Properties

Label 2268.1.h.a
Level $2268$
Weight $1$
Character orbit 2268.h
Analytic conductor $1.132$
Analytic rank $0$
Dimension $8$
Projective image $D_{12}$
CM discriminant -7
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2268.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.13187944865\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{12}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{24} q^{2} + \zeta_{24}^{2} q^{4} + \zeta_{24}^{6} q^{7} -\zeta_{24}^{3} q^{8} +O(q^{10})\) \( q -\zeta_{24} q^{2} + \zeta_{24}^{2} q^{4} + \zeta_{24}^{6} q^{7} -\zeta_{24}^{3} q^{8} + ( -\zeta_{24}^{5} + \zeta_{24}^{7} ) q^{11} -\zeta_{24}^{7} q^{14} + \zeta_{24}^{4} q^{16} + ( \zeta_{24}^{6} - \zeta_{24}^{8} ) q^{22} + ( \zeta_{24}^{3} - \zeta_{24}^{9} ) q^{23} - q^{25} + \zeta_{24}^{8} q^{28} + ( \zeta_{24}^{3} + \zeta_{24}^{9} ) q^{29} -\zeta_{24}^{5} q^{32} + ( \zeta_{24}^{2} - \zeta_{24}^{10} ) q^{37} + \zeta_{24}^{6} q^{43} + ( -\zeta_{24}^{7} + \zeta_{24}^{9} ) q^{44} + ( -\zeta_{24}^{4} + \zeta_{24}^{10} ) q^{46} - q^{49} + \zeta_{24} q^{50} + ( \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{53} -\zeta_{24}^{9} q^{56} + ( -\zeta_{24}^{4} - \zeta_{24}^{10} ) q^{58} + \zeta_{24}^{6} q^{64} + ( \zeta_{24}^{4} + \zeta_{24}^{8} ) q^{67} + ( -\zeta_{24} + \zeta_{24}^{11} ) q^{71} + ( -\zeta_{24}^{3} + \zeta_{24}^{11} ) q^{74} + ( -\zeta_{24} - \zeta_{24}^{11} ) q^{77} + ( -\zeta_{24}^{4} - \zeta_{24}^{8} ) q^{79} -\zeta_{24}^{7} q^{86} + ( \zeta_{24}^{8} - \zeta_{24}^{10} ) q^{88} + ( \zeta_{24}^{5} - \zeta_{24}^{11} ) q^{92} + \zeta_{24} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 4q^{16} + 4q^{22} - 8q^{25} - 4q^{28} - 4q^{46} - 8q^{49} - 4q^{58} - 4q^{88} + O(q^{100}) \)

Character values

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

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2267.1
0.965926 + 0.258819i
0.965926 0.258819i
0.258819 + 0.965926i
0.258819 0.965926i
−0.258819 + 0.965926i
−0.258819 0.965926i
−0.965926 + 0.258819i
−0.965926 0.258819i
−0.965926 0.258819i 0 0.866025 + 0.500000i 0 0 1.00000i −0.707107 0.707107i 0 0
2267.2 −0.965926 + 0.258819i 0 0.866025 0.500000i 0 0 1.00000i −0.707107 + 0.707107i 0 0
2267.3 −0.258819 0.965926i 0 −0.866025 + 0.500000i 0 0 1.00000i 0.707107 + 0.707107i 0 0
2267.4 −0.258819 + 0.965926i 0 −0.866025 0.500000i 0 0 1.00000i 0.707107 0.707107i 0 0
2267.5 0.258819 0.965926i 0 −0.866025 0.500000i 0 0 1.00000i −0.707107 + 0.707107i 0 0
2267.6 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0 0 1.00000i −0.707107 0.707107i 0 0
2267.7 0.965926 0.258819i 0 0.866025 0.500000i 0 0 1.00000i 0.707107 0.707107i 0 0
2267.8 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0 0 1.00000i 0.707107 + 0.707107i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2267.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner
84.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.1.h.a 8
3.b odd 2 1 inner 2268.1.h.a 8
4.b odd 2 1 inner 2268.1.h.a 8
7.b odd 2 1 CM 2268.1.h.a 8
9.c even 3 1 2268.1.s.e 8
9.c even 3 1 2268.1.s.f 8
9.d odd 6 1 2268.1.s.e 8
9.d odd 6 1 2268.1.s.f 8
12.b even 2 1 inner 2268.1.h.a 8
21.c even 2 1 inner 2268.1.h.a 8
28.d even 2 1 inner 2268.1.h.a 8
36.f odd 6 1 2268.1.s.e 8
36.f odd 6 1 2268.1.s.f 8
36.h even 6 1 2268.1.s.e 8
36.h even 6 1 2268.1.s.f 8
63.l odd 6 1 2268.1.s.e 8
63.l odd 6 1 2268.1.s.f 8
63.o even 6 1 2268.1.s.e 8
63.o even 6 1 2268.1.s.f 8
84.h odd 2 1 inner 2268.1.h.a 8
252.s odd 6 1 2268.1.s.e 8
252.s odd 6 1 2268.1.s.f 8
252.bi even 6 1 2268.1.s.e 8
252.bi even 6 1 2268.1.s.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2268.1.h.a 8 1.a even 1 1 trivial
2268.1.h.a 8 3.b odd 2 1 inner
2268.1.h.a 8 4.b odd 2 1 inner
2268.1.h.a 8 7.b odd 2 1 CM
2268.1.h.a 8 12.b even 2 1 inner
2268.1.h.a 8 21.c even 2 1 inner
2268.1.h.a 8 28.d even 2 1 inner
2268.1.h.a 8 84.h odd 2 1 inner
2268.1.s.e 8 9.c even 3 1
2268.1.s.e 8 9.d odd 6 1
2268.1.s.e 8 36.f odd 6 1
2268.1.s.e 8 36.h even 6 1
2268.1.s.e 8 63.l odd 6 1
2268.1.s.e 8 63.o even 6 1
2268.1.s.e 8 252.s odd 6 1
2268.1.s.e 8 252.bi even 6 1
2268.1.s.f 8 9.c even 3 1
2268.1.s.f 8 9.d odd 6 1
2268.1.s.f 8 36.f odd 6 1
2268.1.s.f 8 36.h even 6 1
2268.1.s.f 8 63.l odd 6 1
2268.1.s.f 8 63.o even 6 1
2268.1.s.f 8 252.s odd 6 1
2268.1.s.f 8 252.bi even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2268, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{4} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( T^{8} \)
$7$ \( ( 1 + T^{2} )^{4} \)
$11$ \( ( 1 - 4 T^{2} + T^{4} )^{2} \)
$13$ \( T^{8} \)
$17$ \( T^{8} \)
$19$ \( T^{8} \)
$23$ \( ( -2 + T^{2} )^{4} \)
$29$ \( ( 2 + T^{2} )^{4} \)
$31$ \( T^{8} \)
$37$ \( ( -3 + T^{2} )^{4} \)
$41$ \( T^{8} \)
$43$ \( ( 1 + T^{2} )^{4} \)
$47$ \( T^{8} \)
$53$ \( ( 1 + 4 T^{2} + T^{4} )^{2} \)
$59$ \( T^{8} \)
$61$ \( T^{8} \)
$67$ \( ( 3 + T^{2} )^{4} \)
$71$ \( ( 1 - 4 T^{2} + T^{4} )^{2} \)
$73$ \( T^{8} \)
$79$ \( ( 3 + T^{2} )^{4} \)
$83$ \( T^{8} \)
$89$ \( T^{8} \)
$97$ \( T^{8} \)
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