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Properties

Label 81.9.f.a.71.13
Level $81$
Weight $9$
Character 81.71
Analytic conductor $32.998$
Analytic rank $0$
Dimension $138$
Inner twists $2$

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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] = [81,9,Mod(8,81)] 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("81.8"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 81.f (of order \(18\), degree \(6\), not minimal)

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: no
Analytic conductor: \(32.9976674150\)
Analytic rank: \(0\)
Dimension: \(138\)
Relative dimension: \(23\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 71.13
Character \(\chi\) \(=\) 81.71
Dual form 81.9.f.a.8.13

$q$-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)
 
\(f(q)\) \(=\) \(q+(3.24198 - 0.571649i) q^{2} +(-230.378 + 83.8506i) q^{4} +(-387.667 + 462.003i) q^{5} +(-3862.15 - 1405.71i) q^{7} +(-1428.79 + 824.913i) q^{8} +(-992.706 + 1719.42i) q^{10} +(-11423.3 - 13613.8i) q^{11} +(-882.859 + 5006.94i) q^{13} +(-13324.6 - 2349.49i) q^{14} +(43917.7 - 36851.3i) q^{16} +(-23746.9 - 13710.3i) q^{17} +(92857.3 + 160834. i) q^{19} +(50570.5 - 138941. i) q^{20} +(-44816.5 - 37605.5i) q^{22} +(88410.2 + 242905. i) q^{23} +(4669.78 + 26483.6i) q^{25} +16737.1i q^{26} +1.00762e6 q^{28} +(234351. - 41322.5i) q^{29} +(-618503. + 225117. i) q^{31} +(392799. - 468120. i) q^{32} +(-84824.4 - 30873.6i) q^{34} +(2.14667e6 - 1.23938e6i) q^{35} +(779022. - 1.34931e6i) q^{37} +(392982. + 468338. i) q^{38} +(172782. - 979898. i) q^{40} +(-3.82274e6 - 674052. i) q^{41} +(-1.19676e6 + 1.00420e6i) q^{43} +(3.77320e6 + 2.17846e6i) q^{44} +(425481. + 736954. i) q^{46} +(-407155. + 1.11865e6i) q^{47} +(8.52411e6 + 7.15258e6i) q^{49} +(30278.7 + 83190.0i) q^{50} +(-216444. - 1.22752e6i) q^{52} -1.29944e7i q^{53} +1.07180e7 q^{55} +(6.67780e6 - 1.17748e6i) q^{56} +(736141. - 267934. i) q^{58} +(1.38268e7 - 1.64782e7i) q^{59} +(1.39731e7 + 5.08578e6i) q^{61} +(-1.87649e6 + 1.08339e6i) q^{62} +(-6.33245e6 + 1.09681e7i) q^{64} +(-1.97097e6 - 2.34891e6i) q^{65} +(-1.06177e6 + 6.02157e6i) q^{67} +(6.62036e6 + 1.16735e6i) q^{68} +(6.25098e6 - 5.24520e6i) q^{70} +(-3.07395e7 - 1.77475e7i) q^{71} +(6.89919e6 + 1.19498e7i) q^{73} +(1.75425e6 - 4.81975e6i) q^{74} +(-3.48782e7 - 2.92663e7i) q^{76} +(2.49816e7 + 6.86363e7i) q^{77} +(-7.76297e6 - 4.40260e7i) q^{79} +3.45761e7i q^{80} -1.27786e7 q^{82} +(4.57442e7 - 8.06593e6i) q^{83} +(1.55401e7 - 5.65612e6i) q^{85} +(-3.30581e6 + 3.93971e6i) q^{86} +(2.75517e7 + 1.00280e7i) q^{88} +(3.60126e6 - 2.07919e6i) q^{89} +(1.04480e7 - 1.80965e7i) q^{91} +(-4.07355e7 - 4.85466e7i) q^{92} +(-680516. + 3.85940e6i) q^{94} +(-1.10303e8 - 1.94495e7i) q^{95} +(-1.08572e8 + 9.11031e7i) q^{97} +(3.17238e7 + 1.83157e7i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 138 q + 6 q^{2} - 6 q^{4} + 447 q^{5} - 6 q^{7} + 9 q^{8} - 3 q^{10} - 28668 q^{11} - 6 q^{13} + 120975 q^{14} - 774 q^{16} + 9 q^{17} - 3 q^{19} - 137913 q^{20} - 185478 q^{22} - 68376 q^{23} + 507585 q^{25}+ \cdots - 1293135102 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{17}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.24198 − 0.571649i 0.202624 − 0.0357281i −0.0714149 − 0.997447i \(-0.522751\pi\)
0.274039 + 0.961719i \(0.411640\pi\)
\(3\) 0 0
\(4\) −230.378 + 83.8506i −0.899913 + 0.327541i
\(5\) −387.667 + 462.003i −0.620267 + 0.739206i −0.981116 − 0.193419i \(-0.938042\pi\)
0.360849 + 0.932624i \(0.382487\pi\)
\(6\) 0 0
\(7\) −3862.15 − 1405.71i −1.60856 − 0.585468i −0.627405 − 0.778693i \(-0.715883\pi\)
−0.981155 + 0.193225i \(0.938105\pi\)
\(8\) −1428.79 + 824.913i −0.348826 + 0.201395i
\(9\) 0 0
\(10\) −992.706 + 1719.42i −0.0992706 + 0.171942i
\(11\) −11423.3 − 13613.8i −0.780227 − 0.929839i 0.218716 − 0.975788i \(-0.429813\pi\)
−0.998944 + 0.0459497i \(0.985369\pi\)
\(12\) 0 0
\(13\) −882.859 + 5006.94i −0.0309113 + 0.175307i −0.996355 − 0.0853055i \(-0.972813\pi\)
0.965443 + 0.260612i \(0.0839245\pi\)
\(14\) −13324.6 − 2349.49i −0.346850 − 0.0611591i
\(15\) 0 0
\(16\) 43917.7 − 36851.3i 0.670130 − 0.562306i
\(17\) −23746.9 − 13710.3i −0.284322 − 0.164153i 0.351056 − 0.936354i \(-0.385823\pi\)
−0.635378 + 0.772201i \(0.719156\pi\)
\(18\) 0 0
\(19\) 92857.3 + 160834.i 0.712528 + 1.23413i 0.963905 + 0.266245i \(0.0857831\pi\)
−0.251378 + 0.967889i \(0.580884\pi\)
\(20\) 50570.5 − 138941.i 0.316066 − 0.868384i
\(21\) 0 0
\(22\) −44816.5 − 37605.5i −0.191314 − 0.160532i
\(23\) 88410.2 + 242905.i 0.315930 + 0.868011i 0.991429 + 0.130650i \(0.0417063\pi\)
−0.675498 + 0.737361i \(0.736072\pi\)
\(24\) 0 0
\(25\) 4669.78 + 26483.6i 0.0119546 + 0.0677981i
\(26\) 16737.1i 0.0366258i
\(27\) 0 0
\(28\) 1.00762e6 1.63933
\(29\) 234351. − 41322.5i 0.331341 − 0.0584244i −0.00550273 − 0.999985i \(-0.501752\pi\)
0.336844 + 0.941560i \(0.390640\pi\)
\(30\) 0 0
\(31\) −618503. + 225117.i −0.669723 + 0.243759i −0.654429 − 0.756124i \(-0.727091\pi\)
−0.0152945 + 0.999883i \(0.504869\pi\)
\(32\) 392799. − 468120.i 0.374602 − 0.446434i
\(33\) 0 0
\(34\) −84824.4 − 30873.6i −0.0634754 − 0.0231031i
\(35\) 2.14667e6 − 1.23938e6i 1.43052 − 0.825910i
\(36\) 0 0
\(37\) 779022. − 1.34931e6i 0.415664 − 0.719952i −0.579834 − 0.814735i \(-0.696882\pi\)
0.995498 + 0.0947831i \(0.0302158\pi\)
\(38\) 392982. + 468338.i 0.188468 + 0.224608i
\(39\) 0 0
\(40\) 172782. − 979898.i 0.0674931 − 0.382773i
\(41\) −3.82274e6 − 674052.i −1.35282 − 0.238538i −0.550201 − 0.835033i \(-0.685449\pi\)
−0.802617 + 0.596494i \(0.796560\pi\)
\(42\) 0 0
\(43\) −1.19676e6 + 1.00420e6i −0.350051 + 0.293728i −0.800811 − 0.598918i \(-0.795598\pi\)
0.450759 + 0.892645i \(0.351153\pi\)
\(44\) 3.77320e6 + 2.17846e6i 1.00670 + 0.581217i
\(45\) 0 0
\(46\) 425481. + 736954.i 0.0950274 + 0.164592i
\(47\) −407155. + 1.11865e6i −0.0834390 + 0.229247i −0.974395 − 0.224843i \(-0.927813\pi\)
0.890956 + 0.454089i \(0.150035\pi\)
\(48\) 0 0
\(49\) 8.52411e6 + 7.15258e6i 1.47865 + 1.24073i
\(50\) 30278.7 + 83190.0i 0.00484459 + 0.0133104i
\(51\) 0 0
\(52\) −216444. − 1.22752e6i −0.0296028 − 0.167886i
\(53\) − 1.29944e7i − 1.64684i −0.567432 − 0.823420i \(-0.692063\pi\)
0.567432 − 0.823420i \(-0.307937\pi\)
\(54\) 0 0
\(55\) 1.07180e7 1.17129
\(56\) 6.67780e6 − 1.17748e6i 0.679018 − 0.119729i
\(57\) 0 0
\(58\) 736141. − 267934.i 0.0650503 − 0.0236764i
\(59\) 1.38268e7 − 1.64782e7i 1.14107 − 1.35988i 0.217684 − 0.976019i \(-0.430150\pi\)
0.923391 − 0.383861i \(-0.125406\pi\)
\(60\) 0 0
\(61\) 1.39731e7 + 5.08578e6i 1.00919 + 0.367315i 0.793121 − 0.609064i \(-0.208455\pi\)
0.216068 + 0.976378i \(0.430677\pi\)
\(62\) −1.87649e6 + 1.08339e6i −0.126993 + 0.0733194i
\(63\) 0 0
\(64\) −6.33245e6 + 1.09681e7i −0.377443 + 0.653751i
\(65\) −1.97097e6 − 2.34891e6i −0.110415 − 0.131587i
\(66\) 0 0
\(67\) −1.06177e6 + 6.02157e6i −0.0526901 + 0.298821i −0.999753 − 0.0222278i \(-0.992924\pi\)
0.947063 + 0.321048i \(0.104035\pi\)
\(68\) 6.62036e6 + 1.16735e6i 0.309632 + 0.0545965i
\(69\) 0 0
\(70\) 6.25098e6 − 5.24520e6i 0.260349 − 0.218459i
\(71\) −3.07395e7 − 1.77475e7i −1.20966 − 0.698398i −0.246975 − 0.969022i \(-0.579437\pi\)
−0.962685 + 0.270624i \(0.912770\pi\)
\(72\) 0 0
\(73\) 6.89919e6 + 1.19498e7i 0.242944 + 0.420792i 0.961552 − 0.274624i \(-0.0885534\pi\)
−0.718607 + 0.695416i \(0.755220\pi\)
\(74\) 1.75425e6 − 4.81975e6i 0.0585011 − 0.160730i
\(75\) 0 0
\(76\) −3.48782e7 − 2.92663e7i −1.04544 − 0.877231i
\(77\) 2.49816e7 + 6.86363e7i 0.710652 + 1.95250i
\(78\) 0 0
\(79\) −7.76297e6 − 4.40260e7i −0.199306 − 1.13032i −0.906152 − 0.422952i \(-0.860994\pi\)
0.706846 − 0.707367i \(-0.250117\pi\)
\(80\) 3.45761e7i 0.844144i
\(81\) 0 0
\(82\) −1.27786e7 −0.282636
\(83\) 4.57442e7 − 8.06593e6i 0.963881 − 0.169958i 0.330506 − 0.943804i \(-0.392780\pi\)
0.633374 + 0.773846i \(0.281669\pi\)
\(84\) 0 0
\(85\) 1.55401e7 − 5.65612e6i 0.297699 − 0.108354i
\(86\) −3.30581e6 + 3.93971e6i −0.0604344 + 0.0720230i
\(87\) 0 0
\(88\) 2.75517e7 + 1.00280e7i 0.459428 + 0.167218i
\(89\) 3.60126e6 − 2.07919e6i 0.0573977 − 0.0331386i −0.471027 − 0.882119i \(-0.656116\pi\)
0.528424 + 0.848980i \(0.322783\pi\)
\(90\) 0 0
\(91\) 1.04480e7 − 1.80965e7i 0.152359 − 0.263894i
\(92\) −4.07355e7 − 4.85466e7i −0.568619 − 0.677654i
\(93\) 0 0
\(94\) −680516. + 3.85940e6i −0.00871619 + 0.0494320i
\(95\) −1.10303e8 − 1.94495e7i −1.35424 − 0.238788i
\(96\) 0 0
\(97\) −1.08572e8 + 9.11031e7i −1.22640 + 1.02907i −0.227937 + 0.973676i \(0.573198\pi\)
−0.998464 + 0.0553970i \(0.982358\pi\)
\(98\) 3.17238e7 + 1.83157e7i 0.343938 + 0.198573i
\(99\) 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 81.9.f.a.71.13 138
3.2 odd 2 27.9.f.a.14.11 yes 138
27.2 odd 18 inner 81.9.f.a.8.13 138
27.25 even 9 27.9.f.a.2.11 ✓ 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.11 ✓ 138 27.25 even 9
27.9.f.a.14.11 yes 138 3.2 odd 2
81.9.f.a.8.13 138 27.2 odd 18 inner
81.9.f.a.71.13 138 1.1 even 1 trivial