Properties

Label 5292.2.a.u
Level $5292$
Weight $2$
Character orbit 5292.a
Self dual yes
Analytic conductor $42.257$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5292.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(42.2568327497\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.321.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 756)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} +O(q^{10})\) \( q + \beta_{2} q^{5} + ( -2 + \beta_{1} ) q^{11} + ( -1 - \beta_{2} ) q^{13} + ( 1 + \beta_{1} ) q^{17} + ( 1 - \beta_{1} - \beta_{2} ) q^{19} + ( -4 - \beta_{1} + \beta_{2} ) q^{23} + ( 3 - \beta_{1} - 2 \beta_{2} ) q^{25} + ( 2 - \beta_{1} ) q^{29} + ( -1 - \beta_{2} ) q^{31} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{37} + ( -4 - \beta_{1} - \beta_{2} ) q^{41} + ( -2 - \beta_{1} + 2 \beta_{2} ) q^{43} + ( 4 - 2 \beta_{1} + \beta_{2} ) q^{47} + ( -3 - 3 \beta_{2} ) q^{53} + ( 1 + \beta_{1} - 4 \beta_{2} ) q^{55} + ( 1 + \beta_{1} - \beta_{2} ) q^{59} + ( 1 + 2 \beta_{1} - 2 \beta_{2} ) q^{61} + ( -8 + \beta_{1} + \beta_{2} ) q^{65} + ( -6 + \beta_{1} - \beta_{2} ) q^{67} + ( -5 + \beta_{1} - 3 \beta_{2} ) q^{71} + ( \beta_{1} + 2 \beta_{2} ) q^{73} + ( -4 + 3 \beta_{1} - \beta_{2} ) q^{79} + ( -6 - \beta_{2} ) q^{83} + ( 1 + \beta_{1} - \beta_{2} ) q^{85} + ( -1 + 2 \beta_{1} - 4 \beta_{2} ) q^{89} + ( -9 + 5 \beta_{2} ) q^{95} + ( 6 - 2 \beta_{1} + 2 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{5} + O(q^{10}) \) \( 3q - q^{5} - 5q^{11} - 2q^{13} + 4q^{17} + 3q^{19} - 14q^{23} + 10q^{25} + 5q^{29} - 2q^{31} - 12q^{41} - 9q^{43} + 9q^{47} - 6q^{53} + 8q^{55} + 5q^{59} + 7q^{61} - 24q^{65} - 16q^{67} - 11q^{71} - q^{73} - 8q^{79} - 17q^{83} + 5q^{85} + 3q^{89} - 32q^{95} + 14q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} + \nu - 3 \)
\(\beta_{2}\)\(=\)\( -\nu^{2} + 2 \nu + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{2} + 2 \beta_{1} + 8\)\()/3\)

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
−1.69963
2.46050
0.239123
0 0 0 −4.28799 0 0 0 0 0
1.2 0 0 0 0.866926 0 0 0 0 0
1.3 0 0 0 2.42107 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(1\)

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 5292.2.a.u 3
3.b odd 2 1 5292.2.a.x 3
7.b odd 2 1 5292.2.a.w 3
7.c even 3 2 756.2.k.f yes 6
21.c even 2 1 5292.2.a.v 3
21.h odd 6 2 756.2.k.e 6
63.g even 3 2 2268.2.l.j 6
63.h even 3 2 2268.2.i.k 6
63.j odd 6 2 2268.2.i.j 6
63.n odd 6 2 2268.2.l.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.k.e 6 21.h odd 6 2
756.2.k.f yes 6 7.c even 3 2
2268.2.i.j 6 63.j odd 6 2
2268.2.i.k 6 63.h even 3 2
2268.2.l.j 6 63.g even 3 2
2268.2.l.k 6 63.n odd 6 2
5292.2.a.u 3 1.a even 1 1 trivial
5292.2.a.v 3 21.c even 2 1
5292.2.a.w 3 7.b odd 2 1
5292.2.a.x 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5292))\):

\( T_{5}^{3} + T_{5}^{2} - 12 T_{5} + 9 \)
\( T_{11}^{3} + 5 T_{11}^{2} - 12 T_{11} - 63 \)
\( T_{13}^{3} + 2 T_{13}^{2} - 11 T_{13} - 21 \)
\( T_{17}^{3} - 4 T_{17}^{2} - 15 T_{17} - 9 \)
\( T_{19}^{3} - 3 T_{19}^{2} - 36 T_{19} + 49 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( T^{3} \)
$5$ \( 9 - 12 T + T^{2} + T^{3} \)
$7$ \( T^{3} \)
$11$ \( -63 - 12 T + 5 T^{2} + T^{3} \)
$13$ \( -21 - 11 T + 2 T^{2} + T^{3} \)
$17$ \( -9 - 15 T - 4 T^{2} + T^{3} \)
$19$ \( 49 - 36 T - 3 T^{2} + T^{3} \)
$23$ \( -63 + 39 T + 14 T^{2} + T^{3} \)
$29$ \( 63 - 12 T - 5 T^{2} + T^{3} \)
$31$ \( -21 - 11 T + 2 T^{2} + T^{3} \)
$37$ \( -137 - 57 T + T^{3} \)
$41$ \( -81 + 9 T + 12 T^{2} + T^{3} \)
$43$ \( -281 - 30 T + 9 T^{2} + T^{3} \)
$47$ \( 243 - 54 T - 9 T^{2} + T^{3} \)
$53$ \( -567 - 99 T + 6 T^{2} + T^{3} \)
$59$ \( 81 - 18 T - 5 T^{2} + T^{3} \)
$61$ \( 567 - 89 T - 7 T^{2} + T^{3} \)
$67$ \( 53 + 59 T + 16 T^{2} + T^{3} \)
$71$ \( -189 - 72 T + 11 T^{2} + T^{3} \)
$73$ \( 161 - 82 T + T^{2} + T^{3} \)
$79$ \( -873 - 155 T + 8 T^{2} + T^{3} \)
$83$ \( 99 + 84 T + 17 T^{2} + T^{3} \)
$89$ \( 1323 - 225 T - 3 T^{2} + T^{3} \)
$97$ \( 56 - 40 T - 14 T^{2} + T^{3} \)
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