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Properties

Label 1200.2.a.o.1.1
Level $1200$
Weight $2$
Character 1200.1
Self dual yes
Analytic conductor $9.582$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1200,2,Mod(1,1200)] 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("1200.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1200.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,1,0,0,0,0,0,1,0,4,0,-6] 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: \(9.58204824255\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1200.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{9} +4.00000 q^{11} -6.00000 q^{13} +6.00000 q^{17} +4.00000 q^{19} +1.00000 q^{27} -2.00000 q^{29} +8.00000 q^{31} +4.00000 q^{33} +2.00000 q^{37} -6.00000 q^{39} -6.00000 q^{41} +12.0000 q^{43} +8.00000 q^{47} -7.00000 q^{49} +6.00000 q^{51} -6.00000 q^{53} +4.00000 q^{57} -12.0000 q^{59} +14.0000 q^{61} +4.00000 q^{67} -8.00000 q^{71} +6.00000 q^{73} +8.00000 q^{79} +1.00000 q^{81} -12.0000 q^{83} -2.00000 q^{87} +10.0000 q^{89} +8.00000 q^{93} -2.00000 q^{97} +4.00000 q^{99} +O(q^{100})\)

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
\(5\) 0 0
\(6\) 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
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 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
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 12.0000 1.82998 0.914991 − 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 − 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 − 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 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
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) 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
\(99\) 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 1200.2.a.o.1.1 1
3.2 odd 2 3600.2.a.t.1.1 1
4.3 odd 2 600.2.a.c.1.1 1
5.2 odd 4 1200.2.f.g.49.1 2
5.3 odd 4 1200.2.f.g.49.2 2
5.4 even 2 240.2.a.c.1.1 1
8.3 odd 2 4800.2.a.cd.1.1 1
8.5 even 2 4800.2.a.r.1.1 1
12.11 even 2 1800.2.a.n.1.1 1
15.2 even 4 3600.2.f.c.2449.2 2
15.8 even 4 3600.2.f.c.2449.1 2
15.14 odd 2 720.2.a.d.1.1 1
20.3 even 4 600.2.f.b.49.1 2
20.7 even 4 600.2.f.b.49.2 2
20.19 odd 2 120.2.a.b.1.1 ✓ 1
40.3 even 4 4800.2.f.bc.3649.2 2
40.13 odd 4 4800.2.f.i.3649.1 2
40.19 odd 2 960.2.a.c.1.1 1
40.27 even 4 4800.2.f.bc.3649.1 2
40.29 even 2 960.2.a.j.1.1 1
40.37 odd 4 4800.2.f.i.3649.2 2
60.23 odd 4 1800.2.f.j.649.1 2
60.47 odd 4 1800.2.f.j.649.2 2
60.59 even 2 360.2.a.b.1.1 1
80.19 odd 4 3840.2.k.o.1921.2 2
80.29 even 4 3840.2.k.j.1921.1 2
80.59 odd 4 3840.2.k.o.1921.1 2
80.69 even 4 3840.2.k.j.1921.2 2
120.29 odd 2 2880.2.a.bb.1.1 1
120.59 even 2 2880.2.a.x.1.1 1
140.139 even 2 5880.2.a.a.1.1 1
180.59 even 6 3240.2.q.q.2161.1 2
180.79 odd 6 3240.2.q.g.1081.1 2
180.119 even 6 3240.2.q.q.1081.1 2
180.139 odd 6 3240.2.q.g.2161.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.a.b.1.1 ✓ 1 20.19 odd 2
240.2.a.c.1.1 1 5.4 even 2
360.2.a.b.1.1 1 60.59 even 2
600.2.a.c.1.1 1 4.3 odd 2
600.2.f.b.49.1 2 20.3 even 4
600.2.f.b.49.2 2 20.7 even 4
720.2.a.d.1.1 1 15.14 odd 2
960.2.a.c.1.1 1 40.19 odd 2
960.2.a.j.1.1 1 40.29 even 2
1200.2.a.o.1.1 1 1.1 even 1 trivial
1200.2.f.g.49.1 2 5.2 odd 4
1200.2.f.g.49.2 2 5.3 odd 4
1800.2.a.n.1.1 1 12.11 even 2
1800.2.f.j.649.1 2 60.23 odd 4
1800.2.f.j.649.2 2 60.47 odd 4
2880.2.a.x.1.1 1 120.59 even 2
2880.2.a.bb.1.1 1 120.29 odd 2
3240.2.q.g.1081.1 2 180.79 odd 6
3240.2.q.g.2161.1 2 180.139 odd 6
3240.2.q.q.1081.1 2 180.119 even 6
3240.2.q.q.2161.1 2 180.59 even 6
3600.2.a.t.1.1 1 3.2 odd 2
3600.2.f.c.2449.1 2 15.8 even 4
3600.2.f.c.2449.2 2 15.2 even 4
3840.2.k.j.1921.1 2 80.29 even 4
3840.2.k.j.1921.2 2 80.69 even 4
3840.2.k.o.1921.1 2 80.59 odd 4
3840.2.k.o.1921.2 2 80.19 odd 4
4800.2.a.r.1.1 1 8.5 even 2
4800.2.a.cd.1.1 1 8.3 odd 2
4800.2.f.i.3649.1 2 40.13 odd 4
4800.2.f.i.3649.2 2 40.37 odd 4
4800.2.f.bc.3649.1 2 40.27 even 4
4800.2.f.bc.3649.2 2 40.3 even 4
5880.2.a.a.1.1 1 140.139 even 2