Newform orbit 968.2.c.a

• Label: 968.2.c.a
• Level: $968$
• Weight: $2$
• Character orbit: 968.c
• Analytic conductor: $7.730$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -88
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 968 = 2^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	968.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.72951891566\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-2}) \)
Defining polynomial:	\( x^{2} + 2 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + \beta q^{2} - 2 q^{4} - 2 \beta q^{8} + 3 q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} + 6 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/968\mathbb{Z}\right)^\times\).

\(n\)	\(485\)	\(727\)	\(849\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

485.1

	−	1.41421i
		1.41421i

−

1.41421i

0

−2.00000

0

0

0

2.82843i

3.00000

0

485.2

1.41421i

0

−2.00000

0

0

0

−

2.82843i

3.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
88.b	odd	2	1	CM by \(\Q(\sqrt{-22}) \)
8.b	even	2	1	inner
11.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	968.2.c.a	✓	2
4.b	odd	2	1		3872.2.c.a		2
8.b	even	2	1	inner	968.2.c.a	✓	2
8.d	odd	2	1		3872.2.c.a		2
11.b	odd	2	1	inner	968.2.c.a	✓	2
11.c	even	5	4		968.2.o.a		8
11.d	odd	10	4		968.2.o.a		8
44.c	even	2	1		3872.2.c.a		2
88.b	odd	2	1	CM	968.2.c.a	✓	2
88.g	even	2	1		3872.2.c.a		2
88.o	even	10	4		968.2.o.a		8
88.p	odd	10	4		968.2.o.a		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
968.2.c.a	✓	2	1.a	even	1	1	trivial
968.2.c.a	✓	2	8.b	even	2	1	inner
968.2.c.a	✓	2	11.b	odd	2	1	inner
968.2.c.a	✓	2	88.b	odd	2	1	CM
968.2.o.a		8	11.c	even	5	4
968.2.o.a		8	11.d	odd	10	4
968.2.o.a		8	88.o	even	10	4
968.2.o.a		8	88.p	odd	10	4
3872.2.c.a		2	4.b	odd	2	1
3872.2.c.a		2	8.d	odd	2	1
3872.2.c.a		2	44.c	even	2	1
3872.2.c.a		2	88.g	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(968, [\chi])\):

\( T_{3} \)

\( T_{5} \)

\( T_{7} \)

Hecke characteristic polynomials

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