Properties

Label 2592.2.a.o
Level $2592$
Weight $2$
Character orbit 2592.a
Self dual yes
Analytic conductor $20.697$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \( x^{2} - 6 \) 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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} + ( - \beta - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} + ( - \beta - 1) q^{7} + (\beta - 1) q^{11} + (2 \beta + 1) q^{13} - 2 \beta q^{17} - 4 q^{19} + ( - \beta - 3) q^{23} - 4 q^{25} + (2 \beta - 5) q^{29} + (\beta - 5) q^{31} + ( - \beta - 1) q^{35} + ( - 2 \beta + 4) q^{37} + (2 \beta + 7) q^{41} + (3 \beta - 5) q^{43} + ( - 3 \beta + 1) q^{47} + 2 \beta q^{49} + ( - 2 \beta - 4) q^{53} + (\beta - 1) q^{55} + (3 \beta + 7) q^{59} + ( - 2 \beta - 3) q^{61} + (2 \beta + 1) q^{65} + ( - 3 \beta - 5) q^{67} + (4 \beta - 2) q^{71} - 2 \beta q^{73} - 5 q^{77} + ( - \beta - 11) q^{79} + (\beta - 3) q^{83} - 2 \beta q^{85} + (2 \beta + 8) q^{89} + ( - 3 \beta - 13) q^{91} - 4 q^{95} + ( - 2 \beta + 1) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 2 q^{7} - 2 q^{11} + 2 q^{13} - 8 q^{19} - 6 q^{23} - 8 q^{25} - 10 q^{29} - 10 q^{31} - 2 q^{35} + 8 q^{37} + 14 q^{41} - 10 q^{43} + 2 q^{47} - 8 q^{53} - 2 q^{55} + 14 q^{59} - 6 q^{61} + 2 q^{65} - 10 q^{67} - 4 q^{71} - 10 q^{77} - 22 q^{79} - 6 q^{83} + 16 q^{89} - 26 q^{91} - 8 q^{95} + 2 q^{97}+O(q^{100}) \) 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
2.44949
−2.44949
0 0 0 1.00000 0 −3.44949 0 0 0
1.2 0 0 0 1.00000 0 1.44949 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.o 2
3.b odd 2 1 2592.2.a.j 2
4.b odd 2 1 2592.2.a.s 2
8.b even 2 1 5184.2.a.bj 2
8.d odd 2 1 5184.2.a.bn 2
9.c even 3 2 864.2.i.e 4
9.d odd 6 2 288.2.i.c 4
12.b even 2 1 2592.2.a.n 2
24.f even 2 1 5184.2.a.by 2
24.h odd 2 1 5184.2.a.bu 2
36.f odd 6 2 864.2.i.c 4
36.h even 6 2 288.2.i.e yes 4
72.j odd 6 2 576.2.i.m 4
72.l even 6 2 576.2.i.i 4
72.n even 6 2 1728.2.i.m 4
72.p odd 6 2 1728.2.i.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.c 4 9.d odd 6 2
288.2.i.e yes 4 36.h even 6 2
576.2.i.i 4 72.l even 6 2
576.2.i.m 4 72.j odd 6 2
864.2.i.c 4 36.f odd 6 2
864.2.i.e 4 9.c even 3 2
1728.2.i.k 4 72.p odd 6 2
1728.2.i.m 4 72.n even 6 2
2592.2.a.j 2 3.b odd 2 1
2592.2.a.n 2 12.b even 2 1
2592.2.a.o 2 1.a even 1 1 trivial
2592.2.a.s 2 4.b odd 2 1
5184.2.a.bj 2 8.b even 2 1
5184.2.a.bn 2 8.d odd 2 1
5184.2.a.bu 2 24.h odd 2 1
5184.2.a.by 2 24.f even 2 1

Hecke kernels

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

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