Properties

Label 2304.2.a.h
Level $2304$
Weight $2$
Character orbit 2304.a
Self dual yes
Analytic conductor $18.398$
Analytic rank $1$
Dimension $1$
CM discriminant -8
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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 64)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q + O(q^{10}) \) \( q - 6q^{11} + 6q^{17} + 2q^{19} - 5q^{25} - 6q^{41} - 10q^{43} - 7q^{49} - 6q^{59} - 14q^{67} - 2q^{73} - 18q^{83} + 18q^{89} + 10q^{97} + O(q^{100}) \)

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 0 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 CM by \(\Q(\sqrt{-2}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.2.a.h 1
3.b odd 2 1 256.2.a.d 1
4.b odd 2 1 2304.2.a.i 1
8.b even 2 1 2304.2.a.i 1
8.d odd 2 1 CM 2304.2.a.h 1
12.b even 2 1 256.2.a.a 1
15.d odd 2 1 6400.2.a.a 1
16.e even 4 2 576.2.d.a 2
16.f odd 4 2 576.2.d.a 2
24.f even 2 1 256.2.a.d 1
24.h odd 2 1 256.2.a.a 1
48.i odd 4 2 64.2.b.a 2
48.k even 4 2 64.2.b.a 2
60.h even 2 1 6400.2.a.x 1
96.o even 8 4 1024.2.e.l 4
96.p odd 8 4 1024.2.e.l 4
120.i odd 2 1 6400.2.a.x 1
120.m even 2 1 6400.2.a.a 1
240.t even 4 2 1600.2.d.a 2
240.z odd 4 2 1600.2.f.b 2
240.bb even 4 2 1600.2.f.a 2
240.bd odd 4 2 1600.2.f.a 2
240.bf even 4 2 1600.2.f.b 2
240.bm odd 4 2 1600.2.d.a 2
336.v odd 4 2 3136.2.b.b 2
336.y even 4 2 3136.2.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.2.b.a 2 48.i odd 4 2
64.2.b.a 2 48.k even 4 2
256.2.a.a 1 12.b even 2 1
256.2.a.a 1 24.h odd 2 1
256.2.a.d 1 3.b odd 2 1
256.2.a.d 1 24.f even 2 1
576.2.d.a 2 16.e even 4 2
576.2.d.a 2 16.f odd 4 2
1024.2.e.l 4 96.o even 8 4
1024.2.e.l 4 96.p odd 8 4
1600.2.d.a 2 240.t even 4 2
1600.2.d.a 2 240.bm odd 4 2
1600.2.f.a 2 240.bb even 4 2
1600.2.f.a 2 240.bd odd 4 2
1600.2.f.b 2 240.z odd 4 2
1600.2.f.b 2 240.bf even 4 2
2304.2.a.h 1 1.a even 1 1 trivial
2304.2.a.h 1 8.d odd 2 1 CM
2304.2.a.i 1 4.b odd 2 1
2304.2.a.i 1 8.b even 2 1
3136.2.b.b 2 336.v odd 4 2
3136.2.b.b 2 336.y even 4 2
6400.2.a.a 1 15.d odd 2 1
6400.2.a.a 1 120.m even 2 1
6400.2.a.x 1 60.h even 2 1
6400.2.a.x 1 120.i odd 2 1

Hecke kernels

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

\( T_{5} \)
\( T_{7} \)
\( T_{11} + 6 \)
\( T_{13} \)
\( T_{19} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 5 T^{2} \)
$7$ \( 1 + 7 T^{2} \)
$11$ \( 1 + 6 T + 11 T^{2} \)
$13$ \( 1 + 13 T^{2} \)
$17$ \( 1 - 6 T + 17 T^{2} \)
$19$ \( 1 - 2 T + 19 T^{2} \)
$23$ \( 1 + 23 T^{2} \)
$29$ \( 1 + 29 T^{2} \)
$31$ \( 1 + 31 T^{2} \)
$37$ \( 1 + 37 T^{2} \)
$41$ \( 1 + 6 T + 41 T^{2} \)
$43$ \( 1 + 10 T + 43 T^{2} \)
$47$ \( 1 + 47 T^{2} \)
$53$ \( 1 + 53 T^{2} \)
$59$ \( 1 + 6 T + 59 T^{2} \)
$61$ \( 1 + 61 T^{2} \)
$67$ \( 1 + 14 T + 67 T^{2} \)
$71$ \( 1 + 71 T^{2} \)
$73$ \( 1 + 2 T + 73 T^{2} \)
$79$ \( 1 + 79 T^{2} \)
$83$ \( 1 + 18 T + 83 T^{2} \)
$89$ \( 1 - 18 T + 89 T^{2} \)
$97$ \( 1 - 10 T + 97 T^{2} \)
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