Properties

Label 324.2.a.b
Level $324$
Weight $2$
Character orbit 324.a
Self dual yes
Analytic conductor $2.587$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 324.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.58715302549\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3q^{5} + 2q^{7} + O(q^{10}) \) \( q - 3q^{5} + 2q^{7} + 6q^{11} + 5q^{13} + 3q^{17} + 2q^{19} - 6q^{23} + 4q^{25} - 3q^{29} - 4q^{31} - 6q^{35} + 5q^{37} + 6q^{41} - 10q^{43} - 3q^{49} + 6q^{53} - 18q^{55} + 12q^{59} + 5q^{61} - 15q^{65} + 2q^{67} - 6q^{71} - q^{73} + 12q^{77} - 10q^{79} - 9q^{85} + 3q^{89} + 10q^{91} - 6q^{95} - 10q^{97} + O(q^{100}) \)

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
0
0 0 0 −3.00000 0 2.00000 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 324.2.a.b 1
3.b odd 2 1 324.2.a.d yes 1
4.b odd 2 1 1296.2.a.a 1
5.b even 2 1 8100.2.a.f 1
5.c odd 4 2 8100.2.d.j 2
8.b even 2 1 5184.2.a.bc 1
8.d odd 2 1 5184.2.a.z 1
9.c even 3 2 324.2.e.d 2
9.d odd 6 2 324.2.e.a 2
12.b even 2 1 1296.2.a.j 1
15.d odd 2 1 8100.2.a.a 1
15.e even 4 2 8100.2.d.a 2
24.f even 2 1 5184.2.a.d 1
24.h odd 2 1 5184.2.a.g 1
36.f odd 6 2 1296.2.i.p 2
36.h even 6 2 1296.2.i.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.2.a.b 1 1.a even 1 1 trivial
324.2.a.d yes 1 3.b odd 2 1
324.2.e.a 2 9.d odd 6 2
324.2.e.d 2 9.c even 3 2
1296.2.a.a 1 4.b odd 2 1
1296.2.a.j 1 12.b even 2 1
1296.2.i.d 2 36.h even 6 2
1296.2.i.p 2 36.f odd 6 2
5184.2.a.d 1 24.f even 2 1
5184.2.a.g 1 24.h odd 2 1
5184.2.a.z 1 8.d odd 2 1
5184.2.a.bc 1 8.b even 2 1
8100.2.a.a 1 15.d odd 2 1
8100.2.a.f 1 5.b even 2 1
8100.2.d.a 2 15.e even 4 2
8100.2.d.j 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(324))\):

\( T_{5} + 3 \)
\( T_{7} - 2 \)

Hecke characteristic polynomials

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