Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1800,2,Mod(1,1800)] 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("1800.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1800.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,0,0,0,0,0,0,-4,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(14.3730723638\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 1800.1

$q$-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)
 
\(f(q)\) \(=\) \(q-4.00000 q^{11} +2.00000 q^{13} +2.00000 q^{17} -4.00000 q^{19} -8.00000 q^{23} -6.00000 q^{29} +8.00000 q^{31} -6.00000 q^{37} +6.00000 q^{41} -4.00000 q^{43} -7.00000 q^{49} -2.00000 q^{53} -4.00000 q^{59} -2.00000 q^{61} +4.00000 q^{67} -8.00000 q^{71} -10.0000 q^{73} -8.00000 q^{79} -4.00000 q^{83} +6.00000 q^{89} -2.00000 q^{97} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 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 \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 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 \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 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
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 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
\(51\) 0 0
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 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
\(75\) 0 0
\(76\) 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
\(81\) 0 0
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 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
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 0 0
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 1800.2.a.m.1.1 1
3.2 odd 2 600.2.a.h.1.1 1
4.3 odd 2 3600.2.a.v.1.1 1
5.2 odd 4 1800.2.f.c.649.1 2
5.3 odd 4 1800.2.f.c.649.2 2
5.4 even 2 72.2.a.a.1.1 1
12.11 even 2 1200.2.a.d.1.1 1
15.2 even 4 600.2.f.e.49.1 2
15.8 even 4 600.2.f.e.49.2 2
15.14 odd 2 24.2.a.a.1.1 ✓ 1
20.3 even 4 3600.2.f.r.2449.2 2
20.7 even 4 3600.2.f.r.2449.1 2
20.19 odd 2 144.2.a.b.1.1 1
24.5 odd 2 4800.2.a.q.1.1 1
24.11 even 2 4800.2.a.cc.1.1 1
35.4 even 6 3528.2.s.j.3313.1 2
35.9 even 6 3528.2.s.j.361.1 2
35.19 odd 6 3528.2.s.y.361.1 2
35.24 odd 6 3528.2.s.y.3313.1 2
35.34 odd 2 3528.2.a.d.1.1 1
40.19 odd 2 576.2.a.b.1.1 1
40.29 even 2 576.2.a.d.1.1 1
45.4 even 6 648.2.i.b.217.1 2
45.14 odd 6 648.2.i.g.217.1 2
45.29 odd 6 648.2.i.g.433.1 2
45.34 even 6 648.2.i.b.433.1 2
55.54 odd 2 8712.2.a.u.1.1 1
60.23 odd 4 1200.2.f.b.49.1 2
60.47 odd 4 1200.2.f.b.49.2 2
60.59 even 2 48.2.a.a.1.1 1
80.19 odd 4 2304.2.d.k.1153.1 2
80.29 even 4 2304.2.d.i.1153.1 2
80.59 odd 4 2304.2.d.k.1153.2 2
80.69 even 4 2304.2.d.i.1153.2 2
105.44 odd 6 1176.2.q.i.361.1 2
105.59 even 6 1176.2.q.a.961.1 2
105.74 odd 6 1176.2.q.i.961.1 2
105.89 even 6 1176.2.q.a.361.1 2
105.104 even 2 1176.2.a.i.1.1 1
120.29 odd 2 192.2.a.d.1.1 1
120.53 even 4 4800.2.f.d.3649.1 2
120.59 even 2 192.2.a.b.1.1 1
120.77 even 4 4800.2.f.d.3649.2 2
120.83 odd 4 4800.2.f.bg.3649.2 2
120.107 odd 4 4800.2.f.bg.3649.1 2
140.139 even 2 7056.2.a.q.1.1 1
165.164 even 2 2904.2.a.c.1.1 1
180.59 even 6 1296.2.i.m.865.1 2
180.79 odd 6 1296.2.i.e.433.1 2
180.119 even 6 1296.2.i.m.433.1 2
180.139 odd 6 1296.2.i.e.865.1 2
195.44 even 4 4056.2.c.e.337.2 2
195.164 even 4 4056.2.c.e.337.1 2
195.194 odd 2 4056.2.a.i.1.1 1
240.29 odd 4 768.2.d.e.385.1 2
240.59 even 4 768.2.d.d.385.1 2
240.149 odd 4 768.2.d.e.385.2 2
240.179 even 4 768.2.d.d.385.2 2
255.254 odd 2 6936.2.a.p.1.1 1
285.284 even 2 8664.2.a.j.1.1 1
420.59 odd 6 2352.2.q.r.961.1 2
420.179 even 6 2352.2.q.l.961.1 2
420.299 odd 6 2352.2.q.r.1537.1 2
420.359 even 6 2352.2.q.l.1537.1 2
420.419 odd 2 2352.2.a.i.1.1 1
660.659 odd 2 5808.2.a.s.1.1 1
780.779 even 2 8112.2.a.be.1.1 1
840.419 odd 2 9408.2.a.cc.1.1 1
840.629 even 2 9408.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.2.a.a.1.1 ✓ 1 15.14 odd 2
48.2.a.a.1.1 1 60.59 even 2
72.2.a.a.1.1 1 5.4 even 2
144.2.a.b.1.1 1 20.19 odd 2
192.2.a.b.1.1 1 120.59 even 2
192.2.a.d.1.1 1 120.29 odd 2
576.2.a.b.1.1 1 40.19 odd 2
576.2.a.d.1.1 1 40.29 even 2
600.2.a.h.1.1 1 3.2 odd 2
600.2.f.e.49.1 2 15.2 even 4
600.2.f.e.49.2 2 15.8 even 4
648.2.i.b.217.1 2 45.4 even 6
648.2.i.b.433.1 2 45.34 even 6
648.2.i.g.217.1 2 45.14 odd 6
648.2.i.g.433.1 2 45.29 odd 6
768.2.d.d.385.1 2 240.59 even 4
768.2.d.d.385.2 2 240.179 even 4
768.2.d.e.385.1 2 240.29 odd 4
768.2.d.e.385.2 2 240.149 odd 4
1176.2.a.i.1.1 1 105.104 even 2
1176.2.q.a.361.1 2 105.89 even 6
1176.2.q.a.961.1 2 105.59 even 6
1176.2.q.i.361.1 2 105.44 odd 6
1176.2.q.i.961.1 2 105.74 odd 6
1200.2.a.d.1.1 1 12.11 even 2
1200.2.f.b.49.1 2 60.23 odd 4
1200.2.f.b.49.2 2 60.47 odd 4
1296.2.i.e.433.1 2 180.79 odd 6
1296.2.i.e.865.1 2 180.139 odd 6
1296.2.i.m.433.1 2 180.119 even 6
1296.2.i.m.865.1 2 180.59 even 6
1800.2.a.m.1.1 1 1.1 even 1 trivial
1800.2.f.c.649.1 2 5.2 odd 4
1800.2.f.c.649.2 2 5.3 odd 4
2304.2.d.i.1153.1 2 80.29 even 4
2304.2.d.i.1153.2 2 80.69 even 4
2304.2.d.k.1153.1 2 80.19 odd 4
2304.2.d.k.1153.2 2 80.59 odd 4
2352.2.a.i.1.1 1 420.419 odd 2
2352.2.q.l.961.1 2 420.179 even 6
2352.2.q.l.1537.1 2 420.359 even 6
2352.2.q.r.961.1 2 420.59 odd 6
2352.2.q.r.1537.1 2 420.299 odd 6
2904.2.a.c.1.1 1 165.164 even 2
3528.2.a.d.1.1 1 35.34 odd 2
3528.2.s.j.361.1 2 35.9 even 6
3528.2.s.j.3313.1 2 35.4 even 6
3528.2.s.y.361.1 2 35.19 odd 6
3528.2.s.y.3313.1 2 35.24 odd 6
3600.2.a.v.1.1 1 4.3 odd 2
3600.2.f.r.2449.1 2 20.7 even 4
3600.2.f.r.2449.2 2 20.3 even 4
4056.2.a.i.1.1 1 195.194 odd 2
4056.2.c.e.337.1 2 195.164 even 4
4056.2.c.e.337.2 2 195.44 even 4
4800.2.a.q.1.1 1 24.5 odd 2
4800.2.a.cc.1.1 1 24.11 even 2
4800.2.f.d.3649.1 2 120.53 even 4
4800.2.f.d.3649.2 2 120.77 even 4
4800.2.f.bg.3649.1 2 120.107 odd 4
4800.2.f.bg.3649.2 2 120.83 odd 4
5808.2.a.s.1.1 1 660.659 odd 2
6936.2.a.p.1.1 1 255.254 odd 2
7056.2.a.q.1.1 1 140.139 even 2
8112.2.a.be.1.1 1 780.779 even 2
8664.2.a.j.1.1 1 285.284 even 2
8712.2.a.u.1.1 1 55.54 odd 2
9408.2.a.h.1.1 1 840.629 even 2
9408.2.a.cc.1.1 1 840.419 odd 2