Newform orbit 3600.1.j.a

• Label: 3600.1.j.a
• Level: $3600$
• Weight: $1$
• Character orbit: 3600.j
• Analytic conductor: $1.797$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{2}$
• CM/RM discs: -3, -4, 12
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3600.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.79663404548\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 144)
Projective image:	\(D_{2}\)
Projective field:	Galois closure of \(\Q(\zeta_{12})\)
Artin image:	$D_4:C_2$
Artin field:	Galois closure of 8.0.2916000000.5

$q$-expansion

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

\(f(q)\)	\(=\)	\( q - 2 i q^{13} - 2 i q^{37} - q^{49} + 2 q^{61} - 2 i q^{73} + 2 i q^{97} +O(q^{100}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{49} + 4 q^{61}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times\).

\(n\)	\(577\)	\(901\)	\(2801\)	\(3151\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1999.1

		1.00000i
	−	1.00000i

0

0

0

0

0

0

0

0

0

1999.2

0

0

0

0

0

0

0

0

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3600.1.j.a		2
3.b	odd	2	1	CM	3600.1.j.a		2
4.b	odd	2	1	CM	3600.1.j.a		2
5.b	even	2	1	inner	3600.1.j.a		2
5.c	odd	4	1		144.1.g.a	✓	1
5.c	odd	4	1		3600.1.e.b		1
12.b	even	2	1	RM	3600.1.j.a		2
15.d	odd	2	1	inner	3600.1.j.a		2
15.e	even	4	1		144.1.g.a	✓	1
15.e	even	4	1		3600.1.e.b		1
20.d	odd	2	1	inner	3600.1.j.a		2
20.e	even	4	1		144.1.g.a	✓	1
20.e	even	4	1		3600.1.e.b		1
40.i	odd	4	1		576.1.g.a		1
40.k	even	4	1		576.1.g.a		1
45.k	odd	12	2		1296.1.o.b		2
45.l	even	12	2		1296.1.o.b		2
60.h	even	2	1	inner	3600.1.j.a		2
60.l	odd	4	1		144.1.g.a	✓	1
60.l	odd	4	1		3600.1.e.b		1
80.i	odd	4	1		2304.1.b.a		2
80.j	even	4	1		2304.1.b.a		2
80.s	even	4	1		2304.1.b.a		2
80.t	odd	4	1		2304.1.b.a		2
120.q	odd	4	1		576.1.g.a		1
120.w	even	4	1		576.1.g.a		1
180.v	odd	12	2		1296.1.o.b		2
180.x	even	12	2		1296.1.o.b		2
240.z	odd	4	1		2304.1.b.a		2
240.bb	even	4	1		2304.1.b.a		2
240.bd	odd	4	1		2304.1.b.a		2
240.bf	even	4	1		2304.1.b.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
144.1.g.a	✓	1	5.c	odd	4	1
144.1.g.a	✓	1	15.e	even	4	1
144.1.g.a	✓	1	20.e	even	4	1
144.1.g.a	✓	1	60.l	odd	4	1
576.1.g.a		1	40.i	odd	4	1
576.1.g.a		1	40.k	even	4	1
576.1.g.a		1	120.q	odd	4	1
576.1.g.a		1	120.w	even	4	1
1296.1.o.b		2	45.k	odd	12	2
1296.1.o.b		2	45.l	even	12	2
1296.1.o.b		2	180.v	odd	12	2
1296.1.o.b		2	180.x	even	12	2
2304.1.b.a		2	80.i	odd	4	1
2304.1.b.a		2	80.j	even	4	1
2304.1.b.a		2	80.s	even	4	1
2304.1.b.a		2	80.t	odd	4	1
2304.1.b.a		2	240.z	odd	4	1
2304.1.b.a		2	240.bb	even	4	1
2304.1.b.a		2	240.bd	odd	4	1
2304.1.b.a		2	240.bf	even	4	1
3600.1.e.b		1	5.c	odd	4	1
3600.1.e.b		1	15.e	even	4	1
3600.1.e.b		1	20.e	even	4	1
3600.1.e.b		1	60.l	odd	4	1
3600.1.j.a		2	1.a	even	1	1	trivial
3600.1.j.a		2	3.b	odd	2	1	CM
3600.1.j.a		2	4.b	odd	2	1	CM
3600.1.j.a		2	5.b	even	2	1	inner
3600.1.j.a		2	12.b	even	2	1	RM
3600.1.j.a		2	15.d	odd	2	1	inner
3600.1.j.a		2	20.d	odd	2	1	inner
3600.1.j.a		2	60.h	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} \) acting on \(S_{1}^{\mathrm{new}}(3600, [\chi])\).

Hecke characteristic polynomials

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