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,-4,0,0,1,1] 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: \(1\)
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{12} -4.00000 q^{13} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +3.00000 q^{19} +1.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} -5.00000 q^{25} +4.00000 q^{26} +1.00000 q^{27} -1.00000 q^{29} -6.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} -3.00000 q^{37} -3.00000 q^{38} -4.00000 q^{39} +6.00000 q^{41} +1.00000 q^{43} -1.00000 q^{44} +1.00000 q^{46} +1.00000 q^{47} +1.00000 q^{48} +5.00000 q^{50} +1.00000 q^{51} -4.00000 q^{52} -1.00000 q^{54} +3.00000 q^{57} +1.00000 q^{58} -7.00000 q^{59} -6.00000 q^{61} +6.00000 q^{62} +1.00000 q^{64} +1.00000 q^{66} -4.00000 q^{67} +1.00000 q^{68} -1.00000 q^{69} -15.0000 q^{71} -1.00000 q^{72} -12.0000 q^{73} +3.00000 q^{74} -5.00000 q^{75} +3.00000 q^{76} +4.00000 q^{78} +1.00000 q^{81} -6.00000 q^{82} +16.0000 q^{83} -1.00000 q^{86} -1.00000 q^{87} +1.00000 q^{88} +8.00000 q^{89} -1.00000 q^{92} -6.00000 q^{93} -1.00000 q^{94} -1.00000 q^{96} -7.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)

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
\(5\) 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 \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536 0.121268 − 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) −1.00000 −0.235702
\(19\) 3.00000 0.688247 0.344124 − 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −1.00000 −0.208514 −0.104257 − 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) −1.00000 −0.204124
\(25\) −5.00000 −1.00000
\(26\) 4.00000 0.784465
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.00000 −0.185695 −0.0928477 − 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 − 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −3.00000 −0.493197 −0.246598 − 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) −3.00000 −0.486664
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499 0.0762493 − 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 1.00000 0.145865 0.0729325 − 0.997337i \(-0.476764\pi\)
0.0729325 + 0.997337i \(0.476764\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 5.00000 0.707107
\(51\) 1.00000 0.140028
\(52\) −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
\(57\) 3.00000 0.397360
\(58\) 1.00000 0.131306
\(59\) −7.00000 −0.911322 −0.455661 − 0.890153i \(-0.650597\pi\)
−0.455661 + 0.890153i \(0.650597\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 6.00000 0.762001
\(63\) 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 \(0.578571\pi\)
\(68\) 1.00000 0.121268
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −15.0000 −1.78017 −0.890086 − 0.455792i \(-0.849356\pi\)
−0.890086 + 0.455792i \(0.849356\pi\)
\(72\) −1.00000 −0.117851
\(73\) −12.0000 −1.40449 −0.702247 − 0.711934i \(-0.747820\pi\)
−0.702247 + 0.711934i \(0.747820\pi\)
\(74\) 3.00000 0.348743
\(75\) −5.00000 −0.577350
\(76\) 3.00000 0.344124
\(77\) 0 0
\(78\) 4.00000 0.452911
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 16.0000 1.75623 0.878114 − 0.478451i \(-0.158802\pi\)
0.878114 + 0.478451i \(0.158802\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.00000 −0.107833
\(87\) −1.00000 −0.107211
\(88\) 1.00000 0.106600
\(89\) 8.00000 0.847998 0.423999 − 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) −6.00000 −0.622171
\(94\) −1.00000 −0.103142
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −7.00000 −0.710742 −0.355371 − 0.934725i \(-0.615646\pi\)
−0.355371 + 0.934725i \(0.615646\pi\)
\(98\) 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.j.1.1 1
3.2 odd 2 9702.2.a.bq.1.1 1
7.3 odd 6 462.2.i.d.331.1 yes 2
7.5 odd 6 462.2.i.d.67.1 ✓ 2
7.6 odd 2 3234.2.a.c.1.1 1
21.5 even 6 1386.2.k.f.991.1 2
21.17 even 6 1386.2.k.f.793.1 2
21.20 even 2 9702.2.a.bv.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.i.d.67.1 ✓ 2 7.5 odd 6
462.2.i.d.331.1 yes 2 7.3 odd 6
1386.2.k.f.793.1 2 21.17 even 6
1386.2.k.f.991.1 2 21.5 even 6
3234.2.a.c.1.1 1 7.6 odd 2
3234.2.a.j.1.1 1 1.1 even 1 trivial
9702.2.a.bq.1.1 1 3.2 odd 2
9702.2.a.bv.1.1 1 21.20 even 2