Properties

Label 2304.2.a.l
Level $2304$
Weight $2$
Character orbit 2304.a
Self dual yes
Analytic conductor $18.398$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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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 24)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{5} - 2q^{7} + O(q^{10}) \) \( q + 2q^{5} - 2q^{7} - 4q^{13} + 2q^{17} - 4q^{19} - 4q^{23} - q^{25} + 6q^{29} - 2q^{31} - 4q^{35} - 8q^{37} + 2q^{41} - 4q^{43} - 12q^{47} - 3q^{49} + 6q^{53} - 4q^{59} - 8q^{65} + 12q^{67} - 12q^{71} + 6q^{73} - 10q^{79} + 16q^{83} + 4q^{85} - 10q^{89} + 8q^{91} - 8q^{95} - 2q^{97} + O(q^{100}) \)

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 2.00000 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

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.2.a.l 1
3.b odd 2 1 768.2.a.e 1
4.b odd 2 1 2304.2.a.o 1
8.b even 2 1 2304.2.a.b 1
8.d odd 2 1 2304.2.a.e 1
12.b even 2 1 768.2.a.a 1
16.e even 4 2 288.2.d.b 2
16.f odd 4 2 72.2.d.b 2
24.f even 2 1 768.2.a.h 1
24.h odd 2 1 768.2.a.d 1
48.i odd 4 2 96.2.d.a 2
48.k even 4 2 24.2.d.a 2
80.i odd 4 2 7200.2.d.g 2
80.j even 4 2 1800.2.d.b 2
80.k odd 4 2 1800.2.k.a 2
80.q even 4 2 7200.2.k.d 2
80.s even 4 2 1800.2.d.i 2
80.t odd 4 2 7200.2.d.d 2
144.u even 12 4 648.2.n.k 4
144.v odd 12 4 648.2.n.c 4
144.w odd 12 4 2592.2.r.f 4
144.x even 12 4 2592.2.r.g 4
240.t even 4 2 600.2.k.b 2
240.z odd 4 2 600.2.d.b 2
240.bb even 4 2 2400.2.d.b 2
240.bd odd 4 2 600.2.d.c 2
240.bf even 4 2 2400.2.d.c 2
240.bm odd 4 2 2400.2.k.a 2
336.v odd 4 2 1176.2.c.a 2
336.y even 4 2 4704.2.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.d.a 2 48.k even 4 2
72.2.d.b 2 16.f odd 4 2
96.2.d.a 2 48.i odd 4 2
288.2.d.b 2 16.e even 4 2
600.2.d.b 2 240.z odd 4 2
600.2.d.c 2 240.bd odd 4 2
600.2.k.b 2 240.t even 4 2
648.2.n.c 4 144.v odd 12 4
648.2.n.k 4 144.u even 12 4
768.2.a.a 1 12.b even 2 1
768.2.a.d 1 24.h odd 2 1
768.2.a.e 1 3.b odd 2 1
768.2.a.h 1 24.f even 2 1
1176.2.c.a 2 336.v odd 4 2
1800.2.d.b 2 80.j even 4 2
1800.2.d.i 2 80.s even 4 2
1800.2.k.a 2 80.k odd 4 2
2304.2.a.b 1 8.b even 2 1
2304.2.a.e 1 8.d odd 2 1
2304.2.a.l 1 1.a even 1 1 trivial
2304.2.a.o 1 4.b odd 2 1
2400.2.d.b 2 240.bb even 4 2
2400.2.d.c 2 240.bf even 4 2
2400.2.k.a 2 240.bm odd 4 2
2592.2.r.f 4 144.w odd 12 4
2592.2.r.g 4 144.x even 12 4
4704.2.c.a 2 336.y even 4 2
7200.2.d.d 2 80.t odd 4 2
7200.2.d.g 2 80.i odd 4 2
7200.2.k.d 2 80.q even 4 2

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 2 T + 5 T^{2} \)
$7$ \( 1 + 2 T + 7 T^{2} \)
$11$ \( 1 + 11 T^{2} \)
$13$ \( 1 + 4 T + 13 T^{2} \)
$17$ \( 1 - 2 T + 17 T^{2} \)
$19$ \( 1 + 4 T + 19 T^{2} \)
$23$ \( 1 + 4 T + 23 T^{2} \)
$29$ \( 1 - 6 T + 29 T^{2} \)
$31$ \( 1 + 2 T + 31 T^{2} \)
$37$ \( 1 + 8 T + 37 T^{2} \)
$41$ \( 1 - 2 T + 41 T^{2} \)
$43$ \( 1 + 4 T + 43 T^{2} \)
$47$ \( 1 + 12 T + 47 T^{2} \)
$53$ \( 1 - 6 T + 53 T^{2} \)
$59$ \( 1 + 4 T + 59 T^{2} \)
$61$ \( 1 + 61 T^{2} \)
$67$ \( 1 - 12 T + 67 T^{2} \)
$71$ \( 1 + 12 T + 71 T^{2} \)
$73$ \( 1 - 6 T + 73 T^{2} \)
$79$ \( 1 + 10 T + 79 T^{2} \)
$83$ \( 1 - 16 T + 83 T^{2} \)
$89$ \( 1 + 10 T + 89 T^{2} \)
$97$ \( 1 + 2 T + 97 T^{2} \)
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