Properties

Label 2304.2.a.n
Level $2304$
Weight $2$
Character orbit 2304.a
Self dual yes
Analytic conductor $18.398$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.3975326257\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1152)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{5} + O(q^{10}) \) \( q + 2q^{5} + 4q^{13} + 8q^{17} - q^{25} - 10q^{29} + 12q^{37} - 8q^{41} - 7q^{49} + 14q^{53} + 12q^{61} + 8q^{65} + 6q^{73} + 16q^{85} - 16q^{89} + 18q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 0 2.00000 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)

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 2304.2.a.n 1
3.b odd 2 1 2304.2.a.d 1
4.b odd 2 1 CM 2304.2.a.n 1
8.b even 2 1 2304.2.a.c 1
8.d odd 2 1 2304.2.a.c 1
12.b even 2 1 2304.2.a.d 1
16.e even 4 2 1152.2.d.e yes 2
16.f odd 4 2 1152.2.d.e yes 2
24.f even 2 1 2304.2.a.m 1
24.h odd 2 1 2304.2.a.m 1
48.i odd 4 2 1152.2.d.b 2
48.k even 4 2 1152.2.d.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.d.b 2 48.i odd 4 2
1152.2.d.b 2 48.k even 4 2
1152.2.d.e yes 2 16.e even 4 2
1152.2.d.e yes 2 16.f odd 4 2
2304.2.a.c 1 8.b even 2 1
2304.2.a.c 1 8.d odd 2 1
2304.2.a.d 1 3.b odd 2 1
2304.2.a.d 1 12.b even 2 1
2304.2.a.m 1 24.f even 2 1
2304.2.a.m 1 24.h odd 2 1
2304.2.a.n 1 1.a even 1 1 trivial
2304.2.a.n 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2304))\):

\( T_{5} - 2 \)
\( T_{7} \)
\( T_{11} \)
\( T_{13} - 4 \)
\( T_{19} \)