Properties

Label 576.3.g.e
Level $576$
Weight $3$
Character orbit 576.g
Analytic conductor $15.695$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 576.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.6948632272\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 12)
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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 q^{5} + ( 4 - 8 \zeta_{6} ) q^{7} +O(q^{10})\) \( q -2 q^{5} + ( 4 - 8 \zeta_{6} ) q^{7} + ( -4 + 8 \zeta_{6} ) q^{11} -2 q^{13} -10 q^{17} + ( 12 - 24 \zeta_{6} ) q^{19} + ( -16 + 32 \zeta_{6} ) q^{23} -21 q^{25} -26 q^{29} + ( -4 + 8 \zeta_{6} ) q^{31} + ( -8 + 16 \zeta_{6} ) q^{35} -26 q^{37} -58 q^{41} + ( -28 + 56 \zeta_{6} ) q^{43} + ( 40 - 80 \zeta_{6} ) q^{47} + q^{49} -74 q^{53} + ( 8 - 16 \zeta_{6} ) q^{55} + ( 52 - 104 \zeta_{6} ) q^{59} -26 q^{61} + 4 q^{65} + ( 4 - 8 \zeta_{6} ) q^{67} -46 q^{73} + 48 q^{77} + ( -68 + 136 \zeta_{6} ) q^{79} + ( -28 + 56 \zeta_{6} ) q^{83} + 20 q^{85} -82 q^{89} + ( -8 + 16 \zeta_{6} ) q^{91} + ( -24 + 48 \zeta_{6} ) q^{95} + 2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{5} + O(q^{10}) \) \( 2q - 4q^{5} - 4q^{13} - 20q^{17} - 42q^{25} - 52q^{29} - 52q^{37} - 116q^{41} + 2q^{49} - 148q^{53} - 52q^{61} + 8q^{65} - 92q^{73} + 96q^{77} + 40q^{85} - 164q^{89} + 4q^{97} + O(q^{100}) \)

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
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 −2.00000 0 6.92820i 0 0 0
127.2 0 0 0 −2.00000 0 6.92820i 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
4.b 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.g.e 2
3.b odd 2 1 192.3.g.b 2
4.b odd 2 1 inner 576.3.g.e 2
8.b even 2 1 36.3.d.c 2
8.d odd 2 1 36.3.d.c 2
12.b even 2 1 192.3.g.b 2
16.e even 4 2 2304.3.b.l 4
16.f odd 4 2 2304.3.b.l 4
24.f even 2 1 12.3.d.a 2
24.h odd 2 1 12.3.d.a 2
40.e odd 2 1 900.3.c.e 2
40.f even 2 1 900.3.c.e 2
40.i odd 4 2 900.3.f.c 4
40.k even 4 2 900.3.f.c 4
48.i odd 4 2 768.3.b.c 4
48.k even 4 2 768.3.b.c 4
72.j odd 6 1 324.3.f.d 2
72.j odd 6 1 324.3.f.j 2
72.l even 6 1 324.3.f.d 2
72.l even 6 1 324.3.f.j 2
72.n even 6 1 324.3.f.a 2
72.n even 6 1 324.3.f.g 2
72.p odd 6 1 324.3.f.a 2
72.p odd 6 1 324.3.f.g 2
120.i odd 2 1 300.3.c.b 2
120.m even 2 1 300.3.c.b 2
120.q odd 4 2 300.3.f.a 4
120.w even 4 2 300.3.f.a 4
168.e odd 2 1 588.3.g.b 2
168.i even 2 1 588.3.g.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.3.d.a 2 24.f even 2 1
12.3.d.a 2 24.h odd 2 1
36.3.d.c 2 8.b even 2 1
36.3.d.c 2 8.d odd 2 1
192.3.g.b 2 3.b odd 2 1
192.3.g.b 2 12.b even 2 1
300.3.c.b 2 120.i odd 2 1
300.3.c.b 2 120.m even 2 1
300.3.f.a 4 120.q odd 4 2
300.3.f.a 4 120.w even 4 2
324.3.f.a 2 72.n even 6 1
324.3.f.a 2 72.p odd 6 1
324.3.f.d 2 72.j odd 6 1
324.3.f.d 2 72.l even 6 1
324.3.f.g 2 72.n even 6 1
324.3.f.g 2 72.p odd 6 1
324.3.f.j 2 72.j odd 6 1
324.3.f.j 2 72.l even 6 1
576.3.g.e 2 1.a even 1 1 trivial
576.3.g.e 2 4.b odd 2 1 inner
588.3.g.b 2 168.e odd 2 1
588.3.g.b 2 168.i even 2 1
768.3.b.c 4 48.i odd 4 2
768.3.b.c 4 48.k even 4 2
900.3.c.e 2 40.e odd 2 1
900.3.c.e 2 40.f even 2 1
900.3.f.c 4 40.i odd 4 2
900.3.f.c 4 40.k even 4 2
2304.3.b.l 4 16.e even 4 2
2304.3.b.l 4 16.f odd 4 2

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 2 + T )^{2} \)
$7$ \( 48 + T^{2} \)
$11$ \( 48 + T^{2} \)
$13$ \( ( 2 + T )^{2} \)
$17$ \( ( 10 + T )^{2} \)
$19$ \( 432 + T^{2} \)
$23$ \( 768 + T^{2} \)
$29$ \( ( 26 + T )^{2} \)
$31$ \( 48 + T^{2} \)
$37$ \( ( 26 + T )^{2} \)
$41$ \( ( 58 + T )^{2} \)
$43$ \( 2352 + T^{2} \)
$47$ \( 4800 + T^{2} \)
$53$ \( ( 74 + T )^{2} \)
$59$ \( 8112 + T^{2} \)
$61$ \( ( 26 + T )^{2} \)
$67$ \( 48 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 46 + T )^{2} \)
$79$ \( 13872 + T^{2} \)
$83$ \( 2352 + T^{2} \)
$89$ \( ( 82 + T )^{2} \)
$97$ \( ( -2 + T )^{2} \)
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