Embedded newform 315.3.e.b.244.1

• Label: 315.3.e.b.244.1
• Level: $315$
• Weight: $3$
• Character: 315.244
• 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:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			244.1
Character	\(\chi\)	\(=\)	315.244

$q$-expansion

\(f(q)\)

\(=\)

\(q+4.00000 q^{4} +5.00000 q^{5} +7.00000 q^{7} +O(q^{10})\)

Character values

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

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

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(3\)	0	0
			\(5\)	5.00000	1.00000
\(7\)	7.00000	1.00000
			\(11\)	13.0000	1.18182	0.590909	−	0.806738i	\(-0.298769\pi\)
0.590909	+	0.806738i				\(0.298769\pi\)
\(13\)	−19.0000	−1.46154	−0.730769	−	0.682625i	\(-0.760838\pi\)
			−0.730769	+	0.682625i	\(0.760838\pi\)
\(17\)	−29.0000	−1.70588	−0.852941	−	0.522007i	\(-0.825183\pi\)
			−0.852941	+	0.522007i	\(0.825183\pi\)
\(19\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(23\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(29\)	−23.0000	−0.793103	−0.396552	−	0.918012i	\(-0.629793\pi\)
			−0.396552	+	0.918012i	\(0.629793\pi\)
\(31\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(37\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(41\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(43\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(47\)	31.0000	0.659574	0.329787	−	0.944055i	\(-0.393023\pi\)
			0.329787	+	0.944055i	\(0.393023\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.3.e.b.244.1		1
3.2	odd	2		35.3.c.b.34.1	yes	1
5.4	even	2		315.3.e.a.244.1		1
7.6	odd	2		315.3.e.a.244.1		1
12.11	even	2		560.3.p.a.209.1		1
15.2	even	4		175.3.d.e.76.1		2
15.8	even	4		175.3.d.e.76.2		2
15.14	odd	2		35.3.c.a.34.1	✓	1
21.2	odd	6		245.3.i.a.129.1		2
21.5	even	6		245.3.i.b.129.1		2
21.11	odd	6		245.3.i.a.19.1		2
21.17	even	6		245.3.i.b.19.1		2
21.20	even	2		35.3.c.a.34.1	✓	1
35.34	odd	2	CM	315.3.e.b.244.1		1
60.59	even	2		560.3.p.b.209.1		1
84.83	odd	2		560.3.p.b.209.1		1
105.44	odd	6		245.3.i.b.129.1		2
105.59	even	6		245.3.i.a.19.1		2
105.62	odd	4		175.3.d.e.76.2		2
105.74	odd	6		245.3.i.b.19.1		2
105.83	odd	4		175.3.d.e.76.1		2
105.89	even	6		245.3.i.a.129.1		2
105.104	even	2		35.3.c.b.34.1	yes	1
420.419	odd	2		560.3.p.a.209.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.3.c.a.34.1	✓	1	15.14	odd	2
35.3.c.a.34.1	✓	1	21.20	even	2
35.3.c.b.34.1	yes	1	3.2	odd	2
35.3.c.b.34.1	yes	1	105.104	even	2
175.3.d.e.76.1		2	15.2	even	4
175.3.d.e.76.1		2	105.83	odd	4
175.3.d.e.76.2		2	15.8	even	4
175.3.d.e.76.2		2	105.62	odd	4
245.3.i.a.19.1		2	21.11	odd	6
245.3.i.a.19.1		2	105.59	even	6
245.3.i.a.129.1		2	21.2	odd	6
245.3.i.a.129.1		2	105.89	even	6
245.3.i.b.19.1		2	21.17	even	6
245.3.i.b.19.1		2	105.74	odd	6
245.3.i.b.129.1		2	21.5	even	6
245.3.i.b.129.1		2	105.44	odd	6
315.3.e.a.244.1		1	5.4	even	2
315.3.e.a.244.1		1	7.6	odd	2
315.3.e.b.244.1		1	1.1	even	1	trivial
315.3.e.b.244.1		1	35.34	odd	2	CM
560.3.p.a.209.1		1	12.11	even	2
560.3.p.a.209.1		1	420.419	odd	2
560.3.p.b.209.1		1	60.59	even	2
560.3.p.b.209.1		1	84.83	odd	2