Properties

Label 2304.3.g.l
Level $2304$
Weight $3$
Character orbit 2304.g
Analytic conductor $62.779$
Analytic rank $0$
Dimension $2$
CM discriminant -24
Inner twists $4$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{-6}) \)
Defining polynomial: \(x^{2} + 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1152)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4\sqrt{-6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{5} -\beta q^{7} +O(q^{10})\) \( q + 2 q^{5} -\beta q^{7} -2 \beta q^{11} -21 q^{25} -50 q^{29} -5 \beta q^{31} -2 \beta q^{35} -47 q^{49} + 94 q^{53} -4 \beta q^{55} + 12 \beta q^{59} -50 q^{73} -192 q^{77} + 15 \beta q^{79} + 10 \beta q^{83} -190 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{5} + O(q^{10}) \) \( 2q + 4q^{5} - 42q^{25} - 100q^{29} - 94q^{49} + 188q^{53} - 100q^{73} - 384q^{77} - 380q^{97} + O(q^{100}) \)

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
2.44949i
2.44949i
0 0 0 2.00000 0 9.79796i 0 0 0
1279.2 0 0 0 2.00000 0 9.79796i 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 \(\Q(\sqrt{-6}) \)
4.b odd 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.l 2
3.b odd 2 1 2304.3.g.i 2
4.b odd 2 1 inner 2304.3.g.l 2
8.b even 2 1 2304.3.g.i 2
8.d odd 2 1 2304.3.g.i 2
12.b even 2 1 2304.3.g.i 2
16.e even 4 2 1152.3.b.f 4
16.f odd 4 2 1152.3.b.f 4
24.f even 2 1 inner 2304.3.g.l 2
24.h odd 2 1 CM 2304.3.g.l 2
48.i odd 4 2 1152.3.b.f 4
48.k even 4 2 1152.3.b.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.3.b.f 4 16.e even 4 2
1152.3.b.f 4 16.f odd 4 2
1152.3.b.f 4 48.i odd 4 2
1152.3.b.f 4 48.k even 4 2
2304.3.g.i 2 3.b odd 2 1
2304.3.g.i 2 8.b even 2 1
2304.3.g.i 2 8.d odd 2 1
2304.3.g.i 2 12.b even 2 1
2304.3.g.l 2 1.a even 1 1 trivial
2304.3.g.l 2 4.b odd 2 1 inner
2304.3.g.l 2 24.f even 2 1 inner
2304.3.g.l 2 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 \(S_{3}^{\mathrm{new}}(2304, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -2 + T )^{2} \)
$7$ \( 96 + T^{2} \)
$11$ \( 384 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( 50 + T )^{2} \)
$31$ \( 2400 + T^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( ( -94 + T )^{2} \)
$59$ \( 13824 + T^{2} \)
$61$ \( T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 50 + T )^{2} \)
$79$ \( 21600 + T^{2} \)
$83$ \( 9600 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( 190 + T )^{2} \)
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