Properties

Label 2304.2.a.z
Level $2304$
Weight $2$
Character orbit 2304.a
Self dual yes
Analytic conductor $18.398$
Analytic rank $0$
Dimension $4$
CM discriminant -24
Inner twists $8$

Related objects

Downloads

Learn more about

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: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 1152)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -4 + 2 \beta^{2} ) q^{5} + ( -10 \beta + 2 \beta^{3} ) q^{7} +O(q^{10})\) \( q + ( -4 + 2 \beta^{2} ) q^{5} + ( -10 \beta + 2 \beta^{3} ) q^{7} + ( -12 \beta + 4 \beta^{3} ) q^{11} + 7 q^{25} + ( -12 + 6 \beta^{2} ) q^{29} + ( -10 \beta + 2 \beta^{3} ) q^{31} + ( 36 \beta - 12 \beta^{3} ) q^{35} + 17 q^{49} + ( 4 - 2 \beta^{2} ) q^{53} + ( 40 \beta - 8 \beta^{3} ) q^{55} + ( -24 \beta + 8 \beta^{3} ) q^{59} + 14 q^{73} + ( 32 - 16 \beta^{2} ) q^{77} + ( 30 \beta - 6 \beta^{3} ) q^{79} + ( 12 \beta - 4 \beta^{3} ) q^{83} + 2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 28q^{25} + 68q^{49} + 56q^{73} + 8q^{97} + O(q^{100}) \)

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
0.517638
−0.517638
1.93185
−1.93185
0 0 0 −3.46410 0 −4.89898 0 0 0
1.2 0 0 0 −3.46410 0 4.89898 0 0 0
1.3 0 0 0 3.46410 0 −4.89898 0 0 0
1.4 0 0 0 3.46410 0 4.89898 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
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f 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.z 4
3.b odd 2 1 inner 2304.2.a.z 4
4.b odd 2 1 inner 2304.2.a.z 4
8.b even 2 1 inner 2304.2.a.z 4
8.d odd 2 1 inner 2304.2.a.z 4
12.b even 2 1 inner 2304.2.a.z 4
16.e even 4 2 1152.2.d.g 4
16.f odd 4 2 1152.2.d.g 4
24.f even 2 1 inner 2304.2.a.z 4
24.h odd 2 1 CM 2304.2.a.z 4
48.i odd 4 2 1152.2.d.g 4
48.k even 4 2 1152.2.d.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.d.g 4 16.e even 4 2
1152.2.d.g 4 16.f odd 4 2
1152.2.d.g 4 48.i odd 4 2
1152.2.d.g 4 48.k even 4 2
2304.2.a.z 4 1.a even 1 1 trivial
2304.2.a.z 4 3.b odd 2 1 inner
2304.2.a.z 4 4.b odd 2 1 inner
2304.2.a.z 4 8.b even 2 1 inner
2304.2.a.z 4 8.d odd 2 1 inner
2304.2.a.z 4 12.b even 2 1 inner
2304.2.a.z 4 24.f even 2 1 inner
2304.2.a.z 4 24.h 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2304))\):

\( T_{5}^{2} - 12 \)
\( T_{7}^{2} - 24 \)
\( T_{11}^{2} - 32 \)
\( T_{13} \)
\( T_{19} \)