Properties

Label 432.2.a.g
Level $432$
Weight $2$
Character orbit 432.a
Self dual yes
Analytic conductor $3.450$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 54)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{5} + q^{7} + O(q^{10}) \) \( q + 3q^{5} + q^{7} + 3q^{11} - 4q^{13} - 2q^{19} + 6q^{23} + 4q^{25} + 6q^{29} - 5q^{31} + 3q^{35} + 2q^{37} - 6q^{41} + 10q^{43} - 6q^{47} - 6q^{49} + 9q^{53} + 9q^{55} - 12q^{59} + 8q^{61} - 12q^{65} - 14q^{67} - 7q^{73} + 3q^{77} - 8q^{79} + 3q^{83} - 18q^{89} - 4q^{91} - 6q^{95} - q^{97} + O(q^{100}) \)

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
0
0 0 0 3.00000 0 1.00000 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 432.2.a.g 1
3.b odd 2 1 432.2.a.b 1
4.b odd 2 1 54.2.a.a 1
8.b even 2 1 1728.2.a.d 1
8.d odd 2 1 1728.2.a.c 1
9.c even 3 2 1296.2.i.c 2
9.d odd 6 2 1296.2.i.o 2
12.b even 2 1 54.2.a.b yes 1
20.d odd 2 1 1350.2.a.r 1
20.e even 4 2 1350.2.c.b 2
24.f even 2 1 1728.2.a.y 1
24.h odd 2 1 1728.2.a.z 1
28.d even 2 1 2646.2.a.a 1
36.f odd 6 2 162.2.c.c 2
36.h even 6 2 162.2.c.b 2
44.c even 2 1 6534.2.a.bc 1
52.b odd 2 1 9126.2.a.u 1
60.h even 2 1 1350.2.a.h 1
60.l odd 4 2 1350.2.c.k 2
84.h odd 2 1 2646.2.a.bd 1
132.d odd 2 1 6534.2.a.b 1
156.h even 2 1 9126.2.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.2.a.a 1 4.b odd 2 1
54.2.a.b yes 1 12.b even 2 1
162.2.c.b 2 36.h even 6 2
162.2.c.c 2 36.f odd 6 2
432.2.a.b 1 3.b odd 2 1
432.2.a.g 1 1.a even 1 1 trivial
1296.2.i.c 2 9.c even 3 2
1296.2.i.o 2 9.d odd 6 2
1350.2.a.h 1 60.h even 2 1
1350.2.a.r 1 20.d odd 2 1
1350.2.c.b 2 20.e even 4 2
1350.2.c.k 2 60.l odd 4 2
1728.2.a.c 1 8.d odd 2 1
1728.2.a.d 1 8.b even 2 1
1728.2.a.y 1 24.f even 2 1
1728.2.a.z 1 24.h odd 2 1
2646.2.a.a 1 28.d even 2 1
2646.2.a.bd 1 84.h odd 2 1
6534.2.a.b 1 132.d odd 2 1
6534.2.a.bc 1 44.c even 2 1
9126.2.a.r 1 156.h even 2 1
9126.2.a.u 1 52.b odd 2 1

Hecke kernels

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

\( T_{5} - 3 \)
\( T_{7} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -3 + T \)
$7$ \( -1 + T \)
$11$ \( -3 + T \)
$13$ \( 4 + T \)
$17$ \( T \)
$19$ \( 2 + T \)
$23$ \( -6 + T \)
$29$ \( -6 + T \)
$31$ \( 5 + T \)
$37$ \( -2 + T \)
$41$ \( 6 + T \)
$43$ \( -10 + T \)
$47$ \( 6 + T \)
$53$ \( -9 + T \)
$59$ \( 12 + T \)
$61$ \( -8 + T \)
$67$ \( 14 + T \)
$71$ \( T \)
$73$ \( 7 + T \)
$79$ \( 8 + T \)
$83$ \( -3 + T \)
$89$ \( 18 + T \)
$97$ \( 1 + T \)
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