Embedded newform 861.2.l.a.419.11

• Label: 861.2.l.a.419.11
• Level: $861$
• Weight: $2$
• Character: 861.419
• Analytic conductor: $6.875$
• Analytic rank: $0$
• Dimension: $216$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 861 = 3 \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	861.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.87511961403\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(108\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			419.11
Character	\(\chi\)	\(=\)	861.419
Dual form			861.2.l.a.524.11

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.39321 q^{2} +(-1.20157 - 1.24749i) q^{3} +3.72747 q^{4} +2.99101i q^{5} +(2.87562 + 2.98550i) q^{6} +(1.32481 + 2.29017i) q^{7} -4.13422 q^{8} +(-0.112448 + 2.99789i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q + 192 q^{4} - 4 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/861\mathbb{Z}\right)^\times\).

\(n\)	\(211\)	\(493\)	\(575\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.39321			−1.69226			−0.846129	−	0.532978i	\(-0.821073\pi\)
							−0.846129	+	0.532978i	\(0.821073\pi\)
\(3\)	−1.20157	−	1.24749i	−0.693728	−	0.720237i
							\(5\)			2.99101i			1.33762i	0.743434	+	0.668809i	\(0.233196\pi\)
−0.743434	+	0.668809i	\(0.766804\pi\)
\(7\)	1.32481	+	2.29017i	0.500731	+	0.865603i
							\(11\)	2.90101	−	2.90101i	0.874688	−	0.874688i	−0.118291	−	0.992979i	\(-0.537741\pi\)
0.992979	+	0.118291i	\(0.0377415\pi\)
\(13\)	3.19148	−	3.19148i	0.885158	−	0.885158i	−0.108895	−	0.994053i	\(-0.534731\pi\)
							0.994053	+	0.108895i	\(0.0347314\pi\)
\(17\)	2.63687	−	2.63687i	0.639535	−	0.639535i	−0.310906	−	0.950441i	\(-0.600632\pi\)
							0.950441	+	0.310906i	\(0.100632\pi\)
\(19\)	−3.58437	−	3.58437i	−0.822310	−	0.822310i	0.164129	−	0.986439i	\(-0.447519\pi\)
							−0.986439	+	0.164129i	\(0.947519\pi\)
\(23\)			8.20621i			1.71111i	0.517710	+	0.855556i	\(0.326785\pi\)
							−0.517710	+	0.855556i	\(0.673215\pi\)
\(29\)	0.634060	−	0.634060i	0.117742	−	0.117742i	−0.645781	−	0.763523i	\(-0.723468\pi\)
							0.763523	+	0.645781i	\(0.223468\pi\)
\(31\)			3.92241i			0.704486i	0.935909	+	0.352243i	\(0.114581\pi\)
							−0.935909	+	0.352243i	\(0.885419\pi\)
\(37\)	9.12555			1.50023			0.750115	−	0.661307i	\(-0.229998\pi\)
							0.750115	+	0.661307i	\(0.229998\pi\)
\(41\)	−6.23789	+	1.44524i	−0.974195	+	0.225709i
							\(43\)		−	9.79377i		−	1.49354i	−0.665085	−	0.746768i	\(-0.731605\pi\)
0.665085	−	0.746768i	\(-0.268395\pi\)
\(47\)	6.07132	−	6.07132i	0.885594	−	0.885594i	−0.108503	−	0.994096i	\(-0.534606\pi\)
							0.994096	+	0.108503i	\(0.0346056\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	861.2.l.a.419.11	✓	216
3.2	odd	2	inner	861.2.l.a.419.98	yes	216
7.6	odd	2	inner	861.2.l.a.419.12	yes	216
21.20	even	2	inner	861.2.l.a.419.97	yes	216
41.32	even	4	inner	861.2.l.a.524.97	yes	216
123.32	odd	4	inner	861.2.l.a.524.12	yes	216
287.237	odd	4	inner	861.2.l.a.524.98	yes	216
861.524	even	4	inner	861.2.l.a.524.11	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
861.2.l.a.419.11	✓	216	1.1	even	1	trivial
861.2.l.a.419.12	yes	216	7.6	odd	2	inner
861.2.l.a.419.97	yes	216	21.20	even	2	inner
861.2.l.a.419.98	yes	216	3.2	odd	2	inner
861.2.l.a.524.11	yes	216	861.524	even	4	inner
861.2.l.a.524.12	yes	216	123.32	odd	4	inner
861.2.l.a.524.97	yes	216	41.32	even	4	inner
861.2.l.a.524.98	yes	216	287.237	odd	4	inner