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:

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)

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

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