Newform orbit 1200.3.e.a

• Label: 1200.3.e.a
• Level: $1200$
• Weight: $3$
• Character orbit: 1200.e
• Analytic conductor: $32.698$
• Analytic rank: $0$
• Dimension: $2$
• 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 240)
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 - b * q^3 - 4*b * q^7 - 3 * q^9 - 8*b * q^11 - 24 * q^13 + 24 * q^17 - 16*b * q^19 - 12 * q^21 + 20*b * q^23 + 3*b * q^27 - 10 * q^29 + 8*b * q^31 - 24 * q^33 - 24 * q^37 + 24*b * q^39 - 34 * q^41 + 12*b * q^43 - 4*b * q^47 + q^49 - 24*b * q^51 - 48 * q^53 - 48 * q^57 + 8*b * q^59 + 70 * q^61 + 12*b * q^63 + 52*b * q^67 + 60 * q^69 - 32*b * q^71 - 48 * q^73 - 96 * q^77 - 24*b * q^79 + 9 * q^81 - 52*b * q^83 + 10*b * q^87 + 14 * q^89 + 96*b * q^91 + 24 * q^93 - 96 * q^97 + 24*b * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 6 * q^9 - 48 * q^13 + 48 * q^17 - 24 * q^21 - 20 * q^29 - 48 * q^33 - 48 * q^37 - 68 * q^41 + 2 * q^49 - 96 * q^53 - 96 * q^57 + 140 * q^61 + 120 * q^69 - 96 * q^73 - 192 * q^77 + 18 * q^81 + 28 * q^89 + 48 * q^93 - 192 * q^97

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.a		2
3.b	odd	2	1		3600.3.e.f		2
4.b	odd	2	1	inner	1200.3.e.a		2
5.b	even	2	1		1200.3.e.i		2
5.c	odd	4	2		240.3.j.a	✓	4
12.b	even	2	1		3600.3.e.f		2
15.d	odd	2	1		3600.3.e.x		2
15.e	even	4	2		720.3.j.g		4
20.d	odd	2	1		1200.3.e.i		2
20.e	even	4	2		240.3.j.a	✓	4
40.i	odd	4	2		960.3.j.d		4
40.k	even	4	2		960.3.j.d		4
60.h	even	2	1		3600.3.e.x		2
60.l	odd	4	2		720.3.j.g		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
240.3.j.a	✓	4	5.c	odd	4	2
240.3.j.a	✓	4	20.e	even	4	2
720.3.j.g		4	15.e	even	4	2
720.3.j.g		4	60.l	odd	4	2
960.3.j.d		4	40.i	odd	4	2
960.3.j.d		4	40.k	even	4	2
1200.3.e.a		2	1.a	even	1	1	trivial
1200.3.e.a		2	4.b	odd	2	1	inner
1200.3.e.i		2	5.b	even	2	1
1200.3.e.i		2	20.d	odd	2	1
3600.3.e.f		2	3.b	odd	2	1
3600.3.e.f		2	12.b	even	2	1
3600.3.e.x		2	15.d	odd	2	1
3600.3.e.x		2	60.h	even	2	1

Hecke kernels

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

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

\( T_{11}^{2} + 192 \)

\( T_{13} + 24 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 3 \)
$5$	\( T^{2} \)
$7$	\( T^{2} + 48 \)
$11$	\( T^{2} + 192 \)