Newform orbit 1421.2.b.b

• Label: 1421.2.b.b
• Level: $1421$
• Weight: $2$
• Character orbit: 1421.b
• Analytic conductor: $11.347$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1421 = 7^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1421.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.3467421272\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-5}) \)
Defining polynomial:	\( x^{2} + 5 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 29)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta q^{2} + \beta q^{3} - 3 q^{4} + 3 q^{5} - 5 q^{6} - \beta q^{8} - 2 q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{4} + 6 q^{5} - 10 q^{6} - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1421\mathbb{Z}\right)^\times\).

\(n\)	\(785\)	\(1277\)
\(\chi(n)\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1275.1

	−	2.23607i
		2.23607i

−

2.23607i

−

2.23607i

−3.00000

3.00000

−5.00000

0

2.23607i

−2.00000

−

6.70820i

1275.2

2.23607i

2.23607i

−3.00000

3.00000

−5.00000

0

−

2.23607i

−2.00000

6.70820i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1421.2.b.b		2
7.b	odd	2	1		29.2.b.a	✓	2
21.c	even	2	1		261.2.c.a		2
28.d	even	2	1		464.2.e.a		2
29.b	even	2	1	inner	1421.2.b.b		2
35.c	odd	2	1		725.2.c.c		2
35.f	even	4	2		725.2.d.a		4
56.e	even	2	1		1856.2.e.f		2
56.h	odd	2	1		1856.2.e.g		2
84.h	odd	2	1		4176.2.o.k		2
203.c	odd	2	1		29.2.b.a	✓	2
203.g	even	4	2		841.2.a.b		2
203.n	odd	14	6		841.2.e.g		12
203.o	odd	14	6		841.2.e.g		12
203.r	even	28	12		841.2.d.h		12
609.h	even	2	1		261.2.c.a		2
609.k	odd	4	2		7569.2.a.i		2
812.c	even	2	1		464.2.e.a		2
1015.f	odd	2	1		725.2.c.c		2
1015.l	even	4	2		725.2.d.a		4
1624.i	odd	2	1		1856.2.e.g		2
1624.o	even	2	1		1856.2.e.f		2
2436.j	odd	2	1		4176.2.o.k		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
29.2.b.a	✓	2	7.b	odd	2	1
29.2.b.a	✓	2	203.c	odd	2	1
261.2.c.a		2	21.c	even	2	1
261.2.c.a		2	609.h	even	2	1
464.2.e.a		2	28.d	even	2	1
464.2.e.a		2	812.c	even	2	1
725.2.c.c		2	35.c	odd	2	1
725.2.c.c		2	1015.f	odd	2	1
725.2.d.a		4	35.f	even	4	2
725.2.d.a		4	1015.l	even	4	2
841.2.a.b		2	203.g	even	4	2
841.2.d.h		12	203.r	even	28	12
841.2.e.g		12	203.n	odd	14	6
841.2.e.g		12	203.o	odd	14	6
1421.2.b.b		2	1.a	even	1	1	trivial
1421.2.b.b		2	29.b	even	2	1	inner
1856.2.e.f		2	56.e	even	2	1
1856.2.e.f		2	1624.o	even	2	1
1856.2.e.g		2	56.h	odd	2	1
1856.2.e.g		2	1624.i	odd	2	1
4176.2.o.k		2	84.h	odd	2	1
4176.2.o.k		2	2436.j	odd	2	1
7569.2.a.i		2	609.k	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1421, [\chi])\):

\( T_{2}^{2} + 5 \)

\( T_{3}^{2} + 5 \)

\( T_{5} - 3 \)

Hecke characteristic polynomials

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