Properties

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

Newform invariants

Self dual: yes
Analytic conductor: \(18.3975326257\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 72)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

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

\(f(q)\) \(=\) \( q + 2 \beta q^{5} + 2 q^{7} +O(q^{10})\) \( q + 2 \beta q^{5} + 2 q^{7} + 4 \beta q^{11} + 3 q^{25} -2 \beta q^{29} + 10 q^{31} + 4 \beta q^{35} -3 q^{49} -10 \beta q^{53} + 16 q^{55} -8 \beta q^{59} -14 q^{73} + 8 \beta q^{77} + 10 q^{79} -4 \beta q^{83} + 2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{7} + O(q^{10}) \) \( 2q + 4q^{7} + 6q^{25} + 20q^{31} - 6q^{49} + 32q^{55} - 28q^{73} + 20q^{79} + 4q^{97} + O(q^{100}) \)

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
−1.41421
1.41421
0 0 0 −2.82843 0 2.00000 0 0 0
1.2 0 0 0 2.82843 0 2.00000 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
8.b even 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.y 2
3.b odd 2 1 inner 2304.2.a.y 2
4.b odd 2 1 2304.2.a.q 2
8.b even 2 1 inner 2304.2.a.y 2
8.d odd 2 1 2304.2.a.q 2
12.b even 2 1 2304.2.a.q 2
16.e even 4 2 288.2.d.a 2
16.f odd 4 2 72.2.d.a 2
24.f even 2 1 2304.2.a.q 2
24.h odd 2 1 CM 2304.2.a.y 2
48.i odd 4 2 288.2.d.a 2
48.k even 4 2 72.2.d.a 2
80.i odd 4 2 7200.2.d.p 4
80.j even 4 2 1800.2.d.n 4
80.k odd 4 2 1800.2.k.e 2
80.q even 4 2 7200.2.k.h 2
80.s even 4 2 1800.2.d.n 4
80.t odd 4 2 7200.2.d.p 4
144.u even 12 4 648.2.n.h 4
144.v odd 12 4 648.2.n.h 4
144.w odd 12 4 2592.2.r.i 4
144.x even 12 4 2592.2.r.i 4
240.t even 4 2 1800.2.k.e 2
240.z odd 4 2 1800.2.d.n 4
240.bb even 4 2 7200.2.d.p 4
240.bd odd 4 2 1800.2.d.n 4
240.bf even 4 2 7200.2.d.p 4
240.bm odd 4 2 7200.2.k.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.d.a 2 16.f odd 4 2
72.2.d.a 2 48.k even 4 2
288.2.d.a 2 16.e even 4 2
288.2.d.a 2 48.i odd 4 2
648.2.n.h 4 144.u even 12 4
648.2.n.h 4 144.v odd 12 4
1800.2.d.n 4 80.j even 4 2
1800.2.d.n 4 80.s even 4 2
1800.2.d.n 4 240.z odd 4 2
1800.2.d.n 4 240.bd odd 4 2
1800.2.k.e 2 80.k odd 4 2
1800.2.k.e 2 240.t even 4 2
2304.2.a.q 2 4.b odd 2 1
2304.2.a.q 2 8.d odd 2 1
2304.2.a.q 2 12.b even 2 1
2304.2.a.q 2 24.f even 2 1
2304.2.a.y 2 1.a even 1 1 trivial
2304.2.a.y 2 3.b odd 2 1 inner
2304.2.a.y 2 8.b even 2 1 inner
2304.2.a.y 2 24.h odd 2 1 CM
2592.2.r.i 4 144.w odd 12 4
2592.2.r.i 4 144.x even 12 4
7200.2.d.p 4 80.i odd 4 2
7200.2.d.p 4 80.t odd 4 2
7200.2.d.p 4 240.bb even 4 2
7200.2.d.p 4 240.bf even 4 2
7200.2.k.h 2 80.q even 4 2
7200.2.k.h 2 240.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2304))\):

\( T_{5}^{2} - 8 \)
\( T_{7} - 2 \)
\( T_{11}^{2} - 32 \)
\( T_{13} \)
\( T_{19} \)