Embedded newform 882.2.g.i.667.1

• Label: 882.2.g.i.667.1
• Level: $882$
• Weight: $2$
• Character: 882.667
• Analytic conductor: $7.043$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	882.g (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.04280545828\)
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 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			667.1
Root			\(0.500000 - 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	882.667
Dual form			882.2.g.i.361.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-0.500000 + 0.866025i) q^{5} -1.00000 q^{8} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} - q^{5} - 2 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(785\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.500000	−	0.866025i	0.353553	−	0.612372i
							\(3\)		0			0
\(5\)	−0.500000	+	0.866025i	−0.223607	+	0.387298i								−0.955901	−	0.293691i	\(-0.905116\pi\)
							0.732294	+	0.680989i	\(0.238450\pi\)
\(7\)		0			0
							\(11\)	2.50000	+	4.33013i	0.753778	+	1.30558i	0.945979	+	0.324227i	\(0.105104\pi\)
−0.192201	+	0.981356i	\(0.561563\pi\)
\(13\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(17\)	2.00000	+	3.46410i	0.485071	+	0.840168i	0.999853	−	0.0171533i	\(-0.00546033\pi\)
							−0.514782	+	0.857321i	\(0.672127\pi\)
\(19\)	4.00000	−	6.92820i	0.917663	−	1.58944i	0.114708	−	0.993399i	\(-0.463407\pi\)
							0.802955	−	0.596040i	\(-0.203260\pi\)
\(23\)	−2.00000	+	3.46410i	−0.417029	+	0.722315i	−0.995639	−	0.0932891i	\(-0.970262\pi\)
							0.578610	+	0.815604i	\(0.303595\pi\)
\(29\)	5.00000			0.928477			0.464238	−	0.885710i	\(-0.346328\pi\)
							0.464238	+	0.885710i	\(0.346328\pi\)
\(31\)	1.50000	+	2.59808i	0.269408	+	0.466628i	0.968709	−	0.248199i	\(-0.0798387\pi\)
							−0.699301	+	0.714827i	\(0.746505\pi\)
\(37\)	2.00000	−	3.46410i	0.328798	−	0.569495i	−0.653476	−	0.756948i	\(-0.726690\pi\)
							0.982274	+	0.187453i	\(0.0600231\pi\)
\(41\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(43\)	2.00000			0.304997			0.152499	−	0.988304i	\(-0.451268\pi\)
							0.152499	+	0.988304i	\(0.451268\pi\)
\(47\)	3.00000	−	5.19615i	0.437595	−	0.757937i	−0.559908	−	0.828554i	\(-0.689164\pi\)
							0.997503	+	0.0706177i	\(0.0224970\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.g.i.667.1		2
3.2	odd	2		294.2.e.b.79.1		2
7.2	even	3		882.2.a.d.1.1		1
7.3	odd	6		126.2.g.c.109.1		2
7.4	even	3	inner	882.2.g.i.361.1		2
7.5	odd	6		882.2.a.c.1.1		1
7.6	odd	2		126.2.g.c.37.1		2
12.11	even	2		2352.2.q.u.961.1		2
21.2	odd	6		294.2.a.f.1.1		1
21.5	even	6		294.2.a.e.1.1		1
21.11	odd	6		294.2.e.b.67.1		2
21.17	even	6		42.2.e.a.25.1	✓	2
21.20	even	2		42.2.e.a.37.1	yes	2
28.3	even	6		1008.2.s.k.865.1		2
28.19	even	6		7056.2.a.w.1.1		1
28.23	odd	6		7056.2.a.bl.1.1		1
28.27	even	2		1008.2.s.k.289.1		2
63.13	odd	6		1134.2.h.l.541.1		2
63.20	even	6		1134.2.e.l.919.1		2
63.31	odd	6		1134.2.e.e.865.1		2
63.34	odd	6		1134.2.e.e.919.1		2
63.38	even	6		1134.2.h.e.109.1		2
63.41	even	6		1134.2.h.e.541.1		2
63.52	odd	6		1134.2.h.l.109.1		2
63.59	even	6		1134.2.e.l.865.1		2
84.11	even	6		2352.2.q.u.1537.1		2
84.23	even	6		2352.2.a.f.1.1		1
84.47	odd	6		2352.2.a.t.1.1		1
84.59	odd	6		336.2.q.b.193.1		2
84.83	odd	2		336.2.q.b.289.1		2
105.17	odd	12		1050.2.o.a.949.2		4
105.38	odd	12		1050.2.o.a.949.1		4
105.44	odd	6		7350.2.a.q.1.1		1
105.59	even	6		1050.2.i.l.151.1		2
105.62	odd	4		1050.2.o.a.499.1		4
105.83	odd	4		1050.2.o.a.499.2		4
105.89	even	6		7350.2.a.bl.1.1		1
105.104	even	2		1050.2.i.l.751.1		2
168.5	even	6		9408.2.a.ce.1.1		1
168.59	odd	6		1344.2.q.s.193.1		2
168.83	odd	2		1344.2.q.s.961.1		2
168.101	even	6		1344.2.q.g.193.1		2
168.107	even	6		9408.2.a.cr.1.1		1
168.125	even	2		1344.2.q.g.961.1		2
168.131	odd	6		9408.2.a.q.1.1		1
168.149	odd	6		9408.2.a.z.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.2.e.a.25.1	✓	2	21.17	even	6
42.2.e.a.37.1	yes	2	21.20	even	2
126.2.g.c.37.1		2	7.6	odd	2
126.2.g.c.109.1		2	7.3	odd	6
294.2.a.e.1.1		1	21.5	even	6
294.2.a.f.1.1		1	21.2	odd	6
294.2.e.b.67.1		2	21.11	odd	6
294.2.e.b.79.1		2	3.2	odd	2
336.2.q.b.193.1		2	84.59	odd	6
336.2.q.b.289.1		2	84.83	odd	2
882.2.a.c.1.1		1	7.5	odd	6
882.2.a.d.1.1		1	7.2	even	3
882.2.g.i.361.1		2	7.4	even	3	inner
882.2.g.i.667.1		2	1.1	even	1	trivial
1008.2.s.k.289.1		2	28.27	even	2
1008.2.s.k.865.1		2	28.3	even	6
1050.2.i.l.151.1		2	105.59	even	6
1050.2.i.l.751.1		2	105.104	even	2
1050.2.o.a.499.1		4	105.62	odd	4
1050.2.o.a.499.2		4	105.83	odd	4
1050.2.o.a.949.1		4	105.38	odd	12
1050.2.o.a.949.2		4	105.17	odd	12
1134.2.e.e.865.1		2	63.31	odd	6
1134.2.e.e.919.1		2	63.34	odd	6
1134.2.e.l.865.1		2	63.59	even	6
1134.2.e.l.919.1		2	63.20	even	6
1134.2.h.e.109.1		2	63.38	even	6
1134.2.h.e.541.1		2	63.41	even	6
1134.2.h.l.109.1		2	63.52	odd	6
1134.2.h.l.541.1		2	63.13	odd	6
1344.2.q.g.193.1		2	168.101	even	6
1344.2.q.g.961.1		2	168.125	even	2
1344.2.q.s.193.1		2	168.59	odd	6
1344.2.q.s.961.1		2	168.83	odd	2
2352.2.a.f.1.1		1	84.23	even	6
2352.2.a.t.1.1		1	84.47	odd	6
2352.2.q.u.961.1		2	12.11	even	2
2352.2.q.u.1537.1		2	84.11	even	6
7056.2.a.w.1.1		1	28.19	even	6
7056.2.a.bl.1.1		1	28.23	odd	6
7350.2.a.q.1.1		1	105.44	odd	6
7350.2.a.bl.1.1		1	105.89	even	6
9408.2.a.q.1.1		1	168.131	odd	6
9408.2.a.z.1.1		1	168.149	odd	6
9408.2.a.ce.1.1		1	168.5	even	6
9408.2.a.cr.1.1		1	168.107	even	6