Newform orbit 1260.3.p.c

• Label: 1260.3.p.c
• Level: $1260$
• Weight: $3$
• Character orbit: 1260.p
• Analytic conductor: $34.333$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1260.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(34.3325133094\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\sqrt{2}, \sqrt{-41})\)
Defining polynomial:	\( x^{4} + 40x^{2} + 441 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + ( - \beta_{2} + \beta_1) q^{5} + (\beta_{3} + 2 \beta_{2}) q^{7}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 40x^{2} + 441 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{3} + 19\nu ) / 21 \)
\(\beta_{3}\)	\(=\)	\( \nu^{2} + 20 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times\).

\(n\)	\(281\)	\(631\)	\(757\)	\(1081\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1189.1

−0.707107	−	4.52769i
−0.707107	+	4.52769i
0.707107	−	4.52769i
0.707107	+	4.52769i

0

0

0

−2.12132

−

4.52769i

0

2.82843

+

6.40312i

0

0

0

1189.2

0

0

0

−2.12132

+

4.52769i

0

2.82843

−

6.40312i

0

0

0

1189.3

0

0

0

2.12132

−

4.52769i

0

−2.82843

−

6.40312i

0

0

0

1189.4

0

0

0

2.12132

+

4.52769i

0

−2.82843

+

6.40312i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
7.b	odd	2	1	inner
35.c	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1260.3.p.c		4
3.b	odd	2	1		140.3.h.c	✓	4
5.b	even	2	1	inner	1260.3.p.c		4
7.b	odd	2	1	inner	1260.3.p.c		4
12.b	even	2	1		560.3.p.e		4
15.d	odd	2	1		140.3.h.c	✓	4
15.e	even	4	2		700.3.d.c		4
21.c	even	2	1		140.3.h.c	✓	4
21.g	even	6	2		980.3.n.c		8
21.h	odd	6	2		980.3.n.c		8
35.c	odd	2	1	inner	1260.3.p.c		4
60.h	even	2	1		560.3.p.e		4
84.h	odd	2	1		560.3.p.e		4
105.g	even	2	1		140.3.h.c	✓	4
105.k	odd	4	2		700.3.d.c		4
105.o	odd	6	2		980.3.n.c		8
105.p	even	6	2		980.3.n.c		8
420.o	odd	2	1		560.3.p.e		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
140.3.h.c	✓	4	3.b	odd	2	1
140.3.h.c	✓	4	15.d	odd	2	1
140.3.h.c	✓	4	21.c	even	2	1
140.3.h.c	✓	4	105.g	even	2	1
560.3.p.e		4	12.b	even	2	1
560.3.p.e		4	60.h	even	2	1
560.3.p.e		4	84.h	odd	2	1
560.3.p.e		4	420.o	odd	2	1
700.3.d.c		4	15.e	even	4	2
700.3.d.c		4	105.k	odd	4	2
980.3.n.c		8	21.g	even	6	2
980.3.n.c		8	21.h	odd	6	2
980.3.n.c		8	105.o	odd	6	2
980.3.n.c		8	105.p	even	6	2
1260.3.p.c		4	1.a	even	1	1	trivial
1260.3.p.c		4	5.b	even	2	1	inner
1260.3.p.c		4	7.b	odd	2	1	inner
1260.3.p.c		4	35.c	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 \(S_{3}^{\mathrm{new}}(1260, [\chi])\):

\( T_{11} - 2 \)

\( T_{13}^{2} - 242 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} \)
$3$	\( T^{4} \)
$5$	\( T^{4} + 32T^{2} + 625 \)
$7$	\( T^{4} + 66T^{2} + 2401 \)
$11$	\( (T - 2)^{4} \)