Properties

Label 2304.2.a.w
Level $2304$
Weight $2$
Character orbit 2304.a
Self dual yes
Analytic conductor $18.398$
Analytic rank $0$
Dimension $2$
CM discriminant -3
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: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 576)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

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

\(f(q)\) \(=\) \( q -2 \beta q^{7} +O(q^{10})\) \( q -2 \beta q^{7} -4 \beta q^{13} + 8 q^{19} -5 q^{25} + 6 \beta q^{31} + 4 \beta q^{37} + 8 q^{43} + 5 q^{49} + 4 \beta q^{61} + 16 q^{67} + 10 q^{73} -10 \beta q^{79} + 24 q^{91} -14 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 16q^{19} - 10q^{25} + 16q^{43} + 10q^{49} + 32q^{67} + 20q^{73} + 48q^{91} - 28q^{97} + O(q^{100}) \)

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 0 0 −3.46410 0 0 0
1.2 0 0 0 0 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
8.d odd 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.w 2
3.b odd 2 1 CM 2304.2.a.w 2
4.b odd 2 1 2304.2.a.v 2
8.b even 2 1 2304.2.a.v 2
8.d odd 2 1 inner 2304.2.a.w 2
12.b even 2 1 2304.2.a.v 2
16.e even 4 2 576.2.d.c 4
16.f odd 4 2 576.2.d.c 4
24.f even 2 1 inner 2304.2.a.w 2
24.h odd 2 1 2304.2.a.v 2
48.i odd 4 2 576.2.d.c 4
48.k even 4 2 576.2.d.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.2.d.c 4 16.e even 4 2
576.2.d.c 4 16.f odd 4 2
576.2.d.c 4 48.i odd 4 2
576.2.d.c 4 48.k even 4 2
2304.2.a.v 2 4.b odd 2 1
2304.2.a.v 2 8.b even 2 1
2304.2.a.v 2 12.b even 2 1
2304.2.a.v 2 24.h odd 2 1
2304.2.a.w 2 1.a even 1 1 trivial
2304.2.a.w 2 3.b odd 2 1 CM
2304.2.a.w 2 8.d odd 2 1 inner
2304.2.a.w 2 24.f even 2 1 inner

Hecke kernels

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

\( T_{5} \)
\( T_{7}^{2} - 12 \)
\( T_{11} \)
\( T_{13}^{2} - 48 \)
\( T_{19} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 + 5 T^{2} )^{2} \)
$7$ \( 1 + 2 T^{2} + 49 T^{4} \)
$11$ \( ( 1 + 11 T^{2} )^{2} \)
$13$ \( 1 - 22 T^{2} + 169 T^{4} \)
$17$ \( ( 1 + 17 T^{2} )^{2} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )^{2} \)
$23$ \( ( 1 + 23 T^{2} )^{2} \)
$29$ \( ( 1 + 29 T^{2} )^{2} \)
$31$ \( 1 - 46 T^{2} + 961 T^{4} \)
$37$ \( 1 + 26 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 8 T + 43 T^{2} )^{2} \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( ( 1 + 53 T^{2} )^{2} \)
$59$ \( ( 1 + 59 T^{2} )^{2} \)
$61$ \( 1 + 74 T^{2} + 3721 T^{4} \)
$67$ \( ( 1 - 16 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 10 T + 73 T^{2} )^{2} \)
$79$ \( 1 - 142 T^{2} + 6241 T^{4} \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( ( 1 + 89 T^{2} )^{2} \)
$97$ \( ( 1 + 14 T + 97 T^{2} )^{2} \)
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