Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1014,2,Mod(1,1014)] 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("1014.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1014, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1014.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1,1,1,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(8.09683076496\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1014.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} +6.00000 q^{11} +1.00000 q^{12} -2.00000 q^{14} +3.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{18} +2.00000 q^{19} +3.00000 q^{20} +2.00000 q^{21} -6.00000 q^{22} -6.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} +1.00000 q^{27} +2.00000 q^{28} +3.00000 q^{29} -3.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +6.00000 q^{33} +3.00000 q^{34} +6.00000 q^{35} +1.00000 q^{36} -7.00000 q^{37} -2.00000 q^{38} -3.00000 q^{40} -3.00000 q^{41} -2.00000 q^{42} -10.0000 q^{43} +6.00000 q^{44} +3.00000 q^{45} +6.00000 q^{46} +6.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} -4.00000 q^{50} -3.00000 q^{51} +3.00000 q^{53} -1.00000 q^{54} +18.0000 q^{55} -2.00000 q^{56} +2.00000 q^{57} -3.00000 q^{58} +3.00000 q^{60} -7.00000 q^{61} +4.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -6.00000 q^{66} -10.0000 q^{67} -3.00000 q^{68} -6.00000 q^{69} -6.00000 q^{70} +6.00000 q^{71} -1.00000 q^{72} -13.0000 q^{73} +7.00000 q^{74} +4.00000 q^{75} +2.00000 q^{76} +12.0000 q^{77} -4.00000 q^{79} +3.00000 q^{80} +1.00000 q^{81} +3.00000 q^{82} -6.00000 q^{83} +2.00000 q^{84} -9.00000 q^{85} +10.0000 q^{86} +3.00000 q^{87} -6.00000 q^{88} +18.0000 q^{89} -3.00000 q^{90} -6.00000 q^{92} -4.00000 q^{93} -6.00000 q^{94} +6.00000 q^{95} -1.00000 q^{96} +14.0000 q^{97} +3.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

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
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −3.00000 −0.948683
\(11\) 6.00000 1.80907 0.904534 − 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −2.00000 −0.534522
\(15\) 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 \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 3.00000 0.670820
\(21\) 2.00000 0.436436
\(22\) −6.00000 −1.27920
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) −1.00000 −0.204124
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) 3.00000 0.557086 0.278543 − 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) −3.00000 −0.547723
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) 6.00000 1.04447
\(34\) 3.00000 0.514496
\(35\) 6.00000 1.01419
\(36\) 1.00000 0.166667
\(37\) −7.00000 −1.15079 −0.575396 − 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) −3.00000 −0.468521 −0.234261 − 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) −2.00000 −0.308607
\(43\) −10.0000 −1.52499 −0.762493 − 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 6.00000 0.904534
\(45\) 3.00000 0.447214
\(46\) 6.00000 0.884652
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) −4.00000 −0.565685
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) 3.00000 0.412082 0.206041 − 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) −1.00000 −0.136083
\(55\) 18.0000 2.42712
\(56\) −2.00000 −0.267261
\(57\) 2.00000 0.264906
\(58\) −3.00000 −0.393919
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 3.00000 0.387298
\(61\) −7.00000 −0.896258 −0.448129 − 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 4.00000 0.508001
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −6.00000 −0.738549
\(67\) −10.0000 −1.22169 −0.610847 − 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) −3.00000 −0.363803
\(69\) −6.00000 −0.722315
\(70\) −6.00000 −0.717137
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) −1.00000 −0.117851
\(73\) −13.0000 −1.52153 −0.760767 − 0.649025i \(-0.775177\pi\)
−0.760767 + 0.649025i \(0.775177\pi\)
\(74\) 7.00000 0.813733
\(75\) 4.00000 0.461880
\(76\) 2.00000 0.229416
\(77\) 12.0000 1.36753
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 3.00000 0.335410
\(81\) 1.00000 0.111111
\(82\) 3.00000 0.331295
\(83\) −6.00000 −0.658586 −0.329293 − 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 2.00000 0.218218
\(85\) −9.00000 −0.976187
\(86\) 10.0000 1.07833
\(87\) 3.00000 0.321634
\(88\) −6.00000 −0.639602
\(89\) 18.0000 1.90800 0.953998 − 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) −3.00000 −0.316228
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) −4.00000 −0.414781
\(94\) −6.00000 −0.618853
\(95\) 6.00000 0.615587
\(96\) −1.00000 −0.102062
\(97\) 14.0000 1.42148 0.710742 − 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 3.00000 0.303046
\(99\) 6.00000 0.603023
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 1014.2.a.c.1.1 1
3.2 odd 2 3042.2.a.i.1.1 1
4.3 odd 2 8112.2.a.m.1.1 1
13.2 odd 12 1014.2.i.b.823.1 4
13.3 even 3 78.2.e.a.61.1 yes 2
13.4 even 6 1014.2.e.a.991.1 2
13.5 odd 4 1014.2.b.c.337.2 2
13.6 odd 12 1014.2.i.b.361.2 4
13.7 odd 12 1014.2.i.b.361.1 4
13.8 odd 4 1014.2.b.c.337.1 2
13.9 even 3 78.2.e.a.55.1 ✓ 2
13.10 even 6 1014.2.e.a.529.1 2
13.11 odd 12 1014.2.i.b.823.2 4
13.12 even 2 1014.2.a.f.1.1 1
39.5 even 4 3042.2.b.h.1351.1 2
39.8 even 4 3042.2.b.h.1351.2 2
39.29 odd 6 234.2.h.a.217.1 2
39.35 odd 6 234.2.h.a.55.1 2
39.38 odd 2 3042.2.a.h.1.1 1
52.3 odd 6 624.2.q.g.529.1 2
52.35 odd 6 624.2.q.g.289.1 2
52.51 odd 2 8112.2.a.c.1.1 1
65.3 odd 12 1950.2.z.g.1699.1 4
65.9 even 6 1950.2.i.m.601.1 2
65.22 odd 12 1950.2.z.g.1849.1 4
65.29 even 6 1950.2.i.m.451.1 2
65.42 odd 12 1950.2.z.g.1699.2 4
65.48 odd 12 1950.2.z.g.1849.2 4
156.35 even 6 1872.2.t.c.289.1 2
156.107 even 6 1872.2.t.c.1153.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.e.a.55.1 ✓ 2 13.9 even 3
78.2.e.a.61.1 yes 2 13.3 even 3
234.2.h.a.55.1 2 39.35 odd 6
234.2.h.a.217.1 2 39.29 odd 6
624.2.q.g.289.1 2 52.35 odd 6
624.2.q.g.529.1 2 52.3 odd 6
1014.2.a.c.1.1 1 1.1 even 1 trivial
1014.2.a.f.1.1 1 13.12 even 2
1014.2.b.c.337.1 2 13.8 odd 4
1014.2.b.c.337.2 2 13.5 odd 4
1014.2.e.a.529.1 2 13.10 even 6
1014.2.e.a.991.1 2 13.4 even 6
1014.2.i.b.361.1 4 13.7 odd 12
1014.2.i.b.361.2 4 13.6 odd 12
1014.2.i.b.823.1 4 13.2 odd 12
1014.2.i.b.823.2 4 13.11 odd 12
1872.2.t.c.289.1 2 156.35 even 6
1872.2.t.c.1153.1 2 156.107 even 6
1950.2.i.m.451.1 2 65.29 even 6
1950.2.i.m.601.1 2 65.9 even 6
1950.2.z.g.1699.1 4 65.3 odd 12
1950.2.z.g.1699.2 4 65.42 odd 12
1950.2.z.g.1849.1 4 65.22 odd 12
1950.2.z.g.1849.2 4 65.48 odd 12
3042.2.a.h.1.1 1 39.38 odd 2
3042.2.a.i.1.1 1 3.2 odd 2
3042.2.b.h.1351.1 2 39.5 even 4
3042.2.b.h.1351.2 2 39.8 even 4
8112.2.a.c.1.1 1 52.51 odd 2
8112.2.a.m.1.1 1 4.3 odd 2