Properties

Label 576.4.a.m
Level $576$
Weight $4$
Character orbit 576.a
Self dual yes
Analytic conductor $33.985$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 576.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.9851001633\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 9)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q + 20 q^{7} + O(q^{10}) \) \( q + 20 q^{7} + 70 q^{13} - 56 q^{19} - 125 q^{25} + 308 q^{31} - 110 q^{37} + 520 q^{43} + 57 q^{49} - 182 q^{61} + 880 q^{67} + 1190 q^{73} + 884 q^{79} + 1400 q^{91} - 1330 q^{97} + O(q^{100}) \)

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 0 0 20.0000 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 576.4.a.m 1
3.b odd 2 1 CM 576.4.a.m 1
4.b odd 2 1 576.4.a.l 1
8.b even 2 1 9.4.a.a 1
8.d odd 2 1 144.4.a.d 1
12.b even 2 1 576.4.a.l 1
24.f even 2 1 144.4.a.d 1
24.h odd 2 1 9.4.a.a 1
40.f even 2 1 225.4.a.d 1
40.i odd 4 2 225.4.b.g 2
56.h odd 2 1 441.4.a.f 1
56.j odd 6 2 441.4.e.j 2
56.p even 6 2 441.4.e.i 2
72.j odd 6 2 81.4.c.b 2
72.n even 6 2 81.4.c.b 2
88.b odd 2 1 1089.4.a.g 1
104.e even 2 1 1521.4.a.g 1
120.i odd 2 1 225.4.a.d 1
120.w even 4 2 225.4.b.g 2
168.i even 2 1 441.4.a.f 1
168.s odd 6 2 441.4.e.i 2
168.ba even 6 2 441.4.e.j 2
264.m even 2 1 1089.4.a.g 1
312.b odd 2 1 1521.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.a.a 1 8.b even 2 1
9.4.a.a 1 24.h odd 2 1
81.4.c.b 2 72.j odd 6 2
81.4.c.b 2 72.n even 6 2
144.4.a.d 1 8.d odd 2 1
144.4.a.d 1 24.f even 2 1
225.4.a.d 1 40.f even 2 1
225.4.a.d 1 120.i odd 2 1
225.4.b.g 2 40.i odd 4 2
225.4.b.g 2 120.w even 4 2
441.4.a.f 1 56.h odd 2 1
441.4.a.f 1 168.i even 2 1
441.4.e.i 2 56.p even 6 2
441.4.e.i 2 168.s odd 6 2
441.4.e.j 2 56.j odd 6 2
441.4.e.j 2 168.ba even 6 2
576.4.a.l 1 4.b odd 2 1
576.4.a.l 1 12.b even 2 1
576.4.a.m 1 1.a even 1 1 trivial
576.4.a.m 1 3.b odd 2 1 CM
1089.4.a.g 1 88.b odd 2 1
1089.4.a.g 1 264.m even 2 1
1521.4.a.g 1 104.e even 2 1
1521.4.a.g 1 312.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(576))\):

\( T_{5} \)
\( T_{7} - 20 \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( -20 + T \)
$11$ \( T \)
$13$ \( -70 + T \)
$17$ \( T \)
$19$ \( 56 + T \)
$23$ \( T \)
$29$ \( T \)
$31$ \( -308 + T \)
$37$ \( 110 + T \)
$41$ \( T \)
$43$ \( -520 + T \)
$47$ \( T \)
$53$ \( T \)
$59$ \( T \)
$61$ \( 182 + T \)
$67$ \( -880 + T \)
$71$ \( T \)
$73$ \( -1190 + T \)
$79$ \( -884 + T \)
$83$ \( T \)
$89$ \( T \)
$97$ \( 1330 + T \)
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