Properties

Label 2592.2.a.u
Level $2592$
Weight $2$
Character orbit 2592.a
Self dual yes
Analytic conductor $20.697$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2592 = 2^{5} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2592.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(20.6972242039\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.13068.1
Defining polynomial: \(x^{4} - x^{3} - 6 x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3 \)
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 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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{1} ) q^{5} + \beta_{3} q^{7} +O(q^{10})\) \( q + ( -1 - \beta_{1} ) q^{5} + \beta_{3} q^{7} -\beta_{2} q^{11} + ( 1 - \beta_{1} ) q^{13} -\beta_{1} q^{17} + ( -\beta_{2} + \beta_{3} ) q^{19} + \beta_{3} q^{23} + ( 4 + \beta_{1} ) q^{25} + ( -3 - \beta_{1} ) q^{29} + ( 2 \beta_{2} - \beta_{3} ) q^{31} + ( 2 \beta_{2} - 3 \beta_{3} ) q^{35} + 4 q^{37} + q^{41} + \beta_{2} q^{43} + \beta_{3} q^{47} + ( 8 + 3 \beta_{1} ) q^{49} -4 q^{53} + ( -2 \beta_{2} - \beta_{3} ) q^{55} + ( \beta_{2} + 2 \beta_{3} ) q^{59} + ( 5 - 3 \beta_{1} ) q^{61} + ( 7 - \beta_{1} ) q^{65} + ( -\beta_{2} - 2 \beta_{3} ) q^{67} + 2 \beta_{3} q^{71} + ( 8 + \beta_{1} ) q^{73} + ( -3 + 3 \beta_{1} ) q^{77} + ( -2 \beta_{2} + \beta_{3} ) q^{79} + ( -2 \beta_{2} - \beta_{3} ) q^{83} + 8 q^{85} + ( 8 + 2 \beta_{1} ) q^{89} + ( 2 \beta_{2} - \beta_{3} ) q^{91} -4 \beta_{3} q^{95} + 9 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{5} + O(q^{10}) \) \( 4q - 2q^{5} + 6q^{13} + 2q^{17} + 14q^{25} - 10q^{29} + 16q^{37} + 4q^{41} + 26q^{49} - 16q^{53} + 26q^{61} + 30q^{65} + 30q^{73} - 18q^{77} + 32q^{85} + 28q^{89} + 36q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 6 x^{2} - x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} - \nu^{2} - 7 \nu - 1 \)
\(\beta_{2}\)\(=\)\( \nu^{3} - 6 \nu - 4 \)
\(\beta_{3}\)\(=\)\( 2 \nu^{3} - 3 \nu^{2} - 9 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 3 \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 5 \beta_{2} - 3 \beta_{1} + 18\)\()/6\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 2 \beta_{2} - 3 \beta_{1} + 4\)

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
−1.82405
−0.548230
0.328543
3.04374
0 0 0 −3.37228 0 −4.70285 0 0 0
1.2 0 0 0 −3.37228 0 4.70285 0 0 0
1.3 0 0 0 2.37228 0 −2.20979 0 0 0
1.4 0 0 0 2.37228 0 2.20979 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 2592.2.a.u 4
3.b odd 2 1 2592.2.a.x 4
4.b odd 2 1 inner 2592.2.a.u 4
8.b even 2 1 5184.2.a.cf 4
8.d odd 2 1 5184.2.a.cf 4
9.c even 3 2 288.2.i.f 8
9.d odd 6 2 864.2.i.f 8
12.b even 2 1 2592.2.a.x 4
24.f even 2 1 5184.2.a.cc 4
24.h odd 2 1 5184.2.a.cc 4
36.f odd 6 2 288.2.i.f 8
36.h even 6 2 864.2.i.f 8
72.j odd 6 2 1728.2.i.n 8
72.l even 6 2 1728.2.i.n 8
72.n even 6 2 576.2.i.n 8
72.p odd 6 2 576.2.i.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.f 8 9.c even 3 2
288.2.i.f 8 36.f odd 6 2
576.2.i.n 8 72.n even 6 2
576.2.i.n 8 72.p odd 6 2
864.2.i.f 8 9.d odd 6 2
864.2.i.f 8 36.h even 6 2
1728.2.i.n 8 72.j odd 6 2
1728.2.i.n 8 72.l even 6 2
2592.2.a.u 4 1.a even 1 1 trivial
2592.2.a.u 4 4.b odd 2 1 inner
2592.2.a.x 4 3.b odd 2 1
2592.2.a.x 4 12.b even 2 1
5184.2.a.cc 4 24.f even 2 1
5184.2.a.cc 4 24.h odd 2 1
5184.2.a.cf 4 8.b even 2 1
5184.2.a.cf 4 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2592))\):

\( T_{5}^{2} + T_{5} - 8 \)
\( T_{7}^{4} - 27 T_{7}^{2} + 108 \)
\( T_{11}^{4} - 36 T_{11}^{2} + 27 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -8 + T + T^{2} )^{2} \)
$7$ \( 108 - 27 T^{2} + T^{4} \)
$11$ \( 27 - 36 T^{2} + T^{4} \)
$13$ \( ( -6 - 3 T + T^{2} )^{2} \)
$17$ \( ( -8 - T + T^{2} )^{2} \)
$19$ \( 432 - 45 T^{2} + T^{4} \)
$23$ \( 108 - 27 T^{2} + T^{4} \)
$29$ \( ( -2 + 5 T + T^{2} )^{2} \)
$31$ \( 3888 - 135 T^{2} + T^{4} \)
$37$ \( ( -4 + T )^{4} \)
$41$ \( ( -1 + T )^{4} \)
$43$ \( 27 - 36 T^{2} + T^{4} \)
$47$ \( 108 - 27 T^{2} + T^{4} \)
$53$ \( ( 4 + T )^{4} \)
$59$ \( 7803 - 180 T^{2} + T^{4} \)
$61$ \( ( -32 - 13 T + T^{2} )^{2} \)
$67$ \( 7803 - 180 T^{2} + T^{4} \)
$71$ \( 1728 - 108 T^{2} + T^{4} \)
$73$ \( ( 48 - 15 T + T^{2} )^{2} \)
$79$ \( 3888 - 135 T^{2} + T^{4} \)
$83$ \( 1728 - 207 T^{2} + T^{4} \)
$89$ \( ( 16 - 14 T + T^{2} )^{2} \)
$97$ \( ( -9 + T )^{4} \)
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