Properties

Label 2304.2.a.r
Level $2304$
Weight $2$
Character orbit 2304.a
Self dual yes
Analytic conductor $18.398$
Analytic rank $0$
Dimension $2$
CM no
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 384)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + 2 \beta q^{5} -2 \beta q^{7} +O(q^{10})\) \( q + 2 \beta q^{5} -2 \beta q^{7} -4 q^{11} + 4 \beta q^{13} + 2 q^{17} + 4 q^{19} -4 \beta q^{23} + 3 q^{25} -2 \beta q^{29} + 6 \beta q^{31} -8 q^{35} + 10 q^{41} + 12 q^{43} + 4 \beta q^{47} + q^{49} -2 \beta q^{53} -8 \beta q^{55} + 4 q^{59} -8 \beta q^{61} + 16 q^{65} + 4 q^{67} + 4 \beta q^{71} + 2 q^{73} + 8 \beta q^{77} + 6 \beta q^{79} + 4 q^{83} + 4 \beta q^{85} + 6 q^{89} -16 q^{91} + 8 \beta q^{95} + 14 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 8q^{11} + 4q^{17} + 8q^{19} + 6q^{25} - 16q^{35} + 20q^{41} + 24q^{43} + 2q^{49} + 8q^{59} + 32q^{65} + 8q^{67} + 4q^{73} + 8q^{83} + 12q^{89} - 32q^{91} + 28q^{97} + O(q^{100}) \)

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
−1.41421
1.41421
0 0 0 −2.82843 0 2.82843 0 0 0
1.2 0 0 0 2.82843 0 −2.82843 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 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.2.a.r 2
3.b odd 2 1 768.2.a.l 2
4.b odd 2 1 2304.2.a.x 2
8.b even 2 1 2304.2.a.x 2
8.d odd 2 1 inner 2304.2.a.r 2
12.b even 2 1 768.2.a.i 2
16.e even 4 2 1152.2.d.h 4
16.f odd 4 2 1152.2.d.h 4
24.f even 2 1 768.2.a.l 2
24.h odd 2 1 768.2.a.i 2
48.i odd 4 2 384.2.d.c 4
48.k even 4 2 384.2.d.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.d.c 4 48.i odd 4 2
384.2.d.c 4 48.k even 4 2
768.2.a.i 2 12.b even 2 1
768.2.a.i 2 24.h odd 2 1
768.2.a.l 2 3.b odd 2 1
768.2.a.l 2 24.f even 2 1
1152.2.d.h 4 16.e even 4 2
1152.2.d.h 4 16.f odd 4 2
2304.2.a.r 2 1.a even 1 1 trivial
2304.2.a.r 2 8.d odd 2 1 inner
2304.2.a.x 2 4.b odd 2 1
2304.2.a.x 2 8.b even 2 1

Hecke kernels

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

\( T_{5}^{2} - 8 \)
\( T_{7}^{2} - 8 \)
\( T_{11} + 4 \)
\( T_{13}^{2} - 32 \)
\( T_{19} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 2 T^{2} + 25 T^{4} \)
$7$ \( 1 + 6 T^{2} + 49 T^{4} \)
$11$ \( ( 1 + 4 T + 11 T^{2} )^{2} \)
$13$ \( 1 - 6 T^{2} + 169 T^{4} \)
$17$ \( ( 1 - 2 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 - 4 T + 19 T^{2} )^{2} \)
$23$ \( 1 + 14 T^{2} + 529 T^{4} \)
$29$ \( 1 + 50 T^{2} + 841 T^{4} \)
$31$ \( 1 - 10 T^{2} + 961 T^{4} \)
$37$ \( ( 1 + 37 T^{2} )^{2} \)
$41$ \( ( 1 - 10 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 12 T + 43 T^{2} )^{2} \)
$47$ \( 1 + 62 T^{2} + 2209 T^{4} \)
$53$ \( 1 + 98 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 - 4 T + 59 T^{2} )^{2} \)
$61$ \( 1 - 6 T^{2} + 3721 T^{4} \)
$67$ \( ( 1 - 4 T + 67 T^{2} )^{2} \)
$71$ \( 1 + 110 T^{2} + 5041 T^{4} \)
$73$ \( ( 1 - 2 T + 73 T^{2} )^{2} \)
$79$ \( 1 + 86 T^{2} + 6241 T^{4} \)
$83$ \( ( 1 - 4 T + 83 T^{2} )^{2} \)
$89$ \( ( 1 - 6 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 14 T + 97 T^{2} )^{2} \)
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