Newform orbit 1200.2.h.l

• Label: 1200.2.h.l
• Level: $1200$
• Weight: $2$
• Character orbit: 1200.h
• Analytic conductor: $9.582$
• Analytic rank: $0$
• Dimension: $4$
• CM discriminant: -20
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(9.58204824255\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\sqrt{-2}, \sqrt{-5})\)
Defining polynomial:	\( x^{4} - 4x^{2} + 9 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	no (minimal twist has level 240)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta_1 q^{3} + (\beta_{3} + \beta_1) q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 4x^{2} + 9 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 2 \)
\(\beta_{3}\)	\(=\)	\( \nu^{3} - 2\nu \)

Character values

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

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1151.1

1.58114	+	0.707107i
1.58114	−	0.707107i
−1.58114	+	0.707107i
−1.58114	−	0.707107i

0

−1.58114

−

0.707107i

0

0

0

4.24264i

0

2.00000

+

2.23607i

0

1151.2

0

−1.58114

+

0.707107i

0

0

0

−

4.24264i

0

2.00000

−

2.23607i

0

1151.3

0

1.58114

−

0.707107i

0

0

0

4.24264i

0

2.00000

−

2.23607i

0

1151.4

0

1.58114

+

0.707107i

0

0

0

−

4.24264i

0

2.00000

+

2.23607i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.d	odd	2	1	CM by \(\Q(\sqrt{-5}) \)
3.b	odd	2	1	inner
4.b	odd	2	1	inner
5.b	even	2	1	inner
12.b	even	2	1	inner
15.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	1200.2.h.l		4
3.b	odd	2	1	inner	1200.2.h.l		4
4.b	odd	2	1	inner	1200.2.h.l		4
5.b	even	2	1	inner	1200.2.h.l		4
5.c	odd	4	2		240.2.o.a	✓	4
12.b	even	2	1	inner	1200.2.h.l		4
15.d	odd	2	1	inner	1200.2.h.l		4
15.e	even	4	2		240.2.o.a	✓	4
20.d	odd	2	1	CM	1200.2.h.l		4
20.e	even	4	2		240.2.o.a	✓	4
40.i	odd	4	2		960.2.o.b		4
40.k	even	4	2		960.2.o.b		4
60.h	even	2	1	inner	1200.2.h.l		4
60.l	odd	4	2		240.2.o.a	✓	4
120.q	odd	4	2		960.2.o.b		4
120.w	even	4	2		960.2.o.b		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
240.2.o.a	✓	4	5.c	odd	4	2
240.2.o.a	✓	4	15.e	even	4	2
240.2.o.a	✓	4	20.e	even	4	2
240.2.o.a	✓	4	60.l	odd	4	2
960.2.o.b		4	40.i	odd	4	2
960.2.o.b		4	40.k	even	4	2
960.2.o.b		4	120.q	odd	4	2
960.2.o.b		4	120.w	even	4	2
1200.2.h.l		4	1.a	even	1	1	trivial
1200.2.h.l		4	3.b	odd	2	1	inner
1200.2.h.l		4	4.b	odd	2	1	inner
1200.2.h.l		4	5.b	even	2	1	inner
1200.2.h.l		4	12.b	even	2	1	inner
1200.2.h.l		4	15.d	odd	2	1	inner
1200.2.h.l		4	20.d	odd	2	1	CM
1200.2.h.l		4	60.h	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7}^{2} + 18 \)

\( T_{11} \)

\( T_{13} \)

\( T_{23}^{2} - 90 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} \)
$3$	\( T^{4} - 4T^{2} + 9 \)
$5$	\( T^{4} \)
$7$	\( (T^{2} + 18)^{2} \)
$11$	\( T^{4} \)