Newform orbit 315.3.e.a

• Label: 315.3.e.a
• Level: $315$
• Weight: $3$
• Character orbit: 315.e
• Self dual: yes
• Analytic conductor: $8.583$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -35
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	315.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(8.58312832735\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\)

\(=\)

\( q + 4 q^{4} - 5 q^{5} - 7 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(136\)	\(281\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

244.1

0

0

0

4.00000

−5.00000

0

−7.00000

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
35.c	odd	2	1	CM by \(\Q(\sqrt{-35}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	315.3.e.a		1
3.b	odd	2	1		35.3.c.a	✓	1
5.b	even	2	1		315.3.e.b		1
7.b	odd	2	1		315.3.e.b		1
12.b	even	2	1		560.3.p.b		1
15.d	odd	2	1		35.3.c.b	yes	1
15.e	even	4	2		175.3.d.e		2
21.c	even	2	1		35.3.c.b	yes	1
21.g	even	6	2		245.3.i.a		2
21.h	odd	6	2		245.3.i.b		2
35.c	odd	2	1	CM	315.3.e.a		1
60.h	even	2	1		560.3.p.a		1
84.h	odd	2	1		560.3.p.a		1
105.g	even	2	1		35.3.c.a	✓	1
105.k	odd	4	2		175.3.d.e		2
105.o	odd	6	2		245.3.i.a		2
105.p	even	6	2		245.3.i.b		2
420.o	odd	2	1		560.3.p.b		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
35.3.c.a	✓	1	3.b	odd	2	1
35.3.c.a	✓	1	105.g	even	2	1
35.3.c.b	yes	1	15.d	odd	2	1
35.3.c.b	yes	1	21.c	even	2	1
175.3.d.e		2	15.e	even	4	2
175.3.d.e		2	105.k	odd	4	2
245.3.i.a		2	21.g	even	6	2
245.3.i.a		2	105.o	odd	6	2
245.3.i.b		2	21.h	odd	6	2
245.3.i.b		2	105.p	even	6	2
315.3.e.a		1	1.a	even	1	1	trivial
315.3.e.a		1	35.c	odd	2	1	CM
315.3.e.b		1	5.b	even	2	1
315.3.e.b		1	7.b	odd	2	1
560.3.p.a		1	60.h	even	2	1
560.3.p.a		1	84.h	odd	2	1
560.3.p.b		1	12.b	even	2	1
560.3.p.b		1	420.o	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(315, [\chi])\):

\( T_{2} \)

\( T_{13} - 19 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T \)
$3$	\( T \)
$5$	\( T + 5 \)
$7$	\( T + 7 \)
$11$	\( T - 13 \)