Properties

Label 2304.2.c.j
Level $2304$
Weight $2$
Character orbit 2304.c
Analytic conductor $18.398$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(18.3975326257\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 576)
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 \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{5} -2 \zeta_{24}^{6} q^{7} +O(q^{10})\) \( q + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{5} -2 \zeta_{24}^{6} q^{7} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{11} + ( -4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{13} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{17} + ( 4 - 8 \zeta_{24}^{4} ) q^{19} + ( 6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{23} - q^{25} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{29} -2 \zeta_{24}^{6} q^{31} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{35} + ( 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{37} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{41} + ( 6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{47} + 3 q^{49} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{53} -12 \zeta_{24}^{6} q^{55} + ( 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{61} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{65} + ( -4 + 8 \zeta_{24}^{4} ) q^{67} + ( 6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{71} -4 q^{73} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{77} -14 \zeta_{24}^{6} q^{79} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{83} + ( -12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{85} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{89} + ( -4 + 8 \zeta_{24}^{4} ) q^{91} + ( 12 \zeta_{24} + 12 \zeta_{24}^{3} - 12 \zeta_{24}^{5} ) q^{95} -16 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 8q^{25} + 24q^{49} - 32q^{73} - 128q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2303.1
−0.965926 0.258819i
−0.258819 0.965926i
0.965926 0.258819i
0.258819 0.965926i
0.965926 + 0.258819i
0.258819 + 0.965926i
−0.965926 + 0.258819i
−0.258819 + 0.965926i
0 0 0 2.44949i 0 2.00000i 0 0 0
2303.2 0 0 0 2.44949i 0 2.00000i 0 0 0
2303.3 0 0 0 2.44949i 0 2.00000i 0 0 0
2303.4 0 0 0 2.44949i 0 2.00000i 0 0 0
2303.5 0 0 0 2.44949i 0 2.00000i 0 0 0
2303.6 0 0 0 2.44949i 0 2.00000i 0 0 0
2303.7 0 0 0 2.44949i 0 2.00000i 0 0 0
2303.8 0 0 0 2.44949i 0 2.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2303.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.2.c.j 8
3.b odd 2 1 inner 2304.2.c.j 8
4.b odd 2 1 inner 2304.2.c.j 8
8.b even 2 1 inner 2304.2.c.j 8
8.d odd 2 1 inner 2304.2.c.j 8
12.b even 2 1 inner 2304.2.c.j 8
16.e even 4 2 576.2.f.a 8
16.f odd 4 2 576.2.f.a 8
24.f even 2 1 inner 2304.2.c.j 8
24.h odd 2 1 inner 2304.2.c.j 8
48.i odd 4 2 576.2.f.a 8
48.k even 4 2 576.2.f.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.2.f.a 8 16.e even 4 2
576.2.f.a 8 16.f odd 4 2
576.2.f.a 8 48.i odd 4 2
576.2.f.a 8 48.k even 4 2
2304.2.c.j 8 1.a even 1 1 trivial
2304.2.c.j 8 3.b odd 2 1 inner
2304.2.c.j 8 4.b odd 2 1 inner
2304.2.c.j 8 8.b even 2 1 inner
2304.2.c.j 8 8.d odd 2 1 inner
2304.2.c.j 8 12.b even 2 1 inner
2304.2.c.j 8 24.f even 2 1 inner
2304.2.c.j 8 24.h 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 \(S_{2}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{2} + 6 \)
\( T_{7}^{2} + 4 \)
\( T_{11}^{2} - 24 \)
\( T_{13}^{2} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - 4 T^{2} + 25 T^{4} )^{4} \)
$7$ \( ( 1 - 10 T^{2} + 49 T^{4} )^{4} \)
$11$ \( ( 1 - 2 T^{2} + 121 T^{4} )^{4} \)
$13$ \( ( 1 + 14 T^{2} + 169 T^{4} )^{4} \)
$17$ \( ( 1 - 16 T^{2} + 289 T^{4} )^{4} \)
$19$ \( ( 1 + 10 T^{2} + 361 T^{4} )^{4} \)
$23$ \( ( 1 - 26 T^{2} + 529 T^{4} )^{4} \)
$29$ \( ( 1 - 4 T^{2} + 841 T^{4} )^{4} \)
$31$ \( ( 1 - 58 T^{2} + 961 T^{4} )^{4} \)
$37$ \( ( 1 + 26 T^{2} + 1369 T^{4} )^{4} \)
$41$ \( ( 1 - 64 T^{2} + 1681 T^{4} )^{4} \)
$43$ \( ( 1 - 43 T^{2} )^{8} \)
$47$ \( ( 1 + 22 T^{2} + 2209 T^{4} )^{4} \)
$53$ \( ( 1 - 100 T^{2} + 2809 T^{4} )^{4} \)
$59$ \( ( 1 + 59 T^{2} )^{8} \)
$61$ \( ( 1 + 74 T^{2} + 3721 T^{4} )^{4} \)
$67$ \( ( 1 - 86 T^{2} + 4489 T^{4} )^{4} \)
$71$ \( ( 1 + 70 T^{2} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 + 4 T + 73 T^{2} )^{8} \)
$79$ \( ( 1 + 38 T^{2} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 + 142 T^{2} + 6889 T^{4} )^{4} \)
$89$ \( ( 1 - 160 T^{2} + 7921 T^{4} )^{4} \)
$97$ \( ( 1 + 16 T + 97 T^{2} )^{8} \)
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