Properties

Label 5184.2.a.bk
Level $5184$
Weight $2$
Character orbit 5184.a
Self dual yes
Analytic conductor $41.394$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5184 = 2^{6} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5184.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(41.3944484078\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2592)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{5} -\beta q^{7} +O(q^{10})\) \( q - q^{5} -\beta q^{7} + \beta q^{11} -3 q^{13} + 5 q^{17} -\beta q^{19} -\beta q^{23} -4 q^{25} -5 q^{29} + \beta q^{35} + 5 q^{37} -2 q^{41} -\beta q^{43} + 2 \beta q^{47} + 17 q^{49} -2 q^{53} -\beta q^{55} -2 \beta q^{59} + 13 q^{61} + 3 q^{65} -\beta q^{67} + \beta q^{71} + 3 q^{73} -24 q^{77} + 3 \beta q^{79} -2 \beta q^{83} -5 q^{85} + 13 q^{89} + 3 \beta q^{91} + \beta q^{95} -6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + O(q^{10}) \) \( 2 q - 2 q^{5} - 6 q^{13} + 10 q^{17} - 8 q^{25} - 10 q^{29} + 10 q^{37} - 4 q^{41} + 34 q^{49} - 4 q^{53} + 26 q^{61} + 6 q^{65} + 6 q^{73} - 48 q^{77} - 10 q^{85} + 26 q^{89} - 12 q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.44949
−2.44949
0 0 0 −1.00000 0 −4.89898 0 0 0
1.2 0 0 0 −1.00000 0 4.89898 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5184.2.a.bk 2
3.b odd 2 1 5184.2.a.bv 2
4.b odd 2 1 inner 5184.2.a.bk 2
8.b even 2 1 2592.2.a.r yes 2
8.d odd 2 1 2592.2.a.r yes 2
12.b even 2 1 5184.2.a.bv 2
24.f even 2 1 2592.2.a.m 2
24.h odd 2 1 2592.2.a.m 2
72.j odd 6 2 2592.2.i.bd 4
72.l even 6 2 2592.2.i.bd 4
72.n even 6 2 2592.2.i.z 4
72.p odd 6 2 2592.2.i.z 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2592.2.a.m 2 24.f even 2 1
2592.2.a.m 2 24.h odd 2 1
2592.2.a.r yes 2 8.b even 2 1
2592.2.a.r yes 2 8.d odd 2 1
2592.2.i.z 4 72.n even 6 2
2592.2.i.z 4 72.p odd 6 2
2592.2.i.bd 4 72.j odd 6 2
2592.2.i.bd 4 72.l even 6 2
5184.2.a.bk 2 1.a even 1 1 trivial
5184.2.a.bk 2 4.b odd 2 1 inner
5184.2.a.bv 2 3.b odd 2 1
5184.2.a.bv 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5184))\):

\( T_{5} + 1 \)
\( T_{7}^{2} - 24 \)
\( T_{11}^{2} - 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( -24 + T^{2} \)
$11$ \( -24 + T^{2} \)
$13$ \( ( 3 + T )^{2} \)
$17$ \( ( -5 + T )^{2} \)
$19$ \( -24 + T^{2} \)
$23$ \( -24 + T^{2} \)
$29$ \( ( 5 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( ( -5 + T )^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( -24 + T^{2} \)
$47$ \( -96 + T^{2} \)
$53$ \( ( 2 + T )^{2} \)
$59$ \( -96 + T^{2} \)
$61$ \( ( -13 + T )^{2} \)
$67$ \( -24 + T^{2} \)
$71$ \( -24 + T^{2} \)
$73$ \( ( -3 + T )^{2} \)
$79$ \( -216 + T^{2} \)
$83$ \( -96 + T^{2} \)
$89$ \( ( -13 + T )^{2} \)
$97$ \( ( 6 + T )^{2} \)
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