Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9522,2,Mod(1,9522)] 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("9522.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9522, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9522 = 2 \cdot 3^{2} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9522.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,1,0,1,-2,0,2,1,0,-2,-6,0,-2,2,0,1,0,0,0,-2,0,-6,0,0,-1,-2, 0,2,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(76.0335528047\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 138)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 9522.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +2.00000 q^{7} +1.00000 q^{8} -2.00000 q^{10} -6.00000 q^{11} -2.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} -2.00000 q^{20} -6.00000 q^{22} -1.00000 q^{25} -2.00000 q^{26} +2.00000 q^{28} -6.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} -4.00000 q^{35} -2.00000 q^{40} -10.0000 q^{41} +12.0000 q^{43} -6.00000 q^{44} +8.00000 q^{47} -3.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} +2.00000 q^{53} +12.0000 q^{55} +2.00000 q^{56} -6.00000 q^{58} +12.0000 q^{59} -4.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} +12.0000 q^{67} -4.00000 q^{70} -10.0000 q^{73} -12.0000 q^{77} +6.00000 q^{79} -2.00000 q^{80} -10.0000 q^{82} +14.0000 q^{83} +12.0000 q^{86} -6.00000 q^{88} -4.00000 q^{91} +8.00000 q^{94} +6.00000 q^{97} -3.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) −6.00000 −1.80907 −0.904534 − 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) −6.00000 −1.27920
\(23\) 0 0
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −2.00000 −0.316228
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 12.0000 1.82998 0.914991 − 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 12.0000 1.61808
\(56\) 2.00000 0.267261
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 − 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 − 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −4.00000 −0.478091
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −12.0000 −1.36753
\(78\) 0 0
\(79\) 6.00000 0.675053 0.337526 − 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) −2.00000 −0.223607
\(81\) 0 0
\(82\) −10.0000 −1.10432
\(83\) 14.0000 1.53670 0.768350 − 0.640030i \(-0.221078\pi\)
0.768350 + 0.640030i \(0.221078\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) 0 0
\(88\) −6.00000 −0.639602
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 − 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) −3.00000 −0.303046
\(99\) 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 9522.2.a.i.1.1 1
3.2 odd 2 3174.2.a.b.1.1 1
23.22 odd 2 414.2.a.d.1.1 1
69.68 even 2 138.2.a.a.1.1 ✓ 1
92.91 even 2 3312.2.a.n.1.1 1
276.275 odd 2 1104.2.a.e.1.1 1
345.68 odd 4 3450.2.d.j.2899.2 2
345.137 odd 4 3450.2.d.j.2899.1 2
345.344 even 2 3450.2.a.y.1.1 1
483.482 odd 2 6762.2.a.q.1.1 1
552.275 odd 2 4416.2.a.m.1.1 1
552.413 even 2 4416.2.a.z.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.a.a.1.1 ✓ 1 69.68 even 2
414.2.a.d.1.1 1 23.22 odd 2
1104.2.a.e.1.1 1 276.275 odd 2
3174.2.a.b.1.1 1 3.2 odd 2
3312.2.a.n.1.1 1 92.91 even 2
3450.2.a.y.1.1 1 345.344 even 2
3450.2.d.j.2899.1 2 345.137 odd 4
3450.2.d.j.2899.2 2 345.68 odd 4
4416.2.a.m.1.1 1 552.275 odd 2
4416.2.a.z.1.1 1 552.413 even 2
6762.2.a.q.1.1 1 483.482 odd 2
9522.2.a.i.1.1 1 1.1 even 1 trivial