Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9408,2,Mod(1,9408)] 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("9408.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9408, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9408.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-1,0,-2,0,0,0,1,0,4,0,6,0,2,0,-2,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(75.1232582216\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 9408.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.00000 q^{5} +1.00000 q^{9} +4.00000 q^{11} +6.00000 q^{13} +2.00000 q^{15} -2.00000 q^{17} -4.00000 q^{19} +8.00000 q^{23} -1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{29} -4.00000 q^{33} +10.0000 q^{37} -6.00000 q^{39} +6.00000 q^{41} +4.00000 q^{43} -2.00000 q^{45} +2.00000 q^{51} -6.00000 q^{53} -8.00000 q^{55} +4.00000 q^{57} +4.00000 q^{59} +6.00000 q^{61} -12.0000 q^{65} -4.00000 q^{67} -8.00000 q^{69} +8.00000 q^{71} -10.0000 q^{73} +1.00000 q^{75} +1.00000 q^{81} -4.00000 q^{83} +4.00000 q^{85} -2.00000 q^{87} +6.00000 q^{89} +8.00000 q^{95} +14.0000 q^{97} +4.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 − 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 − 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.0000 1.64399 0.821995 − 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 0 0
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −8.00000 −1.07872
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −12.0000 −1.48842
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 0 0
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 − 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
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 9408.2.a.n.1.1 1
4.3 odd 2 9408.2.a.bw.1.1 1
7.6 odd 2 1344.2.a.q.1.1 1
8.3 odd 2 2352.2.a.l.1.1 1
8.5 even 2 294.2.a.g.1.1 1
21.20 even 2 4032.2.a.e.1.1 1
24.5 odd 2 882.2.a.b.1.1 1
24.11 even 2 7056.2.a.k.1.1 1
28.27 even 2 1344.2.a.i.1.1 1
40.29 even 2 7350.2.a.f.1.1 1
56.3 even 6 2352.2.q.i.961.1 2
56.5 odd 6 294.2.e.c.67.1 2
56.11 odd 6 2352.2.q.n.961.1 2
56.13 odd 2 42.2.a.a.1.1 ✓ 1
56.19 even 6 2352.2.q.i.1537.1 2
56.27 even 2 336.2.a.d.1.1 1
56.37 even 6 294.2.e.a.67.1 2
56.45 odd 6 294.2.e.c.79.1 2
56.51 odd 6 2352.2.q.n.1537.1 2
56.53 even 6 294.2.e.a.79.1 2
84.83 odd 2 4032.2.a.m.1.1 1
112.13 odd 4 5376.2.c.bc.2689.2 2
112.27 even 4 5376.2.c.e.2689.2 2
112.69 odd 4 5376.2.c.bc.2689.1 2
112.83 even 4 5376.2.c.e.2689.1 2
168.5 even 6 882.2.g.h.361.1 2
168.53 odd 6 882.2.g.j.667.1 2
168.83 odd 2 1008.2.a.j.1.1 1
168.101 even 6 882.2.g.h.667.1 2
168.125 even 2 126.2.a.a.1.1 1
168.149 odd 6 882.2.g.j.361.1 2
280.13 even 4 1050.2.g.a.799.1 2
280.69 odd 2 1050.2.a.i.1.1 1
280.139 even 2 8400.2.a.k.1.1 1
280.237 even 4 1050.2.g.a.799.2 2
504.13 odd 6 1134.2.f.g.379.1 2
504.293 even 6 1134.2.f.j.379.1 2
504.349 odd 6 1134.2.f.g.757.1 2
504.461 even 6 1134.2.f.j.757.1 2
616.461 even 2 5082.2.a.d.1.1 1
728.181 odd 2 7098.2.a.f.1.1 1
840.293 odd 4 3150.2.g.r.2899.2 2
840.629 even 2 3150.2.a.bo.1.1 1
840.797 odd 4 3150.2.g.r.2899.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.a.a.1.1 ✓ 1 56.13 odd 2
126.2.a.a.1.1 1 168.125 even 2
294.2.a.g.1.1 1 8.5 even 2
294.2.e.a.67.1 2 56.37 even 6
294.2.e.a.79.1 2 56.53 even 6
294.2.e.c.67.1 2 56.5 odd 6
294.2.e.c.79.1 2 56.45 odd 6
336.2.a.d.1.1 1 56.27 even 2
882.2.a.b.1.1 1 24.5 odd 2
882.2.g.h.361.1 2 168.5 even 6
882.2.g.h.667.1 2 168.101 even 6
882.2.g.j.361.1 2 168.149 odd 6
882.2.g.j.667.1 2 168.53 odd 6
1008.2.a.j.1.1 1 168.83 odd 2
1050.2.a.i.1.1 1 280.69 odd 2
1050.2.g.a.799.1 2 280.13 even 4
1050.2.g.a.799.2 2 280.237 even 4
1134.2.f.g.379.1 2 504.13 odd 6
1134.2.f.g.757.1 2 504.349 odd 6
1134.2.f.j.379.1 2 504.293 even 6
1134.2.f.j.757.1 2 504.461 even 6
1344.2.a.i.1.1 1 28.27 even 2
1344.2.a.q.1.1 1 7.6 odd 2
2352.2.a.l.1.1 1 8.3 odd 2
2352.2.q.i.961.1 2 56.3 even 6
2352.2.q.i.1537.1 2 56.19 even 6
2352.2.q.n.961.1 2 56.11 odd 6
2352.2.q.n.1537.1 2 56.51 odd 6
3150.2.a.bo.1.1 1 840.629 even 2
3150.2.g.r.2899.1 2 840.797 odd 4
3150.2.g.r.2899.2 2 840.293 odd 4
4032.2.a.e.1.1 1 21.20 even 2
4032.2.a.m.1.1 1 84.83 odd 2
5082.2.a.d.1.1 1 616.461 even 2
5376.2.c.e.2689.1 2 112.83 even 4
5376.2.c.e.2689.2 2 112.27 even 4
5376.2.c.bc.2689.1 2 112.69 odd 4
5376.2.c.bc.2689.2 2 112.13 odd 4
7056.2.a.k.1.1 1 24.11 even 2
7098.2.a.f.1.1 1 728.181 odd 2
7350.2.a.f.1.1 1 40.29 even 2
8400.2.a.k.1.1 1 280.139 even 2
9408.2.a.n.1.1 1 1.1 even 1 trivial
9408.2.a.bw.1.1 1 4.3 odd 2