Embedded newform 891.2.e.g.595.1

• Label: 891.2.e.g.595.1
• Level: $891$
• Weight: $2$
• Character: 891.595
• Analytic conductor: $7.115$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 891 = 3^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	891.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.11467082010\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			595.1
Root			\(0.500000 - 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	891.595
Dual form			891.2.e.g.298.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.00000 + 1.73205i) q^{5} +(-2.00000 - 3.46410i) q^{7} +3.00000 q^{8} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + q^{4} - 2 q^{5} - 4 q^{7} + 6 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/891\mathbb{Z}\right)^\times\).

\(n\)	\(244\)	\(650\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.500000	+	0.866025i	0.353553	+	0.612372i	0.986869	−	0.161521i	\(-0.0516399\pi\)
							−0.633316	+	0.773893i	\(0.718307\pi\)
\(3\)		0			0
							\(5\)	−1.00000	+	1.73205i	−0.447214	+	0.774597i	−0.998203	−	0.0599153i	\(-0.980917\pi\)
0.550990	+	0.834512i	\(0.314250\pi\)
\(7\)	−2.00000	−	3.46410i	−0.755929	−	1.30931i	−0.944911	−	0.327327i	\(-0.893852\pi\)
							0.188982	−	0.981981i	\(-0.439481\pi\)
\(11\)	0.500000	+	0.866025i	0.150756	+	0.261116i
							\(13\)	1.00000	−	1.73205i	0.277350	−	0.480384i	−0.693375	−	0.720577i	\(-0.743877\pi\)
0.970725	+	0.240192i	\(0.0772105\pi\)
\(17\)	2.00000			0.485071			0.242536	−	0.970143i	\(-0.422021\pi\)
							0.242536	+	0.970143i	\(0.422021\pi\)
\(19\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(23\)	4.00000	−	6.92820i	0.834058	−	1.44463i	−0.0607377	−	0.998154i	\(-0.519345\pi\)
							0.894795	−	0.446476i	\(-0.147321\pi\)
\(29\)	−3.00000	−	5.19615i	−0.557086	−	0.964901i	−0.997738	−	0.0672232i	\(-0.978586\pi\)
							0.440652	−	0.897678i	\(-0.354747\pi\)
\(31\)	4.00000	−	6.92820i	0.718421	−	1.24434i	−0.243204	−	0.969975i	\(-0.578198\pi\)
							0.961625	−	0.274367i	\(-0.0884683\pi\)
\(37\)	6.00000			0.986394			0.493197	−	0.869918i	\(-0.335828\pi\)
							0.493197	+	0.869918i	\(0.335828\pi\)
\(41\)	−1.00000	+	1.73205i	−0.156174	+	0.270501i	−0.933486	−	0.358614i	\(-0.883249\pi\)
							0.777312	+	0.629115i	\(0.216583\pi\)
\(43\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)
\(47\)	4.00000	+	6.92820i	0.583460	+	1.01058i	0.995066	+	0.0992202i	\(0.0316348\pi\)
							−0.411606	+	0.911362i	\(0.635032\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	891.2.e.g.595.1		2
3.2	odd	2		891.2.e.e.595.1		2
9.2	odd	6		891.2.e.e.298.1		2
9.4	even	3		99.2.a.b.1.1		1
9.5	odd	6		33.2.a.a.1.1	✓	1
9.7	even	3	inner	891.2.e.g.298.1		2
36.23	even	6		528.2.a.g.1.1		1
36.31	odd	6		1584.2.a.o.1.1		1
45.4	even	6		2475.2.a.g.1.1		1
45.13	odd	12		2475.2.c.d.199.2		2
45.14	odd	6		825.2.a.a.1.1		1
45.22	odd	12		2475.2.c.d.199.1		2
45.23	even	12		825.2.c.a.199.1		2
45.32	even	12		825.2.c.a.199.2		2
63.13	odd	6		4851.2.a.b.1.1		1
63.41	even	6		1617.2.a.j.1.1		1
72.5	odd	6		2112.2.a.bb.1.1		1
72.13	even	6		6336.2.a.x.1.1		1
72.59	even	6		2112.2.a.j.1.1		1
72.67	odd	6		6336.2.a.n.1.1		1
99.5	odd	30		363.2.e.e.124.1		4
99.14	odd	30		363.2.e.e.130.1		4
99.32	even	6		363.2.a.b.1.1		1
99.41	even	30		363.2.e.g.130.1		4
99.50	even	30		363.2.e.g.124.1		4
99.59	odd	30		363.2.e.e.148.1		4
99.68	even	30		363.2.e.g.202.1		4
99.76	odd	6		1089.2.a.j.1.1		1
99.86	odd	30		363.2.e.e.202.1		4
99.95	even	30		363.2.e.g.148.1		4
117.77	odd	6		5577.2.a.a.1.1		1
153.50	odd	6		9537.2.a.m.1.1		1
396.131	odd	6		5808.2.a.t.1.1		1
495.329	even	6		9075.2.a.q.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.2.a.a.1.1	✓	1	9.5	odd	6
99.2.a.b.1.1		1	9.4	even	3
363.2.a.b.1.1		1	99.32	even	6
363.2.e.e.124.1		4	99.5	odd	30
363.2.e.e.130.1		4	99.14	odd	30
363.2.e.e.148.1		4	99.59	odd	30
363.2.e.e.202.1		4	99.86	odd	30
363.2.e.g.124.1		4	99.50	even	30
363.2.e.g.130.1		4	99.41	even	30
363.2.e.g.148.1		4	99.95	even	30
363.2.e.g.202.1		4	99.68	even	30
528.2.a.g.1.1		1	36.23	even	6
825.2.a.a.1.1		1	45.14	odd	6
825.2.c.a.199.1		2	45.23	even	12
825.2.c.a.199.2		2	45.32	even	12
891.2.e.e.298.1		2	9.2	odd	6
891.2.e.e.595.1		2	3.2	odd	2
891.2.e.g.298.1		2	9.7	even	3	inner
891.2.e.g.595.1		2	1.1	even	1	trivial
1089.2.a.j.1.1		1	99.76	odd	6
1584.2.a.o.1.1		1	36.31	odd	6
1617.2.a.j.1.1		1	63.41	even	6
2112.2.a.j.1.1		1	72.59	even	6
2112.2.a.bb.1.1		1	72.5	odd	6
2475.2.a.g.1.1		1	45.4	even	6
2475.2.c.d.199.1		2	45.22	odd	12
2475.2.c.d.199.2		2	45.13	odd	12
4851.2.a.b.1.1		1	63.13	odd	6
5577.2.a.a.1.1		1	117.77	odd	6
5808.2.a.t.1.1		1	396.131	odd	6
6336.2.a.n.1.1		1	72.67	odd	6
6336.2.a.x.1.1		1	72.13	even	6
9075.2.a.q.1.1		1	495.329	even	6
9537.2.a.m.1.1		1	153.50	odd	6