Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 5184 = 2^{6} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5184.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(41.3944484078\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Defining polynomial: \(x^{4} + 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 324)
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{5} +O(q^{10})\) \( q -\beta_{2} q^{5} + ( -2 - \beta_{3} ) q^{13} + ( 3 \beta_{1} + \beta_{2} ) q^{17} + ( -9 + \beta_{3} ) q^{25} + ( 3 \beta_{1} - \beta_{2} ) q^{29} + ( 1 + 2 \beta_{3} ) q^{37} + \beta_{1} q^{41} + 7 q^{49} + 5 \beta_{1} q^{53} + ( -5 + 2 \beta_{3} ) q^{61} + ( 13 \beta_{1} + \beta_{2} ) q^{65} + ( 8 + \beta_{3} ) q^{73} + ( 11 + 2 \beta_{3} ) q^{85} + ( -6 \beta_{1} + \beta_{2} ) q^{89} + 8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + O(q^{10}) \) \( 4 q - 8 q^{13} - 36 q^{25} + 4 q^{37} + 28 q^{49} - 20 q^{61} + 32 q^{73} + 44 q^{85} + 32 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 4 x^{2} + 1\):

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5183.1
1.93185i
0.517638i
0.517638i
1.93185i
0 0 0 4.38134i 0 0 0 0 0
5183.2 0 0 0 2.96713i 0 0 0 0 0
5183.3 0 0 0 2.96713i 0 0 0 0 0
5183.4 0 0 0 4.38134i 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
3.b odd 2 1 inner
12.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.c.e 4
3.b odd 2 1 inner 5184.2.c.e 4
4.b odd 2 1 CM 5184.2.c.e 4
8.b even 2 1 324.2.b.a 4
8.d odd 2 1 324.2.b.a 4
12.b even 2 1 inner 5184.2.c.e 4
24.f even 2 1 324.2.b.a 4
24.h odd 2 1 324.2.b.a 4
72.j odd 6 2 324.2.h.e 8
72.l even 6 2 324.2.h.e 8
72.n even 6 2 324.2.h.e 8
72.p odd 6 2 324.2.h.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.2.b.a 4 8.b even 2 1
324.2.b.a 4 8.d odd 2 1
324.2.b.a 4 24.f even 2 1
324.2.b.a 4 24.h odd 2 1
324.2.h.e 8 72.j odd 6 2
324.2.h.e 8 72.l even 6 2
324.2.h.e 8 72.n even 6 2
324.2.h.e 8 72.p odd 6 2
5184.2.c.e 4 1.a even 1 1 trivial
5184.2.c.e 4 3.b odd 2 1 inner
5184.2.c.e 4 4.b odd 2 1 CM
5184.2.c.e 4 12.b even 2 1 inner

Hecke kernels

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

\( T_{5}^{4} + 28 T_{5}^{2} + 169 \)
\( T_{7} \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 169 + 28 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( ( -23 + 4 T + T^{2} )^{2} \)
$17$ \( 1 + 52 T^{2} + T^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( 121 + 76 T^{2} + T^{4} \)
$31$ \( T^{4} \)
$37$ \( ( -107 - 2 T + T^{2} )^{2} \)
$41$ \( ( 2 + T^{2} )^{2} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( ( 50 + T^{2} )^{2} \)
$59$ \( T^{4} \)
$61$ \( ( -83 + 10 T + T^{2} )^{2} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( 37 - 16 T + T^{2} )^{2} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( 5041 + 196 T^{2} + T^{4} \)
$97$ \( ( -8 + T )^{4} \)
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