Newform orbit 3675.1.f.b

• Label: 3675.1.f.b
• Level: $3675$
• Weight: $1$
• Character orbit: 3675.f
• Analytic conductor: $1.834$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{3}$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3675.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.83406392143\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 525)
Projective image:	\(D_{3}\)
Projective field:	Galois closure of 3.1.3675.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\)	\(=\)	\( q + i q^{3} - q^{4} - q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3675\mathbb{Z}\right)^\times\).

\(n\)	\(1177\)	\(1226\)	\(2551\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

2549.1

	−	1.00000i
		1.00000i

0

−

1.00000i

−1.00000

0

0

0

0

−1.00000

0

2549.2

0

1.00000i

−1.00000

0

0

0

0

−1.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
5.b	even	2	1	inner
15.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3675.1.f.b		2
3.b	odd	2	1	CM	3675.1.f.b		2
5.b	even	2	1	inner	3675.1.f.b		2
5.c	odd	4	1		3675.1.c.a		1
5.c	odd	4	1		3675.1.c.c		1
7.b	odd	2	1		3675.1.f.a		2
7.c	even	3	2		525.1.p.a		4
7.d	odd	6	2		3675.1.p.a		4
15.d	odd	2	1	inner	3675.1.f.b		2
15.e	even	4	1		3675.1.c.a		1
15.e	even	4	1		3675.1.c.c		1
21.c	even	2	1		3675.1.f.a		2
21.g	even	6	2		3675.1.p.a		4
21.h	odd	6	2		525.1.p.a		4
35.c	odd	2	1		3675.1.f.a		2
35.f	even	4	1		3675.1.c.b		1
35.f	even	4	1		3675.1.c.d		1
35.i	odd	6	2		3675.1.p.a		4
35.j	even	6	2		525.1.p.a		4
35.k	even	12	2		3675.1.u.a		2
35.k	even	12	2		3675.1.u.b		2
35.l	odd	12	2		525.1.u.a	✓	2
35.l	odd	12	2		525.1.u.b	yes	2
105.g	even	2	1		3675.1.f.a		2
105.k	odd	4	1		3675.1.c.b		1
105.k	odd	4	1		3675.1.c.d		1
105.o	odd	6	2		525.1.p.a		4
105.p	even	6	2		3675.1.p.a		4
105.w	odd	12	2		3675.1.u.a		2
105.w	odd	12	2		3675.1.u.b		2
105.x	even	12	2		525.1.u.a	✓	2
105.x	even	12	2		525.1.u.b	yes	2

    

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3675, [\chi])\):

\( T_{2} \)

\( T_{19} - 1 \)

Hecke characteristic polynomials

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