Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [192,2,Mod(1,192)] 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("192.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(192, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 192.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-1,0,2] 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: \(1.53312771881\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 192.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.00000 q^{5} +1.00000 q^{9} +4.00000 q^{11} +2.00000 q^{13} -2.00000 q^{15} +2.00000 q^{17} -4.00000 q^{19} +8.00000 q^{23} -1.00000 q^{25} -1.00000 q^{27} -6.00000 q^{29} -8.00000 q^{31} -4.00000 q^{33} -6.00000 q^{37} -2.00000 q^{39} -6.00000 q^{41} +4.00000 q^{43} +2.00000 q^{45} -7.00000 q^{49} -2.00000 q^{51} +2.00000 q^{53} +8.00000 q^{55} +4.00000 q^{57} +4.00000 q^{59} +2.00000 q^{61} +4.00000 q^{65} -4.00000 q^{67} -8.00000 q^{69} -8.00000 q^{71} +10.0000 q^{73} +1.00000 q^{75} +8.00000 q^{79} +1.00000 q^{81} -4.00000 q^{83} +4.00000 q^{85} +6.00000 q^{87} -6.00000 q^{89} +8.00000 q^{93} -8.00000 q^{95} +2.00000 q^{97} +4.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 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
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 − 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 − 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
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 192.2.a.b.1.1 1
3.2 odd 2 576.2.a.b.1.1 1
4.3 odd 2 192.2.a.d.1.1 1
5.2 odd 4 4800.2.f.bg.3649.2 2
5.3 odd 4 4800.2.f.bg.3649.1 2
5.4 even 2 4800.2.a.cc.1.1 1
7.6 odd 2 9408.2.a.cc.1.1 1
8.3 odd 2 24.2.a.a.1.1 ✓ 1
8.5 even 2 48.2.a.a.1.1 1
12.11 even 2 576.2.a.d.1.1 1
16.3 odd 4 768.2.d.e.385.2 2
16.5 even 4 768.2.d.d.385.2 2
16.11 odd 4 768.2.d.e.385.1 2
16.13 even 4 768.2.d.d.385.1 2
20.3 even 4 4800.2.f.d.3649.2 2
20.7 even 4 4800.2.f.d.3649.1 2
20.19 odd 2 4800.2.a.q.1.1 1
24.5 odd 2 144.2.a.b.1.1 1
24.11 even 2 72.2.a.a.1.1 1
28.27 even 2 9408.2.a.h.1.1 1
40.3 even 4 600.2.f.e.49.1 2
40.13 odd 4 1200.2.f.b.49.2 2
40.19 odd 2 600.2.a.h.1.1 1
40.27 even 4 600.2.f.e.49.2 2
40.29 even 2 1200.2.a.d.1.1 1
40.37 odd 4 1200.2.f.b.49.1 2
48.5 odd 4 2304.2.d.k.1153.1 2
48.11 even 4 2304.2.d.i.1153.1 2
48.29 odd 4 2304.2.d.k.1153.2 2
48.35 even 4 2304.2.d.i.1153.2 2
56.3 even 6 1176.2.q.a.961.1 2
56.5 odd 6 2352.2.q.r.1537.1 2
56.11 odd 6 1176.2.q.i.961.1 2
56.13 odd 2 2352.2.a.i.1.1 1
56.19 even 6 1176.2.q.a.361.1 2
56.27 even 2 1176.2.a.i.1.1 1
56.37 even 6 2352.2.q.l.1537.1 2
56.45 odd 6 2352.2.q.r.961.1 2
56.51 odd 6 1176.2.q.i.361.1 2
56.53 even 6 2352.2.q.l.961.1 2
72.5 odd 6 1296.2.i.e.865.1 2
72.11 even 6 648.2.i.b.433.1 2
72.13 even 6 1296.2.i.m.865.1 2
72.29 odd 6 1296.2.i.e.433.1 2
72.43 odd 6 648.2.i.g.433.1 2
72.59 even 6 648.2.i.b.217.1 2
72.61 even 6 1296.2.i.m.433.1 2
72.67 odd 6 648.2.i.g.217.1 2
88.21 odd 2 5808.2.a.s.1.1 1
88.43 even 2 2904.2.a.c.1.1 1
104.51 odd 2 4056.2.a.i.1.1 1
104.77 even 2 8112.2.a.be.1.1 1
104.83 even 4 4056.2.c.e.337.2 2
104.99 even 4 4056.2.c.e.337.1 2
120.29 odd 2 3600.2.a.v.1.1 1
120.53 even 4 3600.2.f.r.2449.1 2
120.59 even 2 1800.2.a.m.1.1 1
120.77 even 4 3600.2.f.r.2449.2 2
120.83 odd 4 1800.2.f.c.649.1 2
120.107 odd 4 1800.2.f.c.649.2 2
136.67 odd 2 6936.2.a.p.1.1 1
152.75 even 2 8664.2.a.j.1.1 1
168.11 even 6 3528.2.s.j.3313.1 2
168.59 odd 6 3528.2.s.y.3313.1 2
168.83 odd 2 3528.2.a.d.1.1 1
168.107 even 6 3528.2.s.j.361.1 2
168.125 even 2 7056.2.a.q.1.1 1
168.131 odd 6 3528.2.s.y.361.1 2
264.131 odd 2 8712.2.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.2.a.a.1.1 ✓ 1 8.3 odd 2
48.2.a.a.1.1 1 8.5 even 2
72.2.a.a.1.1 1 24.11 even 2
144.2.a.b.1.1 1 24.5 odd 2
192.2.a.b.1.1 1 1.1 even 1 trivial
192.2.a.d.1.1 1 4.3 odd 2
576.2.a.b.1.1 1 3.2 odd 2
576.2.a.d.1.1 1 12.11 even 2
600.2.a.h.1.1 1 40.19 odd 2
600.2.f.e.49.1 2 40.3 even 4
600.2.f.e.49.2 2 40.27 even 4
648.2.i.b.217.1 2 72.59 even 6
648.2.i.b.433.1 2 72.11 even 6
648.2.i.g.217.1 2 72.67 odd 6
648.2.i.g.433.1 2 72.43 odd 6
768.2.d.d.385.1 2 16.13 even 4
768.2.d.d.385.2 2 16.5 even 4
768.2.d.e.385.1 2 16.11 odd 4
768.2.d.e.385.2 2 16.3 odd 4
1176.2.a.i.1.1 1 56.27 even 2
1176.2.q.a.361.1 2 56.19 even 6
1176.2.q.a.961.1 2 56.3 even 6
1176.2.q.i.361.1 2 56.51 odd 6
1176.2.q.i.961.1 2 56.11 odd 6
1200.2.a.d.1.1 1 40.29 even 2
1200.2.f.b.49.1 2 40.37 odd 4
1200.2.f.b.49.2 2 40.13 odd 4
1296.2.i.e.433.1 2 72.29 odd 6
1296.2.i.e.865.1 2 72.5 odd 6
1296.2.i.m.433.1 2 72.61 even 6
1296.2.i.m.865.1 2 72.13 even 6
1800.2.a.m.1.1 1 120.59 even 2
1800.2.f.c.649.1 2 120.83 odd 4
1800.2.f.c.649.2 2 120.107 odd 4
2304.2.d.i.1153.1 2 48.11 even 4
2304.2.d.i.1153.2 2 48.35 even 4
2304.2.d.k.1153.1 2 48.5 odd 4
2304.2.d.k.1153.2 2 48.29 odd 4
2352.2.a.i.1.1 1 56.13 odd 2
2352.2.q.l.961.1 2 56.53 even 6
2352.2.q.l.1537.1 2 56.37 even 6
2352.2.q.r.961.1 2 56.45 odd 6
2352.2.q.r.1537.1 2 56.5 odd 6
2904.2.a.c.1.1 1 88.43 even 2
3528.2.a.d.1.1 1 168.83 odd 2
3528.2.s.j.361.1 2 168.107 even 6
3528.2.s.j.3313.1 2 168.11 even 6
3528.2.s.y.361.1 2 168.131 odd 6
3528.2.s.y.3313.1 2 168.59 odd 6
3600.2.a.v.1.1 1 120.29 odd 2
3600.2.f.r.2449.1 2 120.53 even 4
3600.2.f.r.2449.2 2 120.77 even 4
4056.2.a.i.1.1 1 104.51 odd 2
4056.2.c.e.337.1 2 104.99 even 4
4056.2.c.e.337.2 2 104.83 even 4
4800.2.a.q.1.1 1 20.19 odd 2
4800.2.a.cc.1.1 1 5.4 even 2
4800.2.f.d.3649.1 2 20.7 even 4
4800.2.f.d.3649.2 2 20.3 even 4
4800.2.f.bg.3649.1 2 5.3 odd 4
4800.2.f.bg.3649.2 2 5.2 odd 4
5808.2.a.s.1.1 1 88.21 odd 2
6936.2.a.p.1.1 1 136.67 odd 2
7056.2.a.q.1.1 1 168.125 even 2
8112.2.a.be.1.1 1 104.77 even 2
8664.2.a.j.1.1 1 152.75 even 2
8712.2.a.u.1.1 1 264.131 odd 2
9408.2.a.h.1.1 1 28.27 even 2
9408.2.a.cc.1.1 1 7.6 odd 2