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 71.4
Character \(\chi\) \(=\) 81.71
Dual form 81.9.f.a.8.4

$q$-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+(-25.1387 + 4.43262i) q^{2} +(371.742 - 135.303i) q^{4} +(546.503 - 651.297i) q^{5} +(-3663.86 - 1333.53i) q^{7} +(-3086.08 + 1781.75i) q^{8} +(-10851.4 + 18795.2i) q^{10} +(7165.64 + 8539.67i) q^{11} +(-6824.34 + 38702.7i) q^{13} +(98015.5 + 17282.8i) q^{14} +(-7898.09 + 6627.28i) q^{16} +(-109614. - 63285.8i) q^{17} +(-62328.9 - 107957. i) q^{19} +(115036. - 316059. i) q^{20} +(-217988. - 182913. i) q^{22} +(84577.6 + 232375. i) q^{23} +(-57691.0 - 327182. i) q^{25} -1.00318e6i q^{26} -1.54244e6 q^{28} +(177273. - 31258.0i) q^{29} +(557486. - 202908. i) q^{31} +(755557. - 900438. i) q^{32} +(3.03608e6 + 1.10504e6i) q^{34} +(-2.87084e6 + 1.65748e6i) q^{35} +(-229672. + 397803. i) q^{37} +(2.04540e6 + 2.43761e6i) q^{38} +(-526104. + 2.98368e6i) q^{40} +(981871. + 173130. i) q^{41} +(-2.11904e6 + 1.77809e6i) q^{43} +(3.81922e6 + 2.20503e6i) q^{44} +(-3.15620e6 - 5.46670e6i) q^{46} +(1.21462e6 - 3.33714e6i) q^{47} +(7.22944e6 + 6.06622e6i) q^{49} +(2.90055e6 + 7.96919e6i) q^{50} +(2.69971e6 + 1.53108e7i) q^{52} +6.97006e6i q^{53} +9.47791e6 q^{55} +(1.36830e7 - 2.41268e6i) q^{56} +(-4.31784e6 + 1.57157e6i) q^{58} +(7.66416e6 - 9.13379e6i) q^{59} +(-4.95543e6 - 1.80363e6i) q^{61} +(-1.31150e7 + 7.57197e6i) q^{62} +(-1.36827e7 + 2.36991e7i) q^{64} +(2.14775e7 + 2.55958e7i) q^{65} +(-5.51669e6 + 3.12867e7i) q^{67} +(-4.93110e7 - 8.69486e6i) q^{68} +(6.48220e7 - 5.43921e7i) q^{70} +(2.14530e7 + 1.23859e7i) q^{71} +(2.29304e6 + 3.97166e6i) q^{73} +(4.01033e6 - 1.10183e7i) q^{74} +(-3.77772e7 - 3.16988e7i) q^{76} +(-1.48659e7 - 4.08438e7i) q^{77} +(1.68902e6 + 9.57891e6i) q^{79} +8.76584e6i q^{80} -2.54503e7 q^{82} +(2.39486e7 - 4.22279e6i) q^{83} +(-1.01122e8 + 3.68056e7i) q^{85} +(4.53882e7 - 5.40916e7i) q^{86} +(-3.73292e7 - 1.35867e7i) q^{88} +(2.53853e7 - 1.46562e7i) q^{89} +(7.66148e7 - 1.32701e8i) q^{91} +(6.28822e7 + 7.49401e7i) q^{92} +(-1.57416e7 + 8.92752e7i) q^{94} +(-1.04375e8 - 1.84041e7i) q^{95} +(8.60715e7 - 7.22226e7i) q^{97} +(-2.08628e8 - 1.20451e8i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −25.1387 + 4.43262i −1.57117 + 0.277039i −0.890302 − 0.455371i \(-0.849507\pi\)
−0.680864 + 0.732410i \(0.738396\pi\)
\(3\) 0 0
\(4\) 371.742 − 135.303i 1.45212 − 0.528528i
\(5\) 546.503 − 651.297i 0.874405 − 1.04208i −0.124352 − 0.992238i \(-0.539685\pi\)
0.998757 − 0.0498377i \(-0.0158704\pi\)
\(6\) 0 0
\(7\) −3663.86 − 1333.53i −1.52597 − 0.555408i −0.563340 − 0.826225i \(-0.690484\pi\)
−0.962631 + 0.270817i \(0.912706\pi\)
\(8\) −3086.08 + 1781.75i −0.753437 + 0.434997i
\(9\) 0 0
\(10\) −10851.4 + 18795.2i −1.08514 + 1.87952i
\(11\) 7165.64 + 8539.67i 0.489423 + 0.583271i 0.953071 − 0.302748i \(-0.0979042\pi\)
−0.463648 + 0.886020i \(0.653460\pi\)
\(12\) 0 0
\(13\) −6824.34 + 38702.7i −0.238939 + 1.35509i 0.595219 + 0.803564i \(0.297065\pi\)
−0.834158 + 0.551526i \(0.814046\pi\)
\(14\) 98015.5 + 17282.8i 2.55142 + 0.449885i
\(15\) 0 0
\(16\) −7898.09 + 6627.28i −0.120515 + 0.101124i
\(17\) −109614. − 63285.8i −1.31242 − 0.757723i −0.329920 − 0.944009i \(-0.607022\pi\)
−0.982496 + 0.186286i \(0.940355\pi\)
\(18\) 0 0
\(19\) −62328.9 − 107957.i −0.478272 − 0.828391i 0.521418 − 0.853302i \(-0.325403\pi\)
−0.999690 + 0.0249102i \(0.992070\pi\)
\(20\) 115036. − 316059.i 0.718974 − 1.97537i
\(21\) 0 0
\(22\) −217988. − 182913.i −0.930553 − 0.780827i
\(23\) 84577.6 + 232375.i 0.302235 + 0.830383i 0.994111 + 0.108367i \(0.0345621\pi\)
−0.691876 + 0.722016i \(0.743216\pi\)
\(24\) 0 0
\(25\) −57691.0 − 327182.i −0.147689 − 0.837586i
\(26\) − 1.00318e6i − 2.19527i
\(27\) 0 0
\(28\) −1.54244e6 −2.50944
\(29\) 177273. − 31258.0i 0.250640 − 0.0441946i −0.0469165 − 0.998899i \(-0.514939\pi\)
0.297556 + 0.954704i \(0.403828\pi\)
\(30\) 0 0
\(31\) 557486. − 202908.i 0.603653 − 0.219712i −0.0220706 − 0.999756i \(-0.507026\pi\)
0.625724 + 0.780045i \(0.284804\pi\)
\(32\) 755557. − 900438.i 0.720555 − 0.858724i
\(33\) 0 0
\(34\) 3.03608e6 + 1.10504e6i 2.27194 + 0.826919i
\(35\) −2.87084e6 + 1.65748e6i −1.91310 + 1.10453i
\(36\) 0 0
\(37\) −229672. + 397803.i −0.122546 + 0.212257i −0.920771 − 0.390103i \(-0.872439\pi\)
0.798225 + 0.602360i \(0.205773\pi\)
\(38\) 2.04540e6 + 2.43761e6i 0.980941 + 1.16904i
\(39\) 0 0
\(40\) −526104. + 2.98368e6i −0.205509 + 1.16550i
\(41\) 981871. + 173130.i 0.347471 + 0.0612686i 0.344660 − 0.938727i \(-0.387994\pi\)
0.00281108 + 0.999996i \(0.499105\pi\)
\(42\) 0 0
\(43\) −2.11904e6 + 1.77809e6i −0.619820 + 0.520091i −0.897747 − 0.440512i \(-0.854797\pi\)
0.277927 + 0.960602i \(0.410353\pi\)
\(44\) 3.81922e6 + 2.20503e6i 1.01898 + 0.588306i
\(45\) 0 0
\(46\) −3.15620e6 − 5.46670e6i −0.704909 − 1.22094i
\(47\) 1.21462e6 − 3.33714e6i 0.248914 − 0.683885i −0.750813 − 0.660515i \(-0.770338\pi\)
0.999727 − 0.0233703i \(-0.00743968\pi\)
\(48\) 0 0
\(49\) 7.22944e6 + 6.06622e6i 1.25407 + 1.05229i
\(50\) 2.90055e6 + 7.96919e6i 0.464088 + 1.27507i
\(51\) 0 0
\(52\) 2.69971e6 + 1.53108e7i 0.369235 + 2.09404i
\(53\) 6.97006e6i 0.883350i 0.897175 + 0.441675i \(0.145616\pi\)
−0.897175 + 0.441675i \(0.854384\pi\)
\(54\) 0 0
\(55\) 9.47791e6 1.03577
\(56\) 1.36830e7 − 2.41268e6i 1.39132 − 0.245328i
\(57\) 0 0
\(58\) −4.31784e6 + 1.57157e6i −0.381553 + 0.138874i
\(59\) 7.66416e6 − 9.13379e6i 0.632494 − 0.753777i −0.350671 − 0.936499i \(-0.614046\pi\)
0.983165 + 0.182722i \(0.0584908\pi\)
\(60\) 0 0
\(61\) −4.95543e6 − 1.80363e6i −0.357900 − 0.130265i 0.156811 − 0.987629i \(-0.449879\pi\)
−0.514711 + 0.857364i \(0.672101\pi\)
\(62\) −1.31150e7 + 7.57197e6i −0.887571 + 0.512439i
\(63\) 0 0
\(64\) −1.36827e7 + 2.36991e7i −0.815551 + 1.41258i
\(65\) 2.14775e7 + 2.55958e7i 1.20318 + 1.43389i
\(66\) 0 0
\(67\) −5.51669e6 + 3.12867e7i −0.273766 + 1.55260i 0.469088 + 0.883151i \(0.344583\pi\)
−0.742854 + 0.669453i \(0.766528\pi\)
\(68\) −4.93110e7 − 8.69486e6i −2.30626 − 0.406656i
\(69\) 0 0
\(70\) 6.48220e7 − 5.43921e7i 2.69979 − 2.26540i
\(71\) 2.14530e7 + 1.23859e7i 0.844217 + 0.487409i 0.858696 − 0.512486i \(-0.171275\pi\)
−0.0144782 + 0.999895i \(0.504609\pi\)
\(72\) 0 0
\(73\) 2.29304e6 + 3.97166e6i 0.0807459 + 0.139856i 0.903571 − 0.428439i \(-0.140936\pi\)
−0.822825 + 0.568295i \(0.807603\pi\)
\(74\) 4.01033e6 − 1.10183e7i 0.133737 − 0.367441i
\(75\) 0 0
\(76\) −3.77772e7 − 3.16988e7i −1.13234 − 0.950143i
\(77\) −1.48659e7 − 4.08438e7i −0.422891 − 1.16188i
\(78\) 0 0
\(79\) 1.68902e6 + 9.57891e6i 0.0433637 + 0.245928i 0.998783 − 0.0493253i \(-0.0157071\pi\)
−0.955419 + 0.295253i \(0.904596\pi\)
\(80\) 8.76584e6i 0.214010i
\(81\) 0 0
\(82\) −2.54503e7 −0.562909
\(83\) 2.39486e7 − 4.22279e6i 0.504625 − 0.0889789i 0.0844600 − 0.996427i \(-0.473083\pi\)
0.420165 + 0.907448i \(0.361972\pi\)
\(84\) 0 0
\(85\) −1.01122e8 + 3.68056e7i −1.93719 + 0.705079i
\(86\) 4.53882e7 − 5.40916e7i 0.829754 − 0.988863i
\(87\) 0 0
\(88\) −3.73292e7 − 1.35867e7i −0.622470 − 0.226561i
\(89\) 2.53853e7 − 1.46562e7i 0.404597 − 0.233594i −0.283869 − 0.958863i \(-0.591618\pi\)
0.688466 + 0.725269i \(0.258285\pi\)
\(90\) 0 0
\(91\) 7.66148e7 − 1.32701e8i 1.11724 − 1.93512i
\(92\) 6.28822e7 + 7.49401e7i 0.877761 + 1.04608i
\(93\) 0 0
\(94\) −1.57416e7 + 8.92752e7i −0.201622 + 1.14346i
\(95\) −1.04375e8 − 1.84041e7i −1.28145 − 0.225954i
\(96\) 0 0
\(97\) 8.60715e7 − 7.22226e7i 0.972238 − 0.815805i −0.0106623 − 0.999943i \(-0.503394\pi\)
0.982900 + 0.184139i \(0.0589495\pi\)
\(98\) −2.08628e8 − 1.20451e8i −2.26187 − 1.30589i
\(99\) 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.71.4 138
3.2 odd 2 27.9.f.a.14.20 yes 138
27.2 odd 18 inner 81.9.f.a.8.4 138
27.25 even 9 27.9.f.a.2.20 ✓ 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.20 ✓ 138 27.25 even 9
27.9.f.a.14.20 yes 138 3.2 odd 2
81.9.f.a.8.4 138 27.2 odd 18 inner
81.9.f.a.71.4 138 1.1 even 1 trivial