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 8.3
Character \(\chi\) \(=\) 81.8
Dual form 81.9.f.a.71.3

$q$-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+(-26.8100 - 4.72733i) q^{2} +(455.867 + 165.922i) q^{4} +(418.443 + 498.682i) q^{5} +(2557.31 - 930.784i) q^{7} +(-5401.88 - 3118.78i) q^{8} +(-8861.04 - 15347.8i) q^{10} +(16649.6 - 19842.2i) q^{11} +(-2324.40 - 13182.3i) q^{13} +(-72961.5 + 12865.1i) q^{14} +(34944.7 + 29322.1i) q^{16} +(114990. - 66389.2i) q^{17} +(-10147.5 + 17576.0i) q^{19} +(108012. + 296761. i) q^{20} +(-540176. + 453262. i) q^{22} +(-93974.9 + 258194. i) q^{23} +(-5756.99 + 32649.5i) q^{25} +364407. i q^{26} +1.32023e6 q^{28} +(-373034. - 65776.0i) q^{29} +(23963.9 + 8722.16i) q^{31} +(228160. + 271911. i) q^{32} +(-3.39671e6 + 1.23630e6i) q^{34} +(1.53425e6 + 885801. i) q^{35} +(-1.41658e6 - 2.45359e6i) q^{37} +(355142. - 423242. i) q^{38} +(-705106. - 3.99885e6i) q^{40} +(1.23941e6 - 218542. i) q^{41} +(390374. + 327563. i) q^{43} +(1.08823e7 - 6.28287e6i) q^{44} +(3.74003e6 - 6.47792e6i) q^{46} +(-2.99482e6 - 8.22820e6i) q^{47} +(1.25737e6 - 1.05506e6i) q^{49} +(308690. - 848118. i) q^{50} +(1.12762e6 - 6.39506e6i) q^{52} +6.12277e6i q^{53} +1.68619e7 q^{55} +(-1.67172e7 - 2.94769e6i) q^{56} +(9.69011e6 + 3.52691e6i) q^{58} +(-1.30891e7 - 1.55989e7i) q^{59} +(3.01877e6 - 1.09874e6i) q^{61} +(-601240. - 347126. i) q^{62} +(-1.06706e7 - 1.84819e7i) q^{64} +(5.60116e6 - 6.67520e6i) q^{65} +(2.53228e6 + 1.43613e7i) q^{67} +(6.34353e7 - 1.11854e7i) q^{68} +(-3.69458e7 - 3.10012e7i) q^{70} +(-1.90737e7 + 1.10122e7i) q^{71} +(-6.96726e6 + 1.20676e7i) q^{73} +(2.63796e7 + 7.24775e7i) q^{74} +(-7.54215e6 + 6.32862e6i) q^{76} +(2.41093e7 - 6.62398e7i) q^{77} +(5.26432e6 - 2.98554e7i) q^{79} +2.96959e7i q^{80} -3.42618e7 q^{82} +(-6.05555e7 - 1.06776e7i) q^{83} +(8.12237e7 + 2.95630e7i) q^{85} +(-8.91743e6 - 1.06274e7i) q^{86} +(-1.51823e8 + 5.52589e7i) q^{88} +(9.16372e7 + 5.29067e7i) q^{89} +(-1.82141e7 - 3.15478e7i) q^{91} +(-8.56800e7 + 1.02110e8i) q^{92} +(4.13937e7 + 2.34755e8i) q^{94} +(-1.30110e7 + 2.29419e6i) q^{95} +(2.45606e7 + 2.06088e7i) q^{97} +(-3.86976e7 + 2.23421e7i) 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 \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −26.8100 − 4.72733i −1.67562 − 0.295458i −0.746544 − 0.665336i \(-0.768288\pi\)
−0.929080 + 0.369878i \(0.879399\pi\)
\(3\) 0 0
\(4\) 455.867 + 165.922i 1.78073 + 0.648133i
\(5\) 418.443 + 498.682i 0.669510 + 0.797890i 0.988717 − 0.149794i \(-0.0478612\pi\)
−0.319208 + 0.947685i \(0.603417\pi\)
\(6\) 0 0
\(7\) 2557.31 − 930.784i 1.06510 − 0.387665i 0.250758 − 0.968050i \(-0.419320\pi\)
0.814342 + 0.580385i \(0.197098\pi\)
\(8\) −5401.88 − 3118.78i −1.31882 − 0.761421i
\(9\) 0 0
\(10\) −8861.04 − 15347.8i −0.886104 − 1.53478i
\(11\) 16649.6 − 19842.2i 1.13719 − 1.35525i 0.211316 − 0.977418i \(-0.432225\pi\)
0.925874 − 0.377832i \(-0.123330\pi\)
\(12\) 0 0
\(13\) −2324.40 − 13182.3i −0.0813838 − 0.461550i −0.998079 − 0.0619620i \(-0.980264\pi\)
0.916695 − 0.399588i \(-0.130847\pi\)
\(14\) −72961.5 + 12865.1i −1.89925 + 0.334889i
\(15\) 0 0
\(16\) 34944.7 + 29322.1i 0.533214 + 0.447420i
\(17\) 114990. − 66389.2i 1.37677 − 0.794881i 0.385004 − 0.922915i \(-0.374200\pi\)
0.991770 + 0.128034i \(0.0408667\pi\)
\(18\) 0 0
\(19\) −10147.5 + 17576.0i −0.0778654 + 0.134867i −0.902329 − 0.431048i \(-0.858144\pi\)
0.824463 + 0.565915i \(0.191477\pi\)
\(20\) 108012. + 296761.i 0.675077 + 1.85476i
\(21\) 0 0
\(22\) −540176. + 453262.i −2.30592 + 1.93490i
\(23\) −93974.9 + 258194.i −0.335815 + 0.922645i 0.650752 + 0.759290i \(0.274454\pi\)
−0.986567 + 0.163355i \(0.947769\pi\)
\(24\) 0 0
\(25\) −5756.99 + 32649.5i −0.0147379 + 0.0835828i
\(26\) 364407.i 0.797430i
\(27\) 0 0
\(28\) 1.32023e6 2.14792
\(29\) −373034. − 65776.0i −0.527420 − 0.0929985i −0.0964036 − 0.995342i \(-0.530734\pi\)
−0.431017 + 0.902344i \(0.641845\pi\)
\(30\) 0 0
\(31\) 23963.9 + 8722.16i 0.0259484 + 0.00944446i 0.354962 − 0.934881i \(-0.384494\pi\)
−0.329013 + 0.944325i \(0.606716\pi\)
\(32\) 228160. + 271911.i 0.217591 + 0.259314i
\(33\) 0 0
\(34\) −3.39671e6 + 1.23630e6i −2.54181 + 0.925143i
\(35\) 1.53425e6 + 885801.i 1.02241 + 0.590288i
\(36\) 0 0
\(37\) −1.41658e6 − 2.45359e6i −0.755849 − 1.30917i −0.944951 − 0.327211i \(-0.893891\pi\)
0.189102 − 0.981957i \(-0.439442\pi\)
\(38\) 355142. − 423242.i 0.170321 − 0.202980i
\(39\) 0 0
\(40\) −705106. − 3.99885e6i −0.275432 − 1.56205i
\(41\) 1.23941e6 − 218542.i 0.438613 − 0.0773393i 0.0500186 − 0.998748i \(-0.484072\pi\)
0.388594 + 0.921409i \(0.372961\pi\)
\(42\) 0 0
\(43\) 390374. + 327563.i 0.114185 + 0.0958122i 0.698092 − 0.716008i \(-0.254033\pi\)
−0.583908 + 0.811820i \(0.698477\pi\)
\(44\) 1.08823e7 − 6.28287e6i 2.90341 − 1.67628i
\(45\) 0 0
\(46\) 3.74003e6 − 6.47792e6i 0.835303 − 1.44679i
\(47\) −2.99482e6 − 8.22820e6i −0.613733 − 1.68622i −0.721825 − 0.692076i \(-0.756696\pi\)
0.108092 − 0.994141i \(-0.465526\pi\)
\(48\) 0 0
\(49\) 1.25737e6 − 1.05506e6i 0.218111 − 0.183017i
\(50\) 308690. − 848118.i 0.0493904 − 0.135699i
\(51\) 0 0
\(52\) 1.12762e6 − 6.39506e6i 0.154223 − 0.874644i
\(53\) 6.12277e6i 0.775970i 0.921666 + 0.387985i \(0.126829\pi\)
−0.921666 + 0.387985i \(0.873171\pi\)
\(54\) 0 0
\(55\) 1.68619e7 1.84270
\(56\) −1.67172e7 − 2.94769e6i −1.69985 − 0.299730i
\(57\) 0 0
\(58\) 9.69011e6 + 3.52691e6i 0.856282 + 0.311661i
\(59\) −1.30891e7 − 1.55989e7i −1.08019 − 1.28732i −0.955457 − 0.295130i \(-0.904637\pi\)
−0.124733 − 0.992190i \(-0.539807\pi\)
\(60\) 0 0
\(61\) 3.01877e6 − 1.09874e6i 0.218027 − 0.0793554i −0.230697 − 0.973026i \(-0.574101\pi\)
0.448724 + 0.893670i \(0.351878\pi\)
\(62\) −601240. − 347126.i −0.0406894 − 0.0234920i
\(63\) 0 0
\(64\) −1.06706e7 − 1.84819e7i −0.636014 − 1.10161i
\(65\) 5.60116e6 − 6.67520e6i 0.313779 − 0.373948i
\(66\) 0 0
\(67\) 2.53228e6 + 1.43613e7i 0.125665 + 0.712680i 0.980911 + 0.194458i \(0.0622947\pi\)
−0.855246 + 0.518222i \(0.826594\pi\)
\(68\) 6.34353e7 − 1.11854e7i 2.96685 − 0.523136i
\(69\) 0 0
\(70\) −3.69458e7 − 3.10012e7i −1.53877 − 1.29118i
\(71\) −1.90737e7 + 1.10122e7i −0.750586 + 0.433351i −0.825906 − 0.563808i \(-0.809336\pi\)
0.0753193 + 0.997159i \(0.476002\pi\)
\(72\) 0 0
\(73\) −6.96726e6 + 1.20676e7i −0.245341 + 0.424943i −0.962227 − 0.272247i \(-0.912233\pi\)
0.716886 + 0.697190i \(0.245567\pi\)
\(74\) 2.63796e7 + 7.24775e7i 0.879715 + 2.41700i
\(75\) 0 0
\(76\) −7.54215e6 + 6.32862e6i −0.226069 + 0.189694i
\(77\) 2.41093e7 − 6.62398e7i 0.685839 − 1.88433i
\(78\) 0 0
\(79\) 5.26432e6 − 2.98554e7i 0.135155 − 0.766505i −0.839596 − 0.543212i \(-0.817208\pi\)
0.974751 − 0.223293i \(-0.0716807\pi\)
\(80\) 2.96959e7i 0.724998i
\(81\) 0 0
\(82\) −3.42618e7 −0.757801
\(83\) −6.05555e7 − 1.06776e7i −1.27597 − 0.224988i −0.505703 − 0.862708i \(-0.668767\pi\)
−0.770269 + 0.637720i \(0.779878\pi\)
\(84\) 0 0
\(85\) 8.12237e7 + 2.95630e7i 1.55599 + 0.566334i
\(86\) −8.91743e6 − 1.06274e7i −0.163022 − 0.194282i
\(87\) 0 0
\(88\) −1.51823e8 + 5.52589e7i −2.53166 + 0.921450i
\(89\) 9.16372e7 + 5.29067e7i 1.46053 + 0.843240i 0.999036 − 0.0439015i \(-0.0139788\pi\)
0.461498 + 0.887141i \(0.347312\pi\)
\(90\) 0 0
\(91\) −1.82141e7 − 3.15478e7i −0.265609 − 0.460048i
\(92\) −8.56800e7 + 1.02110e8i −1.19599 + 1.42533i
\(93\) 0 0
\(94\) 4.13937e7 + 2.34755e8i 0.530180 + 3.00680i
\(95\) −1.30110e7 + 2.29419e6i −0.159741 + 0.0281666i
\(96\) 0 0
\(97\) 2.45606e7 + 2.06088e7i 0.277429 + 0.232790i 0.770876 − 0.636986i \(-0.219819\pi\)
−0.493447 + 0.869776i \(0.664263\pi\)
\(98\) −3.86976e7 + 2.23421e7i −0.419547 + 0.242225i
\(99\) 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.8.3 138
3.2 odd 2 27.9.f.a.2.21 ✓ 138
27.13 even 9 27.9.f.a.14.21 yes 138
27.14 odd 18 inner 81.9.f.a.71.3 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.21 ✓ 138 3.2 odd 2
27.9.f.a.14.21 yes 138 27.13 even 9
81.9.f.a.8.3 138 1.1 even 1 trivial
81.9.f.a.71.3 138 27.14 odd 18 inner