Properties

Label 576.1.g.a
Level 576
Weight 1
Character orbit 576.g
Self dual yes
Analytic conductor 0.287
Analytic rank 0
Dimension 1
Projective image \(D_{2}\)
CM/RM discs -3, -4, 12
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 576.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.287461447277\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 144)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\zeta_{12})\)
Artin image $D_4$
Artin field Galois closure of 4.0.1728.1

$q$-expansion

\(f(q)\) \(=\) \( q + O(q^{10}) \) \( q + 2q^{13} - q^{25} - 2q^{37} + q^{49} - 2q^{61} - 2q^{73} - 2q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
0
0 0 0 0 0 0 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 \(\Q(\sqrt{-3}) \)
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
12.b even 2 1 RM by \(\Q(\sqrt{3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.1.g.a 1
3.b odd 2 1 CM 576.1.g.a 1
4.b odd 2 1 CM 576.1.g.a 1
8.b even 2 1 144.1.g.a 1
8.d odd 2 1 144.1.g.a 1
12.b even 2 1 RM 576.1.g.a 1
16.e even 4 2 2304.1.b.a 2
16.f odd 4 2 2304.1.b.a 2
24.f even 2 1 144.1.g.a 1
24.h odd 2 1 144.1.g.a 1
40.e odd 2 1 3600.1.e.b 1
40.f even 2 1 3600.1.e.b 1
40.i odd 4 2 3600.1.j.a 2
40.k even 4 2 3600.1.j.a 2
48.i odd 4 2 2304.1.b.a 2
48.k even 4 2 2304.1.b.a 2
72.j odd 6 2 1296.1.o.b 2
72.l even 6 2 1296.1.o.b 2
72.n even 6 2 1296.1.o.b 2
72.p odd 6 2 1296.1.o.b 2
120.i odd 2 1 3600.1.e.b 1
120.m even 2 1 3600.1.e.b 1
120.q odd 4 2 3600.1.j.a 2
120.w even 4 2 3600.1.j.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.1.g.a 1 8.b even 2 1
144.1.g.a 1 8.d odd 2 1
144.1.g.a 1 24.f even 2 1
144.1.g.a 1 24.h odd 2 1
576.1.g.a 1 1.a even 1 1 trivial
576.1.g.a 1 3.b odd 2 1 CM
576.1.g.a 1 4.b odd 2 1 CM
576.1.g.a 1 12.b even 2 1 RM
1296.1.o.b 2 72.j odd 6 2
1296.1.o.b 2 72.l even 6 2
1296.1.o.b 2 72.n even 6 2
1296.1.o.b 2 72.p odd 6 2
2304.1.b.a 2 16.e even 4 2
2304.1.b.a 2 16.f odd 4 2
2304.1.b.a 2 48.i odd 4 2
2304.1.b.a 2 48.k even 4 2
3600.1.e.b 1 40.e odd 2 1
3600.1.e.b 1 40.f even 2 1
3600.1.e.b 1 120.i odd 2 1
3600.1.e.b 1 120.m even 2 1
3600.1.j.a 2 40.i odd 4 2
3600.1.j.a 2 40.k even 4 2
3600.1.j.a 2 120.q odd 4 2
3600.1.j.a 2 120.w even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(576, [\chi])\).

Hecke characteristic polynomials

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