Properties

Label 2304.2.a.s
Level $2304$
Weight $2$
Character orbit 2304.a
Self dual yes
Analytic conductor $18.398$
Analytic rank $1$
Dimension $2$
CM no
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 192)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta q^{5} -2 \beta q^{7} +O(q^{10})\) \( q + 2 \beta q^{5} -2 \beta q^{7} -6 q^{17} -4 q^{19} -4 \beta q^{23} + 7 q^{25} -2 \beta q^{29} -2 \beta q^{31} -12 q^{35} + 4 \beta q^{37} -6 q^{41} -4 q^{43} + 4 \beta q^{47} + 5 q^{49} -2 \beta q^{53} -12 q^{59} -4 \beta q^{61} + 4 q^{67} + 4 \beta q^{71} -2 q^{73} + 6 \beta q^{79} -12 \beta q^{85} + 6 q^{89} -8 \beta q^{95} -2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 12q^{17} - 8q^{19} + 14q^{25} - 24q^{35} - 12q^{41} - 8q^{43} + 10q^{49} - 24q^{59} + 8q^{67} - 4q^{73} + 12q^{89} - 4q^{97} + O(q^{100}) \)

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
−1.73205
1.73205
0 0 0 −3.46410 0 3.46410 0 0 0
1.2 0 0 0 3.46410 0 −3.46410 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
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.2.a.s 2
3.b odd 2 1 768.2.a.j 2
4.b odd 2 1 2304.2.a.u 2
8.b even 2 1 2304.2.a.u 2
8.d odd 2 1 inner 2304.2.a.s 2
12.b even 2 1 768.2.a.k 2
16.e even 4 2 576.2.d.b 4
16.f odd 4 2 576.2.d.b 4
24.f even 2 1 768.2.a.j 2
24.h odd 2 1 768.2.a.k 2
48.i odd 4 2 192.2.d.a 4
48.k even 4 2 192.2.d.a 4
240.t even 4 2 4800.2.k.j 4
240.z odd 4 2 4800.2.d.j 4
240.bb even 4 2 4800.2.d.o 4
240.bd odd 4 2 4800.2.d.o 4
240.bf even 4 2 4800.2.d.j 4
240.bm odd 4 2 4800.2.k.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.2.d.a 4 48.i odd 4 2
192.2.d.a 4 48.k even 4 2
576.2.d.b 4 16.e even 4 2
576.2.d.b 4 16.f odd 4 2
768.2.a.j 2 3.b odd 2 1
768.2.a.j 2 24.f even 2 1
768.2.a.k 2 12.b even 2 1
768.2.a.k 2 24.h odd 2 1
2304.2.a.s 2 1.a even 1 1 trivial
2304.2.a.s 2 8.d odd 2 1 inner
2304.2.a.u 2 4.b odd 2 1
2304.2.a.u 2 8.b even 2 1
4800.2.d.j 4 240.z odd 4 2
4800.2.d.j 4 240.bf even 4 2
4800.2.d.o 4 240.bb even 4 2
4800.2.d.o 4 240.bd odd 4 2
4800.2.k.j 4 240.t even 4 2
4800.2.k.j 4 240.bm odd 4 2

Hecke kernels

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

\( T_{5}^{2} - 12 \)
\( T_{7}^{2} - 12 \)
\( T_{11} \)
\( T_{13} \)
\( T_{19} + 4 \)