Newform orbit 2268.2.x.d

• Label: 2268.2.x.d
• Level: $2268$
• Weight: $2$
• Character orbit: 2268.x
• Analytic conductor: $18.110$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -3
• Inner twists: $4$

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:	$2$
Coefficient field:	$\Q(\sqrt{-3})$
Defining polynomial:	$x^{2} - x + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 756)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of a primitive root of unity $\zeta_{6}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + (-3*z + 2) * q^7 + (-z + 2) * q^13 + (10*z - 5) * q^19 + (-5*z + 5) * q^25 + (-6*z + 12) * q^31 + q^37 + (-8*z + 8) * q^43 + (-3*z - 5) * q^49 + (-5*z - 5) * q^61 - 11*z * q^67 + (-2*z + 1) * q^73 + (-13*z + 13) * q^79 + (-5*z + 1) * q^91 + (11*z + 11) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q + q^{7} + 3 q^{13} + 5 q^{25} + 18 q^{31} + 2 q^{37} + 8 q^{43} - 13 q^{49} - 15 q^{61} - 11 q^{67} + 13 q^{79} - 3 q^{91} + 33 q^{97}+O(q^{100})$

Character values

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

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

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

0.500000	−	0.866025i
0.500000	+	0.866025i

0

0

0

0

0

0.500000

+

2.59808i

0

0

0

1889.1

0

0

0

0

0

0.500000

−

2.59808i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3})$
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.d		2
3.b	odd	2	1	CM	2268.2.x.d		2
7.b	odd	2	1		2268.2.x.f		2
9.c	even	3	1		756.2.f.b	✓	2
9.c	even	3	1		2268.2.x.f		2
9.d	odd	6	1		756.2.f.b	✓	2
9.d	odd	6	1		2268.2.x.f		2
21.c	even	2	1		2268.2.x.f		2
36.f	odd	6	1		3024.2.k.c		2
36.h	even	6	1		3024.2.k.c		2
63.l	odd	6	1		756.2.f.b	✓	2
63.l	odd	6	1	inner	2268.2.x.d		2
63.o	even	6	1		756.2.f.b	✓	2
63.o	even	6	1	inner	2268.2.x.d		2
252.s	odd	6	1		3024.2.k.c		2
252.bi	even	6	1		3024.2.k.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
756.2.f.b	✓	2	9.c	even	3	1
756.2.f.b	✓	2	9.d	odd	6	1
756.2.f.b	✓	2	63.l	odd	6	1
756.2.f.b	✓	2	63.o	even	6	1
2268.2.x.d		2	1.a	even	1	1	trivial
2268.2.x.d		2	3.b	odd	2	1	CM
2268.2.x.d		2	63.l	odd	6	1	inner
2268.2.x.d		2	63.o	even	6	1	inner
2268.2.x.f		2	7.b	odd	2	1
2268.2.x.f		2	9.c	even	3	1
2268.2.x.f		2	9.d	odd	6	1
2268.2.x.f		2	21.c	even	2	1
3024.2.k.c		2	36.f	odd	6	1
3024.2.k.c		2	36.h	even	6	1
3024.2.k.c		2	252.s	odd	6	1
3024.2.k.c		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 $S_{2}^{\mathrm{new}}(2268, [\chi])$:

$T_{5}$

$T_{11}$

$T_{13}^{2} - 3T_{13} + 3$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2}$
$5$	$T^{2}$
$7$	$T^{2} - T + 7$
$11$	$T^{2}$