Embedded newform 605.2.m.e.233.4

• Label: 605.2.m.e.233.4
• 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			233.4
Character	\(\chi\)	\(=\)	605.233
Dual form			605.2.m.e.457.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.25897 - 1.15100i) q^{2} +(1.98021 + 0.313634i) q^{3} +(2.60258 - 3.58214i) q^{4} +(-0.867371 + 2.06099i) q^{5} +(4.83422 - 1.57073i) q^{6} +(-0.282837 - 1.78576i) q^{7} +(0.962874 - 6.07934i) q^{8} +(0.969677 + 0.315067i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 10 q^{2} - 4 q^{3} - 2 q^{5} + 20 q^{6} + 10 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.25897	−	1.15100i	1.59733	−	0.813883i	0.597406	−	0.801939i	\(-0.296198\pi\)
							0.999929	−	0.0119437i	\(-0.00380189\pi\)
\(3\)	1.98021	+	0.313634i	1.14327	+	0.181076i	0.699208	−	0.714918i	\(-0.253536\pi\)
							0.444064	+	0.895995i	\(0.353536\pi\)
\(5\)	−0.867371	+	2.06099i	−0.387900	+	0.921701i
							\(7\)	−0.282837	−	1.78576i	−0.106902	−	0.674955i	−0.981695	−	0.190458i	\(-0.939003\pi\)
0.874793	−	0.484497i	\(-0.160997\pi\)
\(11\)		0			0
							\(13\)	1.02411	+	2.00993i	0.284037	+	0.557454i	0.988307	−	0.152478i	\(-0.0487254\pi\)
−0.704270	+	0.709933i	\(0.748725\pi\)
\(17\)	−1.52724	+	2.99738i	−0.370411	+	0.726972i	−0.998698	−	0.0510094i	\(-0.983756\pi\)
							0.628288	+	0.777981i	\(0.283756\pi\)
\(19\)	−1.05647	+	0.767569i	−0.242370	+	0.176092i	−0.702339	−	0.711843i	\(-0.747861\pi\)
							0.459968	+	0.887935i	\(0.347861\pi\)
\(23\)	−3.16488	−	3.16488i	−0.659922	−	0.659922i	0.295439	−	0.955362i	\(-0.404534\pi\)
							−0.955362	+	0.295439i	\(0.904534\pi\)
\(29\)	−2.37569	−	1.72604i	−0.441154	−	0.320517i	0.344939	−	0.938625i	\(-0.387900\pi\)
							−0.786093	+	0.618108i	\(0.787900\pi\)
\(31\)	−1.23925	+	3.81401i	−0.222575	+	0.685016i	0.775953	+	0.630790i	\(0.217269\pi\)
							−0.998529	+	0.0542261i	\(0.982731\pi\)
\(37\)	3.16942	−	0.501987i	0.521050	−	0.0825262i	0.109631	−	0.993972i	\(-0.465033\pi\)
							0.411419	+	0.911446i	\(0.365033\pi\)
\(41\)	−0.766287	−	1.05470i	−0.119674	−	0.164717i	0.744977	−	0.667090i	\(-0.232460\pi\)
							−0.864651	+	0.502373i	\(0.832460\pi\)
\(43\)	4.55431	−	4.55431i	0.694526	−	0.694526i	−0.268698	−	0.963224i	\(-0.586593\pi\)
							0.963224	+	0.268698i	\(0.0865935\pi\)
\(47\)	−1.20824	+	7.62854i	−0.176240	+	1.11274i	0.727957	+	0.685622i	\(0.240470\pi\)
							−0.904198	+	0.427114i	\(0.859530\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.m.e.233.4		32
5.2	odd	4	inner	605.2.m.e.112.4		32
11.2	odd	10		605.2.m.c.118.4		32
11.3	even	5		605.2.m.c.403.1		32
11.4	even	5		605.2.e.b.483.16		32
11.5	even	5		55.2.l.a.28.1	yes	32
11.6	odd	10	inner	605.2.m.e.578.4		32
11.7	odd	10		605.2.e.b.483.1		32
11.8	odd	10		605.2.m.d.403.4		32
11.9	even	5		605.2.m.d.118.1		32
11.10	odd	2		55.2.l.a.13.1	yes	32
33.5	odd	10		495.2.bj.a.28.4		32
33.32	even	2		495.2.bj.a.343.4		32
44.27	odd	10		880.2.cm.a.193.1		32
44.43	even	2		880.2.cm.a.673.1		32
55.2	even	20		605.2.m.c.602.1		32
55.7	even	20		605.2.e.b.362.16		32
55.17	even	20	inner	605.2.m.e.457.4		32
55.27	odd	20		55.2.l.a.17.1	yes	32
55.32	even	4		55.2.l.a.2.1	✓	32
55.37	odd	20		605.2.e.b.362.1		32
55.38	odd	20		275.2.bm.b.182.4		32
55.42	odd	20		605.2.m.d.602.4		32
55.43	even	4		275.2.bm.b.57.4		32
55.47	odd	20		605.2.m.c.282.4		32
55.49	even	10		275.2.bm.b.193.4		32
55.52	even	20		605.2.m.d.282.1		32
55.54	odd	2		275.2.bm.b.68.4		32
165.32	odd	4		495.2.bj.a.442.4		32
165.137	even	20		495.2.bj.a.127.4		32
220.27	even	20		880.2.cm.a.17.1		32
220.87	odd	4		880.2.cm.a.497.1		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.l.a.2.1	✓	32	55.32	even	4
55.2.l.a.13.1	yes	32	11.10	odd	2
55.2.l.a.17.1	yes	32	55.27	odd	20
55.2.l.a.28.1	yes	32	11.5	even	5
275.2.bm.b.57.4		32	55.43	even	4
275.2.bm.b.68.4		32	55.54	odd	2
275.2.bm.b.182.4		32	55.38	odd	20
275.2.bm.b.193.4		32	55.49	even	10
495.2.bj.a.28.4		32	33.5	odd	10
495.2.bj.a.127.4		32	165.137	even	20
495.2.bj.a.343.4		32	33.32	even	2
495.2.bj.a.442.4		32	165.32	odd	4
605.2.e.b.362.1		32	55.37	odd	20
605.2.e.b.362.16		32	55.7	even	20
605.2.e.b.483.1		32	11.7	odd	10
605.2.e.b.483.16		32	11.4	even	5
605.2.m.c.118.4		32	11.2	odd	10
605.2.m.c.282.4		32	55.47	odd	20
605.2.m.c.403.1		32	11.3	even	5
605.2.m.c.602.1		32	55.2	even	20
605.2.m.d.118.1		32	11.9	even	5
605.2.m.d.282.1		32	55.52	even	20
605.2.m.d.403.4		32	11.8	odd	10
605.2.m.d.602.4		32	55.42	odd	20
605.2.m.e.112.4		32	5.2	odd	4	inner
605.2.m.e.233.4		32	1.1	even	1	trivial
605.2.m.e.457.4		32	55.17	even	20	inner
605.2.m.e.578.4		32	11.6	odd	10	inner
880.2.cm.a.17.1		32	220.27	even	20
880.2.cm.a.193.1		32	44.27	odd	10
880.2.cm.a.497.1		32	220.87	odd	4
880.2.cm.a.673.1		32	44.43	even	2