Newform orbit 1200.3.e.h

• Label: 1200.3.e.h
• Level: $1200$
• Weight: $3$
• Character orbit: 1200.e
• Analytic conductor: $32.698$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(32.6976317232\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta q^{3} - 4 \beta q^{7} - 3 q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{9}+O(q^{10}) \)

Character values

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

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

751.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

−

1.73205i

0

0

0

−

6.92820i

0

−3.00000

0

751.2

0

1.73205i

0

0

0

6.92820i

0

−3.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1200.3.e.h		2
3.b	odd	2	1		3600.3.e.t		2
4.b	odd	2	1	inner	1200.3.e.h		2
5.b	even	2	1		48.3.g.a	✓	2
5.c	odd	4	2		1200.3.j.a		4
12.b	even	2	1		3600.3.e.t		2
15.d	odd	2	1		144.3.g.b		2
15.e	even	4	2		3600.3.j.i		4
20.d	odd	2	1		48.3.g.a	✓	2
20.e	even	4	2		1200.3.j.a		4
35.c	odd	2	1		2352.3.m.a		2
40.e	odd	2	1		192.3.g.a		2
40.f	even	2	1		192.3.g.a		2
45.h	odd	6	1		1296.3.o.n		2
45.h	odd	6	1		1296.3.o.p		2
45.j	even	6	1		1296.3.o.a		2
45.j	even	6	1		1296.3.o.c		2
60.h	even	2	1		144.3.g.b		2
60.l	odd	4	2		3600.3.j.i		4
80.k	odd	4	2		768.3.b.b		4
80.q	even	4	2		768.3.b.b		4
120.i	odd	2	1		576.3.g.i		2
120.m	even	2	1		576.3.g.i		2
140.c	even	2	1		2352.3.m.a		2
180.n	even	6	1		1296.3.o.n		2
180.n	even	6	1		1296.3.o.p		2
180.p	odd	6	1		1296.3.o.a		2
180.p	odd	6	1		1296.3.o.c		2
240.t	even	4	2		2304.3.b.n		4
240.bm	odd	4	2		2304.3.b.n		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
48.3.g.a	✓	2	5.b	even	2	1
48.3.g.a	✓	2	20.d	odd	2	1
144.3.g.b		2	15.d	odd	2	1
144.3.g.b		2	60.h	even	2	1
192.3.g.a		2	40.e	odd	2	1
192.3.g.a		2	40.f	even	2	1
576.3.g.i		2	120.i	odd	2	1
576.3.g.i		2	120.m	even	2	1
768.3.b.b		4	80.k	odd	4	2
768.3.b.b		4	80.q	even	4	2
1200.3.e.h		2	1.a	even	1	1	trivial
1200.3.e.h		2	4.b	odd	2	1	inner
1200.3.j.a		4	5.c	odd	4	2
1200.3.j.a		4	20.e	even	4	2
1296.3.o.a		2	45.j	even	6	1
1296.3.o.a		2	180.p	odd	6	1
1296.3.o.c		2	45.j	even	6	1
1296.3.o.c		2	180.p	odd	6	1
1296.3.o.n		2	45.h	odd	6	1
1296.3.o.n		2	180.n	even	6	1
1296.3.o.p		2	45.h	odd	6	1
1296.3.o.p		2	180.n	even	6	1
2304.3.b.n		4	240.t	even	4	2
2304.3.b.n		4	240.bm	odd	4	2
2352.3.m.a		2	35.c	odd	2	1
2352.3.m.a		2	140.c	even	2	1
3600.3.e.t		2	3.b	odd	2	1
3600.3.e.t		2	12.b	even	2	1
3600.3.j.i		4	15.e	even	4	2
3600.3.j.i		4	60.l	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7}^{2} + 48 \)

\( T_{11}^{2} + 432 \)

\( T_{13} - 14 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 3 \)
$5$	\( T^{2} \)
$7$	\( T^{2} + 48 \)
$11$	\( T^{2} + 432 \)