Properties

Label 432.4.a.h
Level 432
Weight 4
Character orbit 432.a
Self dual yes
Analytic conductor 25.489
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 432.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.4888251225\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q + 37q^{7} + O(q^{10}) \) \( q + 37q^{7} - 19q^{13} + 163q^{19} - 125q^{25} - 308q^{31} + 323q^{37} + 520q^{43} + 1026q^{49} + 719q^{61} + 127q^{67} - 919q^{73} + 1387q^{79} - 703q^{91} - 523q^{97} + O(q^{100}) \)

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 37.0000 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 432.4.a.h 1
3.b odd 2 1 CM 432.4.a.h 1
4.b odd 2 1 108.4.a.b 1
8.b even 2 1 1728.4.a.r 1
8.d odd 2 1 1728.4.a.o 1
12.b even 2 1 108.4.a.b 1
24.f even 2 1 1728.4.a.o 1
24.h odd 2 1 1728.4.a.r 1
36.f odd 6 2 324.4.e.e 2
36.h even 6 2 324.4.e.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.a.b 1 4.b odd 2 1
108.4.a.b 1 12.b even 2 1
324.4.e.e 2 36.f odd 6 2
324.4.e.e 2 36.h even 6 2
432.4.a.h 1 1.a even 1 1 trivial
432.4.a.h 1 3.b odd 2 1 CM
1728.4.a.o 1 8.d odd 2 1
1728.4.a.o 1 24.f even 2 1
1728.4.a.r 1 8.b even 2 1
1728.4.a.r 1 24.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(432))\):

\( T_{5} \)
\( T_{7} - 37 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 125 T^{2} \)
$7$ \( 1 - 37 T + 343 T^{2} \)
$11$ \( 1 + 1331 T^{2} \)
$13$ \( 1 + 19 T + 2197 T^{2} \)
$17$ \( 1 + 4913 T^{2} \)
$19$ \( 1 - 163 T + 6859 T^{2} \)
$23$ \( 1 + 12167 T^{2} \)
$29$ \( 1 + 24389 T^{2} \)
$31$ \( 1 + 308 T + 29791 T^{2} \)
$37$ \( 1 - 323 T + 50653 T^{2} \)
$41$ \( 1 + 68921 T^{2} \)
$43$ \( 1 - 520 T + 79507 T^{2} \)
$47$ \( 1 + 103823 T^{2} \)
$53$ \( 1 + 148877 T^{2} \)
$59$ \( 1 + 205379 T^{2} \)
$61$ \( 1 - 719 T + 226981 T^{2} \)
$67$ \( 1 - 127 T + 300763 T^{2} \)
$71$ \( 1 + 357911 T^{2} \)
$73$ \( 1 + 919 T + 389017 T^{2} \)
$79$ \( 1 - 1387 T + 493039 T^{2} \)
$83$ \( 1 + 571787 T^{2} \)
$89$ \( 1 + 704969 T^{2} \)
$97$ \( 1 + 523 T + 912673 T^{2} \)
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