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,-1,2,0,2,1,2] 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 462)
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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} +2.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -2.00000 q^{20} +1.00000 q^{22} -1.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{29} +2.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -2.00000 q^{39} -2.00000 q^{40} +10.0000 q^{41} +4.00000 q^{43} +1.00000 q^{44} -2.00000 q^{45} -4.00000 q^{47} -1.00000 q^{48} -1.00000 q^{50} -2.00000 q^{51} +2.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} -2.00000 q^{55} -2.00000 q^{58} +12.0000 q^{59} +2.00000 q^{60} +2.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} -1.00000 q^{66} +12.0000 q^{67} +2.00000 q^{68} +8.00000 q^{71} +1.00000 q^{72} -6.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} -2.00000 q^{78} -8.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} +8.00000 q^{83} -4.00000 q^{85} +4.00000 q^{86} +2.00000 q^{87} +1.00000 q^{88} +14.0000 q^{89} -2.00000 q^{90} +4.00000 q^{93} -4.00000 q^{94} -1.00000 q^{96} +14.0000 q^{97} +1.00000 q^{99} +O(q^{100})\)

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
\(5\) −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
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 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 \(0.5\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −1.00000 −0.204124
\(25\) −1.00000 −0.200000
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 2.00000 0.365148
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 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
\(39\) −2.00000 −0.320256
\(40\) −2.00000 −0.316228
\(41\) 10.0000 1.56174 0.780869 − 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 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
\(46\) 0 0
\(47\) −4.00000 −0.583460 −0.291730 − 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) −2.00000 −0.280056
\(52\) 2.00000 0.277350
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) −1.00000 −0.136083
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 2.00000 0.258199
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) −1.00000 −0.123091
\(67\) 12.0000 1.46603 0.733017 − 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 1.00000 0.117851
\(73\) −6.00000 −0.702247 −0.351123 − 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −2.00000 −0.232495
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) 8.00000 0.878114 0.439057 − 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) 4.00000 0.431331
\(87\) 2.00000 0.214423
\(88\) 1.00000 0.106600
\(89\) 14.0000 1.48400 0.741999 − 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) 0 0
\(93\) 4.00000 0.414781
\(94\) −4.00000 −0.412568
\(95\) 0 0
\(96\) −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.p.1.1 1
3.2 odd 2 9702.2.a.r.1.1 1
7.6 odd 2 462.2.a.g.1.1 ✓ 1
21.20 even 2 1386.2.a.a.1.1 1
28.27 even 2 3696.2.a.m.1.1 1
77.76 even 2 5082.2.a.n.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.a.g.1.1 ✓ 1 7.6 odd 2
1386.2.a.a.1.1 1 21.20 even 2
3234.2.a.p.1.1 1 1.1 even 1 trivial
3696.2.a.m.1.1 1 28.27 even 2
5082.2.a.n.1.1 1 77.76 even 2
9702.2.a.r.1.1 1 3.2 odd 2