Properties

Label 432.3.e.b
Level $432$
Weight $3$
Character orbit 432.e
Self dual yes
Analytic conductor $11.771$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 432.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(11.7711474204\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q + 13q^{7} + O(q^{10}) \) \( q + 13q^{7} - q^{13} - 11q^{19} + 25q^{25} + 46q^{31} + 47q^{37} + 22q^{43} + 120q^{49} - 121q^{61} + 109q^{67} - 97q^{73} - 131q^{79} - 13q^{91} + 167q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
0
0 0 0 0 0 13.0000 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 432.3.e.b 1
3.b odd 2 1 CM 432.3.e.b 1
4.b odd 2 1 27.3.b.a 1
8.b even 2 1 1728.3.e.d 1
8.d odd 2 1 1728.3.e.a 1
9.c even 3 2 1296.3.q.a 2
9.d odd 6 2 1296.3.q.a 2
12.b even 2 1 27.3.b.a 1
20.d odd 2 1 675.3.c.c 1
20.e even 4 2 675.3.d.c 2
24.f even 2 1 1728.3.e.a 1
24.h odd 2 1 1728.3.e.d 1
36.f odd 6 2 81.3.d.a 2
36.h even 6 2 81.3.d.a 2
60.h even 2 1 675.3.c.c 1
60.l odd 4 2 675.3.d.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.3.b.a 1 4.b odd 2 1
27.3.b.a 1 12.b even 2 1
81.3.d.a 2 36.f odd 6 2
81.3.d.a 2 36.h even 6 2
432.3.e.b 1 1.a even 1 1 trivial
432.3.e.b 1 3.b odd 2 1 CM
675.3.c.c 1 20.d odd 2 1
675.3.c.c 1 60.h even 2 1
675.3.d.c 2 20.e even 4 2
675.3.d.c 2 60.l odd 4 2
1296.3.q.a 2 9.c even 3 2
1296.3.q.a 2 9.d odd 6 2
1728.3.e.a 1 8.d odd 2 1
1728.3.e.a 1 24.f even 2 1
1728.3.e.d 1 8.b even 2 1
1728.3.e.d 1 24.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 \(S_{3}^{\mathrm{new}}(432, [\chi])\):

\( T_{5} \)
\( T_{7} - 13 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( -13 + T \)
$11$ \( T \)
$13$ \( 1 + T \)
$17$ \( T \)
$19$ \( 11 + T \)
$23$ \( T \)
$29$ \( T \)
$31$ \( -46 + T \)
$37$ \( -47 + T \)
$41$ \( T \)
$43$ \( -22 + T \)
$47$ \( T \)
$53$ \( T \)
$59$ \( T \)
$61$ \( 121 + T \)
$67$ \( -109 + T \)
$71$ \( T \)
$73$ \( 97 + T \)
$79$ \( 131 + T \)
$83$ \( T \)
$89$ \( T \)
$97$ \( -167 + T \)
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