Properties

Label 2304.3.g.r
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: \( 2^{8}\cdot 3 \)
Twist minimal: no (minimal twist has level 384)
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.

\(f(q)\) \(=\) \( q + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{5} -12 \zeta_{12}^{3} q^{7} +O(q^{10})\) \( q + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{5} -12 \zeta_{12}^{3} q^{7} + ( 4 - 8 \zeta_{12}^{2} ) q^{11} + ( 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{13} -14 q^{17} + ( 20 - 40 \zeta_{12}^{2} ) q^{19} -24 \zeta_{12}^{3} q^{23} + 23 q^{25} + ( 40 \zeta_{12} - 20 \zeta_{12}^{3} ) q^{29} -12 \zeta_{12}^{3} q^{31} + ( -48 + 96 \zeta_{12}^{2} ) q^{35} + ( 32 \zeta_{12} - 16 \zeta_{12}^{3} ) q^{37} -14 q^{41} + ( -4 + 8 \zeta_{12}^{2} ) q^{43} -72 \zeta_{12}^{3} q^{47} -95 q^{49} + ( 72 \zeta_{12} - 36 \zeta_{12}^{3} ) q^{53} + 48 \zeta_{12}^{3} q^{55} + ( -28 + 56 \zeta_{12}^{2} ) q^{59} + ( 64 \zeta_{12} - 32 \zeta_{12}^{3} ) q^{61} -96 q^{65} + ( 52 - 104 \zeta_{12}^{2} ) q^{67} + 24 \zeta_{12}^{3} q^{71} + 50 q^{73} + ( -96 \zeta_{12} + 48 \zeta_{12}^{3} ) q^{77} -12 \zeta_{12}^{3} q^{79} + ( 12 - 24 \zeta_{12}^{2} ) q^{83} + ( 112 \zeta_{12} - 56 \zeta_{12}^{3} ) q^{85} + 62 q^{89} + ( 96 - 192 \zeta_{12}^{2} ) q^{91} + 240 \zeta_{12}^{3} q^{95} -146 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 56q^{17} + 92q^{25} - 56q^{41} - 380q^{49} - 384q^{65} + 200q^{73} + 248q^{89} - 584q^{97} + O(q^{100}) \)

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 −6.92820 0 12.0000i 0 0 0
1279.2 0 0 0 −6.92820 0 12.0000i 0 0 0
1279.3 0 0 0 6.92820 0 12.0000i 0 0 0
1279.4 0 0 0 6.92820 0 12.0000i 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.r 4
3.b odd 2 1 768.3.g.e 4
4.b odd 2 1 inner 2304.3.g.r 4
8.b even 2 1 inner 2304.3.g.r 4
8.d odd 2 1 inner 2304.3.g.r 4
12.b even 2 1 768.3.g.e 4
16.e even 4 2 1152.3.b.e 4
16.f odd 4 2 1152.3.b.e 4
24.f even 2 1 768.3.g.e 4
24.h odd 2 1 768.3.g.e 4
48.i odd 4 2 384.3.b.b 4
48.k even 4 2 384.3.b.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.3.b.b 4 48.i odd 4 2
384.3.b.b 4 48.k even 4 2
768.3.g.e 4 3.b odd 2 1
768.3.g.e 4 12.b even 2 1
768.3.g.e 4 24.f even 2 1
768.3.g.e 4 24.h odd 2 1
1152.3.b.e 4 16.e even 4 2
1152.3.b.e 4 16.f odd 4 2
2304.3.g.r 4 1.a even 1 1 trivial
2304.3.g.r 4 4.b odd 2 1 inner
2304.3.g.r 4 8.b even 2 1 inner
2304.3.g.r 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 \(S_{3}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{2} - 48 \)
\( T_{7}^{2} + 144 \)
\( T_{11}^{2} + 48 \)
\( T_{13}^{2} - 192 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -48 + T^{2} )^{2} \)
$7$ \( ( 144 + T^{2} )^{2} \)
$11$ \( ( 48 + T^{2} )^{2} \)
$13$ \( ( -192 + T^{2} )^{2} \)
$17$ \( ( 14 + T )^{4} \)
$19$ \( ( 1200 + T^{2} )^{2} \)
$23$ \( ( 576 + T^{2} )^{2} \)
$29$ \( ( -1200 + T^{2} )^{2} \)
$31$ \( ( 144 + T^{2} )^{2} \)
$37$ \( ( -768 + T^{2} )^{2} \)
$41$ \( ( 14 + T )^{4} \)
$43$ \( ( 48 + T^{2} )^{2} \)
$47$ \( ( 5184 + T^{2} )^{2} \)
$53$ \( ( -3888 + T^{2} )^{2} \)
$59$ \( ( 2352 + T^{2} )^{2} \)
$61$ \( ( -3072 + T^{2} )^{2} \)
$67$ \( ( 8112 + T^{2} )^{2} \)
$71$ \( ( 576 + T^{2} )^{2} \)
$73$ \( ( -50 + T )^{4} \)
$79$ \( ( 144 + T^{2} )^{2} \)
$83$ \( ( 432 + T^{2} )^{2} \)
$89$ \( ( -62 + T )^{4} \)
$97$ \( ( 146 + T )^{4} \)
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