Properties

Label 2304.2.k.e
Level $2304$
Weight $2$
Character orbit 2304.k
Analytic conductor $18.398$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.3975326257\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 768)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -2 - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{5} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{7} +O(q^{10})\) \( q + ( -2 - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{5} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{7} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{11} + ( 3 + 3 \zeta_{24}^{6} ) q^{13} + ( -4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{17} + ( -2 \zeta_{24} - 4 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{19} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{23} + ( 4 - 8 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{25} + ( -2 + 6 \zeta_{24}^{2} + 6 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{29} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{31} + ( 2 \zeta_{24} + 6 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{35} + ( -1 + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{37} + ( -2 + 4 \zeta_{24}^{4} - 8 \zeta_{24}^{6} ) q^{41} + ( -2 \zeta_{24}^{3} + 4 \zeta_{24}^{7} ) q^{43} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{47} + q^{49} + ( -4 + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{53} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{55} + ( 8 \zeta_{24}^{3} - 16 \zeta_{24}^{7} ) q^{59} + ( -5 + 4 \zeta_{24}^{2} + 4 \zeta_{24}^{4} - 5 \zeta_{24}^{6} ) q^{61} + ( -6 - 12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{65} + ( 4 \zeta_{24} + 8 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{67} + ( -10 \zeta_{24} + 10 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{71} + 4 \zeta_{24}^{6} q^{73} + ( -8 + 4 \zeta_{24}^{2} + 4 \zeta_{24}^{4} - 8 \zeta_{24}^{6} ) q^{77} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{79} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{83} + ( 8 + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{4} - 8 \zeta_{24}^{6} ) q^{85} + ( 4 - 8 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{89} + ( 6 \zeta_{24}^{3} - 12 \zeta_{24}^{7} ) q^{91} + ( 16 \zeta_{24} + 16 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 12 \zeta_{24}^{7} ) q^{95} + ( 8 - 8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{5} + O(q^{10}) \) \( 8q - 8q^{5} + 24q^{13} + 8q^{29} - 24q^{37} + 8q^{49} - 40q^{53} - 24q^{61} - 48q^{65} - 48q^{77} + 48q^{85} + 64q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{24}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
0.965926 + 0.258819i
−0.965926 0.258819i
0.258819 + 0.965926i
−0.258819 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
0 0 0 −2.73205 + 2.73205i 0 2.44949i 0 0 0
577.2 0 0 0 −2.73205 + 2.73205i 0 2.44949i 0 0 0
577.3 0 0 0 0.732051 0.732051i 0 2.44949i 0 0 0
577.4 0 0 0 0.732051 0.732051i 0 2.44949i 0 0 0
1729.1 0 0 0 −2.73205 2.73205i 0 2.44949i 0 0 0
1729.2 0 0 0 −2.73205 2.73205i 0 2.44949i 0 0 0
1729.3 0 0 0 0.732051 + 0.732051i 0 2.44949i 0 0 0
1729.4 0 0 0 0.732051 + 0.732051i 0 2.44949i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1729.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
16.e even 4 1 inner
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.2.k.e 8
3.b odd 2 1 768.2.j.f yes 8
4.b odd 2 1 inner 2304.2.k.e 8
8.b even 2 1 2304.2.k.l 8
8.d odd 2 1 2304.2.k.l 8
12.b even 2 1 768.2.j.f yes 8
16.e even 4 1 inner 2304.2.k.e 8
16.e even 4 1 2304.2.k.l 8
16.f odd 4 1 inner 2304.2.k.e 8
16.f odd 4 1 2304.2.k.l 8
24.f even 2 1 768.2.j.e 8
24.h odd 2 1 768.2.j.e 8
32.g even 8 1 9216.2.a.bc 4
32.g even 8 1 9216.2.a.bi 4
32.h odd 8 1 9216.2.a.bc 4
32.h odd 8 1 9216.2.a.bi 4
48.i odd 4 1 768.2.j.e 8
48.i odd 4 1 768.2.j.f yes 8
48.k even 4 1 768.2.j.e 8
48.k even 4 1 768.2.j.f yes 8
96.o even 8 1 3072.2.a.l 4
96.o even 8 1 3072.2.a.r 4
96.o even 8 2 3072.2.d.h 8
96.p odd 8 1 3072.2.a.l 4
96.p odd 8 1 3072.2.a.r 4
96.p odd 8 2 3072.2.d.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
768.2.j.e 8 24.f even 2 1
768.2.j.e 8 24.h odd 2 1
768.2.j.e 8 48.i odd 4 1
768.2.j.e 8 48.k even 4 1
768.2.j.f yes 8 3.b odd 2 1
768.2.j.f yes 8 12.b even 2 1
768.2.j.f yes 8 48.i odd 4 1
768.2.j.f yes 8 48.k even 4 1
2304.2.k.e 8 1.a even 1 1 trivial
2304.2.k.e 8 4.b odd 2 1 inner
2304.2.k.e 8 16.e even 4 1 inner
2304.2.k.e 8 16.f odd 4 1 inner
2304.2.k.l 8 8.b even 2 1
2304.2.k.l 8 8.d odd 2 1
2304.2.k.l 8 16.e even 4 1
2304.2.k.l 8 16.f odd 4 1
3072.2.a.l 4 96.o even 8 1
3072.2.a.l 4 96.p odd 8 1
3072.2.a.r 4 96.o even 8 1
3072.2.a.r 4 96.p odd 8 1
3072.2.d.h 8 96.o even 8 2
3072.2.d.h 8 96.p odd 8 2
9216.2.a.bc 4 32.g even 8 1
9216.2.a.bc 4 32.h odd 8 1
9216.2.a.bi 4 32.g even 8 1
9216.2.a.bi 4 32.h odd 8 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{4} + 4 T_{5}^{3} + 8 T_{5}^{2} - 16 T_{5} + 16 \)
\( T_{7}^{2} + 6 \)
\( T_{13}^{2} - 6 T_{13} + 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 + 4 T + 8 T^{2} + 4 T^{3} - 14 T^{4} + 20 T^{5} + 200 T^{6} + 500 T^{7} + 625 T^{8} )^{2} \)
$7$ \( ( 1 - 8 T^{2} + 49 T^{4} )^{4} \)
$11$ \( 1 - 28 T^{4} + 1830 T^{8} - 409948 T^{12} + 214358881 T^{16} \)
$13$ \( ( 1 - 6 T + 18 T^{2} - 78 T^{3} + 169 T^{4} )^{4} \)
$17$ \( ( 1 + 22 T^{2} + 289 T^{4} )^{4} \)
$19$ \( 1 + 292 T^{4} - 25242 T^{8} + 38053732 T^{12} + 16983563041 T^{16} \)
$23$ \( ( 1 - 38 T^{2} + 529 T^{4} )^{4} \)
$29$ \( ( 1 - 4 T + 8 T^{2} + 92 T^{3} - 1646 T^{4} + 2668 T^{5} + 6728 T^{6} - 97556 T^{7} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 + 8 T^{2} + 961 T^{4} )^{4} \)
$37$ \( ( 1 + 12 T + 72 T^{2} + 372 T^{3} + 1886 T^{4} + 13764 T^{5} + 98568 T^{6} + 607836 T^{7} + 1874161 T^{8} )^{2} \)
$41$ \( ( 1 - 12 T^{2} + 326 T^{4} - 20172 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 + 1778 T^{4} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 + 86 T^{2} + 2209 T^{4} )^{4} \)
$53$ \( ( 1 + 20 T + 200 T^{2} + 1940 T^{3} + 16882 T^{4} + 102820 T^{5} + 561800 T^{6} + 2977540 T^{7} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 - 1486 T^{4} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 + 12 T + 72 T^{2} + 660 T^{3} + 6014 T^{4} + 40260 T^{5} + 267912 T^{6} + 2723772 T^{7} + 13845841 T^{8} )^{2} \)
$67$ \( 1 + 7588 T^{4} + 30907110 T^{8} + 152906706148 T^{12} + 406067677556641 T^{16} \)
$71$ \( ( 1 + 20 T^{2} - 2106 T^{4} + 100820 T^{6} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 - 130 T^{2} + 5329 T^{4} )^{4} \)
$79$ \( ( 1 + 152 T^{2} + 6241 T^{4} )^{4} \)
$83$ \( 1 + 17828 T^{4} + 157096038 T^{8} + 846086946788 T^{12} + 2252292232139041 T^{16} \)
$89$ \( ( 1 - 252 T^{2} + 30950 T^{4} - 1996092 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 - 16 T + 210 T^{2} - 1552 T^{3} + 9409 T^{4} )^{4} \)
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