Properties

Label 2304.3.g.q
Level $2304$
Weight $3$
Character orbit 2304.g
Analytic conductor $62.779$
Analytic rank $0$
Dimension $4$
CM no
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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 192)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{5} -10 \zeta_{12}^{3} q^{7} +O(q^{10})\) \( q + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{5} -10 \zeta_{12}^{3} q^{7} + ( -8 + 16 \zeta_{12}^{2} ) q^{11} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{13} -30 q^{17} + ( -4 + 8 \zeta_{12}^{2} ) q^{19} + 12 \zeta_{12}^{3} q^{23} -13 q^{25} + ( 60 \zeta_{12} - 30 \zeta_{12}^{3} ) q^{29} + 14 \zeta_{12}^{3} q^{31} + ( 20 - 40 \zeta_{12}^{2} ) q^{35} + ( -64 \zeta_{12} + 32 \zeta_{12}^{3} ) q^{37} -6 q^{41} + ( -36 + 72 \zeta_{12}^{2} ) q^{43} + 84 \zeta_{12}^{3} q^{47} -51 q^{49} + ( -20 \zeta_{12} + 10 \zeta_{12}^{3} ) q^{53} + 48 \zeta_{12}^{3} q^{55} + ( -36 + 72 \zeta_{12}^{2} ) q^{59} + ( -112 \zeta_{12} + 56 \zeta_{12}^{3} ) q^{61} + 24 q^{65} + ( 28 - 56 \zeta_{12}^{2} ) q^{67} -60 \zeta_{12}^{3} q^{71} + 86 q^{73} + ( 160 \zeta_{12} - 80 \zeta_{12}^{3} ) q^{77} + 38 \zeta_{12}^{3} q^{79} + ( -8 + 16 \zeta_{12}^{2} ) q^{83} + ( -120 \zeta_{12} + 60 \zeta_{12}^{3} ) q^{85} + 78 q^{89} + ( 40 - 80 \zeta_{12}^{2} ) q^{91} + 24 \zeta_{12}^{3} q^{95} + 62 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 120q^{17} - 52q^{25} - 24q^{41} - 204q^{49} + 96q^{65} + 344q^{73} + 312q^{89} + 248q^{97} + O(q^{100}) \)

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
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
0 0 0 −3.46410 0 10.0000i 0 0 0
1279.2 0 0 0 −3.46410 0 10.0000i 0 0 0
1279.3 0 0 0 3.46410 0 10.0000i 0 0 0
1279.4 0 0 0 3.46410 0 10.0000i 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 inner
8.b even 2 1 inner
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.g.q 4
3.b odd 2 1 768.3.g.f 4
4.b odd 2 1 inner 2304.3.g.q 4
8.b even 2 1 inner 2304.3.g.q 4
8.d odd 2 1 inner 2304.3.g.q 4
12.b even 2 1 768.3.g.f 4
16.e even 4 2 576.3.b.a 4
16.f odd 4 2 576.3.b.a 4
24.f even 2 1 768.3.g.f 4
24.h odd 2 1 768.3.g.f 4
48.i odd 4 2 192.3.b.b 4
48.k even 4 2 192.3.b.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.3.b.b 4 48.i odd 4 2
192.3.b.b 4 48.k even 4 2
576.3.b.a 4 16.e even 4 2
576.3.b.a 4 16.f odd 4 2
768.3.g.f 4 3.b odd 2 1
768.3.g.f 4 12.b even 2 1
768.3.g.f 4 24.f even 2 1
768.3.g.f 4 24.h odd 2 1
2304.3.g.q 4 1.a even 1 1 trivial
2304.3.g.q 4 4.b odd 2 1 inner
2304.3.g.q 4 8.b even 2 1 inner
2304.3.g.q 4 8.d odd 2 1 inner

Hecke kernels

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

\( T_{5}^{2} - 12 \)
\( T_{7}^{2} + 100 \)
\( T_{11}^{2} + 192 \)
\( T_{13}^{2} - 48 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -12 + T^{2} )^{2} \)
$7$ \( ( 100 + T^{2} )^{2} \)
$11$ \( ( 192 + T^{2} )^{2} \)
$13$ \( ( -48 + T^{2} )^{2} \)
$17$ \( ( 30 + T )^{4} \)
$19$ \( ( 48 + T^{2} )^{2} \)
$23$ \( ( 144 + T^{2} )^{2} \)
$29$ \( ( -2700 + T^{2} )^{2} \)
$31$ \( ( 196 + T^{2} )^{2} \)
$37$ \( ( -3072 + T^{2} )^{2} \)
$41$ \( ( 6 + T )^{4} \)
$43$ \( ( 3888 + T^{2} )^{2} \)
$47$ \( ( 7056 + T^{2} )^{2} \)
$53$ \( ( -300 + T^{2} )^{2} \)
$59$ \( ( 3888 + T^{2} )^{2} \)
$61$ \( ( -9408 + T^{2} )^{2} \)
$67$ \( ( 2352 + T^{2} )^{2} \)
$71$ \( ( 3600 + T^{2} )^{2} \)
$73$ \( ( -86 + T )^{4} \)
$79$ \( ( 1444 + T^{2} )^{2} \)
$83$ \( ( 192 + T^{2} )^{2} \)
$89$ \( ( -78 + T )^{4} \)
$97$ \( ( -62 + T )^{4} \)
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