Newform orbit 315.2.j.b

• Label: 315.2.j.b
• Level: $315$
• Weight: $2$
• Character orbit: 315.j
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $2$
• 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:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(136\)	\(281\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)	\(1\)

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

0.500000	+	0.866025i
0.500000	−	0.866025i

0

0

1.00000

−

1.73205i

−0.500000

−

0.866025i

0

2.50000

+

0.866025i

0

0

0

226.1

0

0

1.00000

+

1.73205i

−0.500000

+

0.866025i

0

2.50000

−

0.866025i

0

0

0

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.b		2
3.b	odd	2	1		105.2.i.a	✓	2
7.c	even	3	1	inner	315.2.j.b		2
7.c	even	3	1		2205.2.a.f		1
7.d	odd	6	1		2205.2.a.d		1
12.b	even	2	1		1680.2.bg.m		2
15.d	odd	2	1		525.2.i.c		2
15.e	even	4	2		525.2.r.b		4
21.c	even	2	1		735.2.i.c		2
21.g	even	6	1		735.2.a.d		1
21.g	even	6	1		735.2.i.c		2
21.h	odd	6	1		105.2.i.a	✓	2
21.h	odd	6	1		735.2.a.e		1
84.n	even	6	1		1680.2.bg.m		2
105.o	odd	6	1		525.2.i.c		2
105.o	odd	6	1		3675.2.a.h		1
105.p	even	6	1		3675.2.a.i		1
105.x	even	12	2		525.2.r.b		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
105.2.i.a	✓	2	3.b	odd	2	1
105.2.i.a	✓	2	21.h	odd	6	1
315.2.j.b		2	1.a	even	1	1	trivial
315.2.j.b		2	7.c	even	3	1	inner
525.2.i.c		2	15.d	odd	2	1
525.2.i.c		2	105.o	odd	6	1
525.2.r.b		4	15.e	even	4	2
525.2.r.b		4	105.x	even	12	2
735.2.a.d		1	21.g	even	6	1
735.2.a.e		1	21.h	odd	6	1
735.2.i.c		2	21.c	even	2	1
735.2.i.c		2	21.g	even	6	1
1680.2.bg.m		2	12.b	even	2	1
1680.2.bg.m		2	84.n	even	6	1
2205.2.a.d		1	7.d	odd	6	1
2205.2.a.f		1	7.c	even	3	1
3675.2.a.h		1	105.o	odd	6	1
3675.2.a.i		1	105.p	even	6	1

Hecke kernels

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

Hecke characteristic polynomials

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