Newform orbit 77.2.b.a

• Label: 77.2.b.a
• Level: $77$
• Weight: $2$
• Character orbit: 77.b
• Analytic conductor: $0.615$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -7
• 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:	\(2\)
Coefficient field:	\(\Q(\sqrt{-7}) \)
Defining polynomial:	\( x^{2} - x + 2 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-7}\). We also show the integral \(q\)-expansion of the trace form.

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

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

0.500000	+	1.32288i
0.500000	−	1.32288i

−

2.64575i

0

−5.00000

0

0

−

2.64575i

7.93725i

3.00000

0

76.2

2.64575i

0

−5.00000

0

0

2.64575i

−

7.93725i

3.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)
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.a	✓	2
3.b	odd	2	1		693.2.c.a		2
4.b	odd	2	1		1232.2.e.a		2
7.b	odd	2	1	CM	77.2.b.a	✓	2
7.c	even	3	2		539.2.i.a		4
7.d	odd	6	2		539.2.i.a		4
11.b	odd	2	1	inner	77.2.b.a	✓	2
11.c	even	5	4		847.2.l.b		8
11.d	odd	10	4		847.2.l.b		8
21.c	even	2	1		693.2.c.a		2
28.d	even	2	1		1232.2.e.a		2
33.d	even	2	1		693.2.c.a		2
44.c	even	2	1		1232.2.e.a		2
77.b	even	2	1	inner	77.2.b.a	✓	2
77.h	odd	6	2		539.2.i.a		4
77.i	even	6	2		539.2.i.a		4
77.j	odd	10	4		847.2.l.b		8
77.l	even	10	4		847.2.l.b		8
231.h	odd	2	1		693.2.c.a		2
308.g	odd	2	1		1232.2.e.a		2

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 7 \) acting on \(S_{2}^{\mathrm{new}}(77, [\chi])\).

Hecke characteristic polynomials

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