Newform orbit 1200.1.bd.b

• Label: 1200.1.bd.b
• Level: $1200$
• Weight: $1$
• Character orbit: 1200.bd
• Analytic conductor: $0.599$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{4}$
• CM discriminant: -15
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1200.bd (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.598878015160\)
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 4.2.153600.1

$q$-expansion

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

\(f(q)\)	\(=\)	\( q + i q^{2} + q^{3} - q^{4} + i q^{6} - i q^{8} + q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\)	\(401\)	\(577\)	\(751\)	\(901\)
\(\chi(n)\)	\(-1\)	\(i\)	\(-1\)	\(-i\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

443.1

	−	1.00000i
		1.00000i

−

1.00000i

1.00000

−1.00000

0

−

1.00000i

0

1.00000i

1.00000

0

707.1

1.00000i

1.00000

−1.00000

0

1.00000i

0

−

1.00000i

1.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by \(\Q(\sqrt{-15}) \)
80.s	even	4	1	inner
240.bd	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1200.1.bd.b	yes	2
3.b	odd	2	1		1200.1.bd.a	yes	2
5.b	even	2	1		1200.1.bd.a	yes	2
5.c	odd	4	1		1200.1.z.a	✓	2
5.c	odd	4	1		1200.1.z.b	yes	2
15.d	odd	2	1	CM	1200.1.bd.b	yes	2
15.e	even	4	1		1200.1.z.a	✓	2
15.e	even	4	1		1200.1.z.b	yes	2
16.f	odd	4	1		1200.1.z.a	✓	2
48.k	even	4	1		1200.1.z.b	yes	2
80.j	even	4	1		1200.1.bd.a	yes	2
80.k	odd	4	1		1200.1.z.b	yes	2
80.s	even	4	1	inner	1200.1.bd.b	yes	2
240.t	even	4	1		1200.1.z.a	✓	2
240.z	odd	4	1		1200.1.bd.a	yes	2
240.bd	odd	4	1	inner	1200.1.bd.b	yes	2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1200.1.z.a	✓	2	5.c	odd	4	1
1200.1.z.a	✓	2	15.e	even	4	1
1200.1.z.a	✓	2	16.f	odd	4	1
1200.1.z.a	✓	2	240.t	even	4	1
1200.1.z.b	yes	2	5.c	odd	4	1
1200.1.z.b	yes	2	15.e	even	4	1
1200.1.z.b	yes	2	48.k	even	4	1
1200.1.z.b	yes	2	80.k	odd	4	1
1200.1.bd.a	yes	2	3.b	odd	2	1
1200.1.bd.a	yes	2	5.b	even	2	1
1200.1.bd.a	yes	2	80.j	even	4	1
1200.1.bd.a	yes	2	240.z	odd	4	1
1200.1.bd.b	yes	2	1.a	even	1	1	trivial
1200.1.bd.b	yes	2	15.d	odd	2	1	CM
1200.1.bd.b	yes	2	80.s	even	4	1	inner
1200.1.bd.b	yes	2	240.bd	odd	4	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{2} + 2T_{17} + 2 \) acting on \(S_{1}^{\mathrm{new}}(1200, [\chi])\).

Hecke characteristic polynomials

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