Properties

Label 2268.1.s.a
Level $2268$
Weight $1$
Character orbit 2268.s
Analytic conductor $1.132$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -84
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2268.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.13187944865\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 756)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.756.1
Artin image: $C_6\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} + \zeta_{6}^{2} q^{5} + \zeta_{6} q^{7} + q^{8} +O(q^{10})\) \( q -\zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} + \zeta_{6}^{2} q^{5} + \zeta_{6} q^{7} + q^{8} + q^{10} + \zeta_{6} q^{11} -\zeta_{6}^{2} q^{14} -\zeta_{6} q^{16} -2 q^{17} + q^{19} -\zeta_{6} q^{20} -\zeta_{6}^{2} q^{22} -\zeta_{6}^{2} q^{23} - q^{28} + \zeta_{6}^{2} q^{31} + \zeta_{6}^{2} q^{32} + 2 \zeta_{6} q^{34} - q^{35} - q^{37} -\zeta_{6} q^{38} + \zeta_{6}^{2} q^{40} + \zeta_{6}^{2} q^{41} - q^{44} - q^{46} + \zeta_{6}^{2} q^{49} - q^{55} + \zeta_{6} q^{56} + q^{62} + q^{64} -2 \zeta_{6}^{2} q^{68} + \zeta_{6} q^{70} - q^{71} + \zeta_{6} q^{74} + \zeta_{6}^{2} q^{76} + \zeta_{6}^{2} q^{77} + q^{80} + q^{82} -2 \zeta_{6}^{2} q^{85} + \zeta_{6} q^{88} + q^{89} + \zeta_{6} q^{92} + \zeta_{6}^{2} q^{95} + q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{4} - q^{5} + q^{7} + 2q^{8} + O(q^{10}) \) \( 2q - q^{2} - q^{4} - q^{5} + q^{7} + 2q^{8} + 2q^{10} + q^{11} + q^{14} - q^{16} - 4q^{17} + 2q^{19} - q^{20} + q^{22} + q^{23} - 2q^{28} - q^{31} - q^{32} + 2q^{34} - 2q^{35} - 2q^{37} - q^{38} - q^{40} - q^{41} - 2q^{44} - 2q^{46} - q^{49} - 2q^{55} + q^{56} + 2q^{62} + 2q^{64} + 2q^{68} + q^{70} - 2q^{71} + q^{74} - q^{76} - q^{77} + 2q^{80} + 2q^{82} + 2q^{85} + q^{88} + 2q^{89} + q^{92} - q^{95} + 2q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{6}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
755.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i −0.500000 0.866025i 0 0.500000 0.866025i 1.00000 0 1.00000
1511.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i −0.500000 + 0.866025i 0 0.500000 + 0.866025i 1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
84.h odd 2 1 CM by \(\Q(\sqrt{-21}) \)
9.c even 3 1 inner
252.s odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.1.s.a 2
3.b odd 2 1 2268.1.s.d 2
4.b odd 2 1 2268.1.s.c 2
7.b odd 2 1 2268.1.s.b 2
9.c even 3 1 756.1.h.d yes 1
9.c even 3 1 inner 2268.1.s.a 2
9.d odd 6 1 756.1.h.a 1
9.d odd 6 1 2268.1.s.d 2
12.b even 2 1 2268.1.s.b 2
21.c even 2 1 2268.1.s.c 2
28.d even 2 1 2268.1.s.d 2
36.f odd 6 1 756.1.h.b yes 1
36.f odd 6 1 2268.1.s.c 2
36.h even 6 1 756.1.h.c yes 1
36.h even 6 1 2268.1.s.b 2
63.l odd 6 1 756.1.h.c yes 1
63.l odd 6 1 2268.1.s.b 2
63.o even 6 1 756.1.h.b yes 1
63.o even 6 1 2268.1.s.c 2
84.h odd 2 1 CM 2268.1.s.a 2
252.s odd 6 1 756.1.h.d yes 1
252.s odd 6 1 inner 2268.1.s.a 2
252.bi even 6 1 756.1.h.a 1
252.bi even 6 1 2268.1.s.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.1.h.a 1 9.d odd 6 1
756.1.h.a 1 252.bi even 6 1
756.1.h.b yes 1 36.f odd 6 1
756.1.h.b yes 1 63.o even 6 1
756.1.h.c yes 1 36.h even 6 1
756.1.h.c yes 1 63.l odd 6 1
756.1.h.d yes 1 9.c even 3 1
756.1.h.d yes 1 252.s odd 6 1
2268.1.s.a 2 1.a even 1 1 trivial
2268.1.s.a 2 9.c even 3 1 inner
2268.1.s.a 2 84.h odd 2 1 CM
2268.1.s.a 2 252.s odd 6 1 inner
2268.1.s.b 2 7.b odd 2 1
2268.1.s.b 2 12.b even 2 1
2268.1.s.b 2 36.h even 6 1
2268.1.s.b 2 63.l odd 6 1
2268.1.s.c 2 4.b odd 2 1
2268.1.s.c 2 21.c even 2 1
2268.1.s.c 2 36.f odd 6 1
2268.1.s.c 2 63.o even 6 1
2268.1.s.d 2 3.b odd 2 1
2268.1.s.d 2 9.d odd 6 1
2268.1.s.d 2 28.d even 2 1
2268.1.s.d 2 252.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 \(S_{1}^{\mathrm{new}}(2268, [\chi])\):

\( T_{5}^{2} + T_{5} + 1 \)
\( T_{11}^{2} - T_{11} + 1 \)
\( T_{67} \)

Hecke characteristic polynomials

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