Newform orbit 735.2.s.j

• Label: 735.2.s.j
• Level: $735$
• Weight: $2$
• Character orbit: 735.s
• Analytic conductor: $5.869$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	735.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.86900454856\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial:	\( x^{4} - x^{3} - 2x^{2} - 3x + 9 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + ( - \beta_1 + 1) q^{2} + \beta_1 q^{3} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{4} + (\beta_{2} - 1) q^{5} + ( - \beta_{3} - 3 \beta_{2} + \beta_1) q^{6} + (\beta_{3} + 5 \beta_{2} - \beta_1 - 2) q^{8} + (\beta_{3} + 3 \beta_{2}) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{2} + q^{3} + 3 q^{4} - 2 q^{5} - 4 q^{6} + 5 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \)
\(\beta_{3}\)	\(=\)	\( ( -\nu^{3} + 2\nu + 3 ) / 2 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/735\mathbb{Z}\right)^\times\).

\(n\)	\(346\)	\(442\)	\(491\)
\(\chi(n)\)	\(1 - \beta_{2}\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

521.1

1.68614	−	0.396143i
−1.18614	+	1.26217i
1.68614	+	0.396143i
−1.18614	−	1.26217i

−0.686141

+

0.396143i

1.68614

−

0.396143i

−0.686141

+

1.18843i

−0.500000

−

0.866025i

−1.00000

+

0.939764i

0

−

2.67181i

2.68614

−

1.33591i

0.686141

+

0.396143i

521.2

2.18614

−

1.26217i

−1.18614

+

1.26217i

2.18614

−

3.78651i

−0.500000

−

0.866025i

−1.00000

+

4.25639i

0

−

5.98844i

−0.186141

−

2.99422i

−2.18614

−

1.26217i

656.1

−0.686141

−

0.396143i

1.68614

+

0.396143i

−0.686141

−

1.18843i

−0.500000

+

0.866025i

−1.00000

−

0.939764i

0

2.67181i

2.68614

+

1.33591i

0.686141

−

0.396143i

656.2

2.18614

+

1.26217i

−1.18614

−

1.26217i

2.18614

+

3.78651i

−0.500000

+

0.866025i

−1.00000

−

4.25639i

0

5.98844i

−0.186141

+

2.99422i

−2.18614

+

1.26217i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
21.g	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	735.2.s.j		4
3.b	odd	2	1		735.2.s.h		4
7.b	odd	2	1		735.2.s.i		4
7.c	even	3	1		105.2.b.d	yes	4
7.c	even	3	1		735.2.s.g		4
7.d	odd	6	1		105.2.b.c	✓	4
7.d	odd	6	1		735.2.s.h		4
21.c	even	2	1		735.2.s.g		4
21.g	even	6	1		105.2.b.d	yes	4
21.g	even	6	1	inner	735.2.s.j		4
21.h	odd	6	1		105.2.b.c	✓	4
21.h	odd	6	1		735.2.s.i		4
28.f	even	6	1		1680.2.f.h		4
28.g	odd	6	1		1680.2.f.g		4
35.i	odd	6	1		525.2.b.g		4
35.j	even	6	1		525.2.b.e		4
35.k	even	12	2		525.2.g.e		8
35.l	odd	12	2		525.2.g.d		8
84.j	odd	6	1		1680.2.f.g		4
84.n	even	6	1		1680.2.f.h		4
105.o	odd	6	1		525.2.b.g		4
105.p	even	6	1		525.2.b.e		4
105.w	odd	12	2		525.2.g.d		8
105.x	even	12	2		525.2.g.e		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
105.2.b.c	✓	4	7.d	odd	6	1
105.2.b.c	✓	4	21.h	odd	6	1
105.2.b.d	yes	4	7.c	even	3	1
105.2.b.d	yes	4	21.g	even	6	1
525.2.b.e		4	35.j	even	6	1
525.2.b.e		4	105.p	even	6	1
525.2.b.g		4	35.i	odd	6	1
525.2.b.g		4	105.o	odd	6	1
525.2.g.d		8	35.l	odd	12	2
525.2.g.d		8	105.w	odd	12	2
525.2.g.e		8	35.k	even	12	2
525.2.g.e		8	105.x	even	12	2
735.2.s.g		4	7.c	even	3	1
735.2.s.g		4	21.c	even	2	1
735.2.s.h		4	3.b	odd	2	1
735.2.s.h		4	7.d	odd	6	1
735.2.s.i		4	7.b	odd	2	1
735.2.s.i		4	21.h	odd	6	1
735.2.s.j		4	1.a	even	1	1	trivial
735.2.s.j		4	21.g	even	6	1	inner
1680.2.f.g		4	28.g	odd	6	1
1680.2.f.g		4	84.j	odd	6	1
1680.2.f.h		4	28.f	even	6	1
1680.2.f.h		4	84.n	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(735, [\chi])\):

\( T_{2}^{4} - 3T_{2}^{3} + T_{2}^{2} + 6T_{2} + 4 \)

\( T_{13}^{4} + 51T_{13}^{2} + 576 \)

\( T_{17}^{4} + 3T_{17}^{3} + 15T_{17}^{2} - 18T_{17} + 36 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^4 - 3*T^3 + T^2 + 6*T + 4
$3$	T^4 - T^3 - 2*T^2 - 3*T + 9
$5$	\( (T^{2} + T + 1)^{2} \)
$7$	\( T^{4} \)
$11$	T^4 - 3*T^3 + T^2 + 6*T + 4