Properties

Label 576.7.g.h
Level $576$
Weight $7$
Character orbit 576.g
Analytic conductor $132.511$
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: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 576.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(132.511152165\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-15}) \)
Defining polynomial: \(x^{2} - x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 4)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 8\sqrt{-15}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 10 q^{5} + 10 \beta q^{7} +O(q^{10})\) \( q + 10 q^{5} + 10 \beta q^{7} -31 \beta q^{11} -1466 q^{13} + 4766 q^{17} -243 \beta q^{19} + 338 \beta q^{23} -15525 q^{25} + 25498 q^{29} + 1352 \beta q^{31} + 100 \beta q^{35} -1994 q^{37} -29362 q^{41} + 695 \beta q^{43} + 244 \beta q^{47} + 21649 q^{49} -192854 q^{53} -310 \beta q^{55} -2531 \beta q^{59} + 10918 q^{61} -14660 q^{65} + 12721 \beta q^{67} -17178 \beta q^{71} + 288626 q^{73} + 297600 q^{77} -10028 \beta q^{79} -6589 \beta q^{83} + 47660 q^{85} -310738 q^{89} -14660 \beta q^{91} -2430 \beta q^{95} -1457086 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 20q^{5} + O(q^{10}) \) \( 2q + 20q^{5} - 2932q^{13} + 9532q^{17} - 31050q^{25} + 50996q^{29} - 3988q^{37} - 58724q^{41} + 43298q^{49} - 385708q^{53} + 21836q^{61} - 29320q^{65} + 577252q^{73} + 595200q^{77} + 95320q^{85} - 621476q^{89} - 2914172q^{97} + O(q^{100}) \)

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 1.93649i
0.500000 + 1.93649i
0 0 0 10.0000 0 309.839i 0 0 0
127.2 0 0 0 10.0000 0 309.839i 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 576.7.g.h 2
3.b odd 2 1 64.7.c.c 2
4.b odd 2 1 inner 576.7.g.h 2
8.b even 2 1 36.7.d.c 2
8.d odd 2 1 36.7.d.c 2
12.b even 2 1 64.7.c.c 2
24.f even 2 1 4.7.b.a 2
24.h odd 2 1 4.7.b.a 2
48.i odd 4 2 256.7.d.f 4
48.k even 4 2 256.7.d.f 4
120.i odd 2 1 100.7.b.c 2
120.m even 2 1 100.7.b.c 2
120.q odd 4 2 100.7.d.a 4
120.w even 4 2 100.7.d.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.7.b.a 2 24.f even 2 1
4.7.b.a 2 24.h odd 2 1
36.7.d.c 2 8.b even 2 1
36.7.d.c 2 8.d odd 2 1
64.7.c.c 2 3.b odd 2 1
64.7.c.c 2 12.b even 2 1
100.7.b.c 2 120.i odd 2 1
100.7.b.c 2 120.m even 2 1
100.7.d.a 4 120.q odd 4 2
100.7.d.a 4 120.w even 4 2
256.7.d.f 4 48.i odd 4 2
256.7.d.f 4 48.k even 4 2
576.7.g.h 2 1.a even 1 1 trivial
576.7.g.h 2 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 10 \) acting on \(S_{7}^{\mathrm{new}}(576, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -10 + T )^{2} \)
$7$ \( 96000 + T^{2} \)
$11$ \( 922560 + T^{2} \)
$13$ \( ( 1466 + T )^{2} \)
$17$ \( ( -4766 + T )^{2} \)
$19$ \( 56687040 + T^{2} \)
$23$ \( 109674240 + T^{2} \)
$29$ \( ( -25498 + T )^{2} \)
$31$ \( 1754787840 + T^{2} \)
$37$ \( ( 1994 + T )^{2} \)
$41$ \( ( 29362 + T )^{2} \)
$43$ \( 463704000 + T^{2} \)
$47$ \( 57154560 + T^{2} \)
$53$ \( ( 192854 + T )^{2} \)
$59$ \( 6149722560 + T^{2} \)
$61$ \( ( -10918 + T )^{2} \)
$67$ \( 155350887360 + T^{2} \)
$71$ \( 283280336640 + T^{2} \)
$73$ \( ( -288626 + T )^{2} \)
$79$ \( 96538352640 + T^{2} \)
$83$ \( 41678324160 + T^{2} \)
$89$ \( ( 310738 + T )^{2} \)
$97$ \( ( 1457086 + T )^{2} \)
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