Newform orbit 1521.1.j.c

• Label: 1521.1.j.c
• Level: $1521$
• Weight: $1$
• Character orbit: 1521.j
• Analytic conductor: $0.759$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{4}$
• CM discriminant: -39
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1521.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.759077884215\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{4}\)
Projective field:	Galois closure of 4.2.507.1

$q$-expansion

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times\).

\(n\)	\(677\)	\(847\)
\(\chi(n)\)	\(1\)	\(i\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

577.1

	−	1.00000i
		1.00000i

1.00000

−

1.00000i

0

−

1.00000i

−1.00000

+

1.00000i

0

0

0

0

2.00000i

775.1

1.00000

+

1.00000i

0

1.00000i

−1.00000

−

1.00000i

0

0

0

0

−

2.00000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
39.d	odd	2	1	CM by \(\Q(\sqrt{-39}) \)
13.d	odd	4	1	inner
39.f	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1521.1.j.c	yes	2
3.b	odd	2	1		1521.1.j.a	✓	2
13.b	even	2	1		1521.1.j.a	✓	2
13.c	even	3	2		1521.1.bd.a		4
13.d	odd	4	1		1521.1.j.a	✓	2
13.d	odd	4	1	inner	1521.1.j.c	yes	2
13.e	even	6	2		1521.1.bd.d		4
13.f	odd	12	2		1521.1.bd.a		4
13.f	odd	12	2		1521.1.bd.d		4
39.d	odd	2	1	CM	1521.1.j.c	yes	2
39.f	even	4	1		1521.1.j.a	✓	2
39.f	even	4	1	inner	1521.1.j.c	yes	2
39.h	odd	6	2		1521.1.bd.a		4
39.i	odd	6	2		1521.1.bd.d		4
39.k	even	12	2		1521.1.bd.a		4
39.k	even	12	2		1521.1.bd.d		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1521.1.j.a	✓	2	3.b	odd	2	1
1521.1.j.a	✓	2	13.b	even	2	1
1521.1.j.a	✓	2	13.d	odd	4	1
1521.1.j.a	✓	2	39.f	even	4	1
1521.1.j.c	yes	2	1.a	even	1	1	trivial
1521.1.j.c	yes	2	13.d	odd	4	1	inner
1521.1.j.c	yes	2	39.d	odd	2	1	CM
1521.1.j.c	yes	2	39.f	even	4	1	inner
1521.1.bd.a		4	13.c	even	3	2
1521.1.bd.a		4	13.f	odd	12	2
1521.1.bd.a		4	39.h	odd	6	2
1521.1.bd.a		4	39.k	even	12	2
1521.1.bd.d		4	13.e	even	6	2
1521.1.bd.d		4	13.f	odd	12	2
1521.1.bd.d		4	39.i	odd	6	2
1521.1.bd.d		4	39.k	even	12	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 2T_{2} + 2 \) acting on \(S_{1}^{\mathrm{new}}(1521, [\chi])\).

Hecke characteristic polynomials

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