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Properties

Label 5184.2.d.c.2593.3
Level $5184$
Weight $2$
Character 5184.2593
Analytic conductor $41.394$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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(2593,5184)] 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("5184.2593"); 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, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5184 = 2^{6} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5184.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-12,0,0,0,0,0,0,0,-28,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(41)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(41.3944484078\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 576)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2593.3
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 5184.2593
Dual form 5184.2.d.c.2593.2

$q$-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)
 
\(f(q)\) \(=\) \(q+3.46410i q^{5} -3.00000i q^{11} +3.46410i q^{13} -3.00000 q^{17} -7.00000i q^{19} +3.46410 q^{23} -7.00000 q^{25} -6.92820i q^{29} -6.92820 q^{31} +10.3923i q^{37} +3.00000 q^{41} -5.00000i q^{43} +3.46410 q^{47} -7.00000 q^{49} -13.8564i q^{53} +10.3923 q^{55} -9.00000i q^{59} -6.92820i q^{61} -12.0000 q^{65} -5.00000i q^{67} -3.46410 q^{71} -7.00000 q^{73} -17.3205 q^{79} +12.0000i q^{83} -10.3923i q^{85} -6.00000 q^{89} +24.2487 q^{95} +1.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{17} - 28 q^{25} + 12 q^{41} - 28 q^{49} - 48 q^{65} - 28 q^{73} - 24 q^{89} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(5\) 3.46410i 1.54919i 0.632456 + 0.774597i \(0.282047\pi\)
−0.632456 + 0.774597i \(0.717953\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 3.00000i − 0.904534i −0.891883 − 0.452267i \(-0.850615\pi\)
0.891883 − 0.452267i \(-0.149385\pi\)
\(12\) 0 0
\(13\) 3.46410i 0.960769i 0.877058 + 0.480384i \(0.159503\pi\)
−0.877058 + 0.480384i \(0.840497\pi\)
\(14\) 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
\(19\) − 7.00000i − 1.60591i −0.596040 − 0.802955i \(-0.703260\pi\)
0.596040 − 0.802955i \(-0.296740\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.46410 0.722315 0.361158 − 0.932505i \(-0.382382\pi\)
0.361158 + 0.932505i \(0.382382\pi\)
\(24\) 0 0
\(25\) −7.00000 −1.40000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 6.92820i − 1.28654i −0.765641 − 0.643268i \(-0.777578\pi\)
0.765641 − 0.643268i \(-0.222422\pi\)
\(30\) 0 0
\(31\) −6.92820 −1.24434 −0.622171 − 0.782881i \(-0.713749\pi\)
−0.622171 + 0.782881i \(0.713749\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.3923i 1.70848i 0.519875 + 0.854242i \(0.325978\pi\)
−0.519875 + 0.854242i \(0.674022\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 − 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) − 5.00000i − 0.762493i −0.924473 − 0.381246i \(-0.875495\pi\)
0.924473 − 0.381246i \(-0.124505\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.46410 0.505291 0.252646 − 0.967559i \(-0.418699\pi\)
0.252646 + 0.967559i \(0.418699\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 13.8564i − 1.90332i −0.307148 − 0.951662i \(-0.599375\pi\)
0.307148 − 0.951662i \(-0.400625\pi\)
\(54\) 0 0
\(55\) 10.3923 1.40130
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 9.00000i − 1.17170i −0.810419 − 0.585850i \(-0.800761\pi\)
0.810419 − 0.585850i \(-0.199239\pi\)
\(60\) 0 0
\(61\) − 6.92820i − 0.887066i −0.896258 − 0.443533i \(-0.853725\pi\)
0.896258 − 0.443533i \(-0.146275\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −12.0000 −1.48842
\(66\) 0 0
\(67\) − 5.00000i − 0.610847i −0.952217 − 0.305424i \(-0.901202\pi\)
0.952217 − 0.305424i \(-0.0987981\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.46410 −0.411113 −0.205557 − 0.978645i \(-0.565900\pi\)
−0.205557 + 0.978645i \(0.565900\pi\)
\(72\) 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
\(77\) 0 0
\(78\) 0 0
\(79\) −17.3205 −1.94871 −0.974355 − 0.225018i \(-0.927756\pi\)
−0.974355 + 0.225018i \(0.927756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 0 0
\(85\) − 10.3923i − 1.12720i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 24.2487 2.48787
\(96\) 0 0
\(97\) 1.00000 0.101535 0.0507673 − 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) 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.d.c.2593.3 4
3.2 odd 2 5184.2.d.j.2593.2 4
4.3 odd 2 inner 5184.2.d.c.2593.4 4
8.3 odd 2 inner 5184.2.d.c.2593.1 4
8.5 even 2 inner 5184.2.d.c.2593.2 4
9.2 odd 6 1728.2.r.b.1441.2 4
9.4 even 3 576.2.r.b.97.1 yes 4
9.5 odd 6 1728.2.r.a.289.1 4
9.7 even 3 576.2.r.a.481.2 yes 4
12.11 even 2 5184.2.d.j.2593.1 4
24.5 odd 2 5184.2.d.j.2593.3 4
24.11 even 2 5184.2.d.j.2593.4 4
36.7 odd 6 576.2.r.a.481.1 yes 4
36.11 even 6 1728.2.r.b.1441.1 4
36.23 even 6 1728.2.r.a.289.2 4
36.31 odd 6 576.2.r.b.97.2 yes 4
72.5 odd 6 1728.2.r.b.289.2 4
72.11 even 6 1728.2.r.a.1441.2 4
72.13 even 6 576.2.r.a.97.2 yes 4
72.29 odd 6 1728.2.r.a.1441.1 4
72.43 odd 6 576.2.r.b.481.2 yes 4
72.59 even 6 1728.2.r.b.289.1 4
72.61 even 6 576.2.r.b.481.1 yes 4
72.67 odd 6 576.2.r.a.97.1 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
576.2.r.a.97.1 ✓ 4 72.67 odd 6
576.2.r.a.97.2 yes 4 72.13 even 6
576.2.r.a.481.1 yes 4 36.7 odd 6
576.2.r.a.481.2 yes 4 9.7 even 3
576.2.r.b.97.1 yes 4 9.4 even 3
576.2.r.b.97.2 yes 4 36.31 odd 6
576.2.r.b.481.1 yes 4 72.61 even 6
576.2.r.b.481.2 yes 4 72.43 odd 6
1728.2.r.a.289.1 4 9.5 odd 6
1728.2.r.a.289.2 4 36.23 even 6
1728.2.r.a.1441.1 4 72.29 odd 6
1728.2.r.a.1441.2 4 72.11 even 6
1728.2.r.b.289.1 4 72.59 even 6
1728.2.r.b.289.2 4 72.5 odd 6
1728.2.r.b.1441.1 4 36.11 even 6
1728.2.r.b.1441.2 4 9.2 odd 6
5184.2.d.c.2593.1 4 8.3 odd 2 inner
5184.2.d.c.2593.2 4 8.5 even 2 inner
5184.2.d.c.2593.3 4 1.1 even 1 trivial
5184.2.d.c.2593.4 4 4.3 odd 2 inner
5184.2.d.j.2593.1 4 12.11 even 2
5184.2.d.j.2593.2 4 3.2 odd 2
5184.2.d.j.2593.3 4 24.5 odd 2
5184.2.d.j.2593.4 4 24.11 even 2