Properties

Label 2304.2.c.g
Level $2304$
Weight $2$
Character orbit 2304.c
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.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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1152)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta q^{5} + 2 \beta q^{7} +O(q^{10})\) \( q + \beta q^{5} + 2 \beta q^{7} + 4 q^{11} -2 q^{13} + \beta q^{17} + 4 \beta q^{19} -4 q^{23} + 3 q^{25} + 5 \beta q^{29} -6 \beta q^{31} -4 q^{35} -8 q^{37} + 3 \beta q^{41} -8 \beta q^{43} + 12 q^{47} - q^{49} + 9 \beta q^{53} + 4 \beta q^{55} -8 q^{61} -2 \beta q^{65} + 4 \beta q^{67} -4 q^{71} -8 q^{73} + 8 \beta q^{77} -2 \beta q^{79} -12 q^{83} -2 q^{85} -11 \beta q^{89} -4 \beta q^{91} -8 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 8q^{11} - 4q^{13} - 8q^{23} + 6q^{25} - 8q^{35} - 16q^{37} + 24q^{47} - 2q^{49} - 16q^{61} - 8q^{71} - 16q^{73} - 24q^{83} - 4q^{85} - 16q^{95} + O(q^{100}) \)

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 1.41421i 0 2.82843i 0 0 0
2303.2 0 0 0 1.41421i 0 2.82843i 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
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.g 2
3.b odd 2 1 2304.2.c.a 2
4.b odd 2 1 2304.2.c.a 2
8.b even 2 1 2304.2.c.b 2
8.d odd 2 1 2304.2.c.h 2
12.b even 2 1 inner 2304.2.c.g 2
16.e even 4 2 1152.2.f.d yes 4
16.f odd 4 2 1152.2.f.a 4
24.f even 2 1 2304.2.c.b 2
24.h odd 2 1 2304.2.c.h 2
48.i odd 4 2 1152.2.f.a 4
48.k even 4 2 1152.2.f.d yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.f.a 4 16.f odd 4 2
1152.2.f.a 4 48.i odd 4 2
1152.2.f.d yes 4 16.e even 4 2
1152.2.f.d yes 4 48.k even 4 2
2304.2.c.a 2 3.b odd 2 1
2304.2.c.a 2 4.b odd 2 1
2304.2.c.b 2 8.b even 2 1
2304.2.c.b 2 24.f even 2 1
2304.2.c.g 2 1.a even 1 1 trivial
2304.2.c.g 2 12.b even 2 1 inner
2304.2.c.h 2 8.d odd 2 1
2304.2.c.h 2 24.h odd 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 8 T^{2} + 25 T^{4} \)
$7$ \( 1 - 6 T^{2} + 49 T^{4} \)
$11$ \( ( 1 - 4 T + 11 T^{2} )^{2} \)
$13$ \( ( 1 + 2 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 32 T^{2} + 289 T^{4} \)
$19$ \( 1 - 6 T^{2} + 361 T^{4} \)
$23$ \( ( 1 + 4 T + 23 T^{2} )^{2} \)
$29$ \( 1 - 8 T^{2} + 841 T^{4} \)
$31$ \( 1 + 10 T^{2} + 961 T^{4} \)
$37$ \( ( 1 + 8 T + 37 T^{2} )^{2} \)
$41$ \( 1 - 64 T^{2} + 1681 T^{4} \)
$43$ \( 1 + 42 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 - 12 T + 47 T^{2} )^{2} \)
$53$ \( 1 + 56 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 8 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 102 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 4 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 + 8 T + 73 T^{2} )^{2} \)
$79$ \( 1 - 150 T^{2} + 6241 T^{4} \)
$83$ \( ( 1 + 12 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 64 T^{2} + 7921 T^{4} \)
$97$ \( ( 1 + 97 T^{2} )^{2} \)
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