Properties

Label 2592.2.a.n
Level $2592$
Weight $2$
Character orbit 2592.a
Self dual yes
Analytic conductor $20.697$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{5} + ( 1 + \beta ) q^{7} +O(q^{10})\) \( q - q^{5} + ( 1 + \beta ) q^{7} + ( -1 + \beta ) q^{11} + ( 1 + 2 \beta ) q^{13} + 2 \beta q^{17} + 4 q^{19} + ( -3 - \beta ) q^{23} -4 q^{25} + ( 5 - 2 \beta ) q^{29} + ( 5 - \beta ) q^{31} + ( -1 - \beta ) q^{35} + ( 4 - 2 \beta ) q^{37} + ( -7 - 2 \beta ) q^{41} + ( 5 - 3 \beta ) q^{43} + ( 1 - 3 \beta ) q^{47} + 2 \beta q^{49} + ( 4 + 2 \beta ) q^{53} + ( 1 - \beta ) q^{55} + ( 7 + 3 \beta ) q^{59} + ( -3 - 2 \beta ) q^{61} + ( -1 - 2 \beta ) q^{65} + ( 5 + 3 \beta ) q^{67} + ( -2 + 4 \beta ) q^{71} -2 \beta q^{73} + 5 q^{77} + ( 11 + \beta ) q^{79} + ( -3 + \beta ) q^{83} -2 \beta q^{85} + ( -8 - 2 \beta ) q^{89} + ( 13 + 3 \beta ) q^{91} -4 q^{95} + ( 1 - 2 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 2 q^{7} + O(q^{10}) \) \( 2 q - 2 q^{5} + 2 q^{7} - 2 q^{11} + 2 q^{13} + 8 q^{19} - 6 q^{23} - 8 q^{25} + 10 q^{29} + 10 q^{31} - 2 q^{35} + 8 q^{37} - 14 q^{41} + 10 q^{43} + 2 q^{47} + 8 q^{53} + 2 q^{55} + 14 q^{59} - 6 q^{61} - 2 q^{65} + 10 q^{67} - 4 q^{71} + 10 q^{77} + 22 q^{79} - 6 q^{83} - 16 q^{89} + 26 q^{91} - 8 q^{95} + 2 q^{97} + O(q^{100}) \)

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 −1.44949 0 0 0
1.2 0 0 0 −1.00000 0 3.44949 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 2592.2.a.n 2
3.b odd 2 1 2592.2.a.s 2
4.b odd 2 1 2592.2.a.j 2
8.b even 2 1 5184.2.a.by 2
8.d odd 2 1 5184.2.a.bu 2
9.c even 3 2 288.2.i.e yes 4
9.d odd 6 2 864.2.i.c 4
12.b even 2 1 2592.2.a.o 2
24.f even 2 1 5184.2.a.bj 2
24.h odd 2 1 5184.2.a.bn 2
36.f odd 6 2 288.2.i.c 4
36.h even 6 2 864.2.i.e 4
72.j odd 6 2 1728.2.i.k 4
72.l even 6 2 1728.2.i.m 4
72.n even 6 2 576.2.i.i 4
72.p odd 6 2 576.2.i.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.c 4 36.f odd 6 2
288.2.i.e yes 4 9.c even 3 2
576.2.i.i 4 72.n even 6 2
576.2.i.m 4 72.p odd 6 2
864.2.i.c 4 9.d odd 6 2
864.2.i.e 4 36.h even 6 2
1728.2.i.k 4 72.j odd 6 2
1728.2.i.m 4 72.l even 6 2
2592.2.a.j 2 4.b odd 2 1
2592.2.a.n 2 1.a even 1 1 trivial
2592.2.a.o 2 12.b even 2 1
2592.2.a.s 2 3.b odd 2 1
5184.2.a.bj 2 24.f even 2 1
5184.2.a.bn 2 24.h odd 2 1
5184.2.a.bu 2 8.d odd 2 1
5184.2.a.by 2 8.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2592))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( -5 - 2 T + T^{2} \)
$11$ \( -5 + 2 T + T^{2} \)
$13$ \( -23 - 2 T + T^{2} \)
$17$ \( -24 + T^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( 3 + 6 T + T^{2} \)
$29$ \( 1 - 10 T + T^{2} \)
$31$ \( 19 - 10 T + T^{2} \)
$37$ \( -8 - 8 T + T^{2} \)
$41$ \( 25 + 14 T + T^{2} \)
$43$ \( -29 - 10 T + T^{2} \)
$47$ \( -53 - 2 T + T^{2} \)
$53$ \( -8 - 8 T + T^{2} \)
$59$ \( -5 - 14 T + T^{2} \)
$61$ \( -15 + 6 T + T^{2} \)
$67$ \( -29 - 10 T + T^{2} \)
$71$ \( -92 + 4 T + T^{2} \)
$73$ \( -24 + T^{2} \)
$79$ \( 115 - 22 T + T^{2} \)
$83$ \( 3 + 6 T + T^{2} \)
$89$ \( 40 + 16 T + T^{2} \)
$97$ \( -23 - 2 T + T^{2} \)
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