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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [27,9,Mod(2,27)] 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("27.2"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 27.f (of order \(18\), degree \(6\), 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: \(10.9992224717\)
Analytic rank: \(0\)
Dimension: \(138\)
Relative dimension: \(23\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 2.11
Character \(\chi\) \(=\) 27.2
Dual form 27.9.f.a.14.11

$q$-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} +(-71.7460 + 37.5966i) q^{3} +(-230.378 - 83.8506i) q^{4} +(387.667 + 462.003i) q^{5} +(254.092 - 80.8741i) q^{6} +(-3862.15 + 1405.71i) q^{7} +(1428.79 + 824.913i) q^{8} +(3733.99 - 5394.82i) q^{9} +(-992.706 - 1719.42i) q^{10} +(11423.3 - 13613.8i) q^{11} +(19681.2 - 2645.47i) q^{12} +(-882.859 - 5006.94i) q^{13} +(13324.6 - 2349.49i) q^{14} +(-45183.3 - 18571.9i) q^{15} +(43917.7 + 36851.3i) q^{16} +(23746.9 - 13710.3i) q^{17} +(-15189.5 + 15355.4i) q^{18} +(92857.3 - 160834. i) q^{19} +(-50570.5 - 138941. i) q^{20} +(224244. - 246058. i) q^{21} +(-44816.5 + 37605.5i) q^{22} +(-88410.2 + 242905. i) q^{23} +(-133524. - 5466.51i) q^{24} +(4669.78 - 26483.6i) q^{25} +16737.1i q^{26} +(-65071.9 + 527442. i) q^{27} +1.00762e6 q^{28} +(-234351. - 41322.5i) q^{29} +(135867. + 86039.0i) q^{30} +(-618503. - 225117. i) q^{31} +(-392799. - 468120. i) q^{32} +(-307745. + 1.40621e6i) q^{33} +(-84824.4 + 30873.6i) q^{34} +(-2.14667e6 - 1.23938e6i) q^{35} +(-1.31259e6 + 929748. i) q^{36} +(779022. + 1.34931e6i) q^{37} +(-392982. + 468338. i) q^{38} +(251586. + 326036. i) q^{39} +(172782. + 979898. i) q^{40} +(3.82274e6 - 674052. i) q^{41} +(-867655. + 669527. i) q^{42} +(-1.19676e6 - 1.00420e6i) q^{43} +(-3.77320e6 + 2.17846e6i) q^{44} +(3.93997e6 - 366277. i) q^{45} +(425481. - 736954. i) q^{46} +(407155. + 1.11865e6i) q^{47} +(-4.53640e6 - 992779. i) q^{48} +(8.52411e6 - 7.15258e6i) q^{49} +(-30278.7 + 83190.0i) q^{50} +(-1.18828e6 + 1.87646e6i) q^{51} +(-216444. + 1.22752e6i) q^{52} -1.29944e7i q^{53} +(512474. - 1.67276e6i) q^{54} +1.07180e7 q^{55} +(-6.67780e6 - 1.17748e6i) q^{56} +(-615344. + 1.50303e7i) q^{57} +(736141. + 267934. i) q^{58} +(-1.38268e7 - 1.64782e7i) q^{59} +(8.85196e6 + 8.06721e6i) q^{60} +(1.39731e7 - 5.08578e6i) q^{61} +(1.87649e6 + 1.08339e6i) q^{62} +(-6.83769e6 + 2.60845e7i) q^{63} +(-6.33245e6 - 1.09681e7i) q^{64} +(1.97097e6 - 2.34891e6i) q^{65} +(1.80156e6 - 4.38299e6i) q^{66} +(-1.06177e6 - 6.02157e6i) q^{67} +(-6.62036e6 + 1.16735e6i) q^{68} +(-2.78933e6 - 2.07514e7i) q^{69} +(6.25098e6 + 5.24520e6i) q^{70} +(3.07395e7 - 1.77475e7i) q^{71} +(9.78534e6 - 4.62785e6i) q^{72} +(6.89919e6 - 1.19498e7i) q^{73} +(-1.75425e6 - 4.81975e6i) q^{74} +(660657. + 2.07566e6i) q^{75} +(-3.48782e7 + 2.92663e7i) q^{76} +(-2.49816e7 + 6.86363e7i) q^{77} +(-629259. - 1.20082e6i) q^{78} +(-7.76297e6 + 4.40260e7i) q^{79} +3.45761e7i q^{80} +(-1.51614e7 - 4.02884e7i) q^{81} -1.27786e7 q^{82} +(-4.57442e7 - 8.06593e6i) q^{83} +(-7.22930e7 + 3.78832e7i) q^{84} +(1.55401e7 + 5.65612e6i) q^{85} +(3.30581e6 + 3.93971e6i) q^{86} +(1.83674e7 - 5.84610e6i) q^{87} +(2.75517e7 - 1.00280e7i) q^{88} +(-3.60126e6 - 2.07919e6i) q^{89} +(-1.29827e7 - 1.06481e6i) q^{90} +(1.04480e7 + 1.80965e7i) q^{91} +(4.07355e7 - 4.85466e7i) q^{92} +(5.28388e7 - 7.10240e6i) q^{93} +(-680516. - 3.85940e6i) q^{94} +(1.10303e8 - 1.94495e7i) q^{95} +(4.57815e7 + 1.88178e7i) q^{96} +(-1.08572e8 - 9.11031e7i) q^{97} +(-3.17238e7 + 1.83157e7i) q^{98} +(-3.07893e7 - 1.12460e8i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 138 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 447 q^{5} - 774 q^{6} - 6 q^{7} - 9 q^{8} - 12960 q^{9} - 3 q^{10} + 28668 q^{11} + 77421 q^{12} - 6 q^{13} - 120975 q^{14} + 105507 q^{15} - 774 q^{16} - 9 q^{17}+ \cdots + 228876057 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{18}\right)\)

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 \(-0.477249\pi\)
−0.274039 + 0.961719i \(0.588360\pi\)
\(3\) −71.7460 + 37.5966i −0.885754 + 0.464156i
\(4\) −230.378 − 83.8506i −0.899913 − 0.327541i
\(5\) 387.667 + 462.003i 0.620267 + 0.739206i 0.981116 − 0.193419i \(-0.0619576\pi\)
−0.360849 + 0.932624i \(0.617513\pi\)
\(6\) 254.092 − 80.8741i 0.196058 − 0.0624028i
\(7\) −3862.15 + 1405.71i −1.60856 + 0.585468i −0.981155 − 0.193225i \(-0.938105\pi\)
−0.627405 + 0.778693i \(0.715883\pi\)
\(8\) 1428.79 + 824.913i 0.348826 + 0.201395i
\(9\) 3733.99 − 5394.82i 0.569119 − 0.822255i
\(10\) −992.706 − 1719.42i −0.0992706 − 0.171942i
\(11\) 11423.3 − 13613.8i 0.780227 − 0.929839i −0.218716 − 0.975788i \(-0.570187\pi\)
0.998944 + 0.0459497i \(0.0146314\pi\)
\(12\) 19681.2 − 2645.47i 0.949131 − 0.127579i
\(13\) −882.859 − 5006.94i −0.0309113 − 0.175307i 0.965443 − 0.260612i \(-0.0839245\pi\)
−0.996355 + 0.0853055i \(0.972813\pi\)
\(14\) 13324.6 − 2349.49i 0.346850 − 0.0611591i
\(15\) −45183.3 − 18571.9i −0.892510 − 0.366853i
\(16\) 43917.7 + 36851.3i 0.670130 + 0.562306i
\(17\) 23746.9 − 13710.3i 0.284322 − 0.164153i −0.351056 − 0.936354i \(-0.614177\pi\)
0.635378 + 0.772201i \(0.280844\pi\)
\(18\) −15189.5 + 15355.4i −0.144695 + 0.146275i
\(19\) 92857.3 − 160834.i 0.712528 − 1.23413i −0.251378 − 0.967889i \(-0.580884\pi\)
0.963905 − 0.266245i \(-0.0857831\pi\)
\(20\) −50570.5 − 138941.i −0.316066 − 0.868384i
\(21\) 224244. − 246058.i 1.15304 − 1.26520i
\(22\) −44816.5 + 37605.5i −0.191314 + 0.160532i
\(23\) −88410.2 + 242905.i −0.315930 + 0.868011i 0.675498 + 0.737361i \(0.263928\pi\)
−0.991429 + 0.130650i \(0.958294\pi\)
\(24\) −133524. − 5466.51i −0.402452 − 0.0164765i
\(25\) 4669.78 − 26483.6i 0.0119546 − 0.0677981i
\(26\) 16737.1i 0.0366258i
\(27\) −65071.9 + 527442.i −0.122444 + 0.992475i
\(28\) 1.00762e6 1.63933
\(29\) −234351. − 41322.5i −0.331341 − 0.0584244i 0.00550273 − 0.999985i \(-0.498248\pi\)
−0.336844 + 0.941560i \(0.609360\pi\)
\(30\) 135867. + 86039.0i 0.167737 + 0.106221i
\(31\) −618503. − 225117.i −0.669723 − 0.243759i −0.0152945 − 0.999883i \(-0.504869\pi\)
−0.654429 + 0.756124i \(0.727091\pi\)
\(32\) −392799. − 468120.i −0.374602 − 0.446434i
\(33\) −307745. + 1.40621e6i −0.259499 + 1.18576i
\(34\) −84824.4 + 30873.6i −0.0634754 + 0.0231031i
\(35\) −2.14667e6 − 1.23938e6i −1.43052 − 0.825910i
\(36\) −1.31259e6 + 929748.i −0.781480 + 0.553548i
\(37\) 779022. + 1.34931e6i 0.415664 + 0.719952i 0.995498 − 0.0947831i \(-0.0302158\pi\)
−0.579834 + 0.814735i \(0.696882\pi\)
\(38\) −392982. + 468338.i −0.188468 + 0.224608i
\(39\) 251586. + 326036.i 0.108750 + 0.140931i
\(40\) 172782. + 979898.i 0.0674931 + 0.382773i
\(41\) 3.82274e6 − 674052.i 1.35282 − 0.238538i 0.550201 − 0.835033i \(-0.314551\pi\)
0.802617 + 0.596494i \(0.203440\pi\)
\(42\) −867655. + 669527.i −0.278837 + 0.215165i
\(43\) −1.19676e6 − 1.00420e6i −0.350051 − 0.293728i 0.450759 − 0.892645i \(-0.351153\pi\)
−0.800811 + 0.598918i \(0.795598\pi\)
\(44\) −3.77320e6 + 2.17846e6i −1.00670 + 0.581217i
\(45\) 3.93997e6 − 366277.i 0.960821 − 0.0893223i
\(46\) 425481. − 736954.i 0.0950274 − 0.164592i
\(47\) 407155. + 1.11865e6i 0.0834390 + 0.229247i 0.974395 − 0.224843i \(-0.0721869\pi\)
−0.890956 + 0.454089i \(0.849965\pi\)
\(48\) −4.53640e6 − 992779.i −0.854568 − 0.187020i
\(49\) 8.52411e6 − 7.15258e6i 1.47865 − 1.24073i
\(50\) −30278.7 + 83190.0i −0.00484459 + 0.0133104i
\(51\) −1.18828e6 + 1.87646e6i −0.175647 + 0.277369i
\(52\) −216444. + 1.22752e6i −0.0296028 + 0.167886i
\(53\) − 1.29944e7i − 1.64684i −0.567432 − 0.823420i \(-0.692063\pi\)
0.567432 − 0.823420i \(-0.307937\pi\)
\(54\) 512474. − 1.67276e6i 0.0602694 − 0.196725i
\(55\) 1.07180e7 1.17129
\(56\) −6.67780e6 − 1.17748e6i −0.679018 − 0.119729i
\(57\) −615344. + 1.50303e7i −0.0582933 + 1.42386i
\(58\) 736141. + 267934.i 0.0650503 + 0.0236764i
\(59\) −1.38268e7 − 1.64782e7i −1.14107 − 1.35988i −0.923391 − 0.383861i \(-0.874594\pi\)
−0.217684 − 0.976019i \(-0.569850\pi\)
\(60\) 8.85196e6 + 8.06721e6i 0.683022 + 0.622470i
\(61\) 1.39731e7 − 5.08578e6i 1.00919 − 0.367315i 0.216068 − 0.976378i \(-0.430677\pi\)
0.793121 + 0.609064i \(0.208455\pi\)
\(62\) 1.87649e6 + 1.08339e6i 0.126993 + 0.0733194i
\(63\) −6.83769e6 + 2.60845e7i −0.434057 + 1.65585i
\(64\) −6.33245e6 − 1.09681e7i −0.377443 − 0.653751i
\(65\) 1.97097e6 − 2.34891e6i 0.110415 − 0.131587i
\(66\) 1.80156e6 − 4.38299e6i 0.0949455 − 0.230991i
\(67\) −1.06177e6 − 6.02157e6i −0.0526901 − 0.298821i 0.947063 − 0.321048i \(-0.104035\pi\)
−0.999753 + 0.0222278i \(0.992924\pi\)
\(68\) −6.62036e6 + 1.16735e6i −0.309632 + 0.0545965i
\(69\) −2.78933e6 − 2.07514e7i −0.123056 − 0.915485i
\(70\) 6.25098e6 + 5.24520e6i 0.260349 + 0.218459i
\(71\) 3.07395e7 − 1.77475e7i 1.20966 − 0.698398i 0.246975 − 0.969022i \(-0.420563\pi\)
0.962685 + 0.270624i \(0.0872300\pi\)
\(72\) 9.78534e6 − 4.62785e6i 0.364121 − 0.172207i
\(73\) 6.89919e6 − 1.19498e7i 0.242944 − 0.420792i −0.718607 − 0.695416i \(-0.755220\pi\)
0.961552 + 0.274624i \(0.0885534\pi\)
\(74\) −1.75425e6 − 4.81975e6i −0.0585011 − 0.160730i
\(75\) 660657. + 2.07566e6i 0.0208800 + 0.0656012i
\(76\) −3.48782e7 + 2.92663e7i −1.04544 + 0.877231i
\(77\) −2.49816e7 + 6.86363e7i −0.710652 + 1.95250i
\(78\) −629259. − 1.20082e6i −0.0170001 − 0.0324414i
\(79\) −7.76297e6 + 4.40260e7i −0.199306 + 1.13032i 0.706846 + 0.707367i \(0.250117\pi\)
−0.906152 + 0.422952i \(0.860994\pi\)
\(80\) 3.45761e7i 0.844144i
\(81\) −1.51614e7 − 4.02884e7i −0.352208 − 0.935922i
\(82\) −1.27786e7 −0.282636
\(83\) −4.57442e7 − 8.06593e6i −0.963881 − 0.169958i −0.330506 − 0.943804i \(-0.607220\pi\)
−0.633374 + 0.773846i \(0.718331\pi\)
\(84\) −7.22930e7 + 3.78832e7i −1.45204 + 0.760904i
\(85\) 1.55401e7 + 5.65612e6i 0.297699 + 0.108354i
\(86\) 3.30581e6 + 3.93971e6i 0.0604344 + 0.0720230i
\(87\) 1.83674e7 − 5.84610e6i 0.320605 − 0.102044i
\(88\) 2.75517e7 − 1.00280e7i 0.459428 − 0.167218i
\(89\) −3.60126e6 − 2.07919e6i −0.0573977 − 0.0331386i 0.471027 − 0.882119i \(-0.343884\pi\)
−0.528424 + 0.848980i \(0.677217\pi\)
\(90\) −1.29827e7 − 1.06481e6i −0.197877 − 0.0162295i
\(91\) 1.04480e7 + 1.80965e7i 0.152359 + 0.263894i
\(92\) 4.07355e7 − 4.85466e7i 0.568619 − 0.677654i
\(93\) 5.28388e7 − 7.10240e6i 0.706352 − 0.0949453i
\(94\) −680516. − 3.85940e6i −0.00871619 − 0.0494320i
\(95\) 1.10303e8 − 1.94495e7i 1.35424 − 0.238788i
\(96\) 4.57815e7 + 1.88178e7i 0.539020 + 0.221556i
\(97\) −1.08572e8 − 9.11031e7i −1.22640 − 1.02907i −0.998464 − 0.0553970i \(-0.982358\pi\)
−0.227937 − 0.973676i \(-0.573198\pi\)
\(98\) −3.17238e7 + 1.83157e7i −0.343938 + 0.198573i
\(99\) −3.07893e7 − 1.12460e8i −0.320523 − 1.17073i
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 27.9.f.a.2.11 ✓ 138
3.2 odd 2 81.9.f.a.8.13 138
27.13 even 9 81.9.f.a.71.13 138
27.14 odd 18 inner 27.9.f.a.14.11 yes 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.11 ✓ 138 1.1 even 1 trivial
27.9.f.a.14.11 yes 138 27.14 odd 18 inner
81.9.f.a.8.13 138 3.2 odd 2
81.9.f.a.71.13 138 27.13 even 9