Embedded newform 315.2.bz.d.82.6

• Label: 315.2.bz.d.82.6
• Level: $315$
• Weight: $2$
• Character: 315.82
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.bz (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.51528766367\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			82.6
Character	\(\chi\)	\(=\)	315.82
Dual form			315.2.bz.d.73.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.969545 - 0.259789i) q^{2} +(-0.859523 + 0.496246i) q^{4} +(-0.803857 + 2.08658i) q^{5} +(2.42328 - 1.06195i) q^{7} +(-2.12394 + 2.12394i) q^{8} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 12 q^{5} + 8 q^{7} + 24 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(136\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.969545	−	0.259789i	0.685572	−	0.183698i	0.100813	−	0.994905i	\(-0.467856\pi\)
							0.584759	+	0.811207i	\(0.301189\pi\)
\(3\)		0			0
							\(5\)	−0.803857	+	2.08658i	−0.359496	+	0.933147i
\(7\)	2.42328	−	1.06195i	0.915912	−	0.401379i
							\(11\)	1.78283	+	3.08796i	0.537545	+	0.931054i	0.999036	+	0.0439095i	\(0.0139813\pi\)
−0.461491	+	0.887145i	\(0.652685\pi\)
\(13\)	2.78368	+	2.78368i	0.772054	+	0.772054i	0.978465	−	0.206411i	\(-0.0661785\pi\)
							−0.206411	+	0.978465i	\(0.566178\pi\)
\(17\)	0.506489	+	0.135713i	0.122842	+	0.0329153i	0.319716	−	0.947513i	\(-0.396412\pi\)
							−0.196874	+	0.980429i	\(0.563079\pi\)
\(19\)	−2.06188	+	3.57128i	−0.473028	+	0.819308i	−0.999523	−	0.0308699i	\(-0.990172\pi\)
							0.526496	+	0.850178i	\(0.323506\pi\)
\(23\)	−0.668338	−	2.49427i	−0.139358	−	0.520092i	−0.999942	−	0.0107826i	\(-0.996568\pi\)
							0.860584	−	0.509309i	\(-0.170099\pi\)
\(29\)		−	6.14396i		−	1.14091i	−0.821331	−	0.570453i	\(-0.806768\pi\)
							0.821331	−	0.570453i	\(-0.193232\pi\)
\(31\)	−1.71173	+	0.988266i	−0.307435	+	0.177498i	−0.645778	−	0.763525i	\(-0.723467\pi\)
							0.338343	+	0.941023i	\(0.390134\pi\)
\(37\)	0.152803	−	0.0409435i	0.0251207	−	0.00673106i	−0.246237	−	0.969210i	\(-0.579194\pi\)
							0.271357	+	0.962479i	\(0.412527\pi\)
\(41\)		−	8.28475i		−	1.29386i	−0.762549	−	0.646930i	\(-0.776052\pi\)
							0.762549	−	0.646930i	\(-0.223948\pi\)
\(43\)	9.01253	−	9.01253i	1.37440	−	1.37440i	0.520594	−	0.853805i	\(-0.325711\pi\)
							0.853805	−	0.520594i	\(-0.174289\pi\)
\(47\)	1.39091	+	5.19095i	0.202885	+	0.757178i	0.990084	+	0.140478i	\(0.0448640\pi\)
							−0.787198	+	0.616700i	\(0.788469\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.bz.d.82.6		32
3.2	odd	2		105.2.u.a.82.3	yes	32
5.3	odd	4	inner	315.2.bz.d.208.3		32
7.3	odd	6	inner	315.2.bz.d.262.3		32
15.2	even	4		525.2.bc.e.418.3		32
15.8	even	4		105.2.u.a.103.6	yes	32
15.14	odd	2		525.2.bc.e.82.6		32
21.2	odd	6		735.2.m.c.97.12		32
21.5	even	6		735.2.m.c.97.11		32
21.11	odd	6		735.2.v.b.472.6		32
21.17	even	6		105.2.u.a.52.6	✓	32
21.20	even	2		735.2.v.b.607.3		32
35.3	even	12	inner	315.2.bz.d.73.6		32
105.17	odd	12		525.2.bc.e.493.6		32
105.23	even	12		735.2.m.c.538.11		32
105.38	odd	12		105.2.u.a.73.3	yes	32
105.53	even	12		735.2.v.b.178.3		32
105.59	even	6		525.2.bc.e.157.3		32
105.68	odd	12		735.2.m.c.538.12		32
105.83	odd	4		735.2.v.b.313.6		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.u.a.52.6	✓	32	21.17	even	6
105.2.u.a.73.3	yes	32	105.38	odd	12
105.2.u.a.82.3	yes	32	3.2	odd	2
105.2.u.a.103.6	yes	32	15.8	even	4
315.2.bz.d.73.6		32	35.3	even	12	inner
315.2.bz.d.82.6		32	1.1	even	1	trivial
315.2.bz.d.208.3		32	5.3	odd	4	inner
315.2.bz.d.262.3		32	7.3	odd	6	inner
525.2.bc.e.82.6		32	15.14	odd	2
525.2.bc.e.157.3		32	105.59	even	6
525.2.bc.e.418.3		32	15.2	even	4
525.2.bc.e.493.6		32	105.17	odd	12
735.2.m.c.97.11		32	21.5	even	6
735.2.m.c.97.12		32	21.2	odd	6
735.2.m.c.538.11		32	105.23	even	12
735.2.m.c.538.12		32	105.68	odd	12
735.2.v.b.178.3		32	105.53	even	12
735.2.v.b.313.6		32	105.83	odd	4
735.2.v.b.472.6		32	21.11	odd	6
735.2.v.b.607.3		32	21.20	even	2