Newform orbit 3675.1.c.e

• Label: 3675.1.c.e
• Level: $3675$
• Weight: $1$
• Character orbit: 3675.c
• Analytic conductor: $1.834$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{2}$
• CM/RM discs: -15, -35, 21
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3675.c (of order \(2\), degree \(1\), 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 735)
Projective image:	\(D_{2}\)
Projective field:	Galois closure of \(\Q(\sqrt{-15}, \sqrt{21})\)
Artin image:	$D_4:C_2$
Artin field:	Galois closure of 8.0.16544390625.1

$q$-expansion

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

\(f(q)\)	\(=\)	\( q - i q^{3} + q^{4} - q^{9} - i q^{12} + q^{16} - 2 i q^{17} + i q^{27} - q^{36} - 2 i q^{47} - i q^{48} - 2 q^{51} + q^{64} - 2 i q^{68} + 2 q^{79} + q^{81} + 2 i q^{83} +O(q^{100}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 2 q^{9} + 2 q^{16} - 2 q^{36} - 4 q^{51} + 2 q^{64} + 4 q^{79} + 2 q^{81}+O(q^{100}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1226.1

		1.00000i
	−	1.00000i

0

−

1.00000i

1.00000

0

0

0

0

−1.00000

0

1226.2

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
15.d	odd	2	1	CM by \(\Q(\sqrt{-15}) \)
21.c	even	2	1	RM by \(\Q(\sqrt{21}) \)
35.c	odd	2	1	CM by \(\Q(\sqrt{-35}) \)
3.b	odd	2	1	inner
5.b	even	2	1	inner
7.b	odd	2	1	inner
105.g	even	2	1	inner

Twists

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

    

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

Hecke kernels

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

\( T_{2} \)

\( T_{13} \)

\( T_{19} \)

\( T_{37} \)

Hecke characteristic polynomials

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