Embedded newform 605.2.e.b.362.9

• Label: 605.2.e.b.362.9
• Level: $605$
• Weight: $2$
• Character: 605.362
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(i)\)
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			362.9
Character	\(\chi\)	\(=\)	605.362
Dual form			605.2.e.b.483.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.345308 - 0.345308i) q^{2} +(-0.805651 + 0.805651i) q^{3} +1.76152i q^{4} +(-2.18158 - 0.490620i) q^{5} +0.556396i q^{6} +(2.06222 - 2.06222i) q^{7} +(1.29889 + 1.29889i) q^{8} +1.70185i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 4 q^{3} + 8 q^{5}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.345308	−	0.345308i	0.244170	−	0.244170i	−0.574403	−	0.818573i	\(-0.694766\pi\)
							0.818573	+	0.574403i	\(0.194766\pi\)
\(3\)	−0.805651	+	0.805651i	−0.465143	+	0.465143i	−0.900337	−	0.435194i	\(-0.856680\pi\)
							0.435194	+	0.900337i	\(0.356680\pi\)
\(5\)	−2.18158	−	0.490620i	−0.975632	−	0.219412i
							\(7\)	2.06222	−	2.06222i	0.779444	−	0.779444i	−0.200292	−	0.979736i	\(-0.564189\pi\)
0.979736	+	0.200292i	\(0.0641891\pi\)
\(11\)		0			0
							\(13\)	2.06936	+	2.06936i	0.573937	+	0.573937i	0.933226	−	0.359289i	\(-0.116981\pi\)
−0.359289	+	0.933226i	\(0.616981\pi\)
\(17\)	−3.72511	+	3.72511i	−0.903473	+	0.903473i	−0.995735	−	0.0922622i	\(-0.970590\pi\)
							0.0922622	+	0.995735i	\(0.470590\pi\)
\(19\)	−4.09936			−0.940459			−0.470229	−	0.882544i	\(-0.655829\pi\)
							−0.470229	+	0.882544i	\(0.655829\pi\)
\(23\)	−2.12046	+	2.12046i	−0.442147	+	0.442147i	−0.892733	−	0.450586i	\(-0.851215\pi\)
							0.450586	+	0.892733i	\(0.351215\pi\)
\(29\)	−2.64465			−0.491099			−0.245549	−	0.969384i	\(-0.578968\pi\)
							−0.245549	+	0.969384i	\(0.578968\pi\)
\(31\)	−6.74062			−1.21065			−0.605326	−	0.795978i	\(-0.706957\pi\)
							−0.605326	+	0.795978i	\(0.706957\pi\)
\(37\)	0.822560	+	0.822560i	0.135228	+	0.135228i	0.771481	−	0.636253i	\(-0.219516\pi\)
							−0.636253	+	0.771481i	\(0.719516\pi\)
\(41\)			3.55538i			0.555257i	0.960689	+	0.277628i	\(0.0895484\pi\)
							−0.960689	+	0.277628i	\(0.910452\pi\)
\(43\)	5.07292	+	5.07292i	0.773613	+	0.773613i	0.978736	−	0.205123i	\(-0.0657595\pi\)
							−0.205123	+	0.978736i	\(0.565759\pi\)
\(47\)	−2.60335	−	2.60335i	−0.379737	−	0.379737i	0.491270	−	0.871007i	\(-0.336533\pi\)
							−0.871007	+	0.491270i	\(0.836533\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.e.b.362.9		32
5.3	odd	4	inner	605.2.e.b.483.8		32
11.2	odd	10		55.2.l.a.7.3	✓	32
11.3	even	5		605.2.m.d.112.2		32
11.4	even	5		605.2.m.c.457.3		32
11.5	even	5		55.2.l.a.52.2	yes	32
11.6	odd	10		605.2.m.e.602.3		32
11.7	odd	10		605.2.m.d.457.2		32
11.8	odd	10		605.2.m.c.112.3		32
11.9	even	5		605.2.m.e.282.2		32
11.10	odd	2	inner	605.2.e.b.362.8		32
33.2	even	10		495.2.bj.a.172.2		32
33.5	odd	10		495.2.bj.a.217.3		32
44.27	odd	10		880.2.cm.a.657.3		32
44.35	even	10		880.2.cm.a.337.2		32
55.2	even	20		275.2.bm.b.18.3		32
55.3	odd	20		605.2.m.d.233.2		32
55.8	even	20		605.2.m.c.233.3		32
55.13	even	20		55.2.l.a.18.2	yes	32
55.18	even	20		605.2.m.d.578.2		32
55.24	odd	10		275.2.bm.b.7.2		32
55.27	odd	20		275.2.bm.b.118.2		32
55.28	even	20		605.2.m.e.118.2		32
55.38	odd	20		55.2.l.a.8.3	yes	32
55.43	even	4	inner	605.2.e.b.483.9		32
55.48	odd	20		605.2.m.c.578.3		32
55.49	even	10		275.2.bm.b.107.3		32
55.53	odd	20		605.2.m.e.403.3		32
165.38	even	20		495.2.bj.a.118.2		32
165.68	odd	20		495.2.bj.a.73.3		32
220.123	odd	20		880.2.cm.a.513.3		32
220.203	even	20		880.2.cm.a.833.2		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.l.a.7.3	✓	32	11.2	odd	10
55.2.l.a.8.3	yes	32	55.38	odd	20
55.2.l.a.18.2	yes	32	55.13	even	20
55.2.l.a.52.2	yes	32	11.5	even	5
275.2.bm.b.7.2		32	55.24	odd	10
275.2.bm.b.18.3		32	55.2	even	20
275.2.bm.b.107.3		32	55.49	even	10
275.2.bm.b.118.2		32	55.27	odd	20
495.2.bj.a.73.3		32	165.68	odd	20
495.2.bj.a.118.2		32	165.38	even	20
495.2.bj.a.172.2		32	33.2	even	10
495.2.bj.a.217.3		32	33.5	odd	10
605.2.e.b.362.8		32	11.10	odd	2	inner
605.2.e.b.362.9		32	1.1	even	1	trivial
605.2.e.b.483.8		32	5.3	odd	4	inner
605.2.e.b.483.9		32	55.43	even	4	inner
605.2.m.c.112.3		32	11.8	odd	10
605.2.m.c.233.3		32	55.8	even	20
605.2.m.c.457.3		32	11.4	even	5
605.2.m.c.578.3		32	55.48	odd	20
605.2.m.d.112.2		32	11.3	even	5
605.2.m.d.233.2		32	55.3	odd	20
605.2.m.d.457.2		32	11.7	odd	10
605.2.m.d.578.2		32	55.18	even	20
605.2.m.e.118.2		32	55.28	even	20
605.2.m.e.282.2		32	11.9	even	5
605.2.m.e.403.3		32	55.53	odd	20
605.2.m.e.602.3		32	11.6	odd	10
880.2.cm.a.337.2		32	44.35	even	10
880.2.cm.a.513.3		32	220.123	odd	20
880.2.cm.a.657.3		32	44.27	odd	10
880.2.cm.a.833.2		32	220.203	even	20