Properties

Label 432.2.a.d
Level $432$
Weight $2$
Character orbit 432.a
Self dual yes
Analytic conductor $3.450$
Analytic rank $1$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 432.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.44953736732\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 108)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q - 5q^{7} + O(q^{10}) \) \( q - 5q^{7} - 7q^{13} + q^{19} - 5q^{25} + 4q^{31} - q^{37} - 8q^{43} + 18q^{49} - 13q^{61} - 11q^{67} + 17q^{73} + 13q^{79} + 35q^{91} + 5q^{97} + O(q^{100}) \)

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 0 0 −5.00000 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.2.a.d 1
3.b odd 2 1 CM 432.2.a.d 1
4.b odd 2 1 108.2.a.a 1
8.b even 2 1 1728.2.a.m 1
8.d odd 2 1 1728.2.a.p 1
9.c even 3 2 1296.2.i.j 2
9.d odd 6 2 1296.2.i.j 2
12.b even 2 1 108.2.a.a 1
20.d odd 2 1 2700.2.a.b 1
20.e even 4 2 2700.2.d.g 2
24.f even 2 1 1728.2.a.p 1
24.h odd 2 1 1728.2.a.m 1
28.d even 2 1 5292.2.a.j 1
36.f odd 6 2 324.2.e.b 2
36.h even 6 2 324.2.e.b 2
60.h even 2 1 2700.2.a.b 1
60.l odd 4 2 2700.2.d.g 2
84.h odd 2 1 5292.2.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.2.a.a 1 4.b odd 2 1
108.2.a.a 1 12.b even 2 1
324.2.e.b 2 36.f odd 6 2
324.2.e.b 2 36.h even 6 2
432.2.a.d 1 1.a even 1 1 trivial
432.2.a.d 1 3.b odd 2 1 CM
1296.2.i.j 2 9.c even 3 2
1296.2.i.j 2 9.d odd 6 2
1728.2.a.m 1 8.b even 2 1
1728.2.a.m 1 24.h odd 2 1
1728.2.a.p 1 8.d odd 2 1
1728.2.a.p 1 24.f even 2 1
2700.2.a.b 1 20.d odd 2 1
2700.2.a.b 1 60.h even 2 1
2700.2.d.g 2 20.e even 4 2
2700.2.d.g 2 60.l odd 4 2
5292.2.a.j 1 28.d even 2 1
5292.2.a.j 1 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(432))\):

\( T_{5} \)
\( T_{7} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( 5 + T \)
$11$ \( T \)
$13$ \( 7 + T \)
$17$ \( T \)
$19$ \( -1 + T \)
$23$ \( T \)
$29$ \( T \)
$31$ \( -4 + T \)
$37$ \( 1 + T \)
$41$ \( T \)
$43$ \( 8 + T \)
$47$ \( T \)
$53$ \( T \)
$59$ \( T \)
$61$ \( 13 + T \)
$67$ \( 11 + T \)
$71$ \( T \)
$73$ \( -17 + T \)
$79$ \( -13 + T \)
$83$ \( T \)
$89$ \( T \)
$97$ \( -5 + T \)
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