Properties

Label 576.2.s.d
Level $576$
Weight $2$
Character orbit 576.s
Analytic conductor $4.599$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(4.59938315643\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \zeta_{6} ) q^{3} + ( 4 - 2 \zeta_{6} ) q^{5} + ( -2 - 2 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 + \zeta_{6} ) q^{3} + ( 4 - 2 \zeta_{6} ) q^{5} + ( -2 - 2 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} + ( 3 - 3 \zeta_{6} ) q^{11} + 4 \zeta_{6} q^{13} + 6 q^{15} + ( 1 - 2 \zeta_{6} ) q^{17} + ( 1 - 2 \zeta_{6} ) q^{19} -6 \zeta_{6} q^{21} + ( 7 - 7 \zeta_{6} ) q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( 2 + 2 \zeta_{6} ) q^{29} + ( 6 - 3 \zeta_{6} ) q^{33} -12 q^{35} -2 q^{37} + ( -4 + 8 \zeta_{6} ) q^{39} + ( -6 + 3 \zeta_{6} ) q^{41} + ( -3 - 3 \zeta_{6} ) q^{43} + ( 6 + 6 \zeta_{6} ) q^{45} + ( -12 + 12 \zeta_{6} ) q^{47} + 5 \zeta_{6} q^{49} + ( 3 - 3 \zeta_{6} ) q^{51} + ( 6 - 12 \zeta_{6} ) q^{55} + ( 3 - 3 \zeta_{6} ) q^{57} + 15 \zeta_{6} q^{59} + ( 8 - 8 \zeta_{6} ) q^{61} + ( 6 - 12 \zeta_{6} ) q^{63} + ( 8 + 8 \zeta_{6} ) q^{65} + ( 10 - 5 \zeta_{6} ) q^{67} -6 q^{71} -11 q^{73} + ( 14 - 7 \zeta_{6} ) q^{75} + ( -12 + 6 \zeta_{6} ) q^{77} + ( 2 + 2 \zeta_{6} ) q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + ( -12 + 12 \zeta_{6} ) q^{83} -6 \zeta_{6} q^{85} + 6 \zeta_{6} q^{87} + ( 8 - 16 \zeta_{6} ) q^{89} + ( 8 - 16 \zeta_{6} ) q^{91} -6 \zeta_{6} q^{95} + ( -13 + 13 \zeta_{6} ) q^{97} + 9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{3} + 6q^{5} - 6q^{7} + 3q^{9} + O(q^{10}) \) \( 2q + 3q^{3} + 6q^{5} - 6q^{7} + 3q^{9} + 3q^{11} + 4q^{13} + 12q^{15} - 6q^{21} + 7q^{25} + 6q^{29} + 9q^{33} - 24q^{35} - 4q^{37} - 9q^{41} - 9q^{43} + 18q^{45} - 12q^{47} + 5q^{49} + 3q^{51} + 3q^{57} + 15q^{59} + 8q^{61} + 24q^{65} + 15q^{67} - 12q^{71} - 22q^{73} + 21q^{75} - 18q^{77} + 6q^{79} - 9q^{81} - 12q^{83} - 6q^{85} + 6q^{87} - 6q^{95} - 13q^{97} + 18q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(\zeta_{6}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.50000 + 0.866025i 0 3.00000 1.73205i 0 −3.00000 1.73205i 0 1.50000 + 2.59808i 0
383.1 0 1.50000 0.866025i 0 3.00000 + 1.73205i 0 −3.00000 + 1.73205i 0 1.50000 2.59808i 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
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.2.s.d 2
3.b odd 2 1 1728.2.s.a 2
4.b odd 2 1 576.2.s.a 2
8.b even 2 1 144.2.s.a 2
8.d odd 2 1 144.2.s.d yes 2
9.c even 3 1 1728.2.s.b 2
9.c even 3 1 5184.2.c.a 2
9.d odd 6 1 576.2.s.a 2
9.d odd 6 1 5184.2.c.c 2
12.b even 2 1 1728.2.s.b 2
24.f even 2 1 432.2.s.d 2
24.h odd 2 1 432.2.s.c 2
36.f odd 6 1 1728.2.s.a 2
36.f odd 6 1 5184.2.c.c 2
36.h even 6 1 inner 576.2.s.d 2
36.h even 6 1 5184.2.c.a 2
72.j odd 6 1 144.2.s.d yes 2
72.j odd 6 1 1296.2.c.b 2
72.l even 6 1 144.2.s.a 2
72.l even 6 1 1296.2.c.d 2
72.n even 6 1 432.2.s.d 2
72.n even 6 1 1296.2.c.d 2
72.p odd 6 1 432.2.s.c 2
72.p odd 6 1 1296.2.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.s.a 2 8.b even 2 1
144.2.s.a 2 72.l even 6 1
144.2.s.d yes 2 8.d odd 2 1
144.2.s.d yes 2 72.j odd 6 1
432.2.s.c 2 24.h odd 2 1
432.2.s.c 2 72.p odd 6 1
432.2.s.d 2 24.f even 2 1
432.2.s.d 2 72.n even 6 1
576.2.s.a 2 4.b odd 2 1
576.2.s.a 2 9.d odd 6 1
576.2.s.d 2 1.a even 1 1 trivial
576.2.s.d 2 36.h even 6 1 inner
1296.2.c.b 2 72.j odd 6 1
1296.2.c.b 2 72.p odd 6 1
1296.2.c.d 2 72.l even 6 1
1296.2.c.d 2 72.n even 6 1
1728.2.s.a 2 3.b odd 2 1
1728.2.s.a 2 36.f odd 6 1
1728.2.s.b 2 9.c even 3 1
1728.2.s.b 2 12.b even 2 1
5184.2.c.a 2 9.c even 3 1
5184.2.c.a 2 36.h even 6 1
5184.2.c.c 2 9.d odd 6 1
5184.2.c.c 2 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(576, [\chi])\):

\( T_{5}^{2} - 6 T_{5} + 12 \)
\( T_{7}^{2} + 6 T_{7} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 3 - 3 T + T^{2} \)
$5$ \( 12 - 6 T + T^{2} \)
$7$ \( 12 + 6 T + T^{2} \)
$11$ \( 9 - 3 T + T^{2} \)
$13$ \( 16 - 4 T + T^{2} \)
$17$ \( 3 + T^{2} \)
$19$ \( 3 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( 12 - 6 T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( ( 2 + T )^{2} \)
$41$ \( 27 + 9 T + T^{2} \)
$43$ \( 27 + 9 T + T^{2} \)
$47$ \( 144 + 12 T + T^{2} \)
$53$ \( T^{2} \)
$59$ \( 225 - 15 T + T^{2} \)
$61$ \( 64 - 8 T + T^{2} \)
$67$ \( 75 - 15 T + T^{2} \)
$71$ \( ( 6 + T )^{2} \)
$73$ \( ( 11 + T )^{2} \)
$79$ \( 12 - 6 T + T^{2} \)
$83$ \( 144 + 12 T + T^{2} \)
$89$ \( 192 + T^{2} \)
$97$ \( 169 + 13 T + T^{2} \)
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