Properties

Label 576.2.s.e
Level $576$
Weight $2$
Character orbit 576.s
Analytic conductor $4.599$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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: \(4\)
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: \( 3 \)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( -2 + \zeta_{12}^{2} ) q^{5} -3 \zeta_{12} q^{7} + 3 q^{9} +O(q^{10})\) \( q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( -2 + \zeta_{12}^{2} ) q^{5} -3 \zeta_{12} q^{7} + 3 q^{9} + ( 3 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{11} + \zeta_{12}^{2} q^{13} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{15} + ( -2 + 4 \zeta_{12}^{2} ) q^{17} + 6 \zeta_{12}^{3} q^{19} + ( 3 + 3 \zeta_{12}^{2} ) q^{21} + ( 3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{23} + ( -2 + 2 \zeta_{12}^{2} ) q^{25} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} + ( 5 + 5 \zeta_{12}^{2} ) q^{29} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{31} + ( -9 + 9 \zeta_{12}^{2} ) q^{33} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{35} + 4 q^{37} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{39} + ( 6 - 3 \zeta_{12}^{2} ) q^{41} + 3 \zeta_{12} q^{43} + ( -6 + 3 \zeta_{12}^{2} ) q^{45} + ( 3 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{47} + 2 \zeta_{12}^{2} q^{49} -6 \zeta_{12}^{3} q^{51} + ( 6 - 12 \zeta_{12}^{2} ) q^{53} + 9 \zeta_{12}^{3} q^{55} + ( 6 - 12 \zeta_{12}^{2} ) q^{57} + ( 3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{59} + ( -7 + 7 \zeta_{12}^{2} ) q^{61} -9 \zeta_{12} q^{63} + ( -1 - \zeta_{12}^{2} ) q^{65} + ( -9 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{67} -9 \zeta_{12}^{2} q^{69} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{71} + 4 q^{73} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{75} + ( -18 + 9 \zeta_{12}^{2} ) q^{77} + 15 \zeta_{12} q^{79} + 9 q^{81} + ( -3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{83} -6 \zeta_{12}^{2} q^{85} -15 \zeta_{12} q^{87} + ( 2 - 4 \zeta_{12}^{2} ) q^{89} -3 \zeta_{12}^{3} q^{91} + ( 6 - 3 \zeta_{12}^{2} ) q^{93} + ( -6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{95} + ( -1 + \zeta_{12}^{2} ) q^{97} + ( 9 \zeta_{12} - 18 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 6q^{5} + 12q^{9} + O(q^{10}) \) \( 4q - 6q^{5} + 12q^{9} + 2q^{13} + 18q^{21} - 4q^{25} + 30q^{29} - 18q^{33} + 16q^{37} + 18q^{41} - 18q^{45} + 4q^{49} - 14q^{61} - 6q^{65} - 18q^{69} + 16q^{73} - 54q^{77} + 36q^{81} - 12q^{85} + 18q^{93} - 2q^{97} + O(q^{100}) \)

Character values

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

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

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
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 −1.73205 0 −1.50000 + 0.866025i 0 −2.59808 1.50000i 0 3.00000 0
191.2 0 1.73205 0 −1.50000 + 0.866025i 0 2.59808 + 1.50000i 0 3.00000 0
383.1 0 −1.73205 0 −1.50000 0.866025i 0 −2.59808 + 1.50000i 0 3.00000 0
383.2 0 1.73205 0 −1.50000 0.866025i 0 2.59808 1.50000i 0 3.00000 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.d odd 6 1 inner
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.e 4
3.b odd 2 1 1728.2.s.e 4
4.b odd 2 1 inner 576.2.s.e 4
8.b even 2 1 144.2.s.e 4
8.d odd 2 1 144.2.s.e 4
9.c even 3 1 1728.2.s.e 4
9.c even 3 1 5184.2.c.f 4
9.d odd 6 1 inner 576.2.s.e 4
9.d odd 6 1 5184.2.c.f 4
12.b even 2 1 1728.2.s.e 4
24.f even 2 1 432.2.s.e 4
24.h odd 2 1 432.2.s.e 4
36.f odd 6 1 1728.2.s.e 4
36.f odd 6 1 5184.2.c.f 4
36.h even 6 1 inner 576.2.s.e 4
36.h even 6 1 5184.2.c.f 4
72.j odd 6 1 144.2.s.e 4
72.j odd 6 1 1296.2.c.f 4
72.l even 6 1 144.2.s.e 4
72.l even 6 1 1296.2.c.f 4
72.n even 6 1 432.2.s.e 4
72.n even 6 1 1296.2.c.f 4
72.p odd 6 1 432.2.s.e 4
72.p odd 6 1 1296.2.c.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.s.e 4 8.b even 2 1
144.2.s.e 4 8.d odd 2 1
144.2.s.e 4 72.j odd 6 1
144.2.s.e 4 72.l even 6 1
432.2.s.e 4 24.f even 2 1
432.2.s.e 4 24.h odd 2 1
432.2.s.e 4 72.n even 6 1
432.2.s.e 4 72.p odd 6 1
576.2.s.e 4 1.a even 1 1 trivial
576.2.s.e 4 4.b odd 2 1 inner
576.2.s.e 4 9.d odd 6 1 inner
576.2.s.e 4 36.h even 6 1 inner
1296.2.c.f 4 72.j odd 6 1
1296.2.c.f 4 72.l even 6 1
1296.2.c.f 4 72.n even 6 1
1296.2.c.f 4 72.p odd 6 1
1728.2.s.e 4 3.b odd 2 1
1728.2.s.e 4 9.c even 3 1
1728.2.s.e 4 12.b even 2 1
1728.2.s.e 4 36.f odd 6 1
5184.2.c.f 4 9.c even 3 1
5184.2.c.f 4 9.d odd 6 1
5184.2.c.f 4 36.f odd 6 1
5184.2.c.f 4 36.h even 6 1

Hecke kernels

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

\( T_{5}^{2} + 3 T_{5} + 3 \)
\( T_{7}^{4} - 9 T_{7}^{2} + 81 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -3 + T^{2} )^{2} \)
$5$ \( ( 3 + 3 T + T^{2} )^{2} \)
$7$ \( 81 - 9 T^{2} + T^{4} \)
$11$ \( 729 + 27 T^{2} + T^{4} \)
$13$ \( ( 1 - T + T^{2} )^{2} \)
$17$ \( ( 12 + T^{2} )^{2} \)
$19$ \( ( 36 + T^{2} )^{2} \)
$23$ \( 729 + 27 T^{2} + T^{4} \)
$29$ \( ( 75 - 15 T + T^{2} )^{2} \)
$31$ \( 81 - 9 T^{2} + T^{4} \)
$37$ \( ( -4 + T )^{4} \)
$41$ \( ( 27 - 9 T + T^{2} )^{2} \)
$43$ \( 81 - 9 T^{2} + T^{4} \)
$47$ \( 729 + 27 T^{2} + T^{4} \)
$53$ \( ( 108 + T^{2} )^{2} \)
$59$ \( 729 + 27 T^{2} + T^{4} \)
$61$ \( ( 49 + 7 T + T^{2} )^{2} \)
$67$ \( 6561 - 81 T^{2} + T^{4} \)
$71$ \( ( -108 + T^{2} )^{2} \)
$73$ \( ( -4 + T )^{4} \)
$79$ \( 50625 - 225 T^{2} + T^{4} \)
$83$ \( 729 + 27 T^{2} + T^{4} \)
$89$ \( ( 12 + T^{2} )^{2} \)
$97$ \( ( 1 + T + T^{2} )^{2} \)
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