Newform orbit 735.1.o.c

• Label: 735.1.o.c
• Level: $735$
• Weight: $1$
• Character orbit: 735.o
• Analytic conductor: $0.367$
• Analytic rank: $0$
• Dimension: $4$
• Projective image: $D_{4}$
• CM discriminant: -15
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	735.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.366812784285\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial:	\( x^{4} + 2x^{2} + 4 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{4}\)
Projective field:	Galois closure of 4.2.15435.1
Artin image:	$C_6\times D_8$
Artin field:	Galois closure of \(\mathbb{Q}[x]/(x^{48} - \cdots)\)

$q$-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.

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

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

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

Character values

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

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

569.1

0.707107	−	1.22474i
−0.707107	+	1.22474i
0.707107	+	1.22474i
−0.707107	−	1.22474i

−0.707107

+

1.22474i

−0.500000

−

0.866025i

−0.500000

−

0.866025i

0.500000

−

0.866025i

1.41421

0

0

−0.500000

+

0.866025i

0.707107

+

1.22474i

569.2

0.707107

−

1.22474i

−0.500000

−

0.866025i

−0.500000

−

0.866025i

0.500000

−

0.866025i

−1.41421

0

0

−0.500000

+

0.866025i

−0.707107

−

1.22474i

704.1

−0.707107

−

1.22474i

−0.500000

+

0.866025i

−0.500000

+

0.866025i

0.500000

+

0.866025i

1.41421

0

0

−0.500000

−

0.866025i

0.707107

−

1.22474i

704.2

0.707107

+

1.22474i

−0.500000

+

0.866025i

−0.500000

+

0.866025i

0.500000

+

0.866025i

−1.41421

0

0

−0.500000

−

0.866025i

−0.707107

+

1.22474i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by \(\Q(\sqrt{-15}) \)
7.c	even	3	1	inner
21.c	even	2	1	inner
21.g	even	6	1	inner
35.c	odd	2	1	inner
35.i	odd	6	1	inner
105.o	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	735.1.o.c		4
3.b	odd	2	1		735.1.o.d		4
5.b	even	2	1		735.1.o.d		4
5.c	odd	4	2		3675.1.u.f		8
7.b	odd	2	1		735.1.o.d		4
7.c	even	3	1		735.1.f.d	yes	2
7.c	even	3	1	inner	735.1.o.c		4
7.d	odd	6	1		735.1.f.c	✓	2
7.d	odd	6	1		735.1.o.d		4
15.d	odd	2	1	CM	735.1.o.c		4
15.e	even	4	2		3675.1.u.f		8
21.c	even	2	1	inner	735.1.o.c		4
21.g	even	6	1		735.1.f.d	yes	2
21.g	even	6	1	inner	735.1.o.c		4
21.h	odd	6	1		735.1.f.c	✓	2
21.h	odd	6	1		735.1.o.d		4
35.c	odd	2	1	inner	735.1.o.c		4
35.f	even	4	2		3675.1.u.f		8
35.i	odd	6	1		735.1.f.d	yes	2
35.i	odd	6	1	inner	735.1.o.c		4
35.j	even	6	1		735.1.f.c	✓	2
35.j	even	6	1		735.1.o.d		4
35.k	even	12	2		3675.1.c.f		4
35.k	even	12	2		3675.1.u.f		8
35.l	odd	12	2		3675.1.c.f		4
35.l	odd	12	2		3675.1.u.f		8
105.g	even	2	1		735.1.o.d		4
105.k	odd	4	2		3675.1.u.f		8
105.o	odd	6	1		735.1.f.d	yes	2
105.o	odd	6	1	inner	735.1.o.c		4
105.p	even	6	1		735.1.f.c	✓	2
105.p	even	6	1		735.1.o.d		4
105.w	odd	12	2		3675.1.c.f		4
105.w	odd	12	2		3675.1.u.f		8
105.x	even	12	2		3675.1.c.f		4
105.x	even	12	2		3675.1.u.f		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
735.1.f.c	✓	2	7.d	odd	6	1
735.1.f.c	✓	2	21.h	odd	6	1
735.1.f.c	✓	2	35.j	even	6	1
735.1.f.c	✓	2	105.p	even	6	1
735.1.f.d	yes	2	7.c	even	3	1
735.1.f.d	yes	2	21.g	even	6	1
735.1.f.d	yes	2	35.i	odd	6	1
735.1.f.d	yes	2	105.o	odd	6	1
735.1.o.c		4	1.a	even	1	1	trivial
735.1.o.c		4	7.c	even	3	1	inner
735.1.o.c		4	15.d	odd	2	1	CM
735.1.o.c		4	21.c	even	2	1	inner
735.1.o.c		4	21.g	even	6	1	inner
735.1.o.c		4	35.c	odd	2	1	inner
735.1.o.c		4	35.i	odd	6	1	inner
735.1.o.c		4	105.o	odd	6	1	inner
735.1.o.d		4	3.b	odd	2	1
735.1.o.d		4	5.b	even	2	1
735.1.o.d		4	7.b	odd	2	1
735.1.o.d		4	7.d	odd	6	1
735.1.o.d		4	21.h	odd	6	1
735.1.o.d		4	35.j	even	6	1
735.1.o.d		4	105.g	even	2	1
735.1.o.d		4	105.p	even	6	1
3675.1.c.f		4	35.k	even	12	2
3675.1.c.f		4	35.l	odd	12	2
3675.1.c.f		4	105.w	odd	12	2
3675.1.c.f		4	105.x	even	12	2
3675.1.u.f		8	5.c	odd	4	2
3675.1.u.f		8	15.e	even	4	2
3675.1.u.f		8	35.f	even	4	2
3675.1.u.f		8	35.k	even	12	2
3675.1.u.f		8	35.l	odd	12	2
3675.1.u.f		8	105.k	odd	4	2
3675.1.u.f		8	105.w	odd	12	2
3675.1.u.f		8	105.x	even	12	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(735, [\chi])\):

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

\( T_{17} \)

\( T_{167} + 2 \)

Hecke characteristic polynomials

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