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Properties

Label 5184.2.a.br.1.1
Level $5184$
Weight $2$
Character 5184.1
Self dual yes
Analytic conductor $41.394$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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 = [2,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0] 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: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 81)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q-1.73205 q^{5} +2.00000 q^{7} -3.46410 q^{11} +1.00000 q^{13} -5.19615 q^{17} -2.00000 q^{19} -3.46410 q^{23} -2.00000 q^{25} +1.73205 q^{29} +8.00000 q^{31} -3.46410 q^{35} +7.00000 q^{37} +6.92820 q^{41} -2.00000 q^{43} -6.92820 q^{47} -3.00000 q^{49} +6.00000 q^{55} +13.8564 q^{59} +7.00000 q^{61} -1.73205 q^{65} +10.0000 q^{67} +10.3923 q^{71} -7.00000 q^{73} -6.92820 q^{77} +2.00000 q^{79} -13.8564 q^{83} +9.00000 q^{85} +5.19615 q^{89} +2.00000 q^{91} +3.46410 q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{7} + 2 q^{13} - 4 q^{19} - 4 q^{25} + 16 q^{31} + 14 q^{37} - 4 q^{43} - 6 q^{49} + 12 q^{55} + 14 q^{61} + 20 q^{67} - 14 q^{73} + 4 q^{79} + 18 q^{85} + 4 q^{91} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) −1.73205 −0.774597 −0.387298 − 0.921954i \(-0.626592\pi\)
−0.387298 + 0.921954i \(0.626592\pi\)
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.46410 −1.04447 −0.522233 − 0.852803i \(-0.674901\pi\)
−0.522233 + 0.852803i \(0.674901\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 − 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.19615 −1.26025 −0.630126 − 0.776493i \(-0.716997\pi\)
−0.630126 + 0.776493i \(0.716997\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.46410 −0.722315 −0.361158 − 0.932505i \(-0.617618\pi\)
−0.361158 + 0.932505i \(0.617618\pi\)
\(24\) 0 0
\(25\) −2.00000 −0.400000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.73205 0.321634 0.160817 − 0.986984i \(-0.448587\pi\)
0.160817 + 0.986984i \(0.448587\pi\)
\(30\) 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
\(35\) −3.46410 −0.585540
\(36\) 0 0
\(37\) 7.00000 1.15079 0.575396 − 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.92820 1.08200 0.541002 − 0.841021i \(-0.318045\pi\)
0.541002 + 0.841021i \(0.318045\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 − 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.92820 −1.01058 −0.505291 − 0.862949i \(-0.668615\pi\)
−0.505291 + 0.862949i \(0.668615\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 6.00000 0.809040
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 13.8564 1.80395 0.901975 − 0.431788i \(-0.142117\pi\)
0.901975 + 0.431788i \(0.142117\pi\)
\(60\) 0 0
\(61\) 7.00000 0.896258 0.448129 − 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.73205 −0.214834
\(66\) 0 0
\(67\) 10.0000 1.22169 0.610847 − 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.3923 1.23334 0.616670 − 0.787222i \(-0.288481\pi\)
0.616670 + 0.787222i \(0.288481\pi\)
\(72\) 0 0
\(73\) −7.00000 −0.819288 −0.409644 − 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.92820 −0.789542
\(78\) 0 0
\(79\) 2.00000 0.225018 0.112509 − 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −13.8564 −1.52094 −0.760469 − 0.649374i \(-0.775031\pi\)
−0.760469 + 0.649374i \(0.775031\pi\)
\(84\) 0 0
\(85\) 9.00000 0.976187
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.19615 0.550791 0.275396 − 0.961331i \(-0.411191\pi\)
0.275396 + 0.961331i \(0.411191\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.46410 0.355409
\(96\) 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 5184.2.a.br.1.1 2
3.2 odd 2 inner 5184.2.a.br.1.2 2
4.3 odd 2 5184.2.a.bq.1.1 2
8.3 odd 2 1296.2.a.o.1.2 2
8.5 even 2 81.2.a.a.1.1 ✓ 2
12.11 even 2 5184.2.a.bq.1.2 2
24.5 odd 2 81.2.a.a.1.2 yes 2
24.11 even 2 1296.2.a.o.1.1 2
40.13 odd 4 2025.2.b.k.649.3 4
40.29 even 2 2025.2.a.j.1.2 2
40.37 odd 4 2025.2.b.k.649.2 4
56.13 odd 2 3969.2.a.i.1.1 2
72.5 odd 6 81.2.c.b.55.1 4
72.11 even 6 1296.2.i.s.433.2 4
72.13 even 6 81.2.c.b.55.2 4
72.29 odd 6 81.2.c.b.28.1 4
72.43 odd 6 1296.2.i.s.433.1 4
72.59 even 6 1296.2.i.s.865.2 4
72.61 even 6 81.2.c.b.28.2 4
72.67 odd 6 1296.2.i.s.865.1 4
88.21 odd 2 9801.2.a.v.1.2 2
120.29 odd 2 2025.2.a.j.1.1 2
120.53 even 4 2025.2.b.k.649.1 4
120.77 even 4 2025.2.b.k.649.4 4
168.125 even 2 3969.2.a.i.1.2 2
216.5 odd 18 729.2.e.o.649.2 12
216.13 even 18 729.2.e.o.163.2 12
216.29 odd 18 729.2.e.o.568.1 12
216.61 even 18 729.2.e.o.325.1 12
216.77 odd 18 729.2.e.o.406.2 12
216.85 even 18 729.2.e.o.406.1 12
216.101 odd 18 729.2.e.o.325.2 12
216.133 even 18 729.2.e.o.568.2 12
216.149 odd 18 729.2.e.o.163.1 12
216.157 even 18 729.2.e.o.649.1 12
216.173 odd 18 729.2.e.o.82.2 12
216.205 even 18 729.2.e.o.82.1 12
264.197 even 2 9801.2.a.v.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
81.2.a.a.1.1 ✓ 2 8.5 even 2
81.2.a.a.1.2 yes 2 24.5 odd 2
81.2.c.b.28.1 4 72.29 odd 6
81.2.c.b.28.2 4 72.61 even 6
81.2.c.b.55.1 4 72.5 odd 6
81.2.c.b.55.2 4 72.13 even 6
729.2.e.o.82.1 12 216.205 even 18
729.2.e.o.82.2 12 216.173 odd 18
729.2.e.o.163.1 12 216.149 odd 18
729.2.e.o.163.2 12 216.13 even 18
729.2.e.o.325.1 12 216.61 even 18
729.2.e.o.325.2 12 216.101 odd 18
729.2.e.o.406.1 12 216.85 even 18
729.2.e.o.406.2 12 216.77 odd 18
729.2.e.o.568.1 12 216.29 odd 18
729.2.e.o.568.2 12 216.133 even 18
729.2.e.o.649.1 12 216.157 even 18
729.2.e.o.649.2 12 216.5 odd 18
1296.2.a.o.1.1 2 24.11 even 2
1296.2.a.o.1.2 2 8.3 odd 2
1296.2.i.s.433.1 4 72.43 odd 6
1296.2.i.s.433.2 4 72.11 even 6
1296.2.i.s.865.1 4 72.67 odd 6
1296.2.i.s.865.2 4 72.59 even 6
2025.2.a.j.1.1 2 120.29 odd 2
2025.2.a.j.1.2 2 40.29 even 2
2025.2.b.k.649.1 4 120.53 even 4
2025.2.b.k.649.2 4 40.37 odd 4
2025.2.b.k.649.3 4 40.13 odd 4
2025.2.b.k.649.4 4 120.77 even 4
3969.2.a.i.1.1 2 56.13 odd 2
3969.2.a.i.1.2 2 168.125 even 2
5184.2.a.bq.1.1 2 4.3 odd 2
5184.2.a.bq.1.2 2 12.11 even 2
5184.2.a.br.1.1 2 1.1 even 1 trivial
5184.2.a.br.1.2 2 3.2 odd 2 inner
9801.2.a.v.1.1 2 264.197 even 2
9801.2.a.v.1.2 2 88.21 odd 2