Properties

Label 2304.3.g.u
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: \( 2^{8} \)
Twist minimal: no (minimal twist has level 64)
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.

\(f(q)\) \(=\) \( q + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{5} -8 \zeta_{12}^{3} q^{7} +O(q^{10})\) \( q + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{5} -8 \zeta_{12}^{3} q^{7} + ( 2 - 4 \zeta_{12}^{2} ) q^{11} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{13} + 6 q^{17} + ( -14 + 28 \zeta_{12}^{2} ) q^{19} + 24 \zeta_{12}^{3} q^{23} + 23 q^{25} + ( 24 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{29} -32 \zeta_{12}^{3} q^{31} + ( -32 + 64 \zeta_{12}^{2} ) q^{35} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{37} + 66 q^{41} + ( -18 + 36 \zeta_{12}^{2} ) q^{43} -48 \zeta_{12}^{3} q^{47} -15 q^{49} + ( -104 \zeta_{12} + 52 \zeta_{12}^{3} ) q^{53} + 24 \zeta_{12}^{3} q^{55} + ( 18 - 36 \zeta_{12}^{2} ) q^{59} + ( 104 \zeta_{12} - 52 \zeta_{12}^{3} ) q^{61} -48 q^{65} + ( -46 + 92 \zeta_{12}^{2} ) q^{67} -120 \zeta_{12}^{3} q^{71} -58 q^{73} + ( -32 \zeta_{12} + 16 \zeta_{12}^{3} ) q^{77} + 16 \zeta_{12}^{3} q^{79} + ( -34 + 68 \zeta_{12}^{2} ) q^{83} + ( -48 \zeta_{12} + 24 \zeta_{12}^{3} ) q^{85} -102 q^{89} + ( 32 - 64 \zeta_{12}^{2} ) q^{91} -168 \zeta_{12}^{3} q^{95} + 26 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 24q^{17} + 92q^{25} + 264q^{41} - 60q^{49} - 192q^{65} - 232q^{73} - 408q^{89} + 104q^{97} + O(q^{100}) \)

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 8.00000i 0 0 0
1279.2 0 0 0 −6.92820 0 8.00000i 0 0 0
1279.3 0 0 0 6.92820 0 8.00000i 0 0 0
1279.4 0 0 0 6.92820 0 8.00000i 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
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.u 4
3.b odd 2 1 256.3.c.g 4
4.b odd 2 1 inner 2304.3.g.u 4
8.b even 2 1 inner 2304.3.g.u 4
8.d odd 2 1 inner 2304.3.g.u 4
12.b even 2 1 256.3.c.g 4
16.e even 4 2 576.3.b.d 4
16.f odd 4 2 576.3.b.d 4
24.f even 2 1 256.3.c.g 4
24.h odd 2 1 256.3.c.g 4
48.i odd 4 2 64.3.d.a 4
48.k even 4 2 64.3.d.a 4
240.t even 4 2 1600.3.g.c 4
240.z odd 4 1 1600.3.e.a 4
240.z odd 4 1 1600.3.e.f 4
240.bb even 4 1 1600.3.e.a 4
240.bb even 4 1 1600.3.e.f 4
240.bd odd 4 1 1600.3.e.a 4
240.bd odd 4 1 1600.3.e.f 4
240.bf even 4 1 1600.3.e.a 4
240.bf even 4 1 1600.3.e.f 4
240.bm odd 4 2 1600.3.g.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.3.d.a 4 48.i odd 4 2
64.3.d.a 4 48.k even 4 2
256.3.c.g 4 3.b odd 2 1
256.3.c.g 4 12.b even 2 1
256.3.c.g 4 24.f even 2 1
256.3.c.g 4 24.h odd 2 1
576.3.b.d 4 16.e even 4 2
576.3.b.d 4 16.f odd 4 2
1600.3.e.a 4 240.z odd 4 1
1600.3.e.a 4 240.bb even 4 1
1600.3.e.a 4 240.bd odd 4 1
1600.3.e.a 4 240.bf even 4 1
1600.3.e.f 4 240.z odd 4 1
1600.3.e.f 4 240.bb even 4 1
1600.3.e.f 4 240.bd odd 4 1
1600.3.e.f 4 240.bf even 4 1
1600.3.g.c 4 240.t even 4 2
1600.3.g.c 4 240.bm odd 4 2
2304.3.g.u 4 1.a even 1 1 trivial
2304.3.g.u 4 4.b odd 2 1 inner
2304.3.g.u 4 8.b even 2 1 inner
2304.3.g.u 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 \(S_{3}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{2} - 48 \)
\( T_{7}^{2} + 64 \)
\( T_{11}^{2} + 12 \)
\( T_{13}^{2} - 48 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -48 + T^{2} )^{2} \)
$7$ \( ( 64 + T^{2} )^{2} \)
$11$ \( ( 12 + T^{2} )^{2} \)
$13$ \( ( -48 + T^{2} )^{2} \)
$17$ \( ( -6 + T )^{4} \)
$19$ \( ( 588 + T^{2} )^{2} \)
$23$ \( ( 576 + T^{2} )^{2} \)
$29$ \( ( -432 + T^{2} )^{2} \)
$31$ \( ( 1024 + T^{2} )^{2} \)
$37$ \( ( -48 + T^{2} )^{2} \)
$41$ \( ( -66 + T )^{4} \)
$43$ \( ( 972 + T^{2} )^{2} \)
$47$ \( ( 2304 + T^{2} )^{2} \)
$53$ \( ( -8112 + T^{2} )^{2} \)
$59$ \( ( 972 + T^{2} )^{2} \)
$61$ \( ( -8112 + T^{2} )^{2} \)
$67$ \( ( 6348 + T^{2} )^{2} \)
$71$ \( ( 14400 + T^{2} )^{2} \)
$73$ \( ( 58 + T )^{4} \)
$79$ \( ( 256 + T^{2} )^{2} \)
$83$ \( ( 3468 + T^{2} )^{2} \)
$89$ \( ( 102 + T )^{4} \)
$97$ \( ( -26 + T )^{4} \)
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