Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3234,2,Mod(1,3234)] 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("3234.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3234, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,1,1,1,-2,1,0,1,1,-2,-1,1,6,0,-2,1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(25.8236200137\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 66)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3234.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} +6.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -2.00000 q^{20} -1.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} +6.00000 q^{26} +1.00000 q^{27} +6.00000 q^{29} -2.00000 q^{30} +1.00000 q^{32} -1.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} +6.00000 q^{39} -2.00000 q^{40} +6.00000 q^{41} +4.00000 q^{43} -1.00000 q^{44} -2.00000 q^{45} +4.00000 q^{46} +12.0000 q^{47} +1.00000 q^{48} -1.00000 q^{50} -2.00000 q^{51} +6.00000 q^{52} +2.00000 q^{53} +1.00000 q^{54} +2.00000 q^{55} -4.00000 q^{57} +6.00000 q^{58} -12.0000 q^{59} -2.00000 q^{60} +14.0000 q^{61} +1.00000 q^{64} -12.0000 q^{65} -1.00000 q^{66} +4.00000 q^{67} -2.00000 q^{68} +4.00000 q^{69} -12.0000 q^{71} +1.00000 q^{72} +6.00000 q^{73} +6.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} +6.00000 q^{78} -4.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -4.00000 q^{83} +4.00000 q^{85} +4.00000 q^{86} +6.00000 q^{87} -1.00000 q^{88} -10.0000 q^{89} -2.00000 q^{90} +4.00000 q^{92} +12.0000 q^{94} +8.00000 q^{95} +1.00000 q^{96} +14.0000 q^{97} -1.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) −1.00000 −0.301511
\(12\) 1.00000 0.288675
\(13\) 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
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) 6.00000 1.17670
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −2.00000 −0.365148
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −4.00000 −0.648886
\(39\) 6.00000 0.960769
\(40\) −2.00000 −0.316228
\(41\) 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 \(0.401344\pi\)
\(44\) −1.00000 −0.150756
\(45\) −2.00000 −0.298142
\(46\) 4.00000 0.589768
\(47\) 12.0000 1.75038 0.875190 − 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) −2.00000 −0.280056
\(52\) 6.00000 0.832050
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 1.00000 0.136083
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 6.00000 0.787839
\(59\) −12.0000 −1.56227 −0.781133 − 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) −2.00000 −0.258199
\(61\) 14.0000 1.79252 0.896258 − 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −12.0000 −1.48842
\(66\) −1.00000 −0.123091
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −2.00000 −0.242536
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 1.00000 0.117851
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 6.00000 0.697486
\(75\) −1.00000 −0.115470
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 6.00000 0.679366
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −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
\(86\) 4.00000 0.431331
\(87\) 6.00000 0.643268
\(88\) −1.00000 −0.106600
\(89\) −10.0000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) 8.00000 0.820783
\(96\) 1.00000 0.102062
\(97\) 14.0000 1.42148 0.710742 − 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
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 3234.2.a.t.1.1 1
3.2 odd 2 9702.2.a.x.1.1 1
7.6 odd 2 66.2.a.b.1.1 ✓ 1
21.20 even 2 198.2.a.a.1.1 1
28.27 even 2 528.2.a.j.1.1 1
35.13 even 4 1650.2.c.e.199.1 2
35.27 even 4 1650.2.c.e.199.2 2
35.34 odd 2 1650.2.a.k.1.1 1
56.13 odd 2 2112.2.a.r.1.1 1
56.27 even 2 2112.2.a.e.1.1 1
63.13 odd 6 1782.2.e.e.1189.1 2
63.20 even 6 1782.2.e.v.595.1 2
63.34 odd 6 1782.2.e.e.595.1 2
63.41 even 6 1782.2.e.v.1189.1 2
77.6 even 10 726.2.e.o.487.1 4
77.13 even 10 726.2.e.o.565.1 4
77.20 odd 10 726.2.e.g.565.1 4
77.27 odd 10 726.2.e.g.487.1 4
77.41 even 10 726.2.e.o.493.1 4
77.48 odd 10 726.2.e.g.511.1 4
77.62 even 10 726.2.e.o.511.1 4
77.69 odd 10 726.2.e.g.493.1 4
77.76 even 2 726.2.a.c.1.1 1
84.83 odd 2 1584.2.a.f.1.1 1
105.62 odd 4 4950.2.c.p.199.1 2
105.83 odd 4 4950.2.c.p.199.2 2
105.104 even 2 4950.2.a.bu.1.1 1
168.83 odd 2 6336.2.a.cj.1.1 1
168.125 even 2 6336.2.a.bw.1.1 1
231.230 odd 2 2178.2.a.g.1.1 1
308.307 odd 2 5808.2.a.bc.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
66.2.a.b.1.1 ✓ 1 7.6 odd 2
198.2.a.a.1.1 1 21.20 even 2
528.2.a.j.1.1 1 28.27 even 2
726.2.a.c.1.1 1 77.76 even 2
726.2.e.g.487.1 4 77.27 odd 10
726.2.e.g.493.1 4 77.69 odd 10
726.2.e.g.511.1 4 77.48 odd 10
726.2.e.g.565.1 4 77.20 odd 10
726.2.e.o.487.1 4 77.6 even 10
726.2.e.o.493.1 4 77.41 even 10
726.2.e.o.511.1 4 77.62 even 10
726.2.e.o.565.1 4 77.13 even 10
1584.2.a.f.1.1 1 84.83 odd 2
1650.2.a.k.1.1 1 35.34 odd 2
1650.2.c.e.199.1 2 35.13 even 4
1650.2.c.e.199.2 2 35.27 even 4
1782.2.e.e.595.1 2 63.34 odd 6
1782.2.e.e.1189.1 2 63.13 odd 6
1782.2.e.v.595.1 2 63.20 even 6
1782.2.e.v.1189.1 2 63.41 even 6
2112.2.a.e.1.1 1 56.27 even 2
2112.2.a.r.1.1 1 56.13 odd 2
2178.2.a.g.1.1 1 231.230 odd 2
3234.2.a.t.1.1 1 1.1 even 1 trivial
4950.2.a.bu.1.1 1 105.104 even 2
4950.2.c.p.199.1 2 105.62 odd 4
4950.2.c.p.199.2 2 105.83 odd 4
5808.2.a.bc.1.1 1 308.307 odd 2
6336.2.a.bw.1.1 1 168.125 even 2
6336.2.a.cj.1.1 1 168.83 odd 2
9702.2.a.x.1.1 1 3.2 odd 2