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Properties

Label 5184.2.d.e.2593.1
Level $5184$
Weight $2$
Character 5184.2593
Analytic conductor $41.394$
Analytic rank $0$
Dimension $4$
CM discriminant -8
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,-12,0,0,0,0,0,0,0,20,0,0,0,0, 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: \(\Q(i, \sqrt{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 576)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 2593.1
Root \(1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 5184.2593
Dual form 5184.2.d.e.2593.4

$q$-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)
 
\(f(q)\) \(=\) \(q-5.44949i q^{11} +1.89898 q^{17} -8.34847i q^{19} +5.00000 q^{25} -12.7980 q^{41} +2.34847i q^{43} -7.00000 q^{49} +9.24745i q^{59} +14.3485i q^{67} +13.6969 q^{73} -18.0000i q^{83} -18.0000 q^{89} -19.6969 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{17} + 20 q^{25} - 12 q^{41} - 28 q^{49} - 4 q^{73} - 72 q^{89} - 20 q^{97}+O(q^{100}) \) 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
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 5.44949i − 1.64308i −0.570149 − 0.821541i \(-0.693114\pi\)
0.570149 − 0.821541i \(-0.306886\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.89898 0.460570 0.230285 − 0.973123i \(-0.426034\pi\)
0.230285 + 0.973123i \(0.426034\pi\)
\(18\) 0 0
\(19\) − 8.34847i − 1.91527i −0.287984 − 0.957635i \(-0.592985\pi\)
0.287984 − 0.957635i \(-0.407015\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −12.7980 −1.99871 −0.999353 − 0.0359748i \(-0.988546\pi\)
−0.999353 + 0.0359748i \(0.988546\pi\)
\(42\) 0 0
\(43\) 2.34847i 0.358138i 0.983836 + 0.179069i \(0.0573086\pi\)
−0.983836 + 0.179069i \(0.942691\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 9.24745i 1.20392i 0.798528 + 0.601958i \(0.205612\pi\)
−0.798528 + 0.601958i \(0.794388\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 14.3485i 1.75294i 0.481452 + 0.876472i \(0.340109\pi\)
−0.481452 + 0.876472i \(0.659891\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 13.6969 1.60311 0.801553 − 0.597924i \(-0.204008\pi\)
0.801553 + 0.597924i \(0.204008\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 18.0000i − 1.97576i −0.155230 − 0.987878i \(-0.549612\pi\)
0.155230 − 0.987878i \(-0.450388\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −18.0000 −1.90800 −0.953998 − 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −19.6969 −1.99992 −0.999961 − 0.00888289i \(-0.997172\pi\)
−0.999961 + 0.00888289i \(0.997172\pi\)
\(98\) 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.e.2593.1 4
3.2 odd 2 5184.2.d.l.2593.4 4
4.3 odd 2 inner 5184.2.d.e.2593.4 4
8.3 odd 2 CM 5184.2.d.e.2593.1 4
8.5 even 2 inner 5184.2.d.e.2593.4 4
9.2 odd 6 576.2.r.c.481.3 yes 8
9.4 even 3 1728.2.r.c.289.4 8
9.5 odd 6 576.2.r.c.97.2 ✓ 8
9.7 even 3 1728.2.r.c.1441.1 8
12.11 even 2 5184.2.d.l.2593.1 4
24.5 odd 2 5184.2.d.l.2593.1 4
24.11 even 2 5184.2.d.l.2593.4 4
36.7 odd 6 1728.2.r.c.1441.4 8
36.11 even 6 576.2.r.c.481.2 yes 8
36.23 even 6 576.2.r.c.97.3 yes 8
36.31 odd 6 1728.2.r.c.289.1 8
72.5 odd 6 576.2.r.c.97.3 yes 8
72.11 even 6 576.2.r.c.481.3 yes 8
72.13 even 6 1728.2.r.c.289.1 8
72.29 odd 6 576.2.r.c.481.2 yes 8
72.43 odd 6 1728.2.r.c.1441.1 8
72.59 even 6 576.2.r.c.97.2 ✓ 8
72.61 even 6 1728.2.r.c.1441.4 8
72.67 odd 6 1728.2.r.c.289.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
576.2.r.c.97.2 ✓ 8 9.5 odd 6
576.2.r.c.97.2 ✓ 8 72.59 even 6
576.2.r.c.97.3 yes 8 36.23 even 6
576.2.r.c.97.3 yes 8 72.5 odd 6
576.2.r.c.481.2 yes 8 36.11 even 6
576.2.r.c.481.2 yes 8 72.29 odd 6
576.2.r.c.481.3 yes 8 9.2 odd 6
576.2.r.c.481.3 yes 8 72.11 even 6
1728.2.r.c.289.1 8 36.31 odd 6
1728.2.r.c.289.1 8 72.13 even 6
1728.2.r.c.289.4 8 9.4 even 3
1728.2.r.c.289.4 8 72.67 odd 6
1728.2.r.c.1441.1 8 9.7 even 3
1728.2.r.c.1441.1 8 72.43 odd 6
1728.2.r.c.1441.4 8 36.7 odd 6
1728.2.r.c.1441.4 8 72.61 even 6
5184.2.d.e.2593.1 4 1.1 even 1 trivial
5184.2.d.e.2593.1 4 8.3 odd 2 CM
5184.2.d.e.2593.4 4 4.3 odd 2 inner
5184.2.d.e.2593.4 4 8.5 even 2 inner
5184.2.d.l.2593.1 4 12.11 even 2
5184.2.d.l.2593.1 4 24.5 odd 2
5184.2.d.l.2593.4 4 3.2 odd 2
5184.2.d.l.2593.4 4 24.11 even 2