Newform orbit 735.2.s.a

• Label: 735.2.s.a
• Level: $735$
• Weight: $2$
• Character orbit: 735.s
• Analytic conductor: $5.869$
• Analytic rank: $1$
• Dimension: $2$
• 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:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\)	\(346\)	\(442\)	\(491\)
\(\chi(n)\)	\(\zeta_{6}\)	\(1\)	\(-1\)

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

0.500000	+	0.866025i
0.500000	−	0.866025i

−1.50000

+

0.866025i

−1.50000

−

0.866025i

0.500000

−

0.866025i

−0.500000

−

0.866025i

3.00000

0

−

1.73205i

1.50000

+

2.59808i

1.50000

+

0.866025i

656.1

−1.50000

−

0.866025i

−1.50000

+

0.866025i

0.500000

+

0.866025i

−0.500000

+

0.866025i

3.00000

0

1.73205i

1.50000

−

2.59808i

1.50000

−

0.866025i

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.a		2
3.b	odd	2	1		735.2.s.d		2
7.b	odd	2	1		735.2.s.b		2
7.c	even	3	1		105.2.b.b	yes	2
7.c	even	3	1		735.2.s.f		2
7.d	odd	6	1		105.2.b.a	✓	2
7.d	odd	6	1		735.2.s.d		2
21.c	even	2	1		735.2.s.f		2
21.g	even	6	1		105.2.b.b	yes	2
21.g	even	6	1	inner	735.2.s.a		2
21.h	odd	6	1		105.2.b.a	✓	2
21.h	odd	6	1		735.2.s.b		2
28.f	even	6	1		1680.2.f.b		2
28.g	odd	6	1		1680.2.f.c		2
35.i	odd	6	1		525.2.b.a		2
35.j	even	6	1		525.2.b.b		2
35.k	even	12	2		525.2.g.b		4
35.l	odd	12	2		525.2.g.c		4
84.j	odd	6	1		1680.2.f.c		2
84.n	even	6	1		1680.2.f.b		2
105.o	odd	6	1		525.2.b.a		2
105.p	even	6	1		525.2.b.b		2
105.w	odd	12	2		525.2.g.c		4
105.x	even	12	2		525.2.g.b		4

    

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

Hecke kernels

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

\( T_{2}^{2} + 3T_{2} + 3 \)

\( T_{13} \)

\( T_{17}^{2} - 6T_{17} + 36 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 3T + 3 \)
$3$	\( T^{2} + 3T + 3 \)
$5$	\( T^{2} + T + 1 \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 6T + 12 \)