Newform orbit 567.2.f.i

• Label: 567.2.f.i
• Level: $567$
• Weight: $2$
• Character orbit: 567.f
• Analytic conductor: $4.528$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 567 = 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	567.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.52751779461\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial:	\( x^{4} + 7x^{2} + 49 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 189)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + 5 \beta_{2} q^{4} + (\beta_{3} + \beta_1) q^{5} + (\beta_{2} + 1) q^{7} + 3 \beta_{3} q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{4} + 2 q^{7}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 49 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{2} ) / 7 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{3} ) / 7 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(407\)
\(\chi(n)\)	\(1\)	\(\beta_{2}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

190.1

−1.32288	−	2.29129i
1.32288	+	2.29129i
−1.32288	+	2.29129i
1.32288	−	2.29129i

−1.32288

−

2.29129i

0

−2.50000

+

4.33013i

1.32288

−

2.29129i

0

0.500000

+

0.866025i

7.93725

0

−7.00000

190.2

1.32288

+

2.29129i

0

−2.50000

+

4.33013i

−1.32288

+

2.29129i

0

0.500000

+

0.866025i

−7.93725

0

−7.00000

379.1

−1.32288

+

2.29129i

0

−2.50000

−

4.33013i

1.32288

+

2.29129i

0

0.500000

−

0.866025i

7.93725

0

−7.00000

379.2

1.32288

−

2.29129i

0

−2.50000

−

4.33013i

−1.32288

−

2.29129i

0

0.500000

−

0.866025i

−7.93725

0

−7.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
9.c	even	3	1	inner
9.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	567.2.f.i		4
3.b	odd	2	1	inner	567.2.f.i		4
9.c	even	3	1		189.2.a.f	✓	2
9.c	even	3	1	inner	567.2.f.i		4
9.d	odd	6	1		189.2.a.f	✓	2
9.d	odd	6	1	inner	567.2.f.i		4
36.f	odd	6	1		3024.2.a.bi		2
36.h	even	6	1		3024.2.a.bi		2
45.h	odd	6	1		4725.2.a.bb		2
45.j	even	6	1		4725.2.a.bb		2
63.l	odd	6	1		1323.2.a.w		2
63.o	even	6	1		1323.2.a.w		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
189.2.a.f	✓	2	9.c	even	3	1
189.2.a.f	✓	2	9.d	odd	6	1
567.2.f.i		4	1.a	even	1	1	trivial
567.2.f.i		4	3.b	odd	2	1	inner
567.2.f.i		4	9.c	even	3	1	inner
567.2.f.i		4	9.d	odd	6	1	inner
1323.2.a.w		2	63.l	odd	6	1
1323.2.a.w		2	63.o	even	6	1
3024.2.a.bi		2	36.f	odd	6	1
3024.2.a.bi		2	36.h	even	6	1
4725.2.a.bb		2	45.h	odd	6	1
4725.2.a.bb		2	45.j	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(567, [\chi])\):

\( T_{2}^{4} + 7T_{2}^{2} + 49 \)

\( T_{5}^{4} + 7T_{5}^{2} + 49 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} + 7T^{2} + 49 \)
$3$	\( T^{4} \)
$5$	\( T^{4} + 7T^{2} + 49 \)
$7$	\( (T^{2} - T + 1)^{2} \)
$11$	\( T^{4} + 7T^{2} + 49 \)