Properties

Label 576.2.r.c
Level $576$
Weight $2$
Character orbit 576.r
Analytic conductor $4.599$
Analytic rank $0$
Dimension $8$
CM discriminant -8
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(4.59938315643\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_{4}) q^{3} + ( - \beta_{7} - \beta_{2} + \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} - \beta_{4}) q^{3} + ( - \beta_{7} - \beta_{2} + \beta_1 - 1) q^{9} + (2 \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3}) q^{11} + ( - 2 \beta_{7} - \beta_{2} + 3) q^{17} + (3 \beta_{6} - \beta_{5} - 2 \beta_{4} + 3 \beta_{3}) q^{19} - 5 \beta_1 q^{25} + ( - 3 \beta_{5} + 2 \beta_{4}) q^{27} + ( - \beta_{7} - 2 \beta_{2} - 5 \beta_1 + 1) q^{33} + ( - 2 \beta_{7} + 2 \beta_{2} + 3 \beta_1 - 3) q^{41} + ( - \beta_{6} - \beta_{5} + 4 \beta_{4} + 4 \beta_{3}) q^{43} + ( - 7 \beta_1 + 7) q^{49} + (4 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} - 3 \beta_{3}) q^{51} + ( - 2 \beta_{7} - 3 \beta_{2} + 5 \beta_1 - 12) q^{57} + ( - 4 \beta_{6} + 5 \beta_{5} + 5 \beta_{4} - \beta_{3}) q^{59} + ( - 2 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} + 5 \beta_{3}) q^{67} + (6 \beta_{7} + 3 \beta_{2} - 1) q^{73} - 5 \beta_{6} q^{75} + (2 \beta_{7} + 7 \beta_1) q^{81} + (9 \beta_{6} + 9 \beta_{5} - 9 \beta_{4} - 9 \beta_{3}) q^{83} + 18 q^{89} + (3 \beta_{7} + 6 \beta_{2} + 5 \beta_1) q^{97} + ( - 5 \beta_{6} - 6 \beta_{5} + \beta_{4} + 3 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{9} + 24 q^{17} - 20 q^{25} - 12 q^{33} - 12 q^{41} + 28 q^{49} - 76 q^{57} - 8 q^{73} + 28 q^{81} + 144 q^{89} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{24}^{5} + 2\zeta_{24}^{3} - 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3} - \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{6} - \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{6} - \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -2\zeta_{24}^{7} + 2\zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{6} - \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{7} - \beta_{6} - \beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( -\beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} ) / 4 \) Copy content Toggle raw display

Character values

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

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(-1 + \beta_{1}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
−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 −1.57313 + 0.724745i 0 0 0 0 0 1.94949 2.28024i 0
97.2 0 −0.158919 1.72474i 0 0 0 0 0 −2.94949 + 0.548188i 0
97.3 0 0.158919 + 1.72474i 0 0 0 0 0 −2.94949 + 0.548188i 0
97.4 0 1.57313 0.724745i 0 0 0 0 0 1.94949 2.28024i 0
481.1 0 −1.57313 0.724745i 0 0 0 0 0 1.94949 + 2.28024i 0
481.2 0 −0.158919 + 1.72474i 0 0 0 0 0 −2.94949 0.548188i 0
481.3 0 0.158919 1.72474i 0 0 0 0 0 −2.94949 0.548188i 0
481.4 0 1.57313 + 0.724745i 0 0 0 0 0 1.94949 + 2.28024i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 481.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
4.b odd 2 1 inner
8.b even 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner
72.n even 6 1 inner
72.p odd 6 1 inner

Twists

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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{6} - 5 T^{4} + 18 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 30 T^{6} + 891 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{2} - 6 T - 15)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 110 T^{2} + 2809)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} + 6 T^{3} + 123 T^{2} - 522 T + 7569)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 158 T^{6} + 24123 T^{4} + \cdots + 707281 \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} - 318 T^{6} + \cdots + 395254161 \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} - 206 T^{6} + 42411 T^{4} + \cdots + 625 \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{2} + 2 T - 215)^{4} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} - 324 T^{2} + 104976)^{2} \) Copy content Toggle raw display
$89$ \( (T - 18)^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} - 10 T^{3} + 291 T^{2} + \cdots + 36481)^{2} \) Copy content Toggle raw display
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