Newform orbit 77.2.b.b

• Label: 77.2.b.b
• Level: $77$
• Weight: $2$
• Character orbit: 77.b
• Analytic conductor: $0.615$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 77 = 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	77.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.614848095564\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\sqrt{-2}, \sqrt{-5})\)
Defining polynomial:	\( x^{4} - 4x^{2} + 9 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
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.

\(f(q)\)	\(=\)	\( q - \beta_{2} q^{2} + \beta_{3} q^{3} - \beta_{3} q^{5} + ( - \beta_{2} + 2 \beta_1) q^{6} + (\beta_{2} + \beta_1) q^{7} - 2 \beta_{2} q^{8} - 2 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 4x^{2} + 9 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{3} - \nu ) / 3 \)
\(\beta_{3}\)	\(=\)	\( \nu^{2} - 2 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/77\mathbb{Z}\right)^\times\).

\(n\)	\(45\)	\(57\)
\(\chi(n)\)	\(-1\)	\(-1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

76.1

−1.58114	+	0.707107i
1.58114	+	0.707107i
1.58114	−	0.707107i
−1.58114	−	0.707107i

−

1.41421i

−

2.23607i

0

2.23607i

−3.16228

−1.58114

+

2.12132i

−

2.82843i

−2.00000

3.16228

76.2

−

1.41421i

2.23607i

0

−

2.23607i

3.16228

1.58114

+

2.12132i

−

2.82843i

−2.00000

−3.16228

76.3

1.41421i

−

2.23607i

0

2.23607i

3.16228

1.58114

−

2.12132i

2.82843i

−2.00000

−3.16228

76.4

1.41421i

2.23607i

0

−

2.23607i

−3.16228

−1.58114

−

2.12132i

2.82843i

−2.00000

3.16228

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
11.b	odd	2	1	inner
77.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	77.2.b.b	✓	4
3.b	odd	2	1		693.2.c.b		4
4.b	odd	2	1		1232.2.e.c		4
7.b	odd	2	1	inner	77.2.b.b	✓	4
7.c	even	3	2		539.2.i.b		8
7.d	odd	6	2		539.2.i.b		8
11.b	odd	2	1	inner	77.2.b.b	✓	4
11.c	even	5	4		847.2.l.g		16
11.d	odd	10	4		847.2.l.g		16
21.c	even	2	1		693.2.c.b		4
28.d	even	2	1		1232.2.e.c		4
33.d	even	2	1		693.2.c.b		4
44.c	even	2	1		1232.2.e.c		4
77.b	even	2	1	inner	77.2.b.b	✓	4
77.h	odd	6	2		539.2.i.b		8
77.i	even	6	2		539.2.i.b		8
77.j	odd	10	4		847.2.l.g		16
77.l	even	10	4		847.2.l.g		16
231.h	odd	2	1		693.2.c.b		4
308.g	odd	2	1		1232.2.e.c		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
77.2.b.b	✓	4	1.a	even	1	1	trivial
77.2.b.b	✓	4	7.b	odd	2	1	inner
77.2.b.b	✓	4	11.b	odd	2	1	inner
77.2.b.b	✓	4	77.b	even	2	1	inner
539.2.i.b		8	7.c	even	3	2
539.2.i.b		8	7.d	odd	6	2
539.2.i.b		8	77.h	odd	6	2
539.2.i.b		8	77.i	even	6	2
693.2.c.b		4	3.b	odd	2	1
693.2.c.b		4	21.c	even	2	1
693.2.c.b		4	33.d	even	2	1
693.2.c.b		4	231.h	odd	2	1
847.2.l.g		16	11.c	even	5	4
847.2.l.g		16	11.d	odd	10	4
847.2.l.g		16	77.j	odd	10	4
847.2.l.g		16	77.l	even	10	4
1232.2.e.c		4	4.b	odd	2	1
1232.2.e.c		4	28.d	even	2	1
1232.2.e.c		4	44.c	even	2	1
1232.2.e.c		4	308.g	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(77, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} + 2)^{2} \)
$3$	\( (T^{2} + 5)^{2} \)
$5$	\( (T^{2} + 5)^{2} \)
$7$	\( T^{4} + 4T^{2} + 49 \)
$11$	\( (T^{2} + 6 T + 11)^{2} \)