Newform orbit 775.1.d.c

• Label: 775.1.d.c
• Level: $775$
• Weight: $1$
• Character orbit: 775.d
• Self dual: yes
• Analytic conductor: $0.387$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{6}$
• CM discriminant: -31
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 775 = 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	775.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.386775384791\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{3}) \)
Defining polynomial:	\( x^{2} - 3 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 155)
Projective image:	\(D_{6}\)
Projective field:	Galois closure of 6.2.120125.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/775\mathbb{Z}\right)^\times\).

\(n\)	\(251\)	\(652\)
\(\chi(n)\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

526.1

1.73205
−1.73205

−1.73205

0

2.00000

0

0

1.73205

−1.73205

1.00000

0

526.2

1.73205

0

2.00000

0

0

−1.73205

1.73205

1.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
31.b	odd	2	1	CM by \(\Q(\sqrt{-31}) \)
5.b	even	2	1	inner
155.c	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	775.1.d.c		2
5.b	even	2	1	inner	775.1.d.c		2
5.c	odd	4	2		155.1.c.b	✓	2
15.e	even	4	2		1395.1.b.c		2
20.e	even	4	2		2480.1.k.e		2
31.b	odd	2	1	CM	775.1.d.c		2
155.c	odd	2	1	inner	775.1.d.c		2
155.f	even	4	2		155.1.c.b	✓	2
465.m	odd	4	2		1395.1.b.c		2
620.m	odd	4	2		2480.1.k.e		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
155.1.c.b	✓	2	5.c	odd	4	2
155.1.c.b	✓	2	155.f	even	4	2
775.1.d.c		2	1.a	even	1	1	trivial
775.1.d.c		2	5.b	even	2	1	inner
775.1.d.c		2	31.b	odd	2	1	CM
775.1.d.c		2	155.c	odd	2	1	inner
1395.1.b.c		2	15.e	even	4	2
1395.1.b.c		2	465.m	odd	4	2
2480.1.k.e		2	20.e	even	4	2
2480.1.k.e		2	620.m	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3 \) acting on \(S_{1}^{\mathrm{new}}(775, [\chi])\).

Hecke characteristic polynomials

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