Properties

Label 2304.3.g.t
Level $2304$
Weight $3$
Character orbit 2304.g
Analytic conductor $62.779$
Analytic rank $0$
Dimension $4$
CM discriminant -24
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 576)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{5} -5 \beta_{1} q^{7} +O(q^{10})\) \( q -\beta_{2} q^{5} -5 \beta_{1} q^{7} + \beta_{3} q^{11} + 71 q^{25} -3 \beta_{2} q^{29} + 19 \beta_{1} q^{31} + 5 \beta_{3} q^{35} -51 q^{49} + 5 \beta_{2} q^{53} -96 \beta_{1} q^{55} + 6 \beta_{3} q^{59} + 50 q^{73} + 20 \beta_{2} q^{77} -29 \beta_{1} q^{79} -5 \beta_{3} q^{83} -190 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 284q^{25} - 204q^{49} + 200q^{73} - 760q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu^{2} \)\(/3\)
\(\beta_{2}\)\(=\)\((\)\( -4 \nu^{3} + 12 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 8 \nu^{3} + 24 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 2 \beta_{2}\)\()/16\)
\(\nu^{2}\)\(=\)\(3 \beta_{1}\)\(/2\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{3} - 6 \beta_{2}\)\()/16\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1279.1
1.22474 + 1.22474i
1.22474 1.22474i
−1.22474 1.22474i
−1.22474 + 1.22474i
0 0 0 −9.79796 0 10.0000i 0 0 0
1279.2 0 0 0 −9.79796 0 10.0000i 0 0 0
1279.3 0 0 0 9.79796 0 10.0000i 0 0 0
1279.4 0 0 0 9.79796 0 10.0000i 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
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.3.g.t 4
3.b odd 2 1 inner 2304.3.g.t 4
4.b odd 2 1 inner 2304.3.g.t 4
8.b even 2 1 inner 2304.3.g.t 4
8.d odd 2 1 inner 2304.3.g.t 4
12.b even 2 1 inner 2304.3.g.t 4
16.e even 4 2 576.3.b.b 4
16.f odd 4 2 576.3.b.b 4
24.f even 2 1 inner 2304.3.g.t 4
24.h odd 2 1 CM 2304.3.g.t 4
48.i odd 4 2 576.3.b.b 4
48.k even 4 2 576.3.b.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.3.b.b 4 16.e even 4 2
576.3.b.b 4 16.f odd 4 2
576.3.b.b 4 48.i odd 4 2
576.3.b.b 4 48.k even 4 2
2304.3.g.t 4 1.a even 1 1 trivial
2304.3.g.t 4 3.b odd 2 1 inner
2304.3.g.t 4 4.b odd 2 1 inner
2304.3.g.t 4 8.b even 2 1 inner
2304.3.g.t 4 8.d odd 2 1 inner
2304.3.g.t 4 12.b even 2 1 inner
2304.3.g.t 4 24.f even 2 1 inner
2304.3.g.t 4 24.h odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{2} - 96 \)
\( T_{7}^{2} + 100 \)
\( T_{11}^{2} + 384 \)
\( T_{13} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -96 + T^{2} )^{2} \)
$7$ \( ( 100 + T^{2} )^{2} \)
$11$ \( ( 384 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( ( -864 + T^{2} )^{2} \)
$31$ \( ( 1444 + T^{2} )^{2} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( ( -2400 + T^{2} )^{2} \)
$59$ \( ( 13824 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( -50 + T )^{4} \)
$79$ \( ( 3364 + T^{2} )^{2} \)
$83$ \( ( 9600 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( ( 190 + T )^{4} \)
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