Newform orbit 735.2.i.i

• Label: 735.2.i.i
• Level: $735$
• Weight: $2$
• Character orbit: 735.i
• 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.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.86900454856\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{5})\)
Defining polynomial:	\( x^{4} - x^{3} + 2x^{2} + x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 105)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 2x^{2} + x + 1 \) :

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

Character values

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

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

226.1

0.809017	−	1.40126i
−0.309017	+	0.535233i
0.809017	+	1.40126i
−0.309017	−	0.535233i

−1.11803

+

1.93649i

−0.500000

−

0.866025i

−1.50000

−

2.59808i

−0.500000

+

0.866025i

2.23607

0

2.23607

−0.500000

+

0.866025i

−1.11803

−

1.93649i

226.2

1.11803

−

1.93649i

−0.500000

−

0.866025i

−1.50000

−

2.59808i

−0.500000

+

0.866025i

−2.23607

0

−2.23607

−0.500000

+

0.866025i

1.11803

+

1.93649i

361.1

−1.11803

−

1.93649i

−0.500000

+

0.866025i

−1.50000

+

2.59808i

−0.500000

−

0.866025i

2.23607

0

2.23607

−0.500000

−

0.866025i

−1.11803

+

1.93649i

361.2

1.11803

+

1.93649i

−0.500000

+

0.866025i

−1.50000

+

2.59808i

−0.500000

−

0.866025i

−2.23607

0

−2.23607

−0.500000

−

0.866025i

1.11803

−

1.93649i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	735.2.i.i		4
7.b	odd	2	1		735.2.i.k		4
7.c	even	3	1		735.2.a.k		2
7.c	even	3	1	inner	735.2.i.i		4
7.d	odd	6	1		105.2.a.b	✓	2
7.d	odd	6	1		735.2.i.k		4
21.g	even	6	1		315.2.a.d		2
21.h	odd	6	1		2205.2.a.w		2
28.f	even	6	1		1680.2.a.v		2
35.i	odd	6	1		525.2.a.g		2
35.j	even	6	1		3675.2.a.y		2
35.k	even	12	2		525.2.d.c		4
56.j	odd	6	1		6720.2.a.cx		2
56.m	even	6	1		6720.2.a.cs		2
84.j	odd	6	1		5040.2.a.bw		2
105.p	even	6	1		1575.2.a.r		2
105.w	odd	12	2		1575.2.d.d		4
140.s	even	6	1		8400.2.a.cx		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
105.2.a.b	✓	2	7.d	odd	6	1
315.2.a.d		2	21.g	even	6	1
525.2.a.g		2	35.i	odd	6	1
525.2.d.c		4	35.k	even	12	2
735.2.a.k		2	7.c	even	3	1
735.2.i.i		4	1.a	even	1	1	trivial
735.2.i.i		4	7.c	even	3	1	inner
735.2.i.k		4	7.b	odd	2	1
735.2.i.k		4	7.d	odd	6	1
1575.2.a.r		2	105.p	even	6	1
1575.2.d.d		4	105.w	odd	12	2
1680.2.a.v		2	28.f	even	6	1
2205.2.a.w		2	21.h	odd	6	1
3675.2.a.y		2	35.j	even	6	1
5040.2.a.bw		2	84.j	odd	6	1
6720.2.a.cs		2	56.m	even	6	1
6720.2.a.cx		2	56.j	odd	6	1
8400.2.a.cx		2	140.s	even	6	1

Hecke kernels

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

\( T_{2}^{4} + 5T_{2}^{2} + 25 \)

\( T_{13}^{2} - 20 \)

\( T_{17}^{2} + 2T_{17} + 4 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} + 5T^{2} + 25 \)
$3$	\( (T^{2} + T + 1)^{2} \)
$5$	\( (T^{2} + T + 1)^{2} \)
$7$	\( T^{4} \)
$11$	T^4 + 4*T^3 + 32*T^2 - 64*T + 256