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,-1,1,4,-1,0,1,1,4,1,-1,-4,0,-4,1,2] 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} -4.00000 q^{13} -4.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +4.00000 q^{20} +1.00000 q^{22} -6.00000 q^{23} -1.00000 q^{24} +11.0000 q^{25} -4.00000 q^{26} -1.00000 q^{27} +10.0000 q^{29} -4.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{39} +4.00000 q^{40} -2.00000 q^{41} +4.00000 q^{43} +1.00000 q^{44} +4.00000 q^{45} -6.00000 q^{46} +2.00000 q^{47} -1.00000 q^{48} +11.0000 q^{50} -2.00000 q^{51} -4.00000 q^{52} +4.00000 q^{53} -1.00000 q^{54} +4.00000 q^{55} +10.0000 q^{58} -4.00000 q^{60} +8.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -16.0000 q^{65} -1.00000 q^{66} -12.0000 q^{67} +2.00000 q^{68} +6.00000 q^{69} +2.00000 q^{71} +1.00000 q^{72} +6.00000 q^{73} -2.00000 q^{74} -11.0000 q^{75} +4.00000 q^{78} +10.0000 q^{79} +4.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -4.00000 q^{83} +8.00000 q^{85} +4.00000 q^{86} -10.0000 q^{87} +1.00000 q^{88} -10.0000 q^{89} +4.00000 q^{90} -6.00000 q^{92} -8.00000 q^{93} +2.00000 q^{94} -1.00000 q^{96} +2.00000 q^{97} +1.00000 q^{99} +O(q^{100})\)

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
\(5\) 4.00000 1.78885 0.894427 − 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 4.00000 1.26491
\(11\) 1.00000 0.301511
\(12\) −1.00000 −0.288675
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) −4.00000 −1.03280
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 4.00000 0.894427
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) −1.00000 −0.204124
\(25\) 11.0000 2.20000
\(26\) −4.00000 −0.784465
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 10.0000 1.85695 0.928477 − 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) −4.00000 −0.730297
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 4.00000 0.632456
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 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
\(45\) 4.00000 0.596285
\(46\) −6.00000 −0.884652
\(47\) 2.00000 0.291730 0.145865 − 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 11.0000 1.55563
\(51\) −2.00000 −0.280056
\(52\) −4.00000 −0.554700
\(53\) 4.00000 0.549442 0.274721 − 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) −1.00000 −0.136083
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 10.0000 1.31306
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −4.00000 −0.516398
\(61\) 8.00000 1.02430 0.512148 − 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −16.0000 −1.98456
\(66\) −1.00000 −0.123091
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 2.00000 0.242536
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 − 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 1.00000 0.117851
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) −2.00000 −0.232495
\(75\) −11.0000 −1.27017
\(76\) 0 0
\(77\) 0 0
\(78\) 4.00000 0.452911
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 4.00000 0.447214
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 8.00000 0.867722
\(86\) 4.00000 0.431331
\(87\) −10.0000 −1.07211
\(88\) 1.00000 0.106600
\(89\) −10.0000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 4.00000 0.421637
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) −8.00000 −0.829561
\(94\) 2.00000 0.206284
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 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.s.1.1 1
3.2 odd 2 9702.2.a.a.1.1 1
7.6 odd 2 66.2.a.c.1.1 ✓ 1
21.20 even 2 198.2.a.c.1.1 1
28.27 even 2 528.2.a.a.1.1 1
35.13 even 4 1650.2.c.m.199.1 2
35.27 even 4 1650.2.c.m.199.2 2
35.34 odd 2 1650.2.a.c.1.1 1
56.13 odd 2 2112.2.a.n.1.1 1
56.27 even 2 2112.2.a.bd.1.1 1
63.13 odd 6 1782.2.e.l.1189.1 2
63.20 even 6 1782.2.e.n.595.1 2
63.34 odd 6 1782.2.e.l.595.1 2
63.41 even 6 1782.2.e.n.1189.1 2
77.6 even 10 726.2.e.m.487.1 4
77.13 even 10 726.2.e.m.565.1 4
77.20 odd 10 726.2.e.e.565.1 4
77.27 odd 10 726.2.e.e.487.1 4
77.41 even 10 726.2.e.m.493.1 4
77.48 odd 10 726.2.e.e.511.1 4
77.62 even 10 726.2.e.m.511.1 4
77.69 odd 10 726.2.e.e.493.1 4
77.76 even 2 726.2.a.d.1.1 1
84.83 odd 2 1584.2.a.s.1.1 1
105.62 odd 4 4950.2.c.d.199.1 2
105.83 odd 4 4950.2.c.d.199.2 2
105.104 even 2 4950.2.a.bo.1.1 1
168.83 odd 2 6336.2.a.d.1.1 1
168.125 even 2 6336.2.a.c.1.1 1
231.230 odd 2 2178.2.a.m.1.1 1
308.307 odd 2 5808.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
66.2.a.c.1.1 ✓ 1 7.6 odd 2
198.2.a.c.1.1 1 21.20 even 2
528.2.a.a.1.1 1 28.27 even 2
726.2.a.d.1.1 1 77.76 even 2
726.2.e.e.487.1 4 77.27 odd 10
726.2.e.e.493.1 4 77.69 odd 10
726.2.e.e.511.1 4 77.48 odd 10
726.2.e.e.565.1 4 77.20 odd 10
726.2.e.m.487.1 4 77.6 even 10
726.2.e.m.493.1 4 77.41 even 10
726.2.e.m.511.1 4 77.62 even 10
726.2.e.m.565.1 4 77.13 even 10
1584.2.a.s.1.1 1 84.83 odd 2
1650.2.a.c.1.1 1 35.34 odd 2
1650.2.c.m.199.1 2 35.13 even 4
1650.2.c.m.199.2 2 35.27 even 4
1782.2.e.l.595.1 2 63.34 odd 6
1782.2.e.l.1189.1 2 63.13 odd 6
1782.2.e.n.595.1 2 63.20 even 6
1782.2.e.n.1189.1 2 63.41 even 6
2112.2.a.n.1.1 1 56.13 odd 2
2112.2.a.bd.1.1 1 56.27 even 2
2178.2.a.m.1.1 1 231.230 odd 2
3234.2.a.s.1.1 1 1.1 even 1 trivial
4950.2.a.bo.1.1 1 105.104 even 2
4950.2.c.d.199.1 2 105.62 odd 4
4950.2.c.d.199.2 2 105.83 odd 4
5808.2.a.b.1.1 1 308.307 odd 2
6336.2.a.c.1.1 1 168.125 even 2
6336.2.a.d.1.1 1 168.83 odd 2
9702.2.a.a.1.1 1 3.2 odd 2