Show commands: Magma / Pari/GP / SageMath

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 2.10
Character \(\chi\) \(=\) 27.2
Dual form 27.9.f.a.14.10

$q$-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+(-7.66670 - 1.35185i) q^{2} +(72.7671 - 35.5802i) q^{3} +(-183.611 - 66.8288i) q^{4} +(-24.0534 - 28.6657i) q^{5} +(-605.982 + 174.413i) q^{6} +(-2598.90 + 945.921i) q^{7} +(3043.29 + 1757.05i) q^{8} +(4029.09 - 5178.14i) q^{9} +(145.659 + 252.288i) q^{10} +(-4786.80 + 5704.68i) q^{11} +(-15738.6 + 1669.97i) q^{12} +(2684.14 + 15222.5i) q^{13} +(21203.7 - 3738.78i) q^{14} +(-2770.23 - 1230.10i) q^{15} +(17361.5 + 14568.0i) q^{16} +(-124728. + 72011.7i) q^{17} +(-37889.9 + 34252.5i) q^{18} +(-101393. + 175618. i) q^{19} +(2500.76 + 6870.79i) q^{20} +(-155458. + 161301. i) q^{21} +(44410.8 - 37265.1i) q^{22} +(-39698.1 + 109070. i) q^{23} +(283967. + 19574.0i) q^{24} +(67588.2 - 383312. i) q^{25} -120335. i q^{26} +(108946. - 520154. i) q^{27} +540399. q^{28} +(-609347. - 107444. i) q^{29} +(19575.6 + 13175.7i) q^{30} +(1.01617e6 + 369854. i) q^{31} +(-691668. - 824298. i) q^{32} +(-145347. + 585428. i) q^{33} +(1.05360e6 - 383479. i) q^{34} +(89627.8 + 51746.6i) q^{35} +(-1.08583e6 + 681502. i) q^{36} +(-978631. - 1.69504e6i) q^{37} +(1.01476e6 - 1.20934e6i) q^{38} +(736937. + 1.01219e6i) q^{39} +(-22834.5 - 129501. i) q^{40} +(-1.39463e6 + 245911. i) q^{41} +(1.40990e6 - 1.02649e6i) q^{42} +(2.04961e6 + 1.71982e6i) q^{43} +(1.26014e6 - 727544. i) q^{44} +(-245349. + 9054.95i) q^{45} +(451799. - 782539. i) q^{46} +(-1.82238e6 - 5.00694e6i) q^{47} +(1.78168e6 + 442347. i) q^{48} +(1.44340e6 - 1.21116e6i) q^{49} +(-1.03636e6 + 2.84736e6i) q^{50} +(-6.51389e6 + 9.67792e6i) q^{51} +(524465. - 2.97439e6i) q^{52} +3.45452e6i q^{53} +(-1.53842e6 + 3.84059e6i) q^{54} +278668. q^{55} +(-9.57122e6 - 1.68767e6i) q^{56} +(-1.12955e6 + 1.63868e7i) q^{57} +(4.52643e6 + 1.64749e6i) q^{58} +(-1.16111e7 - 1.38375e7i) q^{59} +(426437. + 410989. i) q^{60} +(1.16719e7 - 4.24821e6i) q^{61} +(-7.29065e6 - 4.20926e6i) q^{62} +(-5.57309e6 + 1.72687e7i) q^{63} +(1.28751e6 + 2.23004e6i) q^{64} +(371801. - 443096. i) q^{65} +(1.90574e6 - 4.29181e6i) q^{66} +(-1.92937e6 - 1.09420e7i) q^{67} +(2.77138e7 - 4.88669e6i) q^{68} +(992009. + 9.34915e6i) q^{69} +(-617196. - 517889. i) q^{70} +(-2.34487e7 + 1.35381e7i) q^{71} +(2.13599e7 - 8.67929e6i) q^{72} +(-1.28556e7 + 2.22665e7i) q^{73} +(5.21144e6 + 1.43183e7i) q^{74} +(-8.72012e6 - 3.02973e7i) q^{75} +(3.03531e7 - 2.54693e7i) q^{76} +(7.04421e6 - 1.93538e7i) q^{77} +(-4.28154e6 - 8.75641e6i) q^{78} +(-981583. + 5.56684e6i) q^{79} -848091. i q^{80} +(-1.05795e7 - 4.17264e7i) q^{81} +1.10246e7 q^{82} +(5.90784e6 + 1.04171e6i) q^{83} +(3.93233e7 - 1.92275e7i) q^{84} +(5.06440e6 + 1.84329e6i) q^{85} +(-1.33888e7 - 1.59561e7i) q^{86} +(-4.81633e7 + 1.38623e7i) q^{87} +(-2.45910e7 + 8.95039e6i) q^{88} +(3.29258e7 + 1.90097e7i) q^{89} +(1.89325e6 + 262252. i) q^{90} +(-2.13751e7 - 3.70227e7i) q^{91} +(1.45780e7 - 1.73734e7i) q^{92} +(8.71029e7 - 9.24222e6i) q^{93} +(7.20300e6 + 4.08503e7i) q^{94} +(7.47305e6 - 1.31770e6i) q^{95} +(-7.96594e7 - 3.53720e7i) q^{96} +(9.92181e6 + 8.32538e6i) q^{97} +(-1.27034e7 + 7.33433e6i) q^{98} +(1.02532e7 + 4.77714e7i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.66670 − 1.35185i −0.479169 − 0.0844904i −0.0711532 − 0.997465i \(-0.522668\pi\)
−0.408015 + 0.912975i \(0.633779\pi\)
\(3\) 72.7671 − 35.5802i 0.898359 − 0.439262i
\(4\) −183.611 − 66.8288i −0.717229 − 0.261050i
\(5\) −24.0534 − 28.6657i −0.0384854 − 0.0458652i 0.746458 − 0.665432i \(-0.231753\pi\)
−0.784944 + 0.619567i \(0.787308\pi\)
\(6\) −605.982 + 174.413i −0.467579 + 0.134578i
\(7\) −2598.90 + 945.921i −1.08242 + 0.393970i −0.820809 − 0.571203i \(-0.806477\pi\)
−0.261614 + 0.965173i \(0.584255\pi\)
\(8\) 3043.29 + 1757.05i 0.742991 + 0.428966i
\(9\) 4029.09 − 5178.14i 0.614098 − 0.789230i
\(10\) 145.659 + 252.288i 0.0145659 + 0.0252288i
\(11\) −4786.80 + 5704.68i −0.326945 + 0.389637i −0.904330 − 0.426835i \(-0.859629\pi\)
0.577385 + 0.816472i \(0.304073\pi\)
\(12\) −15738.6 + 1669.97i −0.758998 + 0.0805349i
\(13\) 2684.14 + 15222.5i 0.0939791 + 0.532982i 0.995055 + 0.0993213i \(0.0316672\pi\)
−0.901076 + 0.433661i \(0.857222\pi\)
\(14\) 21203.7 − 3738.78i 0.551949 − 0.0973236i
\(15\) −2770.23 − 1230.10i −0.0547206 − 0.0242982i
\(16\) 17361.5 + 14568.0i 0.264916 + 0.222291i
\(17\) −124728. + 72011.7i −1.49337 + 0.862198i −0.999971 − 0.00760413i \(-0.997580\pi\)
−0.493400 + 0.869802i \(0.664246\pi\)
\(18\) −37889.9 + 34252.5i −0.360939 + 0.326289i
\(19\) −101393. + 175618.i −0.778024 + 1.34758i 0.155054 + 0.987906i \(0.450445\pi\)
−0.933079 + 0.359672i \(0.882889\pi\)
\(20\) 2500.76 + 6870.79i 0.0156298 + 0.0429424i
\(21\) −155458. + 161301.i −0.799348 + 0.829393i
\(22\) 44410.8 − 37265.1i 0.189582 − 0.159078i
\(23\) −39698.1 + 109070.i −0.141860 + 0.389756i −0.990193 − 0.139707i \(-0.955384\pi\)
0.848333 + 0.529462i \(0.177606\pi\)
\(24\) 283967. + 19574.0i 0.855901 + 0.0589977i
\(25\) 67588.2 − 383312.i 0.173026 − 0.981277i
\(26\) − 120335.i − 0.263329i
\(27\) 108946. − 520154.i 0.205001 − 0.978762i
\(28\) 540399. 0.879190
\(29\) −609347. − 107444.i −0.861534 − 0.151912i −0.274612 − 0.961555i \(-0.588549\pi\)
−0.586922 + 0.809643i \(0.699660\pi\)
\(30\) 19575.6 + 13175.7i 0.0241674 + 0.0162663i
\(31\) 1.01617e6 + 369854.i 1.10032 + 0.400483i 0.827433 − 0.561564i \(-0.189800\pi\)
0.272884 + 0.962047i \(0.412022\pi\)
\(32\) −691668. − 824298.i −0.659626 − 0.786112i
\(33\) −145347. + 585428.i −0.122561 + 0.493649i
\(34\) 1.05360e6 − 383479.i 0.788424 − 0.286963i
\(35\) 89627.8 + 51746.6i 0.0597270 + 0.0344834i
\(36\) −1.08583e6 + 681502.i −0.646477 + 0.405748i
\(37\) −978631. − 1.69504e6i −0.522170 − 0.904426i −0.999667 − 0.0257923i \(-0.991789\pi\)
0.477497 − 0.878633i \(-0.341544\pi\)
\(38\) 1.01476e6 − 1.20934e6i 0.486662 − 0.579982i
\(39\) 736937. + 1.01219e6i 0.318546 + 0.437528i
\(40\) −22834.5 − 129501.i −0.00891974 − 0.0505864i
\(41\) −1.39463e6 + 245911.i −0.493541 + 0.0870247i −0.414879 − 0.909877i \(-0.636176\pi\)
−0.0786625 + 0.996901i \(0.525065\pi\)
\(42\) 1.40990e6 − 1.02649e6i 0.453098 − 0.329882i
\(43\) 2.04961e6 + 1.71982e6i 0.599510 + 0.503049i 0.891288 − 0.453437i \(-0.149802\pi\)
−0.291778 + 0.956486i \(0.594247\pi\)
\(44\) 1.26014e6 − 727544.i 0.336209 − 0.194110i
\(45\) −245349. + 9054.95i −0.0598320 + 0.00220819i
\(46\) 451799. − 782539.i 0.100905 − 0.174773i
\(47\) −1.82238e6 − 5.00694e6i −0.373462 − 1.02608i −0.974013 − 0.226492i \(-0.927274\pi\)
0.600551 − 0.799587i \(-0.294948\pi\)
\(48\) 1.78168e6 + 442347.i 0.335633 + 0.0833293i
\(49\) 1.44340e6 − 1.21116e6i 0.250382 − 0.210095i
\(50\) −1.03636e6 + 2.84736e6i −0.165817 + 0.455578i
\(51\) −6.51389e6 + 9.67792e6i −0.962852 + 1.43055i
\(52\) 524465. − 2.97439e6i 0.0717304 − 0.406803i
\(53\) 3.45452e6i 0.437809i 0.975746 + 0.218905i \(0.0702483\pi\)
−0.975746 + 0.218905i \(0.929752\pi\)
\(54\) −1.53842e6 + 3.84059e6i −0.180926 + 0.451671i
\(55\) 278668. 0.0304534
\(56\) −9.57122e6 − 1.68767e6i −0.973230 − 0.171607i
\(57\) −1.12955e6 + 1.63868e7i −0.107005 + 1.55237i
\(58\) 4.52643e6 + 1.64749e6i 0.399985 + 0.145583i
\(59\) −1.16111e7 − 1.38375e7i −0.958217 − 1.14196i −0.989801 − 0.142458i \(-0.954499\pi\)
0.0315836 − 0.999501i \(-0.489945\pi\)
\(60\) 426437. + 410989.i 0.0329041 + 0.0317122i
\(61\) 1.16719e7 − 4.24821e6i 0.842987 − 0.306822i 0.115810 − 0.993271i \(-0.463054\pi\)
0.727178 + 0.686449i \(0.240832\pi\)
\(62\) −7.29065e6 − 4.20926e6i −0.493401 − 0.284865i
\(63\) −5.57309e6 + 1.72687e7i −0.353780 + 1.09622i
\(64\) 1.28751e6 + 2.23004e6i 0.0767418 + 0.132921i
\(65\) 371801. − 443096.i 0.0208285 − 0.0248224i
\(66\) 1.90574e6 − 4.29181e6i 0.100436 − 0.226186i
\(67\) −1.92937e6 − 1.09420e7i −0.0957452 − 0.542998i −0.994516 − 0.104580i \(-0.966650\pi\)
0.898771 − 0.438418i \(-0.144461\pi\)
\(68\) 2.77138e7 − 4.88669e6i 1.29617 − 0.228549i
\(69\) 992009. + 9.34915e6i 0.0437642 + 0.412454i
\(70\) −617196. − 517889.i −0.0257058 − 0.0215697i
\(71\) −2.34487e7 + 1.35381e7i −0.922752 + 0.532751i −0.884512 − 0.466518i \(-0.845508\pi\)
−0.0382397 + 0.999269i \(0.512175\pi\)
\(72\) 2.13599e7 − 8.67929e6i 0.794822 − 0.322964i
\(73\) −1.28556e7 + 2.22665e7i −0.452690 + 0.784082i −0.998552 − 0.0537933i \(-0.982869\pi\)
0.545862 + 0.837875i \(0.316202\pi\)
\(74\) 5.21144e6 + 1.43183e7i 0.173792 + 0.477491i
\(75\) −8.72012e6 − 3.02973e7i −0.275599 − 0.957543i
\(76\) 3.03531e7 − 2.54693e7i 0.909807 − 0.763418i
\(77\) 7.04421e6 − 1.93538e7i 0.200387 − 0.550558i
\(78\) −4.28154e6 − 8.75641e6i −0.115670 − 0.236564i
\(79\) −981583. + 5.56684e6i −0.0252011 + 0.142922i −0.994812 − 0.101729i \(-0.967562\pi\)
0.969611 + 0.244652i \(0.0786736\pi\)
\(80\) − 848091.i − 0.0207054i
\(81\) −1.05795e7 − 4.17264e7i −0.245768 − 0.969329i
\(82\) 1.10246e7 0.243842
\(83\) 5.90784e6 + 1.04171e6i 0.124485 + 0.0219500i 0.235543 − 0.971864i \(-0.424313\pi\)
−0.111059 + 0.993814i \(0.535424\pi\)
\(84\) 3.93233e7 − 1.92275e7i 0.789828 − 0.386195i
\(85\) 5.06440e6 + 1.84329e6i 0.0970179 + 0.0353116i
\(86\) −1.33888e7 − 1.59561e7i −0.244764 − 0.291698i
\(87\) −4.81633e7 + 1.38623e7i −0.840696 + 0.241968i
\(88\) −2.45910e7 + 8.95039e6i −0.410058 + 0.149249i
\(89\) 3.29258e7 + 1.90097e7i 0.524779 + 0.302981i 0.738888 − 0.673829i \(-0.235351\pi\)
−0.214109 + 0.976810i \(0.568685\pi\)
\(90\) 1.89325e6 + 262252.i 0.0288562 + 0.00399713i
\(91\) −2.13751e7 − 3.70227e7i −0.311704 − 0.539887i
\(92\) 1.45780e7 − 1.73734e7i 0.203491 − 0.242512i
\(93\) 8.71029e7 − 9.24222e6i 1.16440 − 0.123551i
\(94\) 7.20300e6 + 4.08503e7i 0.0922576 + 0.523219i
\(95\) 7.47305e6 − 1.31770e6i 0.0917495 − 0.0161779i
\(96\) −7.96594e7 − 3.53720e7i −0.937890 − 0.416462i
\(97\) 9.92181e6 + 8.32538e6i 0.112074 + 0.0940410i 0.697102 − 0.716972i \(-0.254472\pi\)
−0.585029 + 0.811013i \(0.698917\pi\)
\(98\) −1.27034e7 + 7.33433e6i −0.137726 + 0.0795162i
\(99\) 1.02532e7 + 4.77714e7i 0.106738 + 0.497310i
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.2.10 ✓ 138
3.2 odd 2 81.9.f.a.8.14 138
27.13 even 9 81.9.f.a.71.14 138
27.14 odd 18 inner 27.9.f.a.14.10 yes 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.10 ✓ 138 1.1 even 1 trivial
27.9.f.a.14.10 yes 138 27.14 odd 18 inner
81.9.f.a.8.14 138 3.2 odd 2
81.9.f.a.71.14 138 27.13 even 9