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,-1,1,0,1,0,-1,1,0,-1,-1,4,0,0,1,6] 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{12} +4.00000 q^{13} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{22} +6.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} -4.00000 q^{26} -1.00000 q^{27} +6.00000 q^{29} -8.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} -10.0000 q^{37} -4.00000 q^{38} -4.00000 q^{39} -6.00000 q^{41} +8.00000 q^{43} -1.00000 q^{44} -6.00000 q^{46} +6.00000 q^{47} -1.00000 q^{48} +5.00000 q^{50} -6.00000 q^{51} +4.00000 q^{52} +1.00000 q^{54} -4.00000 q^{57} -6.00000 q^{58} -8.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -1.00000 q^{66} -4.00000 q^{67} +6.00000 q^{68} -6.00000 q^{69} +6.00000 q^{71} -1.00000 q^{72} -2.00000 q^{73} +10.0000 q^{74} +5.00000 q^{75} +4.00000 q^{76} +4.00000 q^{78} +14.0000 q^{79} +1.00000 q^{81} +6.00000 q^{82} +12.0000 q^{83} -8.00000 q^{86} -6.00000 q^{87} +1.00000 q^{88} +6.00000 q^{89} +6.00000 q^{92} +8.00000 q^{93} -6.00000 q^{94} +1.00000 q^{96} -14.0000 q^{97} -1.00000 q^{99} +O(q^{100})\)

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
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −1.00000 −0.288675
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 1.00000 0.204124
\(25\) −5.00000 −1.00000
\(26\) −4.00000 −0.784465
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.00000 0.174078
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −4.00000 −0.648886
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 5.00000 0.707107
\(51\) −6.00000 −0.840168
\(52\) 4.00000 0.554700
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) −6.00000 −0.787839
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.00000 −0.123091
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 6.00000 0.727607
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) −1.00000 −0.117851
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 10.0000 1.16248
\(75\) 5.00000 0.577350
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 4.00000 0.452911
\(79\) 14.0000 1.57512 0.787562 − 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) −6.00000 −0.643268
\(88\) 1.00000 0.106600
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) 8.00000 0.829561
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −14.0000 −1.42148 −0.710742 − 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 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.d.1.1 1
3.2 odd 2 9702.2.a.bu.1.1 1
7.6 odd 2 66.2.a.a.1.1 ✓ 1
21.20 even 2 198.2.a.e.1.1 1
28.27 even 2 528.2.a.d.1.1 1
35.13 even 4 1650.2.c.d.199.2 2
35.27 even 4 1650.2.c.d.199.1 2
35.34 odd 2 1650.2.a.m.1.1 1
56.13 odd 2 2112.2.a.i.1.1 1
56.27 even 2 2112.2.a.v.1.1 1
63.13 odd 6 1782.2.e.s.1189.1 2
63.20 even 6 1782.2.e.f.595.1 2
63.34 odd 6 1782.2.e.s.595.1 2
63.41 even 6 1782.2.e.f.1189.1 2
77.6 even 10 726.2.e.b.487.1 4
77.13 even 10 726.2.e.b.565.1 4
77.20 odd 10 726.2.e.k.565.1 4
77.27 odd 10 726.2.e.k.487.1 4
77.41 even 10 726.2.e.b.493.1 4
77.48 odd 10 726.2.e.k.511.1 4
77.62 even 10 726.2.e.b.511.1 4
77.69 odd 10 726.2.e.k.493.1 4
77.76 even 2 726.2.a.i.1.1 1
84.83 odd 2 1584.2.a.h.1.1 1
105.62 odd 4 4950.2.c.r.199.2 2
105.83 odd 4 4950.2.c.r.199.1 2
105.104 even 2 4950.2.a.g.1.1 1
168.83 odd 2 6336.2.a.bf.1.1 1
168.125 even 2 6336.2.a.bj.1.1 1
231.230 odd 2 2178.2.a.b.1.1 1
308.307 odd 2 5808.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
66.2.a.a.1.1 ✓ 1 7.6 odd 2
198.2.a.e.1.1 1 21.20 even 2
528.2.a.d.1.1 1 28.27 even 2
726.2.a.i.1.1 1 77.76 even 2
726.2.e.b.487.1 4 77.6 even 10
726.2.e.b.493.1 4 77.41 even 10
726.2.e.b.511.1 4 77.62 even 10
726.2.e.b.565.1 4 77.13 even 10
726.2.e.k.487.1 4 77.27 odd 10
726.2.e.k.493.1 4 77.69 odd 10
726.2.e.k.511.1 4 77.48 odd 10
726.2.e.k.565.1 4 77.20 odd 10
1584.2.a.h.1.1 1 84.83 odd 2
1650.2.a.m.1.1 1 35.34 odd 2
1650.2.c.d.199.1 2 35.27 even 4
1650.2.c.d.199.2 2 35.13 even 4
1782.2.e.f.595.1 2 63.20 even 6
1782.2.e.f.1189.1 2 63.41 even 6
1782.2.e.s.595.1 2 63.34 odd 6
1782.2.e.s.1189.1 2 63.13 odd 6
2112.2.a.i.1.1 1 56.13 odd 2
2112.2.a.v.1.1 1 56.27 even 2
2178.2.a.b.1.1 1 231.230 odd 2
3234.2.a.d.1.1 1 1.1 even 1 trivial
4950.2.a.g.1.1 1 105.104 even 2
4950.2.c.r.199.1 2 105.83 odd 4
4950.2.c.r.199.2 2 105.62 odd 4
5808.2.a.l.1.1 1 308.307 odd 2
6336.2.a.bf.1.1 1 168.83 odd 2
6336.2.a.bj.1.1 1 168.125 even 2
9702.2.a.bu.1.1 1 3.2 odd 2