Properties

Label 2304.3.g.j
Level $2304$
Weight $3$
Character orbit 2304.g
Analytic conductor $62.779$
Analytic rank $0$
Dimension $2$
CM discriminant -8
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +O(q^{10})\) \( q -14 i q^{11} -2 q^{17} + 34 i q^{19} -25 q^{25} -46 q^{41} + 14 i q^{43} + 49 q^{49} + 82 i q^{59} -62 i q^{67} + 142 q^{73} + 158 i q^{83} + 146 q^{89} -94 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 4q^{17} - 50q^{25} - 92q^{41} + 98q^{49} + 284q^{73} + 292q^{89} - 188q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1279.1
1.00000i
1.00000i
0 0 0 0 0 0 0 0 0
1279.2 0 0 0 0 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
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 2304.3.g.j 2
3.b odd 2 1 256.3.c.e 2
4.b odd 2 1 inner 2304.3.g.j 2
8.b even 2 1 inner 2304.3.g.j 2
8.d odd 2 1 CM 2304.3.g.j 2
12.b even 2 1 256.3.c.e 2
16.e even 4 1 72.3.b.a 1
16.e even 4 1 288.3.b.a 1
16.f odd 4 1 72.3.b.a 1
16.f odd 4 1 288.3.b.a 1
24.f even 2 1 256.3.c.e 2
24.h odd 2 1 256.3.c.e 2
48.i odd 4 1 8.3.d.a 1
48.i odd 4 1 32.3.d.a 1
48.k even 4 1 8.3.d.a 1
48.k even 4 1 32.3.d.a 1
240.t even 4 1 200.3.g.a 1
240.t even 4 1 800.3.g.a 1
240.z odd 4 1 200.3.e.a 2
240.z odd 4 1 800.3.e.a 2
240.bb even 4 1 200.3.e.a 2
240.bb even 4 1 800.3.e.a 2
240.bd odd 4 1 200.3.e.a 2
240.bd odd 4 1 800.3.e.a 2
240.bf even 4 1 200.3.e.a 2
240.bf even 4 1 800.3.e.a 2
240.bm odd 4 1 200.3.g.a 1
240.bm odd 4 1 800.3.g.a 1
336.v odd 4 1 392.3.g.a 1
336.v odd 4 1 1568.3.g.a 1
336.y even 4 1 392.3.g.a 1
336.y even 4 1 1568.3.g.a 1
336.bo even 12 2 392.3.k.b 2
336.br odd 12 2 392.3.k.b 2
336.bt odd 12 2 392.3.k.d 2
336.bu even 12 2 392.3.k.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.3.d.a 1 48.i odd 4 1
8.3.d.a 1 48.k even 4 1
32.3.d.a 1 48.i odd 4 1
32.3.d.a 1 48.k even 4 1
72.3.b.a 1 16.e even 4 1
72.3.b.a 1 16.f odd 4 1
200.3.e.a 2 240.z odd 4 1
200.3.e.a 2 240.bb even 4 1
200.3.e.a 2 240.bd odd 4 1
200.3.e.a 2 240.bf even 4 1
200.3.g.a 1 240.t even 4 1
200.3.g.a 1 240.bm odd 4 1
256.3.c.e 2 3.b odd 2 1
256.3.c.e 2 12.b even 2 1
256.3.c.e 2 24.f even 2 1
256.3.c.e 2 24.h odd 2 1
288.3.b.a 1 16.e even 4 1
288.3.b.a 1 16.f odd 4 1
392.3.g.a 1 336.v odd 4 1
392.3.g.a 1 336.y even 4 1
392.3.k.b 2 336.bo even 12 2
392.3.k.b 2 336.br odd 12 2
392.3.k.d 2 336.bt odd 12 2
392.3.k.d 2 336.bu even 12 2
800.3.e.a 2 240.z odd 4 1
800.3.e.a 2 240.bb even 4 1
800.3.e.a 2 240.bd odd 4 1
800.3.e.a 2 240.bf even 4 1
800.3.g.a 1 240.t even 4 1
800.3.g.a 1 240.bm odd 4 1
1568.3.g.a 1 336.v odd 4 1
1568.3.g.a 1 336.y even 4 1
2304.3.g.j 2 1.a even 1 1 trivial
2304.3.g.j 2 4.b odd 2 1 inner
2304.3.g.j 2 8.b even 2 1 inner
2304.3.g.j 2 8.d odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5} \)
\( T_{7} \)
\( T_{11}^{2} + 196 \)
\( T_{13} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( 196 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( 2 + T )^{2} \)
$19$ \( 1156 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( ( 46 + T )^{2} \)
$43$ \( 196 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( 6724 + T^{2} \)
$61$ \( T^{2} \)
$67$ \( 3844 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -142 + T )^{2} \)
$79$ \( T^{2} \)
$83$ \( 24964 + T^{2} \)
$89$ \( ( -146 + T )^{2} \)
$97$ \( ( 94 + T )^{2} \)
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