Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9,18,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, 18); 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, 18, names="a")
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,528] 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: \(16.4899878610\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1)
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)
 
\(f(q)\) \(=\) \(q+528.000 q^{2} +147712. q^{4} +1.02585e6 q^{5} +3.22599e6 q^{7} +8.78592e6 q^{8} +5.41649e8 q^{10} +7.53618e8 q^{11} +2.54106e9 q^{13} +1.70332e9 q^{14} -1.47219e10 q^{16} +5.42974e9 q^{17} +1.48750e9 q^{19} +1.51530e11 q^{20} +3.97910e11 q^{22} +3.17092e11 q^{23} +2.89429e11 q^{25} +1.34168e12 q^{26} +4.76518e11 q^{28} -2.43341e12 q^{29} -8.84972e12 q^{31} -8.92477e12 q^{32} +2.86690e12 q^{34} +3.30938e12 q^{35} +1.26917e13 q^{37} +7.85400e11 q^{38} +9.01304e12 q^{40} -4.88642e13 q^{41} -9.10200e13 q^{43} +1.11318e14 q^{44} +1.67424e14 q^{46} +4.93050e13 q^{47} -2.22223e14 q^{49} +1.52818e14 q^{50} +3.75346e14 q^{52} -2.29405e13 q^{53} +7.73099e14 q^{55} +2.83433e13 q^{56} -1.28484e15 q^{58} -3.26951e13 q^{59} -1.30829e15 q^{61} -4.67265e15 q^{62} -2.78265e15 q^{64} +2.60675e15 q^{65} +5.19614e15 q^{67} +8.02038e14 q^{68} +1.74735e15 q^{70} +3.70949e15 q^{71} +3.40237e15 q^{73} +6.70119e15 q^{74} +2.19722e14 q^{76} +2.43117e15 q^{77} +2.36653e15 q^{79} -1.51025e16 q^{80} -2.58003e16 q^{82} +2.97668e16 q^{83} +5.57010e15 q^{85} -4.80585e16 q^{86} +6.62123e15 q^{88} -2.91672e16 q^{89} +8.19745e15 q^{91} +4.68383e16 q^{92} +2.60330e16 q^{94} +1.52595e15 q^{95} -6.37699e16 q^{97} -1.17334e17 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 528.000 1.45841 0.729204 − 0.684297i \(-0.239891\pi\)
0.729204 + 0.684297i \(0.239891\pi\)
\(3\) 0 0
\(4\) 147712. 1.12695
\(5\) 1.02585e6 1.17446 0.587231 − 0.809420i \(-0.300218\pi\)
0.587231 + 0.809420i \(0.300218\pi\)
\(6\) 0 0
\(7\) 3.22599e6 0.211510 0.105755 − 0.994392i \(-0.466274\pi\)
0.105755 + 0.994392i \(0.466274\pi\)
\(8\) 8.78592e6 0.185149
\(9\) 0 0
\(10\) 5.41649e8 1.71284
\(11\) 7.53618e8 1.06002 0.530009 − 0.847992i \(-0.322188\pi\)
0.530009 + 0.847992i \(0.322188\pi\)
\(12\) 0 0
\(13\) 2.54106e9 0.863967 0.431984 − 0.901881i \(-0.357814\pi\)
0.431984 + 0.901881i \(0.357814\pi\)
\(14\) 1.70332e9 0.308467
\(15\) 0 0
\(16\) −1.47219e10 −0.856930
\(17\) 5.42974e9 0.188783 0.0943916 − 0.995535i \(-0.469909\pi\)
0.0943916 + 0.995535i \(0.469909\pi\)
\(18\) 0 0
\(19\) 1.48750e9 0.0200933 0.0100467 − 0.999950i \(-0.496802\pi\)
0.0100467 + 0.999950i \(0.496802\pi\)
\(20\) 1.51530e11 1.32356
\(21\) 0 0
\(22\) 3.97910e11 1.54594
\(23\) 3.17092e11 0.844303 0.422152 − 0.906525i \(-0.361275\pi\)
0.422152 + 0.906525i \(0.361275\pi\)
\(24\) 0 0
\(25\) 2.89429e11 0.379360
\(26\) 1.34168e12 1.26002
\(27\) 0 0
\(28\) 4.76518e11 0.238361
\(29\) −2.43341e12 −0.903301 −0.451650 − 0.892195i \(-0.649165\pi\)
−0.451650 + 0.892195i \(0.649165\pi\)
\(30\) 0 0
\(31\) −8.84972e12 −1.86361 −0.931805 − 0.362958i \(-0.881767\pi\)
−0.931805 + 0.362958i \(0.881767\pi\)
\(32\) −8.92477e12 −1.43490
\(33\) 0 0
\(34\) 2.86690e12 0.275323
\(35\) 3.30938e12 0.248410
\(36\) 0 0
\(37\) 1.26917e13 0.594023 0.297012 − 0.954874i \(-0.404010\pi\)
0.297012 + 0.954874i \(0.404010\pi\)
\(38\) 7.85400e11 0.0293042
\(39\) 0 0
\(40\) 9.01304e12 0.217451
\(41\) −4.88642e13 −0.955713 −0.477857 − 0.878438i \(-0.658586\pi\)
−0.477857 + 0.878438i \(0.658586\pi\)
\(42\) 0 0
\(43\) −9.10200e13 −1.18756 −0.593779 − 0.804628i \(-0.702365\pi\)
−0.593779 + 0.804628i \(0.702365\pi\)
\(44\) 1.11318e14 1.19459
\(45\) 0 0
\(46\) 1.67424e14 1.23134
\(47\) 4.93050e13 0.302036 0.151018 − 0.988531i \(-0.451745\pi\)
0.151018 + 0.988531i \(0.451745\pi\)
\(48\) 0 0
\(49\) −2.22223e14 −0.955264
\(50\) 1.52818e14 0.553262
\(51\) 0 0
\(52\) 3.75346e14 0.973650
\(53\) −2.29405e13 −0.0506124 −0.0253062 − 0.999680i \(-0.508056\pi\)
−0.0253062 + 0.999680i \(0.508056\pi\)
\(54\) 0 0
\(55\) 7.73099e14 1.24495
\(56\) 2.83433e13 0.0391609
\(57\) 0 0
\(58\) −1.28484e15 −1.31738
\(59\) −3.26951e13 −0.0289895 −0.0144947 − 0.999895i \(-0.504614\pi\)
−0.0144947 + 0.999895i \(0.504614\pi\)
\(60\) 0 0
\(61\) −1.30829e15 −0.873774 −0.436887 − 0.899516i \(-0.643919\pi\)
−0.436887 + 0.899516i \(0.643919\pi\)
\(62\) −4.67265e15 −2.71790
\(63\) 0 0
\(64\) −2.78265e15 −1.23574
\(65\) 2.60675e15 1.01470
\(66\) 0 0
\(67\) 5.19614e15 1.56331 0.781655 − 0.623711i \(-0.214376\pi\)
0.781655 + 0.623711i \(0.214376\pi\)
\(68\) 8.02038e14 0.212750
\(69\) 0 0
\(70\) 1.74735e15 0.362283
\(71\) 3.70949e15 0.681739 0.340870 − 0.940111i \(-0.389279\pi\)
0.340870 + 0.940111i \(0.389279\pi\)
\(72\) 0 0
\(73\) 3.40237e15 0.493785 0.246892 − 0.969043i \(-0.420591\pi\)
0.246892 + 0.969043i \(0.420591\pi\)
\(74\) 6.70119e15 0.866328
\(75\) 0 0
\(76\) 2.19722e14 0.0226442
\(77\) 2.43117e15 0.224204
\(78\) 0 0
\(79\) 2.36653e15 0.175502 0.0877511 − 0.996142i \(-0.472032\pi\)
0.0877511 + 0.996142i \(0.472032\pi\)
\(80\) −1.51025e16 −1.00643
\(81\) 0 0
\(82\) −2.58003e16 −1.39382
\(83\) 2.97668e16 1.45067 0.725333 − 0.688398i \(-0.241686\pi\)
0.725333 + 0.688398i \(0.241686\pi\)
\(84\) 0 0
\(85\) 5.57010e15 0.221719
\(86\) −4.80585e16 −1.73194
\(87\) 0 0
\(88\) 6.62123e15 0.196262
\(89\) −2.91672e16 −0.785379 −0.392690 − 0.919671i \(-0.628455\pi\)
−0.392690 + 0.919671i \(0.628455\pi\)
\(90\) 0 0
\(91\) 8.19745e15 0.182737
\(92\) 4.68383e16 0.951490
\(93\) 0 0
\(94\) 2.60330e16 0.440492
\(95\) 1.52595e15 0.0235988
\(96\) 0 0
\(97\) −6.37699e16 −0.826144 −0.413072 − 0.910698i \(-0.635544\pi\)
−0.413072 + 0.910698i \(0.635544\pi\)
\(98\) −1.17334e17 −1.39316
\(99\) 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 9.18.a.b.1.1 1
3.2 odd 2 1.18.a.a.1.1 ✓ 1
12.11 even 2 16.18.a.b.1.1 1
15.2 even 4 25.18.b.a.24.1 2
15.8 even 4 25.18.b.a.24.2 2
15.14 odd 2 25.18.a.a.1.1 1
21.20 even 2 49.18.a.a.1.1 1
24.5 odd 2 64.18.a.d.1.1 1
24.11 even 2 64.18.a.b.1.1 1
33.32 even 2 121.18.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.18.a.a.1.1 ✓ 1 3.2 odd 2
9.18.a.b.1.1 1 1.1 even 1 trivial
16.18.a.b.1.1 1 12.11 even 2
25.18.a.a.1.1 1 15.14 odd 2
25.18.b.a.24.1 2 15.2 even 4
25.18.b.a.24.2 2 15.8 even 4
49.18.a.a.1.1 1 21.20 even 2
64.18.a.b.1.1 1 24.11 even 2
64.18.a.d.1.1 1 24.5 odd 2
121.18.a.b.1.1 1 33.32 even 2