Properties

Label 5184.2.a.bp
Level $5184$
Weight $2$
Character orbit 5184.a
Self dual yes
Analytic conductor $41.394$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{5} + ( 2 - \beta ) q^{7} +O(q^{10})\) \( q -\beta q^{5} + ( 2 - \beta ) q^{7} + q^{11} + ( -2 - \beta ) q^{13} + ( 3 - \beta ) q^{17} + ( -3 - \beta ) q^{19} + ( -2 - \beta ) q^{23} + ( 3 + \beta ) q^{25} + ( -2 + \beta ) q^{29} + ( 4 - \beta ) q^{31} + ( 8 - \beta ) q^{35} + ( -4 + 2 \beta ) q^{37} + ( 7 - 2 \beta ) q^{41} + ( -3 - 2 \beta ) q^{43} + ( -2 + \beta ) q^{47} + ( 5 - 3 \beta ) q^{49} + ( -4 - 2 \beta ) q^{53} -\beta q^{55} + 7 q^{59} + \beta q^{61} + ( 8 + 3 \beta ) q^{65} + ( -1 - 2 \beta ) q^{67} + 4 q^{71} + ( -5 + 3 \beta ) q^{73} + ( 2 - \beta ) q^{77} + ( -4 + \beta ) q^{79} + ( 12 + \beta ) q^{83} + ( 8 - 2 \beta ) q^{85} -6 q^{89} + ( 4 + \beta ) q^{91} + ( 8 + 4 \beta ) q^{95} + ( -3 - 2 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{5} + 3q^{7} + O(q^{10}) \) \( 2q - q^{5} + 3q^{7} + 2q^{11} - 5q^{13} + 5q^{17} - 7q^{19} - 5q^{23} + 7q^{25} - 3q^{29} + 7q^{31} + 15q^{35} - 6q^{37} + 12q^{41} - 8q^{43} - 3q^{47} + 7q^{49} - 10q^{53} - q^{55} + 14q^{59} + q^{61} + 19q^{65} - 4q^{67} + 8q^{71} - 7q^{73} + 3q^{77} - 7q^{79} + 25q^{83} + 14q^{85} - 12q^{89} + 9q^{91} + 20q^{95} - 8q^{97} + O(q^{100}) \)

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
3.37228
−2.37228
0 0 0 −3.37228 0 −1.37228 0 0 0
1.2 0 0 0 2.37228 0 4.37228 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

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5184.2.a.bp 2
3.b odd 2 1 5184.2.a.bt 2
4.b odd 2 1 5184.2.a.bo 2
8.b even 2 1 648.2.a.g 2
8.d odd 2 1 1296.2.a.p 2
9.c even 3 2 1728.2.i.i 4
9.d odd 6 2 576.2.i.j 4
12.b even 2 1 5184.2.a.bs 2
24.f even 2 1 1296.2.a.n 2
24.h odd 2 1 648.2.a.f 2
36.f odd 6 2 1728.2.i.j 4
36.h even 6 2 576.2.i.l 4
72.j odd 6 2 72.2.i.b 4
72.l even 6 2 144.2.i.d 4
72.n even 6 2 216.2.i.b 4
72.p odd 6 2 432.2.i.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.i.b 4 72.j odd 6 2
144.2.i.d 4 72.l even 6 2
216.2.i.b 4 72.n even 6 2
432.2.i.d 4 72.p odd 6 2
576.2.i.j 4 9.d odd 6 2
576.2.i.l 4 36.h even 6 2
648.2.a.f 2 24.h odd 2 1
648.2.a.g 2 8.b even 2 1
1296.2.a.n 2 24.f even 2 1
1296.2.a.p 2 8.d odd 2 1
1728.2.i.i 4 9.c even 3 2
1728.2.i.j 4 36.f odd 6 2
5184.2.a.bo 2 4.b odd 2 1
5184.2.a.bp 2 1.a even 1 1 trivial
5184.2.a.bs 2 12.b even 2 1
5184.2.a.bt 2 3.b odd 2 1

Hecke kernels

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

\( T_{5}^{2} + T_{5} - 8 \)
\( T_{7}^{2} - 3 T_{7} - 6 \)
\( T_{11} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -8 + T + T^{2} \)
$7$ \( -6 - 3 T + T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( -2 + 5 T + T^{2} \)
$17$ \( -2 - 5 T + T^{2} \)
$19$ \( 4 + 7 T + T^{2} \)
$23$ \( -2 + 5 T + T^{2} \)
$29$ \( -6 + 3 T + T^{2} \)
$31$ \( 4 - 7 T + T^{2} \)
$37$ \( -24 + 6 T + T^{2} \)
$41$ \( 3 - 12 T + T^{2} \)
$43$ \( -17 + 8 T + T^{2} \)
$47$ \( -6 + 3 T + T^{2} \)
$53$ \( -8 + 10 T + T^{2} \)
$59$ \( ( -7 + T )^{2} \)
$61$ \( -8 - T + T^{2} \)
$67$ \( -29 + 4 T + T^{2} \)
$71$ \( ( -4 + T )^{2} \)
$73$ \( -62 + 7 T + T^{2} \)
$79$ \( 4 + 7 T + T^{2} \)
$83$ \( 148 - 25 T + T^{2} \)
$89$ \( ( 6 + T )^{2} \)
$97$ \( -17 + 8 T + T^{2} \)
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