Properties

Label 5184.2.d.l
Level $5184$
Weight $2$
Character orbit 5184.d
Analytic conductor $41.394$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 5184 = 2^{6} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5184.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(41.3944484078\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 576)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +O(q^{10})\) \( q + ( \beta_{1} + \beta_{2} ) q^{11} + ( 3 - \beta_{3} ) q^{17} + ( \beta_{1} - 3 \beta_{2} ) q^{19} + 5 q^{25} + ( 3 + 2 \beta_{3} ) q^{41} + ( -4 \beta_{1} + 3 \beta_{2} ) q^{43} -7 q^{49} + ( 4 \beta_{1} - 5 \beta_{2} ) q^{59} + ( 2 \beta_{1} + 3 \beta_{2} ) q^{67} + ( -1 + 3 \beta_{3} ) q^{73} + 9 \beta_{1} q^{83} + 18 q^{89} + ( -5 - 3 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + O(q^{10}) \) \( 4 q + 12 q^{17} + 20 q^{25} + 12 q^{41} - 28 q^{49} - 4 q^{73} + 72 q^{89} - 20 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu^{2} \)\(/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + \nu^{2} + 3 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{3} + 6 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 2 \beta_{2} - \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\(3 \beta_{1}\)\(/2\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{3} + 6 \beta_{2} - 3 \beta_{1}\)\()/4\)

Character values

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

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

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2593.1
1.22474 1.22474i
−1.22474 + 1.22474i
−1.22474 1.22474i
1.22474 + 1.22474i
0 0 0 0 0 0 0 0 0
2593.2 0 0 0 0 0 0 0 0 0
2593.3 0 0 0 0 0 0 0 0 0
2593.4 0 0 0 0 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
4.b odd 2 1 inner
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5184.2.d.l 4
3.b odd 2 1 5184.2.d.e 4
4.b odd 2 1 inner 5184.2.d.l 4
8.b even 2 1 inner 5184.2.d.l 4
8.d odd 2 1 CM 5184.2.d.l 4
9.c even 3 2 576.2.r.c 8
9.d odd 6 2 1728.2.r.c 8
12.b even 2 1 5184.2.d.e 4
24.f even 2 1 5184.2.d.e 4
24.h odd 2 1 5184.2.d.e 4
36.f odd 6 2 576.2.r.c 8
36.h even 6 2 1728.2.r.c 8
72.j odd 6 2 1728.2.r.c 8
72.l even 6 2 1728.2.r.c 8
72.n even 6 2 576.2.r.c 8
72.p odd 6 2 576.2.r.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.2.r.c 8 9.c even 3 2
576.2.r.c 8 36.f odd 6 2
576.2.r.c 8 72.n even 6 2
576.2.r.c 8 72.p odd 6 2
1728.2.r.c 8 9.d odd 6 2
1728.2.r.c 8 36.h even 6 2
1728.2.r.c 8 72.j odd 6 2
1728.2.r.c 8 72.l even 6 2
5184.2.d.e 4 3.b odd 2 1
5184.2.d.e 4 12.b even 2 1
5184.2.d.e 4 24.f even 2 1
5184.2.d.e 4 24.h odd 2 1
5184.2.d.l 4 1.a even 1 1 trivial
5184.2.d.l 4 4.b odd 2 1 inner
5184.2.d.l 4 8.b even 2 1 inner
5184.2.d.l 4 8.d odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5184, [\chi])\):

\( T_{5} \)
\( T_{7} \)
\( T_{17}^{2} - 6 T_{17} - 15 \)
\( T_{23} \)
\( T_{41}^{2} - 6 T_{41} - 87 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( 9 + 30 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( -15 - 6 T + T^{2} )^{2} \)
$19$ \( 2809 + 110 T^{2} + T^{4} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( -87 - 6 T + T^{2} )^{2} \)
$43$ \( 841 + 158 T^{2} + T^{4} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( 19881 + 318 T^{2} + T^{4} \)
$61$ \( T^{4} \)
$67$ \( 25 + 206 T^{2} + T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( -215 + 2 T + T^{2} )^{2} \)
$79$ \( T^{4} \)
$83$ \( ( 324 + T^{2} )^{2} \)
$89$ \( ( -18 + T )^{4} \)
$97$ \( ( -191 + 10 T + T^{2} )^{2} \)
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