Newform orbit 775.2.e.e

• Label: 775.2.e.e
• Level: $775$
• Weight: $2$
• Character orbit: 775.e
• Analytic conductor: $6.188$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 775 = 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	775.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.18840615665\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial:	\( x^{4} + 2x^{2} + 4 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 31)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + (\beta_{3} + 1) q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{3} + (2 \beta_{3} + 1) q^{4} + ( - 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{6} + (\beta_{3} - \beta_{2} + \beta_1) q^{7} + (\beta_{3} + 3) q^{8} + 2 \beta_1 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 6 q^{6} + 2 q^{7} + 12 q^{8}+O(q^{10}) \)

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

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

Character values

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

\(n\)	\(251\)	\(652\)
\(\chi(n)\)	\(-1 - \beta_{2}\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

501.1

0.707107	+	1.22474i
−0.707107	−	1.22474i
0.707107	−	1.22474i
−0.707107	+	1.22474i

−0.414214

0.207107

−

0.358719i

−1.82843

0

−0.0857864

+

0.148586i

−0.207107

+

0.358719i

1.58579

1.41421

+

2.44949i

0

501.2

2.41421

−1.20711

+

2.09077i

3.82843

0

−2.91421

+

5.04757i

1.20711

−

2.09077i

4.41421

−1.41421

−

2.44949i

0

676.1

−0.414214

0.207107

+

0.358719i

−1.82843

0

−0.0857864

−

0.148586i

−0.207107

−

0.358719i

1.58579

1.41421

−

2.44949i

0

676.2

2.41421

−1.20711

−

2.09077i

3.82843

0

−2.91421

−

5.04757i

1.20711

+

2.09077i

4.41421

−1.41421

+

2.44949i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
31.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	775.2.e.e		4
5.b	even	2	1		31.2.c.a	✓	4
5.c	odd	4	2		775.2.o.d		8
15.d	odd	2	1		279.2.h.c		4
20.d	odd	2	1		496.2.i.h		4
31.c	even	3	1	inner	775.2.e.e		4
155.c	odd	2	1		961.2.c.a		4
155.i	odd	6	1		961.2.a.c		2
155.i	odd	6	1		961.2.c.a		4
155.j	even	6	1		31.2.c.a	✓	4
155.j	even	6	1		961.2.a.a		2
155.m	odd	10	4		961.2.g.r		16
155.n	even	10	4		961.2.g.o		16
155.o	odd	12	2		775.2.o.d		8
155.u	even	30	4		961.2.d.l		8
155.u	even	30	4		961.2.g.o		16
155.v	odd	30	4		961.2.d.i		8
155.v	odd	30	4		961.2.g.r		16
465.t	even	6	1		8649.2.a.k		2
465.u	odd	6	1		279.2.h.c		4
465.u	odd	6	1		8649.2.a.l		2
620.o	odd	6	1		496.2.i.h		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
31.2.c.a	✓	4	5.b	even	2	1
31.2.c.a	✓	4	155.j	even	6	1
279.2.h.c		4	15.d	odd	2	1
279.2.h.c		4	465.u	odd	6	1
496.2.i.h		4	20.d	odd	2	1
496.2.i.h		4	620.o	odd	6	1
775.2.e.e		4	1.a	even	1	1	trivial
775.2.e.e		4	31.c	even	3	1	inner
775.2.o.d		8	5.c	odd	4	2
775.2.o.d		8	155.o	odd	12	2
961.2.a.a		2	155.j	even	6	1
961.2.a.c		2	155.i	odd	6	1
961.2.c.a		4	155.c	odd	2	1
961.2.c.a		4	155.i	odd	6	1
961.2.d.i		8	155.v	odd	30	4
961.2.d.l		8	155.u	even	30	4
961.2.g.o		16	155.n	even	10	4
961.2.g.o		16	155.u	even	30	4
961.2.g.r		16	155.m	odd	10	4
961.2.g.r		16	155.v	odd	30	4
8649.2.a.k		2	465.t	even	6	1
8649.2.a.l		2	465.u	odd	6	1

Hecke kernels

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

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} - 2 T - 1)^{2} \)
$3$	\( T^{4} + 2 T^{3} + 5 T^{2} - 2 T + 1 \)
$5$	\( T^{4} \)
$7$	\( T^{4} - 2 T^{3} + 5 T^{2} + 2 T + 1 \)
$11$	\( T^{4} - 2 T^{3} + 21 T^{2} + 34 T + 289 \)