Embedded newform 605.2.m.e.233.2

• Label: 605.2.m.e.233.2
• 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			233.2
Character	\(\chi\)	\(=\)	605.233
Dual form			605.2.m.e.457.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.513072 - 0.261423i) q^{2} +(0.760272 + 0.120415i) q^{3} +(-0.980670 + 1.34978i) q^{4} +(2.23511 + 0.0653109i) q^{5} +(0.421554 - 0.136971i) q^{6} +(0.186656 + 1.17850i) q^{7} +(-0.330452 + 2.08639i) q^{8} +(-2.28966 - 0.743954i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 10 q^{2} - 4 q^{3} - 2 q^{5} + 20 q^{6} + 10 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.513072	−	0.261423i	0.362797	−	0.184854i	−0.263084	−	0.964773i	\(-0.584740\pi\)
							0.625881	+	0.779919i	\(0.284740\pi\)
\(3\)	0.760272	+	0.120415i	0.438943	+	0.0695218i	0.371995	−	0.928235i	\(-0.378674\pi\)
							0.0669482	+	0.997756i	\(0.478674\pi\)
\(5\)	2.23511	+	0.0653109i	0.999573	+	0.0292079i
							\(7\)	0.186656	+	1.17850i	0.0705492	+	0.445430i	0.997525	+	0.0703096i	\(0.0223987\pi\)
−0.926976	+	0.375121i	\(0.877601\pi\)
\(11\)		0			0
							\(13\)	2.75941	+	5.41565i	0.765322	+	1.50203i	0.862108	+	0.506724i	\(0.169144\pi\)
−0.0967857	+	0.995305i	\(0.530856\pi\)
\(17\)	−0.168355	+	0.330416i	−0.0408321	+	0.0801376i	−0.910521	−	0.413462i	\(-0.864319\pi\)
							0.869689	+	0.493600i	\(0.164319\pi\)
\(19\)	1.11392	−	0.809308i	0.255550	−	0.185668i	−0.452633	−	0.891697i	\(-0.649515\pi\)
							0.708183	+	0.706029i	\(0.249515\pi\)
\(23\)	3.48057	+	3.48057i	0.725749	+	0.725749i	0.969770	−	0.244021i	\(-0.0784665\pi\)
							−0.244021	+	0.969770i	\(0.578467\pi\)
\(29\)	−4.95629	−	3.60096i	−0.920361	−	0.668681i	0.0232531	−	0.999730i	\(-0.492598\pi\)
							−0.943614	+	0.331048i	\(0.892598\pi\)
\(31\)	−0.764516	+	2.35294i	−0.137311	+	0.422600i	−0.995942	−	0.0899935i	\(-0.971315\pi\)
							0.858631	+	0.512594i	\(0.171315\pi\)
\(37\)	−2.93925	+	0.465532i	−0.483210	+	0.0765329i	−0.393287	−	0.919416i	\(-0.628662\pi\)
							−0.0899229	+	0.995949i	\(0.528662\pi\)
\(41\)	3.60047	+	4.95563i	0.562300	+	0.773939i	0.991617	−	0.129215i	\(-0.0412457\pi\)
							−0.429317	+	0.903154i	\(0.641246\pi\)
\(43\)	6.75396	−	6.75396i	1.02997	−	1.02997i	0.0304328	−	0.999537i	\(-0.490311\pi\)
							0.999537	−	0.0304328i	\(-0.00968857\pi\)
\(47\)	−0.199725	+	1.26101i	−0.0291328	+	0.183937i	−0.997964	−	0.0637861i	\(-0.979682\pi\)
							0.968831	+	0.247724i	\(0.0796825\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.m.e.233.2		32
5.2	odd	4	inner	605.2.m.e.112.2		32
11.2	odd	10		605.2.m.c.118.2		32
11.3	even	5		605.2.m.c.403.3		32
11.4	even	5		605.2.e.b.483.10		32
11.5	even	5		55.2.l.a.28.3	yes	32
11.6	odd	10	inner	605.2.m.e.578.2		32
11.7	odd	10		605.2.e.b.483.7		32
11.8	odd	10		605.2.m.d.403.2		32
11.9	even	5		605.2.m.d.118.3		32
11.10	odd	2		55.2.l.a.13.3	yes	32
33.5	odd	10		495.2.bj.a.28.2		32
33.32	even	2		495.2.bj.a.343.2		32
44.27	odd	10		880.2.cm.a.193.2		32
44.43	even	2		880.2.cm.a.673.2		32
55.2	even	20		605.2.m.c.602.3		32
55.7	even	20		605.2.e.b.362.10		32
55.17	even	20	inner	605.2.m.e.457.2		32
55.27	odd	20		55.2.l.a.17.3	yes	32
55.32	even	4		55.2.l.a.2.3	✓	32
55.37	odd	20		605.2.e.b.362.7		32
55.38	odd	20		275.2.bm.b.182.2		32
55.42	odd	20		605.2.m.d.602.2		32
55.43	even	4		275.2.bm.b.57.2		32
55.47	odd	20		605.2.m.c.282.2		32
55.49	even	10		275.2.bm.b.193.2		32
55.52	even	20		605.2.m.d.282.3		32
55.54	odd	2		275.2.bm.b.68.2		32
165.32	odd	4		495.2.bj.a.442.2		32
165.137	even	20		495.2.bj.a.127.2		32
220.27	even	20		880.2.cm.a.17.2		32
220.87	odd	4		880.2.cm.a.497.2		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.l.a.2.3	✓	32	55.32	even	4
55.2.l.a.13.3	yes	32	11.10	odd	2
55.2.l.a.17.3	yes	32	55.27	odd	20
55.2.l.a.28.3	yes	32	11.5	even	5
275.2.bm.b.57.2		32	55.43	even	4
275.2.bm.b.68.2		32	55.54	odd	2
275.2.bm.b.182.2		32	55.38	odd	20
275.2.bm.b.193.2		32	55.49	even	10
495.2.bj.a.28.2		32	33.5	odd	10
495.2.bj.a.127.2		32	165.137	even	20
495.2.bj.a.343.2		32	33.32	even	2
495.2.bj.a.442.2		32	165.32	odd	4
605.2.e.b.362.7		32	55.37	odd	20
605.2.e.b.362.10		32	55.7	even	20
605.2.e.b.483.7		32	11.7	odd	10
605.2.e.b.483.10		32	11.4	even	5
605.2.m.c.118.2		32	11.2	odd	10
605.2.m.c.282.2		32	55.47	odd	20
605.2.m.c.403.3		32	11.3	even	5
605.2.m.c.602.3		32	55.2	even	20
605.2.m.d.118.3		32	11.9	even	5
605.2.m.d.282.3		32	55.52	even	20
605.2.m.d.403.2		32	11.8	odd	10
605.2.m.d.602.2		32	55.42	odd	20
605.2.m.e.112.2		32	5.2	odd	4	inner
605.2.m.e.233.2		32	1.1	even	1	trivial
605.2.m.e.457.2		32	55.17	even	20	inner
605.2.m.e.578.2		32	11.6	odd	10	inner
880.2.cm.a.17.2		32	220.27	even	20
880.2.cm.a.193.2		32	44.27	odd	10
880.2.cm.a.497.2		32	220.87	odd	4
880.2.cm.a.673.2		32	44.43	even	2