Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9,22,Mod(1,9)] 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("9.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1728] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(25.1529609858\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 9.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1728.00 q^{2} +888832. q^{4} +4.15128e7 q^{5} +5.38430e8 q^{7} +2.08798e9 q^{8} -7.17341e10 q^{10} +6.41130e10 q^{11} -1.30980e11 q^{13} -9.30407e11 q^{14} -5.47204e12 q^{16} -8.24203e12 q^{17} +1.34921e13 q^{19} +3.68979e13 q^{20} -1.10787e14 q^{22} +2.33185e14 q^{23} +1.24647e15 q^{25} +2.26334e14 q^{26} +4.78574e14 q^{28} +2.02456e15 q^{29} -6.86919e15 q^{31} +5.07688e15 q^{32} +1.42422e16 q^{34} +2.23517e16 q^{35} +3.44400e15 q^{37} -2.33144e16 q^{38} +8.66777e16 q^{40} +2.18424e16 q^{41} -7.17928e16 q^{43} +5.69857e16 q^{44} -4.02943e17 q^{46} -2.83545e17 q^{47} -2.68639e17 q^{49} -2.15391e18 q^{50} -1.16419e17 q^{52} +2.17229e18 q^{53} +2.66151e18 q^{55} +1.12423e18 q^{56} -3.49844e18 q^{58} -1.53483e18 q^{59} +4.31159e18 q^{61} +1.18700e19 q^{62} +2.70285e18 q^{64} -5.43735e18 q^{65} +9.24391e18 q^{67} -7.32578e18 q^{68} -3.86238e19 q^{70} +2.03874e19 q^{71} +1.66178e19 q^{73} -5.95123e18 q^{74} +1.19922e19 q^{76} +3.45204e19 q^{77} +6.79403e19 q^{79} -2.27160e20 q^{80} -3.77437e19 q^{82} -3.95037e19 q^{83} -3.42149e20 q^{85} +1.24058e20 q^{86} +1.33867e20 q^{88} -4.16117e19 q^{89} -7.05236e19 q^{91} +2.07262e20 q^{92} +4.89965e20 q^{94} +5.60095e20 q^{95} +5.71815e19 q^{97} +4.64209e20 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1728.00 −1.19324 −0.596621 − 0.802523i \(-0.703491\pi\)
−0.596621 + 0.802523i \(0.703491\pi\)
\(3\) 0 0
\(4\) 888832. 0.423828
\(5\) 4.15128e7 1.90106 0.950532 − 0.310627i \(-0.100539\pi\)
0.950532 + 0.310627i \(0.100539\pi\)
\(6\) 0 0
\(7\) 5.38430e8 0.720443 0.360222 − 0.932867i \(-0.382701\pi\)
0.360222 + 0.932867i \(0.382701\pi\)
\(8\) 2.08798e9 0.687513
\(9\) 0 0
\(10\) −7.17341e10 −2.26843
\(11\) 6.41130e10 0.745286 0.372643 − 0.927975i \(-0.378452\pi\)
0.372643 + 0.927975i \(0.378452\pi\)
\(12\) 0 0
\(13\) −1.30980e11 −0.263512 −0.131756 − 0.991282i \(-0.542062\pi\)
−0.131756 + 0.991282i \(0.542062\pi\)
\(14\) −9.30407e11 −0.859663
\(15\) 0 0
\(16\) −5.47204e12 −1.24420
\(17\) −8.24203e12 −0.991563 −0.495782 − 0.868447i \(-0.665118\pi\)
−0.495782 + 0.868447i \(0.665118\pi\)
\(18\) 0 0
\(19\) 1.34921e13 0.504855 0.252428 − 0.967616i \(-0.418771\pi\)
0.252428 + 0.967616i \(0.418771\pi\)
\(20\) 3.68979e13 0.805724
\(21\) 0 0
\(22\) −1.10787e14 −0.889308
\(23\) 2.33185e14 1.17370 0.586851 − 0.809695i \(-0.300367\pi\)
0.586851 + 0.809695i \(0.300367\pi\)
\(24\) 0 0
\(25\) 1.24647e15 2.61404
\(26\) 2.26334e14 0.314434
\(27\) 0 0
\(28\) 4.78574e14 0.305344
\(29\) 2.02456e15 0.893618 0.446809 − 0.894629i \(-0.352560\pi\)
0.446809 + 0.894629i \(0.352560\pi\)
\(30\) 0 0
\(31\) −6.86919e15 −1.50525 −0.752624 − 0.658451i \(-0.771212\pi\)
−0.752624 + 0.658451i \(0.771212\pi\)
\(32\) 5.07688e15 0.797117
\(33\) 0 0
\(34\) 1.42422e16 1.18318
\(35\) 2.23517e16 1.36961
\(36\) 0 0
\(37\) 3.44400e15 0.117746 0.0588728 − 0.998265i \(-0.481249\pi\)
0.0588728 + 0.998265i \(0.481249\pi\)
\(38\) −2.33144e16 −0.602415
\(39\) 0 0
\(40\) 8.66777e16 1.30701
\(41\) 2.18424e16 0.254138 0.127069 − 0.991894i \(-0.459443\pi\)
0.127069 + 0.991894i \(0.459443\pi\)
\(42\) 0 0
\(43\) −7.17928e16 −0.506597 −0.253298 − 0.967388i \(-0.581515\pi\)
−0.253298 + 0.967388i \(0.581515\pi\)
\(44\) 5.69857e16 0.315873
\(45\) 0 0
\(46\) −4.02943e17 −1.40051
\(47\) −2.83545e17 −0.786310 −0.393155 − 0.919472i \(-0.628617\pi\)
−0.393155 + 0.919472i \(0.628617\pi\)
\(48\) 0 0
\(49\) −2.68639e17 −0.480962
\(50\) −2.15391e18 −3.11919
\(51\) 0 0
\(52\) −1.16419e17 −0.111684
\(53\) 2.17229e18 1.70616 0.853081 − 0.521779i \(-0.174731\pi\)
0.853081 + 0.521779i \(0.174731\pi\)
\(54\) 0 0
\(55\) 2.66151e18 1.41684
\(56\) 1.12423e18 0.495314
\(57\) 0 0
\(58\) −3.49844e18 −1.06630
\(59\) −1.53483e18 −0.390944 −0.195472 − 0.980709i \(-0.562624\pi\)
−0.195472 + 0.980709i \(0.562624\pi\)
\(60\) 0 0
\(61\) 4.31159e18 0.773881 0.386940 − 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(62\) 1.18700e19 1.79613
\(63\) 0 0
\(64\) 2.70285e18 0.293044
\(65\) −5.43735e18 −0.500953
\(66\) 0 0
\(67\) 9.24391e18 0.619541 0.309771 − 0.950811i \(-0.399748\pi\)
0.309771 + 0.950811i \(0.399748\pi\)
\(68\) −7.32578e18 −0.420252
\(69\) 0 0
\(70\) −3.86238e19 −1.63427
\(71\) 2.03874e19 0.743273 0.371636 − 0.928378i \(-0.378797\pi\)
0.371636 + 0.928378i \(0.378797\pi\)
\(72\) 0 0
\(73\) 1.66178e19 0.452566 0.226283 − 0.974062i \(-0.427343\pi\)
0.226283 + 0.974062i \(0.427343\pi\)
\(74\) −5.95123e18 −0.140499
\(75\) 0 0
\(76\) 1.19922e19 0.213972
\(77\) 3.45204e19 0.536936
\(78\) 0 0
\(79\) 6.79403e19 0.807315 0.403658 − 0.914910i \(-0.367739\pi\)
0.403658 + 0.914910i \(0.367739\pi\)
\(80\) −2.27160e20 −2.36530
\(81\) 0 0
\(82\) −3.77437e19 −0.303249
\(83\) −3.95037e19 −0.279459 −0.139730 − 0.990190i \(-0.544623\pi\)
−0.139730 + 0.990190i \(0.544623\pi\)
\(84\) 0 0
\(85\) −3.42149e20 −1.88503
\(86\) 1.24058e20 0.604493
\(87\) 0 0
\(88\) 1.33867e20 0.512394
\(89\) −4.16117e19 −0.141456 −0.0707278 − 0.997496i \(-0.522532\pi\)
−0.0707278 + 0.997496i \(0.522532\pi\)
\(90\) 0 0
\(91\) −7.05236e19 −0.189845
\(92\) 2.07262e20 0.497448
\(93\) 0 0
\(94\) 4.89965e20 0.938259
\(95\) 5.60095e20 0.959762
\(96\) 0 0
\(97\) 5.71815e19 0.0787322 0.0393661 − 0.999225i \(-0.487466\pi\)
0.0393661 + 0.999225i \(0.487466\pi\)
\(98\) 4.64209e20 0.573904
\(99\) 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 9.22.a.a.1.1 1
3.2 odd 2 3.22.a.b.1.1 ✓ 1
12.11 even 2 48.22.a.d.1.1 1
15.2 even 4 75.22.b.b.49.2 2
15.8 even 4 75.22.b.b.49.1 2
15.14 odd 2 75.22.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.22.a.b.1.1 ✓ 1 3.2 odd 2
9.22.a.a.1.1 1 1.1 even 1 trivial
48.22.a.d.1.1 1 12.11 even 2
75.22.a.a.1.1 1 15.14 odd 2
75.22.b.b.49.1 2 15.8 even 4
75.22.b.b.49.2 2 15.2 even 4