Newform orbit 1089.1.c.a

• Label: 1089.1.c.a
• Level: $1089$
• Weight: $1$
• Character orbit: 1089.c
• Analytic conductor: $0.543$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{4}$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1089.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.543481798757\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-2}) \)
Defining polynomial:	\( x^{2} + 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{4}\)
Projective field:	Galois closure of 4.0.3993.1
Artin image:	$\SD_{16}$
Artin field:	Galois closure of 8.2.14206147659.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + q^4 - b * q^7 + b * q^13 + q^16 - b * q^19 - q^25 - b * q^28 + b * q^43 - q^49 + b * q^52 + b * q^61 + q^64 + b * q^73 - b * q^76 - b * q^79 + 2 * q^91 - 2 * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 2 q^{16} - 2 q^{25} - 2 q^{49} + 2 q^{64} + 4 q^{91} - 4 q^{97}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\)	\(244\)	\(848\)
\(\chi(n)\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

604.1

		1.41421i
	−	1.41421i

0

0

1.00000

0

0

−

1.41421i

0

0

0

604.2

0

0

1.00000

0

0

1.41421i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
11.b	odd	2	1	inner
33.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1089.1.c.a	✓	2
3.b	odd	2	1	CM	1089.1.c.a	✓	2
11.b	odd	2	1	inner	1089.1.c.a	✓	2
11.c	even	5	4		1089.1.k.a		8
11.d	odd	10	4		1089.1.k.a		8
33.d	even	2	1	inner	1089.1.c.a	✓	2
33.f	even	10	4		1089.1.k.a		8
33.h	odd	10	4		1089.1.k.a		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1089.1.c.a	✓	2	1.a	even	1	1	trivial
1089.1.c.a	✓	2	3.b	odd	2	1	CM
1089.1.c.a	✓	2	11.b	odd	2	1	inner
1089.1.c.a	✓	2	33.d	even	2	1	inner
1089.1.k.a		8	11.c	even	5	4
1089.1.k.a		8	11.d	odd	10	4
1089.1.k.a		8	33.f	even	10	4
1089.1.k.a		8	33.h	odd	10	4

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1089, [\chi])\).

Hecke characteristic polynomials

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