Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [768,2,Mod(385,768)] 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("768.385"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(768, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,-2,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.13251087523\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 385.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 768.385
Dual form 768.2.d.e.385.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} -2.00000i q^{5} -1.00000 q^{9} +4.00000i q^{11} +2.00000i q^{13} +2.00000 q^{15} +2.00000 q^{17} +4.00000i q^{19} +8.00000 q^{23} +1.00000 q^{25} -1.00000i q^{27} -6.00000i q^{29} +8.00000 q^{31} -4.00000 q^{33} +6.00000i q^{37} -2.00000 q^{39} +6.00000 q^{41} +4.00000i q^{43} +2.00000i q^{45} -7.00000 q^{49} +2.00000i q^{51} -2.00000i q^{53} +8.00000 q^{55} -4.00000 q^{57} +4.00000i q^{59} +2.00000i q^{61} +4.00000 q^{65} +4.00000i q^{67} +8.00000i q^{69} -8.00000 q^{71} -10.0000 q^{73} +1.00000i q^{75} -8.00000 q^{79} +1.00000 q^{81} +4.00000i q^{83} -4.00000i q^{85} +6.00000 q^{87} +6.00000 q^{89} +8.00000i q^{93} +8.00000 q^{95} +2.00000 q^{97} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9} + 4 q^{15} + 4 q^{17} + 16 q^{23} + 2 q^{25} + 16 q^{31} - 8 q^{33} - 4 q^{39} + 12 q^{41} - 14 q^{49} + 16 q^{55} - 8 q^{57} + 8 q^{65} - 16 q^{71} - 20 q^{73} - 16 q^{79} + 2 q^{81} + 12 q^{87}+ \cdots + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) − 2.00000i − 0.894427i −0.894427 − 0.447214i \(-0.852416\pi\)
0.894427 − 0.447214i \(-0.147584\pi\)
\(6\) 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
\(11\) 4.00000i 1.20605i 0.797724 + 0.603023i \(0.206037\pi\)
−0.797724 + 0.603023i \(0.793963\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 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
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 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
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) − 6.00000i − 1.11417i −0.830455 − 0.557086i \(-0.811919\pi\)
0.830455 − 0.557086i \(-0.188081\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 2.00000i 0.298142i
\(46\) 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
\(51\) 2.00000i 0.280056i
\(52\) 0 0
\(53\) − 2.00000i − 0.274721i −0.990521 − 0.137361i \(-0.956138\pi\)
0.990521 − 0.137361i \(-0.0438619\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) 4.00000i 0.520756i 0.965507 + 0.260378i \(0.0838471\pi\)
−0.965507 + 0.260378i \(0.916153\pi\)
\(60\) 0 0
\(61\) 2.00000i 0.256074i 0.991769 + 0.128037i \(0.0408676\pi\)
−0.991769 + 0.128037i \(0.959132\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) 8.00000i 0.963087i
\(70\) 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 \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 1.00000i 0.115470i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) − 4.00000i − 0.433861i
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 8.00000i 0.829561i
\(94\) 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
\(99\) − 4.00000i − 0.402015i
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 768.2.d.e.385.2 2
3.2 odd 2 2304.2.d.i.1153.2 2
4.3 odd 2 768.2.d.d.385.1 2
8.3 odd 2 768.2.d.d.385.2 2
8.5 even 2 inner 768.2.d.e.385.1 2
12.11 even 2 2304.2.d.k.1153.2 2
16.3 odd 4 48.2.a.a.1.1 1
16.5 even 4 192.2.a.d.1.1 1
16.11 odd 4 192.2.a.b.1.1 1
16.13 even 4 24.2.a.a.1.1 ✓ 1
24.5 odd 2 2304.2.d.i.1153.1 2
24.11 even 2 2304.2.d.k.1153.1 2
48.5 odd 4 576.2.a.d.1.1 1
48.11 even 4 576.2.a.b.1.1 1
48.29 odd 4 72.2.a.a.1.1 1
48.35 even 4 144.2.a.b.1.1 1
80.3 even 4 1200.2.f.b.49.2 2
80.13 odd 4 600.2.f.e.49.1 2
80.19 odd 4 1200.2.a.d.1.1 1
80.27 even 4 4800.2.f.bg.3649.2 2
80.29 even 4 600.2.a.h.1.1 1
80.37 odd 4 4800.2.f.d.3649.1 2
80.43 even 4 4800.2.f.bg.3649.1 2
80.53 odd 4 4800.2.f.d.3649.2 2
80.59 odd 4 4800.2.a.cc.1.1 1
80.67 even 4 1200.2.f.b.49.1 2
80.69 even 4 4800.2.a.q.1.1 1
80.77 odd 4 600.2.f.e.49.2 2
112.3 even 12 2352.2.q.r.961.1 2
112.13 odd 4 1176.2.a.i.1.1 1
112.19 even 12 2352.2.q.r.1537.1 2
112.27 even 4 9408.2.a.cc.1.1 1
112.45 odd 12 1176.2.q.a.961.1 2
112.51 odd 12 2352.2.q.l.1537.1 2
112.61 odd 12 1176.2.q.a.361.1 2
112.67 odd 12 2352.2.q.l.961.1 2
112.69 odd 4 9408.2.a.h.1.1 1
112.83 even 4 2352.2.a.i.1.1 1
112.93 even 12 1176.2.q.i.361.1 2
112.109 even 12 1176.2.q.i.961.1 2
144.13 even 12 648.2.i.g.217.1 2
144.29 odd 12 648.2.i.b.433.1 2
144.61 even 12 648.2.i.g.433.1 2
144.67 odd 12 1296.2.i.m.865.1 2
144.77 odd 12 648.2.i.b.217.1 2
144.83 even 12 1296.2.i.e.433.1 2
144.115 odd 12 1296.2.i.m.433.1 2
144.131 even 12 1296.2.i.e.865.1 2
176.109 odd 4 2904.2.a.c.1.1 1
176.131 even 4 5808.2.a.s.1.1 1
208.51 odd 4 8112.2.a.be.1.1 1
208.77 even 4 4056.2.a.i.1.1 1
208.109 odd 4 4056.2.c.e.337.2 2
208.125 odd 4 4056.2.c.e.337.1 2
240.29 odd 4 1800.2.a.m.1.1 1
240.77 even 4 1800.2.f.c.649.2 2
240.83 odd 4 3600.2.f.r.2449.1 2
240.173 even 4 1800.2.f.c.649.1 2
240.179 even 4 3600.2.a.v.1.1 1
240.227 odd 4 3600.2.f.r.2449.2 2
272.237 even 4 6936.2.a.p.1.1 1
304.189 odd 4 8664.2.a.j.1.1 1
336.83 odd 4 7056.2.a.q.1.1 1
336.125 even 4 3528.2.a.d.1.1 1
336.173 even 12 3528.2.s.y.361.1 2
336.221 odd 12 3528.2.s.j.3313.1 2
336.269 even 12 3528.2.s.y.3313.1 2
336.317 odd 12 3528.2.s.j.361.1 2
528.461 even 4 8712.2.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.2.a.a.1.1 ✓ 1 16.13 even 4
48.2.a.a.1.1 1 16.3 odd 4
72.2.a.a.1.1 1 48.29 odd 4
144.2.a.b.1.1 1 48.35 even 4
192.2.a.b.1.1 1 16.11 odd 4
192.2.a.d.1.1 1 16.5 even 4
576.2.a.b.1.1 1 48.11 even 4
576.2.a.d.1.1 1 48.5 odd 4
600.2.a.h.1.1 1 80.29 even 4
600.2.f.e.49.1 2 80.13 odd 4
600.2.f.e.49.2 2 80.77 odd 4
648.2.i.b.217.1 2 144.77 odd 12
648.2.i.b.433.1 2 144.29 odd 12
648.2.i.g.217.1 2 144.13 even 12
648.2.i.g.433.1 2 144.61 even 12
768.2.d.d.385.1 2 4.3 odd 2
768.2.d.d.385.2 2 8.3 odd 2
768.2.d.e.385.1 2 8.5 even 2 inner
768.2.d.e.385.2 2 1.1 even 1 trivial
1176.2.a.i.1.1 1 112.13 odd 4
1176.2.q.a.361.1 2 112.61 odd 12
1176.2.q.a.961.1 2 112.45 odd 12
1176.2.q.i.361.1 2 112.93 even 12
1176.2.q.i.961.1 2 112.109 even 12
1200.2.a.d.1.1 1 80.19 odd 4
1200.2.f.b.49.1 2 80.67 even 4
1200.2.f.b.49.2 2 80.3 even 4
1296.2.i.e.433.1 2 144.83 even 12
1296.2.i.e.865.1 2 144.131 even 12
1296.2.i.m.433.1 2 144.115 odd 12
1296.2.i.m.865.1 2 144.67 odd 12
1800.2.a.m.1.1 1 240.29 odd 4
1800.2.f.c.649.1 2 240.173 even 4
1800.2.f.c.649.2 2 240.77 even 4
2304.2.d.i.1153.1 2 24.5 odd 2
2304.2.d.i.1153.2 2 3.2 odd 2
2304.2.d.k.1153.1 2 24.11 even 2
2304.2.d.k.1153.2 2 12.11 even 2
2352.2.a.i.1.1 1 112.83 even 4
2352.2.q.l.961.1 2 112.67 odd 12
2352.2.q.l.1537.1 2 112.51 odd 12
2352.2.q.r.961.1 2 112.3 even 12
2352.2.q.r.1537.1 2 112.19 even 12
2904.2.a.c.1.1 1 176.109 odd 4
3528.2.a.d.1.1 1 336.125 even 4
3528.2.s.j.361.1 2 336.317 odd 12
3528.2.s.j.3313.1 2 336.221 odd 12
3528.2.s.y.361.1 2 336.173 even 12
3528.2.s.y.3313.1 2 336.269 even 12
3600.2.a.v.1.1 1 240.179 even 4
3600.2.f.r.2449.1 2 240.83 odd 4
3600.2.f.r.2449.2 2 240.227 odd 4
4056.2.a.i.1.1 1 208.77 even 4
4056.2.c.e.337.1 2 208.125 odd 4
4056.2.c.e.337.2 2 208.109 odd 4
4800.2.a.q.1.1 1 80.69 even 4
4800.2.a.cc.1.1 1 80.59 odd 4
4800.2.f.d.3649.1 2 80.37 odd 4
4800.2.f.d.3649.2 2 80.53 odd 4
4800.2.f.bg.3649.1 2 80.43 even 4
4800.2.f.bg.3649.2 2 80.27 even 4
5808.2.a.s.1.1 1 176.131 even 4
6936.2.a.p.1.1 1 272.237 even 4
7056.2.a.q.1.1 1 336.83 odd 4
8112.2.a.be.1.1 1 208.51 odd 4
8664.2.a.j.1.1 1 304.189 odd 4
8712.2.a.u.1.1 1 528.461 even 4
9408.2.a.h.1.1 1 112.69 odd 4
9408.2.a.cc.1.1 1 112.27 even 4