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Properties

Label 1584.2.a.o.1.1
Level $1584$
Weight $2$
Character 1584.1
Self dual yes
Analytic conductor $12.648$
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] = [1584,2,Mod(1,1584)] 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("1584.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1584, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1584.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,2,0,-4,0,0,0,1,0,-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: \(12.6483036802\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1584.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{5} -4.00000 q^{7} +1.00000 q^{11} -2.00000 q^{13} +2.00000 q^{17} +8.00000 q^{23} -1.00000 q^{25} +6.00000 q^{29} +8.00000 q^{31} -8.00000 q^{35} +6.00000 q^{37} +2.00000 q^{41} +8.00000 q^{47} +9.00000 q^{49} -6.00000 q^{53} +2.00000 q^{55} -4.00000 q^{59} +6.00000 q^{61} -4.00000 q^{65} +4.00000 q^{67} -14.0000 q^{73} -4.00000 q^{77} +4.00000 q^{79} +12.0000 q^{83} +4.00000 q^{85} +6.00000 q^{89} +8.00000 q^{91} +2.00000 q^{97} +O(q^{100})\)

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
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 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
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 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
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 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
\(35\) −8.00000 −1.35225
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 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
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 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
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 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 \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −14.0000 −1.63858 −0.819288 − 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 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
\(91\) 8.00000 0.838628
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 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 1584.2.a.o.1.1 1
3.2 odd 2 528.2.a.g.1.1 1
4.3 odd 2 99.2.a.b.1.1 1
8.3 odd 2 6336.2.a.x.1.1 1
8.5 even 2 6336.2.a.n.1.1 1
12.11 even 2 33.2.a.a.1.1 ✓ 1
20.3 even 4 2475.2.c.d.199.2 2
20.7 even 4 2475.2.c.d.199.1 2
20.19 odd 2 2475.2.a.g.1.1 1
24.5 odd 2 2112.2.a.j.1.1 1
24.11 even 2 2112.2.a.bb.1.1 1
28.27 even 2 4851.2.a.b.1.1 1
33.32 even 2 5808.2.a.t.1.1 1
36.7 odd 6 891.2.e.g.595.1 2
36.11 even 6 891.2.e.e.595.1 2
36.23 even 6 891.2.e.e.298.1 2
36.31 odd 6 891.2.e.g.298.1 2
44.43 even 2 1089.2.a.j.1.1 1
60.23 odd 4 825.2.c.a.199.1 2
60.47 odd 4 825.2.c.a.199.2 2
60.59 even 2 825.2.a.a.1.1 1
84.83 odd 2 1617.2.a.j.1.1 1
132.35 odd 10 363.2.e.g.202.1 4
132.47 even 10 363.2.e.e.130.1 4
132.59 even 10 363.2.e.e.148.1 4
132.71 even 10 363.2.e.e.124.1 4
132.83 odd 10 363.2.e.g.124.1 4
132.95 odd 10 363.2.e.g.148.1 4
132.107 odd 10 363.2.e.g.130.1 4
132.119 even 10 363.2.e.e.202.1 4
132.131 odd 2 363.2.a.b.1.1 1
156.155 even 2 5577.2.a.a.1.1 1
204.203 even 2 9537.2.a.m.1.1 1
660.659 odd 2 9075.2.a.q.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.a.a.1.1 ✓ 1 12.11 even 2
99.2.a.b.1.1 1 4.3 odd 2
363.2.a.b.1.1 1 132.131 odd 2
363.2.e.e.124.1 4 132.71 even 10
363.2.e.e.130.1 4 132.47 even 10
363.2.e.e.148.1 4 132.59 even 10
363.2.e.e.202.1 4 132.119 even 10
363.2.e.g.124.1 4 132.83 odd 10
363.2.e.g.130.1 4 132.107 odd 10
363.2.e.g.148.1 4 132.95 odd 10
363.2.e.g.202.1 4 132.35 odd 10
528.2.a.g.1.1 1 3.2 odd 2
825.2.a.a.1.1 1 60.59 even 2
825.2.c.a.199.1 2 60.23 odd 4
825.2.c.a.199.2 2 60.47 odd 4
891.2.e.e.298.1 2 36.23 even 6
891.2.e.e.595.1 2 36.11 even 6
891.2.e.g.298.1 2 36.31 odd 6
891.2.e.g.595.1 2 36.7 odd 6
1089.2.a.j.1.1 1 44.43 even 2
1584.2.a.o.1.1 1 1.1 even 1 trivial
1617.2.a.j.1.1 1 84.83 odd 2
2112.2.a.j.1.1 1 24.5 odd 2
2112.2.a.bb.1.1 1 24.11 even 2
2475.2.a.g.1.1 1 20.19 odd 2
2475.2.c.d.199.1 2 20.7 even 4
2475.2.c.d.199.2 2 20.3 even 4
4851.2.a.b.1.1 1 28.27 even 2
5577.2.a.a.1.1 1 156.155 even 2
5808.2.a.t.1.1 1 33.32 even 2
6336.2.a.n.1.1 1 8.5 even 2
6336.2.a.x.1.1 1 8.3 odd 2
9075.2.a.q.1.1 1 660.659 odd 2
9537.2.a.m.1.1 1 204.203 even 2