Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9,14,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, 14); 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, 14, names="a")
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,12] 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: \(9.65078360567\)
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)
 
\(f(q)\) \(=\) \(q+12.0000 q^{2} -8048.00 q^{4} +30210.0 q^{5} +235088. q^{7} -194880. q^{8} +362520. q^{10} +1.11829e7 q^{11} +8.04961e6 q^{13} +2.82106e6 q^{14} +6.35907e7 q^{16} +1.17495e8 q^{17} -2.14061e8 q^{19} -2.43130e8 q^{20} +1.34195e8 q^{22} -8.30556e8 q^{23} -3.08059e8 q^{25} +9.65954e7 q^{26} -1.89199e9 q^{28} +1.25240e9 q^{29} +6.15935e9 q^{31} +2.35954e9 q^{32} +1.40994e9 q^{34} +7.10201e9 q^{35} -5.49819e9 q^{37} -2.56874e9 q^{38} -5.88732e9 q^{40} +4.67869e9 q^{41} +7.11501e9 q^{43} -9.00000e10 q^{44} -9.96667e9 q^{46} +2.95288e10 q^{47} -4.16226e10 q^{49} -3.69671e9 q^{50} -6.47833e10 q^{52} +2.04125e11 q^{53} +3.37836e11 q^{55} -4.58139e10 q^{56} +1.50288e10 q^{58} +2.99098e10 q^{59} -1.34392e11 q^{61} +7.39122e10 q^{62} -4.92620e11 q^{64} +2.43179e11 q^{65} +3.48519e11 q^{67} -9.45597e11 q^{68} +8.52241e10 q^{70} -1.31434e12 q^{71} -1.17888e12 q^{73} -6.59783e10 q^{74} +1.72277e12 q^{76} +2.62897e12 q^{77} -1.07242e12 q^{79} +1.92107e12 q^{80} +5.61443e10 q^{82} -1.12403e12 q^{83} +3.54951e12 q^{85} +8.53802e10 q^{86} -2.17933e12 q^{88} -2.23561e12 q^{89} +1.89237e12 q^{91} +6.68431e12 q^{92} +3.54345e11 q^{94} -6.46679e12 q^{95} -1.42153e13 q^{97} -4.99472e11 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 12.0000 0.132583 0.0662913 − 0.997800i \(-0.478883\pi\)
0.0662913 + 0.997800i \(0.478883\pi\)
\(3\) 0 0
\(4\) −8048.00 −0.982422
\(5\) 30210.0 0.864661 0.432330 − 0.901715i \(-0.357691\pi\)
0.432330 + 0.901715i \(0.357691\pi\)
\(6\) 0 0
\(7\) 235088. 0.755254 0.377627 − 0.925958i \(-0.376740\pi\)
0.377627 + 0.925958i \(0.376740\pi\)
\(8\) −194880. −0.262834
\(9\) 0 0
\(10\) 362520. 0.114639
\(11\) 1.11829e7 1.90328 0.951639 − 0.307218i \(-0.0993981\pi\)
0.951639 + 0.307218i \(0.0993981\pi\)
\(12\) 0 0
\(13\) 8.04961e6 0.462534 0.231267 − 0.972890i \(-0.425713\pi\)
0.231267 + 0.972890i \(0.425713\pi\)
\(14\) 2.82106e6 0.100134
\(15\) 0 0
\(16\) 6.35907e7 0.947575
\(17\) 1.17495e8 1.18059 0.590296 − 0.807187i \(-0.299011\pi\)
0.590296 + 0.807187i \(0.299011\pi\)
\(18\) 0 0
\(19\) −2.14061e8 −1.04385 −0.521927 − 0.852990i \(-0.674787\pi\)
−0.521927 + 0.852990i \(0.674787\pi\)
\(20\) −2.43130e8 −0.849462
\(21\) 0 0
\(22\) 1.34195e8 0.252341
\(23\) −8.30556e8 −1.16987 −0.584935 − 0.811080i \(-0.698880\pi\)
−0.584935 + 0.811080i \(0.698880\pi\)
\(24\) 0 0
\(25\) −3.08059e8 −0.252362
\(26\) 9.65954e7 0.0613239
\(27\) 0 0
\(28\) −1.89199e9 −0.741978
\(29\) 1.25240e9 0.390981 0.195491 − 0.980706i \(-0.437370\pi\)
0.195491 + 0.980706i \(0.437370\pi\)
\(30\) 0 0
\(31\) 6.15935e9 1.24648 0.623238 − 0.782032i \(-0.285817\pi\)
0.623238 + 0.782032i \(0.285817\pi\)
\(32\) 2.35954e9 0.388466
\(33\) 0 0
\(34\) 1.40994e9 0.156526
\(35\) 7.10201e9 0.653039
\(36\) 0 0
\(37\) −5.49819e9 −0.352297 −0.176148 − 0.984364i \(-0.556364\pi\)
−0.176148 + 0.984364i \(0.556364\pi\)
\(38\) −2.56874e9 −0.138397
\(39\) 0 0
\(40\) −5.88732e9 −0.227263
\(41\) 4.67869e9 0.153826 0.0769129 − 0.997038i \(-0.475494\pi\)
0.0769129 + 0.997038i \(0.475494\pi\)
\(42\) 0 0
\(43\) 7.11501e9 0.171645 0.0858224 − 0.996310i \(-0.472648\pi\)
0.0858224 + 0.996310i \(0.472648\pi\)
\(44\) −9.00000e10 −1.86982
\(45\) 0 0
\(46\) −9.96667e9 −0.155104
\(47\) 2.95288e10 0.399585 0.199793 − 0.979838i \(-0.435973\pi\)
0.199793 + 0.979838i \(0.435973\pi\)
\(48\) 0 0
\(49\) −4.16226e10 −0.429591
\(50\) −3.69671e9 −0.0334588
\(51\) 0 0
\(52\) −6.47833e10 −0.454403
\(53\) 2.04125e11 1.26504 0.632518 − 0.774545i \(-0.282021\pi\)
0.632518 + 0.774545i \(0.282021\pi\)
\(54\) 0 0
\(55\) 3.37836e11 1.64569
\(56\) −4.58139e10 −0.198507
\(57\) 0 0
\(58\) 1.50288e10 0.0518373
\(59\) 2.99098e10 0.0923157 0.0461579 − 0.998934i \(-0.485302\pi\)
0.0461579 + 0.998934i \(0.485302\pi\)
\(60\) 0 0
\(61\) −1.34392e11 −0.333987 −0.166993 − 0.985958i \(-0.553406\pi\)
−0.166993 + 0.985958i \(0.553406\pi\)
\(62\) 7.39122e10 0.165261
\(63\) 0 0
\(64\) −4.92620e11 −0.896071
\(65\) 2.43179e11 0.399935
\(66\) 0 0
\(67\) 3.48519e11 0.470695 0.235348 − 0.971911i \(-0.424377\pi\)
0.235348 + 0.971911i \(0.424377\pi\)
\(68\) −9.45597e11 −1.15984
\(69\) 0 0
\(70\) 8.52241e10 0.0865815
\(71\) −1.31434e12 −1.21766 −0.608831 − 0.793300i \(-0.708361\pi\)
−0.608831 + 0.793300i \(0.708361\pi\)
\(72\) 0 0
\(73\) −1.17888e12 −0.911737 −0.455868 − 0.890047i \(-0.650671\pi\)
−0.455868 + 0.890047i \(0.650671\pi\)
\(74\) −6.59783e10 −0.0467084
\(75\) 0 0
\(76\) 1.72277e12 1.02551
\(77\) 2.62897e12 1.43746
\(78\) 0 0
\(79\) −1.07242e12 −0.496351 −0.248176 − 0.968715i \(-0.579831\pi\)
−0.248176 + 0.968715i \(0.579831\pi\)
\(80\) 1.92107e12 0.819330
\(81\) 0 0
\(82\) 5.61443e10 0.0203946
\(83\) −1.12403e12 −0.377371 −0.188685 − 0.982038i \(-0.560423\pi\)
−0.188685 + 0.982038i \(0.560423\pi\)
\(84\) 0 0
\(85\) 3.54951e12 1.02081
\(86\) 8.53802e10 0.0227571
\(87\) 0 0
\(88\) −2.17933e12 −0.500247
\(89\) −2.23561e12 −0.476827 −0.238414 − 0.971164i \(-0.576627\pi\)
−0.238414 + 0.971164i \(0.576627\pi\)
\(90\) 0 0
\(91\) 1.89237e12 0.349330
\(92\) 6.68431e12 1.14931
\(93\) 0 0
\(94\) 3.54345e11 0.0529780
\(95\) −6.46679e12 −0.902580
\(96\) 0 0
\(97\) −1.42153e13 −1.73276 −0.866380 − 0.499385i \(-0.833559\pi\)
−0.866380 + 0.499385i \(0.833559\pi\)
\(98\) −4.99472e11 −0.0569563
\(99\) 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 9.14.a.a.1.1 1
3.2 odd 2 3.14.a.a.1.1 ✓ 1
4.3 odd 2 144.14.a.k.1.1 1
12.11 even 2 48.14.a.c.1.1 1
15.2 even 4 75.14.b.b.49.1 2
15.8 even 4 75.14.b.b.49.2 2
15.14 odd 2 75.14.a.a.1.1 1
21.20 even 2 147.14.a.a.1.1 1
24.5 odd 2 192.14.a.j.1.1 1
24.11 even 2 192.14.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.14.a.a.1.1 ✓ 1 3.2 odd 2
9.14.a.a.1.1 1 1.1 even 1 trivial
48.14.a.c.1.1 1 12.11 even 2
75.14.a.a.1.1 1 15.14 odd 2
75.14.b.b.49.1 2 15.2 even 4
75.14.b.b.49.2 2 15.8 even 4
144.14.a.k.1.1 1 4.3 odd 2
147.14.a.a.1.1 1 21.20 even 2
192.14.a.e.1.1 1 24.11 even 2
192.14.a.j.1.1 1 24.5 odd 2