Properties

Label 2268.2.x.j
Level $2268$
Weight $2$
Character orbit 2268.x
Analytic conductor $18.110$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.x (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.1100711784\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 756)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{5} + ( -\zeta_{24} - 2 \zeta_{24}^{3} + \zeta_{24}^{4} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{7} +O(q^{10})\) \( q + ( \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{5} + ( -\zeta_{24} - 2 \zeta_{24}^{3} + \zeta_{24}^{4} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{7} + ( 3 \zeta_{24} - 3 \zeta_{24}^{7} ) q^{11} + ( -2 \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{13} + ( -2 \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{17} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{19} + ( -6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{23} + 2 \zeta_{24}^{4} q^{25} + ( 3 \zeta_{24} - 3 \zeta_{24}^{7} ) q^{29} + ( -6 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{31} + ( -3 \zeta_{24} + 2 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{35} + q^{37} + ( -\zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{41} -7 \zeta_{24}^{4} q^{43} + ( 7 \zeta_{24}^{2} + 7 \zeta_{24}^{6} ) q^{47} + ( 5 - 4 \zeta_{24} - 2 \zeta_{24}^{3} - 5 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{49} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{55} + ( 5 \zeta_{24}^{2} - 10 \zeta_{24}^{6} ) q^{59} + ( -\zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{61} + ( -3 \zeta_{24} + 3 \zeta_{24}^{7} ) q^{65} + ( 10 - 10 \zeta_{24}^{4} ) q^{67} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{71} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{73} + ( -6 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 12 \zeta_{24}^{6} - 3 \zeta_{24}^{7} ) q^{77} -5 \zeta_{24}^{4} q^{79} + ( -7 \zeta_{24}^{2} - 7 \zeta_{24}^{6} ) q^{83} + ( -3 + 3 \zeta_{24}^{4} ) q^{85} + ( -12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{89} + ( 6 - \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{91} + ( -3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{95} + ( \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{7} + O(q^{10}) \) \( 8q + 4q^{7} + 8q^{25} + 8q^{37} - 28q^{43} + 20q^{49} + 40q^{67} - 20q^{79} - 12q^{85} + 48q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(-1\) \(1\) \(1 - \zeta_{24}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
377.1
−0.258819 + 0.965926i
0.258819 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
0 0 0 −0.866025 + 1.50000i 0 −1.62132 + 2.09077i 0 0 0
377.2 0 0 0 −0.866025 + 1.50000i 0 2.62132 0.358719i 0 0 0
377.3 0 0 0 0.866025 1.50000i 0 −1.62132 + 2.09077i 0 0 0
377.4 0 0 0 0.866025 1.50000i 0 2.62132 0.358719i 0 0 0
1889.1 0 0 0 −0.866025 1.50000i 0 −1.62132 2.09077i 0 0 0
1889.2 0 0 0 −0.866025 1.50000i 0 2.62132 + 0.358719i 0 0 0
1889.3 0 0 0 0.866025 + 1.50000i 0 −1.62132 2.09077i 0 0 0
1889.4 0 0 0 0.866025 + 1.50000i 0 2.62132 + 0.358719i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1889.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
21.c even 2 1 inner
63.l odd 6 1 inner
63.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.x.j 8
3.b odd 2 1 inner 2268.2.x.j 8
7.b odd 2 1 inner 2268.2.x.j 8
9.c even 3 1 756.2.f.d 4
9.c even 3 1 inner 2268.2.x.j 8
9.d odd 6 1 756.2.f.d 4
9.d odd 6 1 inner 2268.2.x.j 8
21.c even 2 1 inner 2268.2.x.j 8
36.f odd 6 1 3024.2.k.h 4
36.h even 6 1 3024.2.k.h 4
63.l odd 6 1 756.2.f.d 4
63.l odd 6 1 inner 2268.2.x.j 8
63.o even 6 1 756.2.f.d 4
63.o even 6 1 inner 2268.2.x.j 8
252.s odd 6 1 3024.2.k.h 4
252.bi even 6 1 3024.2.k.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.f.d 4 9.c even 3 1
756.2.f.d 4 9.d odd 6 1
756.2.f.d 4 63.l odd 6 1
756.2.f.d 4 63.o even 6 1
2268.2.x.j 8 1.a even 1 1 trivial
2268.2.x.j 8 3.b odd 2 1 inner
2268.2.x.j 8 7.b odd 2 1 inner
2268.2.x.j 8 9.c even 3 1 inner
2268.2.x.j 8 9.d odd 6 1 inner
2268.2.x.j 8 21.c even 2 1 inner
2268.2.x.j 8 63.l odd 6 1 inner
2268.2.x.j 8 63.o even 6 1 inner
3024.2.k.h 4 36.f odd 6 1
3024.2.k.h 4 36.h even 6 1
3024.2.k.h 4 252.s odd 6 1
3024.2.k.h 4 252.bi even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2268, [\chi])\):

\( T_{5}^{4} + 3 T_{5}^{2} + 9 \)
\( T_{11}^{4} - 18 T_{11}^{2} + 324 \)
\( T_{13}^{4} - 6 T_{13}^{2} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 9 + 3 T^{2} + T^{4} )^{2} \)
$7$ \( ( 49 - 14 T - 3 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$11$ \( ( 324 - 18 T^{2} + T^{4} )^{2} \)
$13$ \( ( 36 - 6 T^{2} + T^{4} )^{2} \)
$17$ \( ( -3 + T^{2} )^{4} \)
$19$ \( ( 6 + T^{2} )^{4} \)
$23$ \( ( 5184 - 72 T^{2} + T^{4} )^{2} \)
$29$ \( ( 324 - 18 T^{2} + T^{4} )^{2} \)
$31$ \( ( 2916 - 54 T^{2} + T^{4} )^{2} \)
$37$ \( ( -1 + T )^{8} \)
$41$ \( ( 9 + 3 T^{2} + T^{4} )^{2} \)
$43$ \( ( 49 + 7 T + T^{2} )^{4} \)
$47$ \( ( 21609 + 147 T^{2} + T^{4} )^{2} \)
$53$ \( T^{8} \)
$59$ \( ( 5625 + 75 T^{2} + T^{4} )^{2} \)
$61$ \( ( 36 - 6 T^{2} + T^{4} )^{2} \)
$67$ \( ( 100 - 10 T + T^{2} )^{4} \)
$71$ \( ( 72 + T^{2} )^{4} \)
$73$ \( ( 96 + T^{2} )^{4} \)
$79$ \( ( 25 + 5 T + T^{2} )^{4} \)
$83$ \( ( 21609 + 147 T^{2} + T^{4} )^{2} \)
$89$ \( ( -108 + T^{2} )^{4} \)
$97$ \( ( 36 - 6 T^{2} + T^{4} )^{2} \)
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