Properties

Label 576.1.o.a
Level $576$
Weight $1$
Character orbit 576.o
Analytic conductor $0.287$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.287461447277\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 288)
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.5184.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{12}^{3} q^{3} -\zeta_{12}^{2} q^{5} -\zeta_{12} q^{7} - q^{9} +O(q^{10})\) \( q -\zeta_{12}^{3} q^{3} -\zeta_{12}^{2} q^{5} -\zeta_{12} q^{7} - q^{9} -\zeta_{12} q^{11} + \zeta_{12}^{2} q^{13} + \zeta_{12}^{5} q^{15} + \zeta_{12}^{4} q^{21} -\zeta_{12}^{5} q^{23} + \zeta_{12}^{3} q^{27} -\zeta_{12}^{4} q^{29} -\zeta_{12}^{5} q^{31} + \zeta_{12}^{4} q^{33} + \zeta_{12}^{3} q^{35} -\zeta_{12}^{5} q^{39} -\zeta_{12}^{2} q^{41} + \zeta_{12} q^{43} + \zeta_{12}^{2} q^{45} -\zeta_{12} q^{47} + \zeta_{12}^{3} q^{55} -\zeta_{12}^{5} q^{59} -\zeta_{12}^{4} q^{61} + \zeta_{12} q^{63} -\zeta_{12}^{4} q^{65} + \zeta_{12}^{5} q^{67} -\zeta_{12}^{2} q^{69} + 2 \zeta_{12}^{3} q^{71} + \zeta_{12}^{2} q^{77} + \zeta_{12} q^{79} + q^{81} + \zeta_{12} q^{83} -\zeta_{12} q^{87} -\zeta_{12}^{3} q^{91} -\zeta_{12}^{2} q^{93} + \zeta_{12}^{4} q^{97} + \zeta_{12} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{5} - 4 q^{9} + O(q^{10}) \) \( 4 q - 2 q^{5} - 4 q^{9} + 2 q^{13} - 2 q^{21} + 2 q^{29} - 2 q^{33} - 2 q^{41} + 2 q^{45} + 2 q^{61} + 2 q^{65} - 2 q^{69} + 2 q^{77} + 4 q^{81} - 2 q^{93} - 2 q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
−0.866025 + 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 1.00000i 0 −0.500000 + 0.866025i 0 0.866025 0.500000i 0 −1.00000 0
319.2 0 1.00000i 0 −0.500000 + 0.866025i 0 −0.866025 + 0.500000i 0 −1.00000 0
511.1 0 1.00000i 0 −0.500000 0.866025i 0 −0.866025 0.500000i 0 −1.00000 0
511.2 0 1.00000i 0 −0.500000 0.866025i 0 0.866025 + 0.500000i 0 −1.00000 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
4.b odd 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.1.o.a 4
3.b odd 2 1 1728.1.o.a 4
4.b odd 2 1 inner 576.1.o.a 4
8.b even 2 1 288.1.o.a 4
8.d odd 2 1 288.1.o.a 4
9.c even 3 1 inner 576.1.o.a 4
9.d odd 6 1 1728.1.o.a 4
12.b even 2 1 1728.1.o.a 4
16.e even 4 1 2304.1.t.a 4
16.e even 4 1 2304.1.t.b 4
16.f odd 4 1 2304.1.t.a 4
16.f odd 4 1 2304.1.t.b 4
24.f even 2 1 864.1.o.a 4
24.h odd 2 1 864.1.o.a 4
36.f odd 6 1 inner 576.1.o.a 4
36.h even 6 1 1728.1.o.a 4
72.j odd 6 1 864.1.o.a 4
72.j odd 6 1 2592.1.g.b 2
72.l even 6 1 864.1.o.a 4
72.l even 6 1 2592.1.g.b 2
72.n even 6 1 288.1.o.a 4
72.n even 6 1 2592.1.g.a 2
72.p odd 6 1 288.1.o.a 4
72.p odd 6 1 2592.1.g.a 2
144.v odd 12 1 2304.1.t.a 4
144.v odd 12 1 2304.1.t.b 4
144.x even 12 1 2304.1.t.a 4
144.x even 12 1 2304.1.t.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.1.o.a 4 8.b even 2 1
288.1.o.a 4 8.d odd 2 1
288.1.o.a 4 72.n even 6 1
288.1.o.a 4 72.p odd 6 1
576.1.o.a 4 1.a even 1 1 trivial
576.1.o.a 4 4.b odd 2 1 inner
576.1.o.a 4 9.c even 3 1 inner
576.1.o.a 4 36.f odd 6 1 inner
864.1.o.a 4 24.f even 2 1
864.1.o.a 4 24.h odd 2 1
864.1.o.a 4 72.j odd 6 1
864.1.o.a 4 72.l even 6 1
1728.1.o.a 4 3.b odd 2 1
1728.1.o.a 4 9.d odd 6 1
1728.1.o.a 4 12.b even 2 1
1728.1.o.a 4 36.h even 6 1
2304.1.t.a 4 16.e even 4 1
2304.1.t.a 4 16.f odd 4 1
2304.1.t.a 4 144.v odd 12 1
2304.1.t.a 4 144.x even 12 1
2304.1.t.b 4 16.e even 4 1
2304.1.t.b 4 16.f odd 4 1
2304.1.t.b 4 144.v odd 12 1
2304.1.t.b 4 144.x even 12 1
2592.1.g.a 2 72.n even 6 1
2592.1.g.a 2 72.p odd 6 1
2592.1.g.b 2 72.j odd 6 1
2592.1.g.b 2 72.l even 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( ( 1 + T + T^{2} )^{2} \)
$7$ \( 1 - T^{2} + T^{4} \)
$11$ \( 1 - T^{2} + T^{4} \)
$13$ \( ( 1 - T + T^{2} )^{2} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( 1 - T^{2} + T^{4} \)
$29$ \( ( 1 - T + T^{2} )^{2} \)
$31$ \( 1 - T^{2} + T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( 1 + T + T^{2} )^{2} \)
$43$ \( 1 - T^{2} + T^{4} \)
$47$ \( 1 - T^{2} + T^{4} \)
$53$ \( T^{4} \)
$59$ \( 1 - T^{2} + T^{4} \)
$61$ \( ( 1 - T + T^{2} )^{2} \)
$67$ \( 1 - T^{2} + T^{4} \)
$71$ \( ( 4 + T^{2} )^{2} \)
$73$ \( T^{4} \)
$79$ \( 1 - T^{2} + T^{4} \)
$83$ \( 1 - T^{2} + T^{4} \)
$89$ \( T^{4} \)
$97$ \( ( 1 + T + T^{2} )^{2} \)
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