Properties

Label 2304.2.k.h
Level $2304$
Weight $2$
Character orbit 2304.k
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.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: \( 2^{10} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( -2 \zeta_{24} - 2 \zeta_{24}^{5} ) q^{5} + ( -\zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{7} +O(q^{10})\) \( q + ( -2 \zeta_{24} - 2 \zeta_{24}^{5} ) q^{5} + ( -\zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{7} + ( 2 - 4 \zeta_{24}^{2} - 4 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{11} + ( -1 + \zeta_{24}^{6} ) q^{13} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{17} + ( 6 \zeta_{24} - 6 \zeta_{24}^{5} ) q^{19} + ( -4 + 8 \zeta_{24}^{4} ) q^{23} + 7 \zeta_{24}^{6} q^{25} + ( -2 \zeta_{24}^{3} + 4 \zeta_{24}^{7} ) q^{29} + ( -\zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{31} + ( 2 + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{35} + ( 7 + 7 \zeta_{24}^{6} ) q^{37} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{41} -6 \zeta_{24}^{3} q^{43} + ( 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{47} + 5 q^{49} + ( 2 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{53} + ( -12 \zeta_{24} + 12 \zeta_{24}^{3} + 12 \zeta_{24}^{5} ) q^{55} + ( -4 + 8 \zeta_{24}^{2} + 8 \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{59} + ( -5 + 5 \zeta_{24}^{6} ) q^{61} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{65} + ( 8 \zeta_{24} - 8 \zeta_{24}^{5} ) q^{67} + ( 8 - 16 \zeta_{24}^{4} ) q^{71} + 12 \zeta_{24}^{6} q^{73} + ( 4 \zeta_{24}^{3} - 8 \zeta_{24}^{7} ) q^{77} + ( -11 \zeta_{24} - 11 \zeta_{24}^{3} + 11 \zeta_{24}^{5} ) q^{79} + ( -6 - 12 \zeta_{24}^{2} + 12 \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{83} + ( 12 + 12 \zeta_{24}^{6} ) q^{85} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{89} -2 \zeta_{24}^{3} q^{91} + ( -24 \zeta_{24}^{2} + 12 \zeta_{24}^{6} ) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 8q^{13} + 56q^{37} + 40q^{49} - 40q^{61} + 96q^{85} + O(q^{100}) \)

Character values

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

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

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
0.258819 0.965926i
−0.258819 + 0.965926i
−0.965926 + 0.258819i
0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
−0.258819 0.965926i
0 0 0 −2.44949 + 2.44949i 0 1.41421i 0 0 0
577.2 0 0 0 −2.44949 + 2.44949i 0 1.41421i 0 0 0
577.3 0 0 0 2.44949 2.44949i 0 1.41421i 0 0 0
577.4 0 0 0 2.44949 2.44949i 0 1.41421i 0 0 0
1729.1 0 0 0 −2.44949 2.44949i 0 1.41421i 0 0 0
1729.2 0 0 0 −2.44949 2.44949i 0 1.41421i 0 0 0
1729.3 0 0 0 2.44949 + 2.44949i 0 1.41421i 0 0 0
1729.4 0 0 0 2.44949 + 2.44949i 0 1.41421i 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
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
16.e even 4 1 inner
16.f odd 4 1 inner
48.i odd 4 1 inner
48.k even 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.h 8
3.b odd 2 1 inner 2304.2.k.h 8
4.b odd 2 1 inner 2304.2.k.h 8
8.b even 2 1 2304.2.k.i yes 8
8.d odd 2 1 2304.2.k.i yes 8
12.b even 2 1 inner 2304.2.k.h 8
16.e even 4 1 inner 2304.2.k.h 8
16.e even 4 1 2304.2.k.i yes 8
16.f odd 4 1 inner 2304.2.k.h 8
16.f odd 4 1 2304.2.k.i yes 8
24.f even 2 1 2304.2.k.i yes 8
24.h odd 2 1 2304.2.k.i yes 8
32.g even 8 1 9216.2.a.be 4
32.g even 8 1 9216.2.a.bh 4
32.h odd 8 1 9216.2.a.be 4
32.h odd 8 1 9216.2.a.bh 4
48.i odd 4 1 inner 2304.2.k.h 8
48.i odd 4 1 2304.2.k.i yes 8
48.k even 4 1 inner 2304.2.k.h 8
48.k even 4 1 2304.2.k.i yes 8
96.o even 8 1 9216.2.a.be 4
96.o even 8 1 9216.2.a.bh 4
96.p odd 8 1 9216.2.a.be 4
96.p odd 8 1 9216.2.a.bh 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2304.2.k.h 8 1.a even 1 1 trivial
2304.2.k.h 8 3.b odd 2 1 inner
2304.2.k.h 8 4.b odd 2 1 inner
2304.2.k.h 8 12.b even 2 1 inner
2304.2.k.h 8 16.e even 4 1 inner
2304.2.k.h 8 16.f odd 4 1 inner
2304.2.k.h 8 48.i odd 4 1 inner
2304.2.k.h 8 48.k even 4 1 inner
2304.2.k.i yes 8 8.b even 2 1
2304.2.k.i yes 8 8.d odd 2 1
2304.2.k.i yes 8 16.e even 4 1
2304.2.k.i yes 8 16.f odd 4 1
2304.2.k.i yes 8 24.f even 2 1
2304.2.k.i yes 8 24.h odd 2 1
2304.2.k.i yes 8 48.i odd 4 1
2304.2.k.i yes 8 48.k even 4 1
9216.2.a.be 4 32.g even 8 1
9216.2.a.be 4 32.h odd 8 1
9216.2.a.be 4 96.o even 8 1
9216.2.a.be 4 96.p odd 8 1
9216.2.a.bh 4 32.g even 8 1
9216.2.a.bh 4 32.h odd 8 1
9216.2.a.bh 4 96.o even 8 1
9216.2.a.bh 4 96.p odd 8 1

Hecke kernels

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

\( T_{5}^{4} + 144 \)
\( T_{7}^{2} + 2 \)
\( T_{13}^{2} + 2 T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - 4 T + 8 T^{2} - 20 T^{3} + 25 T^{4} )^{2}( 1 + 4 T + 8 T^{2} + 20 T^{3} + 25 T^{4} )^{2} \)
$7$ \( ( 1 - 12 T^{2} + 49 T^{4} )^{4} \)
$11$ \( ( 1 - 238 T^{4} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )^{4}( 1 + 6 T + 13 T^{2} )^{4} \)
$17$ \( ( 1 + 10 T^{2} + 289 T^{4} )^{4} \)
$19$ \( ( 1 - 718 T^{4} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 + 2 T^{2} + 529 T^{4} )^{4} \)
$29$ \( ( 1 + 434 T^{4} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 + 60 T^{2} + 961 T^{4} )^{4} \)
$37$ \( ( 1 - 12 T + 37 T^{2} )^{4}( 1 - 2 T + 37 T^{2} )^{4} \)
$41$ \( ( 1 - 58 T^{2} + 1681 T^{4} )^{4} \)
$43$ \( ( 1 - 1198 T^{4} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 + 46 T^{2} + 2209 T^{4} )^{4} \)
$53$ \( ( 1 - 20 T + 200 T^{2} - 1060 T^{3} + 2809 T^{4} )^{2}( 1 + 20 T + 200 T^{2} + 1060 T^{3} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 - 6478 T^{4} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 + 10 T + 50 T^{2} + 610 T^{3} + 3721 T^{4} )^{4} \)
$67$ \( ( 1 - 4078 T^{4} + 20151121 T^{8} )^{2} \)
$71$ \( ( 1 + 50 T^{2} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 - 2 T^{2} + 5329 T^{4} )^{4} \)
$79$ \( ( 1 - 84 T^{2} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 - 11278 T^{4} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 - 82 T^{2} + 7921 T^{4} )^{4} \)
$97$ \( ( 1 + 97 T^{2} )^{8} \)
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