\(46\) −2.65948e82 −1.31306
\(47\) −3.44728e82 −0.586992 −0.293496 − 0.955960i \(-0.594819\pi\)
−0.293496 + 0.955960i \(0.594819\pi\)
\(48\) 0 0
\(49\) −3.41563e83 −0.739206
\(50\) 4.82566e83 0.384186
\(51\) 0 0
\(52\) −3.52813e84 −0.403069
\(53\) −2.95920e85 −1.31680 −0.658399 − 0.752669i \(-0.728766\pi\)
−0.658399 + 0.752669i \(0.728766\pi\)
\(54\) 0 0
\(55\) −1.36196e86 −0.968749
\(56\) 1.88016e86 0.548136
\(57\) 0 0
\(58\) −3.29780e86 −0.169254
\(59\) −7.60447e87 −1.67454 −0.837268 − 0.546792i \(-0.815849\pi\)
−0.837268 + 0.546792i \(0.815849\pi\)
\(60\) 0 0
\(61\) 8.96058e87 0.378887 0.189443 − 0.981892i \(-0.439332\pi\)
0.189443 + 0.981892i \(0.439332\pi\)
\(62\) −4.16727e88 −0.787885
\(63\) 0 0
\(64\) 2.39899e89 0.942148
\(65\) −3.33663e89 −0.608271
\(66\) 0 0
\(67\) 2.62496e90 1.06763 0.533813 − 0.845602i \(-0.320759\pi\)
0.533813 + 0.845602i \(0.320759\pi\)
\(68\) −1.15555e90 −0.225735
\(69\) 0 0
\(70\) 5.58700e90 0.259912
\(71\) −6.35730e91 −1.46550 −0.732748 − 0.680501i \(-0.761762\pi\)
−0.732748 + 0.680501i \(0.761762\pi\)
\(72\) 0 0
\(73\) −3.15102e92 −1.83641 −0.918205 − 0.396105i \(-0.870362\pi\)
−0.918205 + 0.396105i \(0.870362\pi\)
\(74\) −2.01993e92 −0.600296
\(75\) 0 0
\(76\) 1.29882e92 0.103104
\(77\) 1.72149e93 0.715506
\(78\) 0 0
\(79\) −1.44064e94 −1.68273 −0.841365 − 0.540468i \(-0.818247\pi\)
−0.841365 + 0.540468i \(0.818247\pi\)
\(80\) 3.66198e93 0.229489
\(81\) 0 0
\(82\) 1.47819e93 0.0272863
\(83\) −6.97978e94 −0.707093 −0.353546 − 0.935417i \(-0.615024\pi\)
−0.353546 + 0.935417i \(0.615024\pi\)
\(84\) 0 0
\(85\) −1.09283e95 −0.340657
\(86\) −5.08455e95 −0.888340
\(87\) 0 0
\(88\) 2.68594e96 1.50385
\(89\) 1.17382e96 0.375660 0.187830 − 0.982202i \(-0.439855\pi\)
0.187830 + 0.982202i \(0.439855\pi\)
\(90\) 0 0
\(91\) 4.21744e96 0.449262
\(92\) −1.31789e97 −0.817301
\(93\) 0 0
\(94\) 2.02019e97 0.432079
\(95\) 1.22833e97 0.155595
\(96\) 0 0
\(97\) −3.50742e98 −1.58412 −0.792059 − 0.610445i \(-0.790991\pi\)
−0.792059 + 0.610445i \(0.790991\pi\)
\(98\) 2.00164e98 0.544122
\(99\) 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.1.3 8
3.2 odd 2 1.100.a.a.1.6 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.100.a.a.1.6 ✓ 8 3.2 odd 2
9.100.a.d.1.3 8 1.1 even 1 trivial