Properties

Label 2304.2.a.k
Level $2304$
Weight $2$
Character orbit 2304.a
Self dual yes
Analytic conductor $18.398$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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 384)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 4q^{7} + O(q^{10}) \) \( q + 4q^{7} + 4q^{11} + 4q^{13} + 2q^{17} + 4q^{19} - 8q^{23} - 5q^{25} + 8q^{29} + 4q^{31} - 4q^{37} - 6q^{41} - 4q^{43} - 8q^{47} + 9q^{49} + 8q^{53} - 12q^{59} + 12q^{61} - 12q^{67} + 8q^{71} - 6q^{73} + 16q^{77} + 4q^{79} - 4q^{83} + 6q^{89} + 16q^{91} - 2q^{97} + O(q^{100}) \)

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 0 0 4.00000 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

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.2.a.k 1
3.b odd 2 1 768.2.a.g 1
4.b odd 2 1 2304.2.a.f 1
8.b even 2 1 2304.2.a.j 1
8.d odd 2 1 2304.2.a.g 1
12.b even 2 1 768.2.a.b 1
16.e even 4 2 1152.2.d.a 2
16.f odd 4 2 1152.2.d.f 2
24.f even 2 1 768.2.a.f 1
24.h odd 2 1 768.2.a.c 1
48.i odd 4 2 384.2.d.a 2
48.k even 4 2 384.2.d.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.d.a 2 48.i odd 4 2
384.2.d.b yes 2 48.k even 4 2
768.2.a.b 1 12.b even 2 1
768.2.a.c 1 24.h odd 2 1
768.2.a.f 1 24.f even 2 1
768.2.a.g 1 3.b odd 2 1
1152.2.d.a 2 16.e even 4 2
1152.2.d.f 2 16.f odd 4 2
2304.2.a.f 1 4.b odd 2 1
2304.2.a.g 1 8.d odd 2 1
2304.2.a.j 1 8.b even 2 1
2304.2.a.k 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2304))\):

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