Newform orbit 1089.3.c.b

• Label: 1089.3.c.b
• Level: $1089$
• Weight: $3$
• Character orbit: 1089.c
• Analytic conductor: $29.673$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(29.6731007888\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\sqrt{-2}, \sqrt{-11})\)
Defining polynomial:	\( x^{4} - 2x^{3} + 11x^{2} - 10x + 3 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta_{3} q^{2} - 7 q^{4} - 2 \beta_1 q^{5} + 7 \beta_{2} q^{7} + 3 \beta_{3} q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 28 q^{4}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} + 11x^{2} - 10x + 3 \) :

\(\beta_{1}\)	\(=\)	\( \nu^{2} - \nu + 5 \)
\(\beta_{2}\)	\(=\)	\( ( 2\nu^{3} - 3\nu^{2} + 19\nu - 9 ) / 3 \)
\(\beta_{3}\)	\(=\)	\( ( 4\nu^{3} - 6\nu^{2} + 44\nu - 21 ) / 3 \)

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

0.500000	+	0.244099i
0.500000	+	3.07253i
0.500000	−	0.244099i
0.500000	−	3.07253i

−

3.31662i

0

−7.00000

−9.38083

0

9.89949i

9.94987i

0

31.1127i

604.2

−

3.31662i

0

−7.00000

9.38083

0

−

9.89949i

9.94987i

0

−

31.1127i

604.3

3.31662i

0

−7.00000

−9.38083

0

−

9.89949i

−

9.94987i

0

−

31.1127i

604.4

3.31662i

0

−7.00000

9.38083

0

9.89949i

−

9.94987i

0

31.1127i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
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	1089.3.c.b	✓	4
3.b	odd	2	1	inner	1089.3.c.b	✓	4
11.b	odd	2	1	inner	1089.3.c.b	✓	4
33.d	even	2	1	inner	1089.3.c.b	✓	4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1089.3.c.b	✓	4	1.a	even	1	1	trivial
1089.3.c.b	✓	4	3.b	odd	2	1	inner
1089.3.c.b	✓	4	11.b	odd	2	1	inner
1089.3.c.b	✓	4	33.d	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 11 \) acting on \(S_{3}^{\mathrm{new}}(1089, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} + 11)^{2} \)
$3$	\( T^{4} \)
$5$	\( (T^{2} - 88)^{2} \)
$7$	\( (T^{2} + 98)^{2} \)
$11$	\( T^{4} \)