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,-2,0,-3,1,-3] 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -3.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{18} -2.00000 q^{19} +3.00000 q^{20} +1.00000 q^{22} +3.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -6.00000 q^{29} +3.00000 q^{30} +4.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} +3.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} +2.00000 q^{38} +2.00000 q^{39} -3.00000 q^{40} +3.00000 q^{41} +2.00000 q^{43} -1.00000 q^{44} +3.00000 q^{45} -3.00000 q^{46} +9.00000 q^{47} -1.00000 q^{48} -4.00000 q^{50} +3.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -3.00000 q^{55} +2.00000 q^{57} +6.00000 q^{58} +12.0000 q^{59} -3.00000 q^{60} -5.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -6.00000 q^{65} -1.00000 q^{66} +5.00000 q^{67} -3.00000 q^{68} -3.00000 q^{69} +12.0000 q^{71} -1.00000 q^{72} +16.0000 q^{73} -2.00000 q^{74} -4.00000 q^{75} -2.00000 q^{76} -2.00000 q^{78} +17.0000 q^{79} +3.00000 q^{80} +1.00000 q^{81} -3.00000 q^{82} -9.00000 q^{83} -9.00000 q^{85} -2.00000 q^{86} +6.00000 q^{87} +1.00000 q^{88} +6.00000 q^{89} -3.00000 q^{90} +3.00000 q^{92} -4.00000 q^{93} -9.00000 q^{94} -6.00000 q^{95} +1.00000 q^{96} -17.0000 q^{97} -1.00000 q^{99} +O(q^{100})\)

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
\(5\) 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
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −3.00000 −0.774597
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 − 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) −1.00000 −0.235702
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 3.00000 0.670820
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 3.00000 0.625543 0.312772 − 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 1.00000 0.204124
\(25\) 4.00000 0.800000
\(26\) 2.00000 0.392232
\(27\) −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
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 2.00000 0.324443
\(39\) 2.00000 0.320256
\(40\) −3.00000 −0.474342
\(41\) 3.00000 0.468521 0.234261 − 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 − 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) −1.00000 −0.150756
\(45\) 3.00000 0.447214
\(46\) −3.00000 −0.442326
\(47\) 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
\(51\) 3.00000 0.420084
\(52\) −2.00000 −0.277350
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 1.00000 0.136083
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 6.00000 0.787839
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) −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
\(65\) −6.00000 −0.744208
\(66\) −1.00000 −0.123091
\(67\) 5.00000 0.610847 0.305424 − 0.952217i \(-0.401202\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) −3.00000 −0.363803
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) −1.00000 −0.117851
\(73\) 16.0000 1.87266 0.936329 − 0.351123i \(-0.114200\pi\)
0.936329 + 0.351123i \(0.114200\pi\)
\(74\) −2.00000 −0.232495
\(75\) −4.00000 −0.461880
\(76\) −2.00000 −0.229416
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) 17.0000 1.91265 0.956325 − 0.292306i \(-0.0944227\pi\)
0.956325 + 0.292306i \(0.0944227\pi\)
\(80\) 3.00000 0.335410
\(81\) 1.00000 0.111111
\(82\) −3.00000 −0.331295
\(83\) −9.00000 −0.987878 −0.493939 − 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) 0 0
\(85\) −9.00000 −0.976187
\(86\) −2.00000 −0.215666
\(87\) 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
\(92\) 3.00000 0.312772
\(93\) −4.00000 −0.414781
\(94\) −9.00000 −0.928279
\(95\) −6.00000 −0.615587
\(96\) 1.00000 0.102062
\(97\) −17.0000 −1.72609 −0.863044 − 0.505128i \(-0.831445\pi\)
−0.863044 + 0.505128i \(0.831445\pi\)
\(98\) 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.g.1.1 1
3.2 odd 2 9702.2.a.bd.1.1 1
7.3 odd 6 462.2.i.c.331.1 yes 2
7.5 odd 6 462.2.i.c.67.1 ✓ 2
7.6 odd 2 3234.2.a.h.1.1 1
21.5 even 6 1386.2.k.c.991.1 2
21.17 even 6 1386.2.k.c.793.1 2
21.20 even 2 9702.2.a.cf.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.i.c.67.1 ✓ 2 7.5 odd 6
462.2.i.c.331.1 yes 2 7.3 odd 6
1386.2.k.c.793.1 2 21.17 even 6
1386.2.k.c.991.1 2 21.5 even 6
3234.2.a.g.1.1 1 1.1 even 1 trivial
3234.2.a.h.1.1 1 7.6 odd 2
9702.2.a.bd.1.1 1 3.2 odd 2
9702.2.a.cf.1.1 1 21.20 even 2