Properties

Label 5184.2.a.bx
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: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 288)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} + \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} + \beta q^{7} + \beta q^{11} + 3 q^{13} + 4 q^{17} + 4 \beta q^{19} + 5 \beta q^{23} - 4 q^{25} - q^{29} - 3 \beta q^{31} + \beta q^{35} + 8 q^{37} + 5 q^{41} - 5 \beta q^{43} - 7 \beta q^{47} - 4 q^{49} + 8 q^{53} + \beta q^{55} + \beta q^{59} + 7 q^{61} + 3 q^{65} - 5 \beta q^{67} - 2 \beta q^{71} - 12 q^{73} + 3 q^{77} + 3 \beta q^{79} - 5 \beta q^{83} + 4 q^{85} - 4 q^{89} + 3 \beta q^{91} + 4 \beta q^{95} - 3 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} + 6 q^{13} + 8 q^{17} - 8 q^{25} - 2 q^{29} + 16 q^{37} + 10 q^{41} - 8 q^{49} + 16 q^{53} + 14 q^{61} + 6 q^{65} - 24 q^{73} + 6 q^{77} + 8 q^{85} - 8 q^{89} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
−1.73205
1.73205
0 0 0 1.00000 0 −1.73205 0 0 0
1.2 0 0 0 1.00000 0 1.73205 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.bx 2
3.b odd 2 1 5184.2.a.bl 2
4.b odd 2 1 inner 5184.2.a.bx 2
8.b even 2 1 2592.2.a.l 2
8.d odd 2 1 2592.2.a.l 2
9.c even 3 2 576.2.i.k 4
9.d odd 6 2 1728.2.i.l 4
12.b even 2 1 5184.2.a.bl 2
24.f even 2 1 2592.2.a.p 2
24.h odd 2 1 2592.2.a.p 2
36.f odd 6 2 576.2.i.k 4
36.h even 6 2 1728.2.i.l 4
72.j odd 6 2 864.2.i.d 4
72.l even 6 2 864.2.i.d 4
72.n even 6 2 288.2.i.d 4
72.p odd 6 2 288.2.i.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.d 4 72.n even 6 2
288.2.i.d 4 72.p odd 6 2
576.2.i.k 4 9.c even 3 2
576.2.i.k 4 36.f odd 6 2
864.2.i.d 4 72.j odd 6 2
864.2.i.d 4 72.l even 6 2
1728.2.i.l 4 9.d odd 6 2
1728.2.i.l 4 36.h even 6 2
2592.2.a.l 2 8.b even 2 1
2592.2.a.l 2 8.d odd 2 1
2592.2.a.p 2 24.f even 2 1
2592.2.a.p 2 24.h odd 2 1
5184.2.a.bl 2 3.b odd 2 1
5184.2.a.bl 2 12.b even 2 1
5184.2.a.bx 2 1.a even 1 1 trivial
5184.2.a.bx 2 4.b odd 2 1 inner

Hecke kernels

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

\( T_{5} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} - 3 \) Copy content Toggle raw display
\( T_{11}^{2} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 3 \) Copy content Toggle raw display
$11$ \( T^{2} - 3 \) Copy content Toggle raw display
$13$ \( (T - 3)^{2} \) Copy content Toggle raw display
$17$ \( (T - 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 48 \) Copy content Toggle raw display
$23$ \( T^{2} - 75 \) Copy content Toggle raw display
$29$ \( (T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 27 \) Copy content Toggle raw display
$37$ \( (T - 8)^{2} \) Copy content Toggle raw display
$41$ \( (T - 5)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 75 \) Copy content Toggle raw display
$47$ \( T^{2} - 147 \) Copy content Toggle raw display
$53$ \( (T - 8)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 3 \) Copy content Toggle raw display
$61$ \( (T - 7)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 75 \) Copy content Toggle raw display
$71$ \( T^{2} - 12 \) Copy content Toggle raw display
$73$ \( (T + 12)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 27 \) Copy content Toggle raw display
$83$ \( T^{2} - 75 \) Copy content Toggle raw display
$89$ \( (T + 4)^{2} \) Copy content Toggle raw display
$97$ \( (T + 3)^{2} \) Copy content Toggle raw display
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