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Properties

Label 5184.2.a.o.1.1
Level $5184$
Weight $2$
Character 5184.1
Self dual yes
Analytic conductor $41.394$
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] = [5184,2,Mod(1,5184)] 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("5184.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5184, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5184 = 2^{6} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5184.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,0,0,-2,0,0,0,-3,0,-2,0,0,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(41.3944484078\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 18)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5184.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{7} -3.00000 q^{11} -2.00000 q^{13} -3.00000 q^{17} -1.00000 q^{19} +6.00000 q^{23} -5.00000 q^{25} -6.00000 q^{29} +4.00000 q^{31} +4.00000 q^{37} +9.00000 q^{41} -1.00000 q^{43} +6.00000 q^{47} -3.00000 q^{49} -12.0000 q^{53} +3.00000 q^{59} -8.00000 q^{61} +5.00000 q^{67} +12.0000 q^{71} +11.0000 q^{73} +6.00000 q^{77} +4.00000 q^{79} +12.0000 q^{83} +6.00000 q^{89} +4.00000 q^{91} +5.00000 q^{97} +O(q^{100})\)

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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 − 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 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
\(17\) −3.00000 −0.727607 −0.363803 − 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 − 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 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
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 − 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.00000 1.40556 0.702782 − 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 − 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −12.0000 −1.64833 −0.824163 − 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.00000 0.390567 0.195283 − 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.00000 0.610847 0.305424 − 0.952217i \(-0.401202\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 11.0000 1.28745 0.643726 − 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.00000 0.683763
\(78\) 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
\(85\) 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
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 5.00000 0.507673 0.253837 − 0.967247i \(-0.418307\pi\)
0.253837 + 0.967247i \(0.418307\pi\)
\(98\) 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 5184.2.a.o.1.1 1
3.2 odd 2 5184.2.a.p.1.1 1
4.3 odd 2 5184.2.a.r.1.1 1
8.3 odd 2 162.2.a.c.1.1 1
8.5 even 2 1296.2.a.g.1.1 1
9.2 odd 6 1728.2.i.f.577.1 2
9.4 even 3 576.2.i.a.385.1 2
9.5 odd 6 1728.2.i.f.1153.1 2
9.7 even 3 576.2.i.a.193.1 2
12.11 even 2 5184.2.a.q.1.1 1
24.5 odd 2 1296.2.a.f.1.1 1
24.11 even 2 162.2.a.b.1.1 1
36.7 odd 6 576.2.i.g.193.1 2
36.11 even 6 1728.2.i.e.577.1 2
36.23 even 6 1728.2.i.e.1153.1 2
36.31 odd 6 576.2.i.g.385.1 2
40.3 even 4 4050.2.c.c.649.1 2
40.19 odd 2 4050.2.a.c.1.1 1
40.27 even 4 4050.2.c.c.649.2 2
56.27 even 2 7938.2.a.x.1.1 1
72.5 odd 6 432.2.i.b.289.1 2
72.11 even 6 54.2.c.a.37.1 2
72.13 even 6 144.2.i.c.97.1 2
72.29 odd 6 432.2.i.b.145.1 2
72.43 odd 6 18.2.c.a.13.1 yes 2
72.59 even 6 54.2.c.a.19.1 2
72.61 even 6 144.2.i.c.49.1 2
72.67 odd 6 18.2.c.a.7.1 ✓ 2
120.59 even 2 4050.2.a.v.1.1 1
120.83 odd 4 4050.2.c.r.649.2 2
120.107 odd 4 4050.2.c.r.649.1 2
168.83 odd 2 7938.2.a.i.1.1 1
360.43 even 12 450.2.j.e.49.1 4
360.59 even 6 1350.2.e.c.451.1 2
360.67 even 12 450.2.j.e.349.1 4
360.83 odd 12 1350.2.j.a.199.2 4
360.139 odd 6 450.2.e.i.151.1 2
360.187 even 12 450.2.j.e.49.2 4
360.203 odd 12 1350.2.j.a.1099.1 4
360.227 odd 12 1350.2.j.a.199.1 4
360.259 odd 6 450.2.e.i.301.1 2
360.283 even 12 450.2.j.e.349.2 4
360.299 even 6 1350.2.e.c.901.1 2
360.347 odd 12 1350.2.j.a.1099.2 4
504.11 even 6 2646.2.e.b.1549.1 2
504.59 odd 6 2646.2.h.i.667.1 2
504.67 odd 6 882.2.h.c.79.1 2
504.83 odd 6 2646.2.f.g.1765.1 2
504.115 even 6 882.2.e.g.373.1 2
504.131 odd 6 2646.2.e.c.2125.1 2
504.139 even 6 882.2.f.d.295.1 2
504.187 even 6 882.2.h.b.67.1 2
504.227 odd 6 2646.2.e.c.1549.1 2
504.275 even 6 2646.2.e.b.2125.1 2
504.283 even 6 882.2.h.b.79.1 2
504.299 odd 6 2646.2.h.i.361.1 2
504.331 odd 6 882.2.h.c.67.1 2
504.347 even 6 2646.2.h.h.667.1 2
504.355 even 6 882.2.e.g.655.1 2
504.403 odd 6 882.2.e.i.373.1 2
504.419 odd 6 2646.2.f.g.883.1 2
504.443 even 6 2646.2.h.h.361.1 2
504.475 even 6 882.2.f.d.589.1 2
504.499 odd 6 882.2.e.i.655.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.2.c.a.7.1 ✓ 2 72.67 odd 6
18.2.c.a.13.1 yes 2 72.43 odd 6
54.2.c.a.19.1 2 72.59 even 6
54.2.c.a.37.1 2 72.11 even 6
144.2.i.c.49.1 2 72.61 even 6
144.2.i.c.97.1 2 72.13 even 6
162.2.a.b.1.1 1 24.11 even 2
162.2.a.c.1.1 1 8.3 odd 2
432.2.i.b.145.1 2 72.29 odd 6
432.2.i.b.289.1 2 72.5 odd 6
450.2.e.i.151.1 2 360.139 odd 6
450.2.e.i.301.1 2 360.259 odd 6
450.2.j.e.49.1 4 360.43 even 12
450.2.j.e.49.2 4 360.187 even 12
450.2.j.e.349.1 4 360.67 even 12
450.2.j.e.349.2 4 360.283 even 12
576.2.i.a.193.1 2 9.7 even 3
576.2.i.a.385.1 2 9.4 even 3
576.2.i.g.193.1 2 36.7 odd 6
576.2.i.g.385.1 2 36.31 odd 6
882.2.e.g.373.1 2 504.115 even 6
882.2.e.g.655.1 2 504.355 even 6
882.2.e.i.373.1 2 504.403 odd 6
882.2.e.i.655.1 2 504.499 odd 6
882.2.f.d.295.1 2 504.139 even 6
882.2.f.d.589.1 2 504.475 even 6
882.2.h.b.67.1 2 504.187 even 6
882.2.h.b.79.1 2 504.283 even 6
882.2.h.c.67.1 2 504.331 odd 6
882.2.h.c.79.1 2 504.67 odd 6
1296.2.a.f.1.1 1 24.5 odd 2
1296.2.a.g.1.1 1 8.5 even 2
1350.2.e.c.451.1 2 360.59 even 6
1350.2.e.c.901.1 2 360.299 even 6
1350.2.j.a.199.1 4 360.227 odd 12
1350.2.j.a.199.2 4 360.83 odd 12
1350.2.j.a.1099.1 4 360.203 odd 12
1350.2.j.a.1099.2 4 360.347 odd 12
1728.2.i.e.577.1 2 36.11 even 6
1728.2.i.e.1153.1 2 36.23 even 6
1728.2.i.f.577.1 2 9.2 odd 6
1728.2.i.f.1153.1 2 9.5 odd 6
2646.2.e.b.1549.1 2 504.11 even 6
2646.2.e.b.2125.1 2 504.275 even 6
2646.2.e.c.1549.1 2 504.227 odd 6
2646.2.e.c.2125.1 2 504.131 odd 6
2646.2.f.g.883.1 2 504.419 odd 6
2646.2.f.g.1765.1 2 504.83 odd 6
2646.2.h.h.361.1 2 504.443 even 6
2646.2.h.h.667.1 2 504.347 even 6
2646.2.h.i.361.1 2 504.299 odd 6
2646.2.h.i.667.1 2 504.59 odd 6
4050.2.a.c.1.1 1 40.19 odd 2
4050.2.a.v.1.1 1 120.59 even 2
4050.2.c.c.649.1 2 40.3 even 4
4050.2.c.c.649.2 2 40.27 even 4
4050.2.c.r.649.1 2 120.107 odd 4
4050.2.c.r.649.2 2 120.83 odd 4
5184.2.a.o.1.1 1 1.1 even 1 trivial
5184.2.a.p.1.1 1 3.2 odd 2
5184.2.a.q.1.1 1 12.11 even 2
5184.2.a.r.1.1 1 4.3 odd 2
7938.2.a.i.1.1 1 168.83 odd 2
7938.2.a.x.1.1 1 56.27 even 2