Embedded newform 605.2.m.e.233.1

• Label: 605.2.m.e.233.1
• Level: $605$
• Weight: $2$
• Character: 605.233
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.m (of order \(20\), degree \(8\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(4\) over \(\Q(\zeta_{20})\)
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			233.1
Character	\(\chi\)	\(=\)	605.233
Dual form			605.2.m.e.457.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30617 + 0.665529i) q^{2} +(-0.822224 - 0.130227i) q^{3} +(0.0875924 - 0.120561i) q^{4} +(-0.233059 - 2.22389i) q^{5} +(1.16064 - 0.377114i) q^{6} +(-0.659422 - 4.16343i) q^{7} +(0.424477 - 2.68004i) q^{8} +(-2.19408 - 0.712899i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 10 q^{2} - 4 q^{3} - 2 q^{5} + 20 q^{6} + 10 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.30617	+	0.665529i	−0.923605	+	0.470600i	−0.850057	−	0.526691i	\(-0.823432\pi\)
							−0.0735483	+	0.997292i	\(0.523432\pi\)
\(3\)	−0.822224	−	0.130227i	−0.474711	−	0.0751869i	−0.0855054	−	0.996338i	\(-0.527250\pi\)
							−0.389206	+	0.921151i	\(0.627250\pi\)
\(5\)	−0.233059	−	2.22389i	−0.104227	−	0.994554i
							\(7\)	−0.659422	−	4.16343i	−0.249238	−	1.57363i	−0.721655	−	0.692253i	\(-0.756618\pi\)
0.472417	−	0.881375i	\(-0.343382\pi\)
\(11\)		0			0
							\(13\)	0.420250	+	0.824787i	0.116556	+	0.228755i	0.941913	−	0.335857i	\(-0.109026\pi\)
−0.825357	+	0.564612i	\(0.809026\pi\)
\(17\)	−0.875188	+	1.71765i	−0.212264	+	0.416592i	−0.972449	−	0.233115i	\(-0.925108\pi\)
							0.760185	+	0.649707i	\(0.225108\pi\)
\(19\)	−4.39439	+	3.19271i	−1.00814	+	0.732458i	−0.963818	−	0.266560i	\(-0.914113\pi\)
							−0.0443230	+	0.999017i	\(0.514113\pi\)
\(23\)	1.95998	+	1.95998i	0.408685	+	0.408685i	0.881280	−	0.472595i	\(-0.156683\pi\)
							−0.472595	+	0.881280i	\(0.656683\pi\)
\(29\)	0.810497	+	0.588860i	0.150505	+	0.109349i	0.660490	−	0.750835i	\(-0.270349\pi\)
							−0.509984	+	0.860184i	\(0.670349\pi\)
\(31\)	0.131006	−	0.403196i	0.0235294	−	0.0724161i	−0.938602	−	0.345001i	\(-0.887879\pi\)
							0.962132	+	0.272585i	\(0.0878786\pi\)
\(37\)	−4.87226	+	0.771690i	−0.800995	+	0.126865i	−0.543494	−	0.839413i	\(-0.682899\pi\)
							−0.257501	+	0.966278i	\(0.582899\pi\)
\(41\)	−0.339428	−	0.467182i	−0.0530097	−	0.0729616i	0.781689	−	0.623668i	\(-0.214358\pi\)
							−0.834699	+	0.550707i	\(0.814358\pi\)
\(43\)	5.05373	−	5.05373i	0.770687	−	0.770687i	−0.207540	−	0.978227i	\(-0.566546\pi\)
							0.978227	+	0.207540i	\(0.0665456\pi\)
\(47\)	−0.186094	+	1.17495i	−0.0271446	+	0.171384i	−0.997536	−	0.0701524i	\(-0.977651\pi\)
							0.970392	+	0.241537i	\(0.0776514\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.m.e.233.1		32
5.2	odd	4	inner	605.2.m.e.112.1		32
11.2	odd	10		605.2.m.c.118.1		32
11.3	even	5		605.2.m.c.403.4		32
11.4	even	5		605.2.e.b.483.4		32
11.5	even	5		55.2.l.a.28.4	yes	32
11.6	odd	10	inner	605.2.m.e.578.1		32
11.7	odd	10		605.2.e.b.483.13		32
11.8	odd	10		605.2.m.d.403.1		32
11.9	even	5		605.2.m.d.118.4		32
11.10	odd	2		55.2.l.a.13.4	yes	32
33.5	odd	10		495.2.bj.a.28.1		32
33.32	even	2		495.2.bj.a.343.1		32
44.27	odd	10		880.2.cm.a.193.3		32
44.43	even	2		880.2.cm.a.673.3		32
55.2	even	20		605.2.m.c.602.4		32
55.7	even	20		605.2.e.b.362.4		32
55.17	even	20	inner	605.2.m.e.457.1		32
55.27	odd	20		55.2.l.a.17.4	yes	32
55.32	even	4		55.2.l.a.2.4	✓	32
55.37	odd	20		605.2.e.b.362.13		32
55.38	odd	20		275.2.bm.b.182.1		32
55.42	odd	20		605.2.m.d.602.1		32
55.43	even	4		275.2.bm.b.57.1		32
55.47	odd	20		605.2.m.c.282.1		32
55.49	even	10		275.2.bm.b.193.1		32
55.52	even	20		605.2.m.d.282.4		32
55.54	odd	2		275.2.bm.b.68.1		32
165.32	odd	4		495.2.bj.a.442.1		32
165.137	even	20		495.2.bj.a.127.1		32
220.27	even	20		880.2.cm.a.17.3		32
220.87	odd	4		880.2.cm.a.497.3		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.l.a.2.4	✓	32	55.32	even	4
55.2.l.a.13.4	yes	32	11.10	odd	2
55.2.l.a.17.4	yes	32	55.27	odd	20
55.2.l.a.28.4	yes	32	11.5	even	5
275.2.bm.b.57.1		32	55.43	even	4
275.2.bm.b.68.1		32	55.54	odd	2
275.2.bm.b.182.1		32	55.38	odd	20
275.2.bm.b.193.1		32	55.49	even	10
495.2.bj.a.28.1		32	33.5	odd	10
495.2.bj.a.127.1		32	165.137	even	20
495.2.bj.a.343.1		32	33.32	even	2
495.2.bj.a.442.1		32	165.32	odd	4
605.2.e.b.362.4		32	55.7	even	20
605.2.e.b.362.13		32	55.37	odd	20
605.2.e.b.483.4		32	11.4	even	5
605.2.e.b.483.13		32	11.7	odd	10
605.2.m.c.118.1		32	11.2	odd	10
605.2.m.c.282.1		32	55.47	odd	20
605.2.m.c.403.4		32	11.3	even	5
605.2.m.c.602.4		32	55.2	even	20
605.2.m.d.118.4		32	11.9	even	5
605.2.m.d.282.4		32	55.52	even	20
605.2.m.d.403.1		32	11.8	odd	10
605.2.m.d.602.1		32	55.42	odd	20
605.2.m.e.112.1		32	5.2	odd	4	inner
605.2.m.e.233.1		32	1.1	even	1	trivial
605.2.m.e.457.1		32	55.17	even	20	inner
605.2.m.e.578.1		32	11.6	odd	10	inner
880.2.cm.a.17.3		32	220.27	even	20
880.2.cm.a.193.3		32	44.27	odd	10
880.2.cm.a.497.3		32	220.87	odd	4
880.2.cm.a.673.3		32	44.43	even	2