Newform orbit 315.2.j.e

• Label: 315.2.j.e
• Level: $315$
• Weight: $2$
• Character orbit: 315.j
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.j (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.51528766367\)
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:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 35)
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.

\(f(q)\)	\(=\)	\( q + (\beta_{2} + \beta_1 + 1) q^{2} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{4} + (\beta_{2} + 1) q^{5} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{7} + (\beta_{3} - 3) q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{7} - 12 q^{8}+O(q^{10}) \)

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 \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(136\)	\(281\)
\(\chi(n)\)	\(1\)	\(-1 - \beta_{2}\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

46.1

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

−0.207107

−

0.358719i

0

0.914214

−

1.58346i

0.500000

+

0.866025i

0

−1.62132

−

2.09077i

−1.58579

0

0.207107

−

0.358719i

46.2

1.20711

+

2.09077i

0

−1.91421

+

3.31552i

0.500000

+

0.866025i

0

2.62132

+

0.358719i

−4.41421

0

−1.20711

+

2.09077i

226.1

−0.207107

+

0.358719i

0

0.914214

+

1.58346i

0.500000

−

0.866025i

0

−1.62132

+

2.09077i

−1.58579

0

0.207107

+

0.358719i

226.2

1.20711

−

2.09077i

0

−1.91421

−

3.31552i

0.500000

−

0.866025i

0

2.62132

−

0.358719i

−4.41421

0

−1.20711

−

2.09077i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	315.2.j.e		4
3.b	odd	2	1		35.2.e.a	✓	4
7.c	even	3	1	inner	315.2.j.e		4
7.c	even	3	1		2205.2.a.n		2
7.d	odd	6	1		2205.2.a.q		2
12.b	even	2	1		560.2.q.k		4
15.d	odd	2	1		175.2.e.c		4
15.e	even	4	2		175.2.k.a		8
21.c	even	2	1		245.2.e.e		4
21.g	even	6	1		245.2.a.g		2
21.g	even	6	1		245.2.e.e		4
21.h	odd	6	1		35.2.e.a	✓	4
21.h	odd	6	1		245.2.a.h		2
84.j	odd	6	1		3920.2.a.bv		2
84.n	even	6	1		560.2.q.k		4
84.n	even	6	1		3920.2.a.bq		2
105.o	odd	6	1		175.2.e.c		4
105.o	odd	6	1		1225.2.a.k		2
105.p	even	6	1		1225.2.a.m		2
105.w	odd	12	2		1225.2.b.h		4
105.x	even	12	2		175.2.k.a		8
105.x	even	12	2		1225.2.b.g		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
35.2.e.a	✓	4	3.b	odd	2	1
35.2.e.a	✓	4	21.h	odd	6	1
175.2.e.c		4	15.d	odd	2	1
175.2.e.c		4	105.o	odd	6	1
175.2.k.a		8	15.e	even	4	2
175.2.k.a		8	105.x	even	12	2
245.2.a.g		2	21.g	even	6	1
245.2.a.h		2	21.h	odd	6	1
245.2.e.e		4	21.c	even	2	1
245.2.e.e		4	21.g	even	6	1
315.2.j.e		4	1.a	even	1	1	trivial
315.2.j.e		4	7.c	even	3	1	inner
560.2.q.k		4	12.b	even	2	1
560.2.q.k		4	84.n	even	6	1
1225.2.a.k		2	105.o	odd	6	1
1225.2.a.m		2	105.p	even	6	1
1225.2.b.g		4	105.x	even	12	2
1225.2.b.h		4	105.w	odd	12	2
2205.2.a.n		2	7.c	even	3	1
2205.2.a.q		2	7.d	odd	6	1
3920.2.a.bq		2	84.n	even	6	1
3920.2.a.bv		2	84.j	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 2T_{2}^{3} + 5T_{2}^{2} + 2T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} - 2 T^{3} + 5 T^{2} + 2 T + 1 \)
$3$	\( T^{4} \)
$5$	\( (T^{2} - T + 1)^{2} \)
$7$	\( T^{4} - 2 T^{3} - 3 T^{2} - 14 T + 49 \)
$11$	\( T^{4} - 4 T^{3} + 20 T^{2} + 16 T + 16 \)