Properties

Label 2304.2.c.e
Level $2304$
Weight $2$
Character orbit 2304.c
Analytic conductor $18.398$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.3975326257\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1152)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + 3 \beta q^{5} +O(q^{10})\) \( q + 3 \beta q^{5} + 6 q^{13} + 5 \beta q^{17} -13 q^{25} + 3 \beta q^{29} + 12 q^{37} -\beta q^{41} + 7 q^{49} -9 \beta q^{53} -12 q^{61} + 18 \beta q^{65} -16 q^{73} -30 q^{85} + 13 \beta q^{89} -8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 12q^{13} - 26q^{25} + 24q^{37} + 14q^{49} - 24q^{61} - 32q^{73} - 60q^{85} - 16q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2303.1
1.41421i
1.41421i
0 0 0 4.24264i 0 0 0 0 0
2303.2 0 0 0 4.24264i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
3.b odd 2 1 inner
12.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.c.e 2
3.b odd 2 1 inner 2304.2.c.e 2
4.b odd 2 1 CM 2304.2.c.e 2
8.b even 2 1 2304.2.c.c 2
8.d odd 2 1 2304.2.c.c 2
12.b even 2 1 inner 2304.2.c.e 2
16.e even 4 2 1152.2.f.c 4
16.f odd 4 2 1152.2.f.c 4
24.f even 2 1 2304.2.c.c 2
24.h odd 2 1 2304.2.c.c 2
48.i odd 4 2 1152.2.f.c 4
48.k even 4 2 1152.2.f.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.f.c 4 16.e even 4 2
1152.2.f.c 4 16.f odd 4 2
1152.2.f.c 4 48.i odd 4 2
1152.2.f.c 4 48.k even 4 2
2304.2.c.c 2 8.b even 2 1
2304.2.c.c 2 8.d odd 2 1
2304.2.c.c 2 24.f even 2 1
2304.2.c.c 2 24.h odd 2 1
2304.2.c.e 2 1.a even 1 1 trivial
2304.2.c.e 2 3.b odd 2 1 inner
2304.2.c.e 2 4.b odd 2 1 CM
2304.2.c.e 2 12.b 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 \(S_{2}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{2} + 18 \)
\( T_{7} \)
\( T_{11} \)
\( T_{13} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 8 T^{2} + 25 T^{4} \)
$7$ \( ( 1 - 7 T^{2} )^{2} \)
$11$ \( ( 1 + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 6 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 16 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - 19 T^{2} )^{2} \)
$23$ \( ( 1 + 23 T^{2} )^{2} \)
$29$ \( 1 - 40 T^{2} + 841 T^{4} \)
$31$ \( ( 1 - 31 T^{2} )^{2} \)
$37$ \( ( 1 - 12 T + 37 T^{2} )^{2} \)
$41$ \( 1 - 80 T^{2} + 1681 T^{4} \)
$43$ \( ( 1 - 43 T^{2} )^{2} \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( 1 + 56 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 12 T + 61 T^{2} )^{2} \)
$67$ \( ( 1 - 67 T^{2} )^{2} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( ( 1 + 16 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 - 79 T^{2} )^{2} \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( 1 + 160 T^{2} + 7921 T^{4} \)
$97$ \( ( 1 + 8 T + 97 T^{2} )^{2} \)
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