Properties

Label 2304.3.g.f
Level $2304$
Weight $3$
Character orbit 2304.g
Self dual yes
Analytic conductor $62.779$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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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: yes
Analytic conductor: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 128)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q + 8q^{5} + O(q^{10}) \) \( q + 8q^{5} + 24q^{13} - 30q^{17} + 39q^{25} + 40q^{29} + 24q^{37} + 18q^{41} + 49q^{49} - 56q^{53} + 120q^{61} + 192q^{65} - 110q^{73} - 240q^{85} + 78q^{89} - 130q^{97} + O(q^{100}) \)

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
0
0 0 0 8.00000 0 0 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 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.3.g.f 1
3.b odd 2 1 256.3.c.a 1
4.b odd 2 1 CM 2304.3.g.f 1
8.b even 2 1 2304.3.g.a 1
8.d odd 2 1 2304.3.g.a 1
12.b even 2 1 256.3.c.a 1
16.e even 4 2 1152.3.b.a 2
16.f odd 4 2 1152.3.b.a 2
24.f even 2 1 256.3.c.b 1
24.h odd 2 1 256.3.c.b 1
48.i odd 4 2 128.3.d.a 2
48.k even 4 2 128.3.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.3.d.a 2 48.i odd 4 2
128.3.d.a 2 48.k even 4 2
256.3.c.a 1 3.b odd 2 1
256.3.c.a 1 12.b even 2 1
256.3.c.b 1 24.f even 2 1
256.3.c.b 1 24.h odd 2 1
1152.3.b.a 2 16.e even 4 2
1152.3.b.a 2 16.f odd 4 2
2304.3.g.a 1 8.b even 2 1
2304.3.g.a 1 8.d odd 2 1
2304.3.g.f 1 1.a even 1 1 trivial
2304.3.g.f 1 4.b 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 \(S_{3}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5} - 8 \)
\( T_{7} \)
\( T_{11} \)
\( T_{13} - 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -8 + T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( -24 + T \)
$17$ \( 30 + T \)
$19$ \( T \)
$23$ \( T \)
$29$ \( -40 + T \)
$31$ \( T \)
$37$ \( -24 + T \)
$41$ \( -18 + T \)
$43$ \( T \)
$47$ \( T \)
$53$ \( 56 + T \)
$59$ \( T \)
$61$ \( -120 + T \)
$67$ \( T \)
$71$ \( T \)
$73$ \( 110 + T \)
$79$ \( T \)
$83$ \( T \)
$89$ \( -78 + T \)
$97$ \( 130 + T \)
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