Properties

Label 2304.3.b.a
Level $2304$
Weight $3$
Character orbit 2304.b
Analytic conductor $62.779$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.b (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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 288)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 4 i q^{5} -4 i q^{7} +O(q^{10})\) \( q + 4 i q^{5} -4 i q^{7} -16 q^{11} -2 i q^{13} -24 q^{17} + 24 q^{19} + 32 i q^{23} + 9 q^{25} -44 i q^{29} + 52 i q^{31} + 16 q^{35} -18 i q^{37} + 8 q^{41} -56 q^{43} -32 i q^{47} + 33 q^{49} -36 i q^{53} -64 i q^{55} + 32 q^{59} -62 i q^{61} + 8 q^{65} + 80 q^{67} -128 i q^{71} + 66 q^{73} + 64 i q^{77} + 20 i q^{79} -16 q^{83} -96 i q^{85} -144 q^{89} -8 q^{91} + 96 i q^{95} + 94 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 32q^{11} - 48q^{17} + 48q^{19} + 18q^{25} + 32q^{35} + 16q^{41} - 112q^{43} + 66q^{49} + 64q^{59} + 16q^{65} + 160q^{67} + 132q^{73} - 32q^{83} - 288q^{89} - 16q^{91} + 188q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
1.00000i
1.00000i
0 0 0 4.00000i 0 4.00000i 0 0 0
127.2 0 0 0 4.00000i 0 4.00000i 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 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.3.b.a 2
3.b odd 2 1 2304.3.b.i 2
4.b odd 2 1 2304.3.b.h 2
8.b even 2 1 2304.3.b.h 2
8.d odd 2 1 inner 2304.3.b.a 2
12.b even 2 1 2304.3.b.b 2
16.e even 4 1 288.3.g.a 2
16.e even 4 1 576.3.g.h 2
16.f odd 4 1 288.3.g.a 2
16.f odd 4 1 576.3.g.h 2
24.f even 2 1 2304.3.b.i 2
24.h odd 2 1 2304.3.b.b 2
48.i odd 4 1 288.3.g.c yes 2
48.i odd 4 1 576.3.g.d 2
48.k even 4 1 288.3.g.c yes 2
48.k even 4 1 576.3.g.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.3.g.a 2 16.e even 4 1
288.3.g.a 2 16.f odd 4 1
288.3.g.c yes 2 48.i odd 4 1
288.3.g.c yes 2 48.k even 4 1
576.3.g.d 2 48.i odd 4 1
576.3.g.d 2 48.k even 4 1
576.3.g.h 2 16.e even 4 1
576.3.g.h 2 16.f odd 4 1
2304.3.b.a 2 1.a even 1 1 trivial
2304.3.b.a 2 8.d odd 2 1 inner
2304.3.b.b 2 12.b even 2 1
2304.3.b.b 2 24.h odd 2 1
2304.3.b.h 2 4.b odd 2 1
2304.3.b.h 2 8.b even 2 1
2304.3.b.i 2 3.b odd 2 1
2304.3.b.i 2 24.f even 2 1

Hecke kernels

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

\( T_{5}^{2} + 16 \)
\( T_{7}^{2} + 16 \)
\( T_{11} + 16 \)
\( T_{17} + 24 \)
\( T_{19} - 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 16 + T^{2} \)
$7$ \( 16 + T^{2} \)
$11$ \( ( 16 + T )^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( ( 24 + T )^{2} \)
$19$ \( ( -24 + T )^{2} \)
$23$ \( 1024 + T^{2} \)
$29$ \( 1936 + T^{2} \)
$31$ \( 2704 + T^{2} \)
$37$ \( 324 + T^{2} \)
$41$ \( ( -8 + T )^{2} \)
$43$ \( ( 56 + T )^{2} \)
$47$ \( 1024 + T^{2} \)
$53$ \( 1296 + T^{2} \)
$59$ \( ( -32 + T )^{2} \)
$61$ \( 3844 + T^{2} \)
$67$ \( ( -80 + T )^{2} \)
$71$ \( 16384 + T^{2} \)
$73$ \( ( -66 + T )^{2} \)
$79$ \( 400 + T^{2} \)
$83$ \( ( 16 + T )^{2} \)
$89$ \( ( 144 + T )^{2} \)
$97$ \( ( -94 + T )^{2} \)
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