Properties

Label 576.3.h.b
Level $576$
Weight $3$
Character orbit 576.h
Analytic conductor $15.695$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 576.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.6948632272\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{5} + ( 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{7} +O(q^{10})\) \( q + ( -3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{5} + ( 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{7} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{11} + ( 6 - 12 \zeta_{24}^{4} ) q^{13} + ( -15 \zeta_{24} + 15 \zeta_{24}^{3} + 15 \zeta_{24}^{5} ) q^{17} + 20 \zeta_{24}^{6} q^{19} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - 12 \zeta_{24}^{7} ) q^{23} + 29 q^{25} + ( 15 \zeta_{24} + 15 \zeta_{24}^{3} + 15 \zeta_{24}^{5} - 30 \zeta_{24}^{7} ) q^{29} + ( -60 \zeta_{24}^{2} + 30 \zeta_{24}^{6} ) q^{31} + ( -54 \zeta_{24} - 54 \zeta_{24}^{3} + 54 \zeta_{24}^{5} ) q^{35} + ( -24 + 48 \zeta_{24}^{4} ) q^{37} + ( -51 \zeta_{24} + 51 \zeta_{24}^{3} + 51 \zeta_{24}^{5} ) q^{41} + 40 \zeta_{24}^{6} q^{43} + ( 30 \zeta_{24} - 30 \zeta_{24}^{3} + 30 \zeta_{24}^{5} + 60 \zeta_{24}^{7} ) q^{47} + 59 q^{49} + ( -15 \zeta_{24} - 15 \zeta_{24}^{3} - 15 \zeta_{24}^{5} + 30 \zeta_{24}^{7} ) q^{53} + ( 72 \zeta_{24}^{2} - 36 \zeta_{24}^{6} ) q^{55} + ( -24 \zeta_{24} - 24 \zeta_{24}^{3} + 24 \zeta_{24}^{5} ) q^{59} + ( -54 \zeta_{24} + 54 \zeta_{24}^{3} + 54 \zeta_{24}^{5} ) q^{65} + 100 \zeta_{24}^{6} q^{67} + ( -30 \zeta_{24} + 30 \zeta_{24}^{3} - 30 \zeta_{24}^{5} - 60 \zeta_{24}^{7} ) q^{71} + 20 q^{73} + ( -36 \zeta_{24} - 36 \zeta_{24}^{3} - 36 \zeta_{24}^{5} + 72 \zeta_{24}^{7} ) q^{77} + ( 60 \zeta_{24}^{2} - 30 \zeta_{24}^{6} ) q^{79} + ( -90 \zeta_{24} - 90 \zeta_{24}^{3} + 90 \zeta_{24}^{5} ) q^{83} + ( 90 - 180 \zeta_{24}^{4} ) q^{85} + ( -9 \zeta_{24} + 9 \zeta_{24}^{3} + 9 \zeta_{24}^{5} ) q^{89} -108 \zeta_{24}^{6} q^{91} + ( -60 \zeta_{24} + 60 \zeta_{24}^{3} - 60 \zeta_{24}^{5} - 120 \zeta_{24}^{7} ) q^{95} + 40 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 232q^{25} + 472q^{49} + 160q^{73} + 320q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
0.258819 0.965926i
0.258819 + 0.965926i
0.965926 + 0.258819i
0.965926 0.258819i
−0.258819 + 0.965926i
−0.258819 0.965926i
−0.965926 0.258819i
−0.965926 + 0.258819i
0 0 0 −7.34847 0 −10.3923 0 0 0
161.2 0 0 0 −7.34847 0 −10.3923 0 0 0
161.3 0 0 0 −7.34847 0 10.3923 0 0 0
161.4 0 0 0 −7.34847 0 10.3923 0 0 0
161.5 0 0 0 7.34847 0 −10.3923 0 0 0
161.6 0 0 0 7.34847 0 −10.3923 0 0 0
161.7 0 0 0 7.34847 0 10.3923 0 0 0
161.8 0 0 0 7.34847 0 10.3923 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.3.h.b 8
3.b odd 2 1 inner 576.3.h.b 8
4.b odd 2 1 inner 576.3.h.b 8
8.b even 2 1 inner 576.3.h.b 8
8.d odd 2 1 inner 576.3.h.b 8
12.b even 2 1 inner 576.3.h.b 8
16.e even 4 1 2304.3.e.g 4
16.e even 4 1 2304.3.e.j 4
16.f odd 4 1 2304.3.e.g 4
16.f odd 4 1 2304.3.e.j 4
24.f even 2 1 inner 576.3.h.b 8
24.h odd 2 1 inner 576.3.h.b 8
48.i odd 4 1 2304.3.e.g 4
48.i odd 4 1 2304.3.e.j 4
48.k even 4 1 2304.3.e.g 4
48.k even 4 1 2304.3.e.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.3.h.b 8 1.a even 1 1 trivial
576.3.h.b 8 3.b odd 2 1 inner
576.3.h.b 8 4.b odd 2 1 inner
576.3.h.b 8 8.b even 2 1 inner
576.3.h.b 8 8.d odd 2 1 inner
576.3.h.b 8 12.b even 2 1 inner
576.3.h.b 8 24.f even 2 1 inner
576.3.h.b 8 24.h odd 2 1 inner
2304.3.e.g 4 16.e even 4 1
2304.3.e.g 4 16.f odd 4 1
2304.3.e.g 4 48.i odd 4 1
2304.3.e.g 4 48.k even 4 1
2304.3.e.j 4 16.e even 4 1
2304.3.e.j 4 16.f odd 4 1
2304.3.e.j 4 48.i odd 4 1
2304.3.e.j 4 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 54 \) acting on \(S_{3}^{\mathrm{new}}(576, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( -54 + T^{2} )^{4} \)
$7$ \( ( -108 + T^{2} )^{4} \)
$11$ \( ( -72 + T^{2} )^{4} \)
$13$ \( ( 108 + T^{2} )^{4} \)
$17$ \( ( 450 + T^{2} )^{4} \)
$19$ \( ( 400 + T^{2} )^{4} \)
$23$ \( ( 216 + T^{2} )^{4} \)
$29$ \( ( -1350 + T^{2} )^{4} \)
$31$ \( ( -2700 + T^{2} )^{4} \)
$37$ \( ( 1728 + T^{2} )^{4} \)
$41$ \( ( 5202 + T^{2} )^{4} \)
$43$ \( ( 1600 + T^{2} )^{4} \)
$47$ \( ( 5400 + T^{2} )^{4} \)
$53$ \( ( -1350 + T^{2} )^{4} \)
$59$ \( ( -1152 + T^{2} )^{4} \)
$61$ \( T^{8} \)
$67$ \( ( 10000 + T^{2} )^{4} \)
$71$ \( ( 5400 + T^{2} )^{4} \)
$73$ \( ( -20 + T )^{8} \)
$79$ \( ( -2700 + T^{2} )^{4} \)
$83$ \( ( -16200 + T^{2} )^{4} \)
$89$ \( ( 162 + T^{2} )^{4} \)
$97$ \( ( -40 + T )^{8} \)
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