Properties

Label 2304.2.a.t
Level $2304$
Weight $2$
Character orbit 2304.a
Self dual yes
Analytic conductor $18.398$
Analytic rank $1$
Dimension $2$
CM discriminant -8
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 128)
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.

\(f(q)\) \(=\) \( q +O(q^{10})\) \( q + 2 \beta q^{11} -6 q^{17} -6 \beta q^{19} -5 q^{25} -6 q^{41} + 6 \beta q^{43} -7 q^{49} + 10 \beta q^{59} -6 \beta q^{67} + 2 q^{73} -2 \beta q^{83} -18 q^{89} -10 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 12q^{17} - 10q^{25} - 12q^{41} - 14q^{49} + 4q^{73} - 36q^{89} - 20q^{97} + O(q^{100}) \)

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 0 0 0 0 0 0
1.2 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 \(\Q(\sqrt{-2}) \)
4.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.t 2
3.b odd 2 1 256.2.a.e 2
4.b odd 2 1 inner 2304.2.a.t 2
8.b even 2 1 inner 2304.2.a.t 2
8.d odd 2 1 CM 2304.2.a.t 2
12.b even 2 1 256.2.a.e 2
15.d odd 2 1 6400.2.a.by 2
16.e even 4 2 1152.2.d.c 2
16.f odd 4 2 1152.2.d.c 2
24.f even 2 1 256.2.a.e 2
24.h odd 2 1 256.2.a.e 2
48.i odd 4 2 128.2.b.a 2
48.k even 4 2 128.2.b.a 2
60.h even 2 1 6400.2.a.by 2
96.o even 8 2 1024.2.e.a 2
96.o even 8 2 1024.2.e.f 2
96.p odd 8 2 1024.2.e.a 2
96.p odd 8 2 1024.2.e.f 2
120.i odd 2 1 6400.2.a.by 2
120.m even 2 1 6400.2.a.by 2
240.t even 4 2 3200.2.d.c 2
240.z odd 4 2 3200.2.f.o 4
240.bb even 4 2 3200.2.f.o 4
240.bd odd 4 2 3200.2.f.o 4
240.bf even 4 2 3200.2.f.o 4
240.bm odd 4 2 3200.2.d.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.b.a 2 48.i odd 4 2
128.2.b.a 2 48.k even 4 2
256.2.a.e 2 3.b odd 2 1
256.2.a.e 2 12.b even 2 1
256.2.a.e 2 24.f even 2 1
256.2.a.e 2 24.h odd 2 1
1024.2.e.a 2 96.o even 8 2
1024.2.e.a 2 96.p odd 8 2
1024.2.e.f 2 96.o even 8 2
1024.2.e.f 2 96.p odd 8 2
1152.2.d.c 2 16.e even 4 2
1152.2.d.c 2 16.f odd 4 2
2304.2.a.t 2 1.a even 1 1 trivial
2304.2.a.t 2 4.b odd 2 1 inner
2304.2.a.t 2 8.b even 2 1 inner
2304.2.a.t 2 8.d odd 2 1 CM
3200.2.d.c 2 240.t even 4 2
3200.2.d.c 2 240.bm odd 4 2
3200.2.f.o 4 240.z odd 4 2
3200.2.f.o 4 240.bb even 4 2
3200.2.f.o 4 240.bd odd 4 2
3200.2.f.o 4 240.bf even 4 2
6400.2.a.by 2 15.d odd 2 1
6400.2.a.by 2 60.h even 2 1
6400.2.a.by 2 120.i odd 2 1
6400.2.a.by 2 120.m even 2 1

Hecke kernels

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

\( T_{5} \)
\( T_{7} \)
\( T_{11}^{2} - 8 \)
\( T_{13} \)
\( T_{19}^{2} - 72 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 + 5 T^{2} )^{2} \)
$7$ \( ( 1 + 7 T^{2} )^{2} \)
$11$ \( 1 + 14 T^{2} + 121 T^{4} \)
$13$ \( ( 1 + 13 T^{2} )^{2} \)
$17$ \( ( 1 + 6 T + 17 T^{2} )^{2} \)
$19$ \( 1 - 34 T^{2} + 361 T^{4} \)
$23$ \( ( 1 + 23 T^{2} )^{2} \)
$29$ \( ( 1 + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 31 T^{2} )^{2} \)
$37$ \( ( 1 + 37 T^{2} )^{2} \)
$41$ \( ( 1 + 6 T + 41 T^{2} )^{2} \)
$43$ \( 1 + 14 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( ( 1 + 53 T^{2} )^{2} \)
$59$ \( 1 - 82 T^{2} + 3481 T^{4} \)
$61$ \( ( 1 + 61 T^{2} )^{2} \)
$67$ \( 1 + 62 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 2 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 + 79 T^{2} )^{2} \)
$83$ \( 1 + 158 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 18 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 + 10 T + 97 T^{2} )^{2} \)
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