Newform orbit 1200.3.l.s

• Label: 1200.3.l.s
• Level: $1200$
• Weight: $3$
• Character orbit: 1200.l
• 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.l (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(32.6976317232\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-11}) \)
Defining polynomial:	\( x^{2} - x + 3 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 75)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{-11})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (-b + 3) * q^3 + (-5*b + 6) * q^9 + (-10*b + 5) * q^11 - 10 * q^13 + (2*b - 1) * q^17 - 7 * q^19 + (-12*b + 6) * q^23 + (-16*b + 3) * q^27 + (20*b - 10) * q^29 - 42 * q^31 + (-25*b - 15) * q^33 + 40 * q^37 + (10*b - 30) * q^39 + (-10*b + 5) * q^41 - 50 * q^43 + (28*b - 14) * q^47 - 49 * q^49 + (5*b + 3) * q^51 + (-28*b + 14) * q^53 + (7*b - 21) * q^57 + (-40*b + 20) * q^59 - 8 * q^61 + 45 * q^67 + (-30*b - 18) * q^69 + (-20*b + 10) * q^71 + 35 * q^73 - 12 * q^79 + (-35*b - 39) * q^81 + (-42*b + 21) * q^83 + (50*b + 30) * q^87 + (-90*b + 45) * q^89 + (42*b - 126) * q^93 + 70 * q^97 + (-35*b - 120) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 5 * q^3 + 7 * q^9 - 20 * q^13 - 14 * q^19 - 10 * q^27 - 84 * q^31 - 55 * q^33 + 80 * q^37 - 50 * q^39 - 100 * q^43 - 98 * q^49 + 11 * q^51 - 35 * q^57 - 16 * q^61 + 90 * q^67 - 66 * q^69 + 70 * q^73 - 24 * q^79 - 113 * q^81 + 110 * q^87 - 210 * q^93 + 140 * q^97 - 275 * q^99

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

401.1

0.500000	+	1.65831i
0.500000	−	1.65831i

0

2.50000

−

1.65831i

0

0

0

0

0

3.50000

−

8.29156i

0

401.2

0

2.50000

+

1.65831i

0

0

0

0

0

3.50000

+

8.29156i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.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.l.s		2
3.b	odd	2	1	inner	1200.3.l.s		2
4.b	odd	2	1		75.3.c.c	✓	2
5.b	even	2	1		1200.3.l.f		2
5.c	odd	4	2		1200.3.c.d		4
12.b	even	2	1		75.3.c.c	✓	2
15.d	odd	2	1		1200.3.l.f		2
15.e	even	4	2		1200.3.c.d		4
20.d	odd	2	1		75.3.c.f	yes	2
20.e	even	4	2		75.3.d.c		4
60.h	even	2	1		75.3.c.f	yes	2
60.l	odd	4	2		75.3.d.c		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
75.3.c.c	✓	2	4.b	odd	2	1
75.3.c.c	✓	2	12.b	even	2	1
75.3.c.f	yes	2	20.d	odd	2	1
75.3.c.f	yes	2	60.h	even	2	1
75.3.d.c		4	20.e	even	4	2
75.3.d.c		4	60.l	odd	4	2
1200.3.c.d		4	5.c	odd	4	2
1200.3.c.d		4	15.e	even	4	2
1200.3.l.f		2	5.b	even	2	1
1200.3.l.f		2	15.d	odd	2	1
1200.3.l.s		2	1.a	even	1	1	trivial
1200.3.l.s		2	3.b	odd	2	1	inner

Hecke kernels

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

\( T_{7} \)

\( T_{11}^{2} + 275 \)

\( T_{13} + 10 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 5T + 9 \)
$5$	\( T^{2} \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 275 \)