Properties

Label 576.4.a.r
Level $576$
Weight $4$
Character orbit 576.a
Self dual yes
Analytic conductor $33.985$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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 6)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 6 q^{5} + 16 q^{7} + O(q^{10}) \) \( q + 6 q^{5} + 16 q^{7} - 12 q^{11} - 38 q^{13} + 126 q^{17} + 20 q^{19} + 168 q^{23} - 89 q^{25} + 30 q^{29} + 88 q^{31} + 96 q^{35} - 254 q^{37} - 42 q^{41} - 52 q^{43} - 96 q^{47} - 87 q^{49} + 198 q^{53} - 72 q^{55} + 660 q^{59} + 538 q^{61} - 228 q^{65} + 884 q^{67} + 792 q^{71} + 218 q^{73} - 192 q^{77} + 520 q^{79} + 492 q^{83} + 756 q^{85} - 810 q^{89} - 608 q^{91} + 120 q^{95} + 1154 q^{97} + O(q^{100}) \)

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 6.00000 0 16.0000 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

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 576.4.a.r 1
3.b odd 2 1 192.4.a.c 1
4.b odd 2 1 576.4.a.q 1
8.b even 2 1 144.4.a.c 1
8.d odd 2 1 18.4.a.a 1
12.b even 2 1 192.4.a.i 1
24.f even 2 1 6.4.a.a 1
24.h odd 2 1 48.4.a.c 1
40.e odd 2 1 450.4.a.h 1
40.k even 4 2 450.4.c.e 2
48.i odd 4 2 768.4.d.c 2
48.k even 4 2 768.4.d.n 2
56.e even 2 1 882.4.a.n 1
56.k odd 6 2 882.4.g.i 2
56.m even 6 2 882.4.g.f 2
72.l even 6 2 162.4.c.f 2
72.p odd 6 2 162.4.c.c 2
88.g even 2 1 2178.4.a.e 1
120.i odd 2 1 1200.4.a.b 1
120.m even 2 1 150.4.a.i 1
120.q odd 4 2 150.4.c.d 2
120.w even 4 2 1200.4.f.j 2
168.e odd 2 1 294.4.a.e 1
168.i even 2 1 2352.4.a.e 1
168.v even 6 2 294.4.e.h 2
168.be odd 6 2 294.4.e.g 2
264.p odd 2 1 726.4.a.f 1
312.h even 2 1 1014.4.a.g 1
312.w odd 4 2 1014.4.b.d 2
408.h even 2 1 1734.4.a.d 1
456.l odd 2 1 2166.4.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 24.f even 2 1
18.4.a.a 1 8.d odd 2 1
48.4.a.c 1 24.h odd 2 1
144.4.a.c 1 8.b even 2 1
150.4.a.i 1 120.m even 2 1
150.4.c.d 2 120.q odd 4 2
162.4.c.c 2 72.p odd 6 2
162.4.c.f 2 72.l even 6 2
192.4.a.c 1 3.b odd 2 1
192.4.a.i 1 12.b even 2 1
294.4.a.e 1 168.e odd 2 1
294.4.e.g 2 168.be odd 6 2
294.4.e.h 2 168.v even 6 2
450.4.a.h 1 40.e odd 2 1
450.4.c.e 2 40.k even 4 2
576.4.a.q 1 4.b odd 2 1
576.4.a.r 1 1.a even 1 1 trivial
726.4.a.f 1 264.p odd 2 1
768.4.d.c 2 48.i odd 4 2
768.4.d.n 2 48.k even 4 2
882.4.a.n 1 56.e even 2 1
882.4.g.f 2 56.m even 6 2
882.4.g.i 2 56.k odd 6 2
1014.4.a.g 1 312.h even 2 1
1014.4.b.d 2 312.w odd 4 2
1200.4.a.b 1 120.i odd 2 1
1200.4.f.j 2 120.w even 4 2
1734.4.a.d 1 408.h even 2 1
2166.4.a.i 1 456.l odd 2 1
2178.4.a.e 1 88.g even 2 1
2352.4.a.e 1 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(576))\):

\( T_{5} - 6 \)
\( T_{7} - 16 \)
\( T_{11} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -6 + T \)
$7$ \( -16 + T \)
$11$ \( 12 + T \)
$13$ \( 38 + T \)
$17$ \( -126 + T \)
$19$ \( -20 + T \)
$23$ \( -168 + T \)
$29$ \( -30 + T \)
$31$ \( -88 + T \)
$37$ \( 254 + T \)
$41$ \( 42 + T \)
$43$ \( 52 + T \)
$47$ \( 96 + T \)
$53$ \( -198 + T \)
$59$ \( -660 + T \)
$61$ \( -538 + T \)
$67$ \( -884 + T \)
$71$ \( -792 + T \)
$73$ \( -218 + T \)
$79$ \( -520 + T \)
$83$ \( -492 + T \)
$89$ \( 810 + T \)
$97$ \( -1154 + T \)
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