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,-1,1,1,3,-1,0,-1,1,-3,-1,1,6,0,3,1,-5] 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} +6.00000 q^{13} +3.00000 q^{15} +1.00000 q^{16} -5.00000 q^{17} -1.00000 q^{18} +6.00000 q^{19} +3.00000 q^{20} +1.00000 q^{22} +5.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} -6.00000 q^{26} +1.00000 q^{27} -6.00000 q^{29} -3.00000 q^{30} +4.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} +5.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -6.00000 q^{38} +6.00000 q^{39} -3.00000 q^{40} +5.00000 q^{41} -10.0000 q^{43} -1.00000 q^{44} +3.00000 q^{45} -5.00000 q^{46} +9.00000 q^{47} +1.00000 q^{48} -4.00000 q^{50} -5.00000 q^{51} +6.00000 q^{52} +2.00000 q^{53} -1.00000 q^{54} -3.00000 q^{55} +6.00000 q^{57} +6.00000 q^{58} -12.0000 q^{59} +3.00000 q^{60} -5.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +18.0000 q^{65} +1.00000 q^{66} +5.00000 q^{67} -5.00000 q^{68} +5.00000 q^{69} +4.00000 q^{71} -1.00000 q^{72} +12.0000 q^{73} +2.00000 q^{74} +4.00000 q^{75} +6.00000 q^{76} -6.00000 q^{78} -1.00000 q^{79} +3.00000 q^{80} +1.00000 q^{81} -5.00000 q^{82} +1.00000 q^{83} -15.0000 q^{85} +10.0000 q^{86} -6.00000 q^{87} +1.00000 q^{88} +6.00000 q^{89} -3.00000 q^{90} +5.00000 q^{92} +4.00000 q^{93} -9.00000 q^{94} +18.0000 q^{95} -1.00000 q^{96} +9.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)

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
\(5\) 3.00000 1.34164 0.670820 − 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −3.00000 −0.948683
\(11\) −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
\(15\) 3.00000 0.774597
\(16\) 1.00000 0.250000
\(17\) −5.00000 −1.21268 −0.606339 − 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) −1.00000 −0.235702
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 3.00000 0.670820
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 5.00000 1.04257 0.521286 − 0.853382i \(-0.325452\pi\)
0.521286 + 0.853382i \(0.325452\pi\)
\(24\) −1.00000 −0.204124
\(25\) 4.00000 0.800000
\(26\) −6.00000 −1.17670
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) −3.00000 −0.547723
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) 5.00000 0.857493
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −6.00000 −0.973329
\(39\) 6.00000 0.960769
\(40\) −3.00000 −0.474342
\(41\) 5.00000 0.780869 0.390434 − 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 − 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) −1.00000 −0.150756
\(45\) 3.00000 0.447214
\(46\) −5.00000 −0.737210
\(47\) 9.00000 1.31278 0.656392 − 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −4.00000 −0.565685
\(51\) −5.00000 −0.700140
\(52\) 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
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 6.00000 0.794719
\(58\) 6.00000 0.787839
\(59\) −12.0000 −1.56227 −0.781133 − 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 3.00000 0.387298
\(61\) −5.00000 −0.640184 −0.320092 − 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 18.0000 2.23263
\(66\) 1.00000 0.123091
\(67\) 5.00000 0.610847 0.305424 − 0.952217i \(-0.401202\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) −5.00000 −0.606339
\(69\) 5.00000 0.601929
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 − 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) −1.00000 −0.117851
\(73\) 12.0000 1.40449 0.702247 − 0.711934i \(-0.252180\pi\)
0.702247 + 0.711934i \(0.252180\pi\)
\(74\) 2.00000 0.232495
\(75\) 4.00000 0.461880
\(76\) 6.00000 0.688247
\(77\) 0 0
\(78\) −6.00000 −0.679366
\(79\) −1.00000 −0.112509 −0.0562544 − 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 3.00000 0.335410
\(81\) 1.00000 0.111111
\(82\) −5.00000 −0.552158
\(83\) 1.00000 0.109764 0.0548821 − 0.998493i \(-0.482522\pi\)
0.0548821 + 0.998493i \(0.482522\pi\)
\(84\) 0 0
\(85\) −15.0000 −1.62698
\(86\) 10.0000 1.07833
\(87\) −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 \(0.396989\pi\)
\(90\) −3.00000 −0.316228
\(91\) 0 0
\(92\) 5.00000 0.521286
\(93\) 4.00000 0.414781
\(94\) −9.00000 −0.928279
\(95\) 18.0000 1.84676
\(96\) −1.00000 −0.102062
\(97\) 9.00000 0.913812 0.456906 − 0.889515i \(-0.348958\pi\)
0.456906 + 0.889515i \(0.348958\pi\)
\(98\) 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.o.1.1 1
3.2 odd 2 9702.2.a.be.1.1 1
7.2 even 3 462.2.i.a.67.1 ✓ 2
7.4 even 3 462.2.i.a.331.1 yes 2
7.6 odd 2 3234.2.a.a.1.1 1
21.2 odd 6 1386.2.k.j.991.1 2
21.11 odd 6 1386.2.k.j.793.1 2
21.20 even 2 9702.2.a.ce.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.i.a.67.1 ✓ 2 7.2 even 3
462.2.i.a.331.1 yes 2 7.4 even 3
1386.2.k.j.793.1 2 21.11 odd 6
1386.2.k.j.991.1 2 21.2 odd 6
3234.2.a.a.1.1 1 7.6 odd 2
3234.2.a.o.1.1 1 1.1 even 1 trivial
9702.2.a.be.1.1 1 3.2 odd 2
9702.2.a.ce.1.1 1 21.20 even 2