Embedded newform 882.2.f.d.295.1

• Label: 882.2.f.d.295.1
• Level: $882$
• Weight: $2$
• Character: 882.295
• Analytic conductor: $7.043$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	882.f (of order \(3\), degree \(2\), 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 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			295.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	882.295
Dual form			882.2.f.d.589.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(1.50000 + 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(-1.50000 + 0.866025i) q^{6} +1.00000 q^{8} +(1.50000 + 2.59808i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 3 q^{3} - q^{4} - 3 q^{6} + 2 q^{8} + 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(785\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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
							\(3\)	1.50000	+	0.866025i	0.866025	+	0.500000i
\(5\)		0			0									0.866025	−	0.500000i	\(-0.166667\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)
\(7\)		0			0
							\(11\)	1.50000	−	2.59808i	0.452267	−	0.783349i	−0.546259	−	0.837616i	\(-0.683949\pi\)
0.998526	+	0.0542666i	\(0.0172821\pi\)
\(13\)	1.00000	+	1.73205i	0.277350	+	0.480384i	0.970725	−	0.240192i	\(-0.0772105\pi\)
							−0.693375	+	0.720577i	\(0.743877\pi\)
\(17\)	3.00000			0.727607			0.363803	−	0.931476i	\(-0.381478\pi\)
							0.363803	+	0.931476i	\(0.381478\pi\)
\(19\)	1.00000			0.229416			0.114708	−	0.993399i	\(-0.463407\pi\)
							0.114708	+	0.993399i	\(0.463407\pi\)
\(23\)	3.00000	+	5.19615i	0.625543	+	1.08347i	0.988436	+	0.151642i	\(0.0484560\pi\)
							−0.362892	+	0.931831i	\(0.618211\pi\)
\(29\)	−3.00000	+	5.19615i	−0.557086	+	0.964901i	0.440652	+	0.897678i	\(0.354747\pi\)
							−0.997738	+	0.0672232i	\(0.978586\pi\)
\(31\)	−2.00000	−	3.46410i	−0.359211	−	0.622171i	0.628619	−	0.777714i	\(-0.283621\pi\)
							−0.987829	+	0.155543i	\(0.950287\pi\)
\(37\)	−4.00000			−0.657596			−0.328798	−	0.944400i	\(-0.606644\pi\)
							−0.328798	+	0.944400i	\(0.606644\pi\)
\(41\)	4.50000	+	7.79423i	0.702782	+	1.21725i	0.967486	+	0.252924i	\(0.0813924\pi\)
							−0.264704	+	0.964330i	\(0.585274\pi\)
\(43\)	0.500000	−	0.866025i	0.0762493	−	0.132068i	−0.825380	−	0.564578i	\(-0.809039\pi\)
							0.901629	+	0.432511i	\(0.142372\pi\)
\(47\)	−3.00000	+	5.19615i	−0.437595	+	0.757937i	−0.997503	−	0.0706177i	\(-0.977503\pi\)
							0.559908	+	0.828554i	\(0.310836\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.f.d.295.1		2
3.2	odd	2		2646.2.f.g.883.1		2
7.2	even	3		882.2.e.g.655.1		2
7.3	odd	6		882.2.h.c.79.1		2
7.4	even	3		882.2.h.b.79.1		2
7.5	odd	6		882.2.e.i.655.1		2
7.6	odd	2		18.2.c.a.7.1	✓	2
9.2	odd	6		7938.2.a.i.1.1		1
9.4	even	3	inner	882.2.f.d.589.1		2
9.5	odd	6		2646.2.f.g.1765.1		2
9.7	even	3		7938.2.a.x.1.1		1
21.2	odd	6		2646.2.e.c.2125.1		2
21.5	even	6		2646.2.e.b.2125.1		2
21.11	odd	6		2646.2.h.i.667.1		2
21.17	even	6		2646.2.h.h.667.1		2
21.20	even	2		54.2.c.a.19.1		2
28.27	even	2		144.2.i.c.97.1		2
35.13	even	4		450.2.j.e.349.2		4
35.27	even	4		450.2.j.e.349.1		4
35.34	odd	2		450.2.e.i.151.1		2
56.13	odd	2		576.2.i.g.385.1		2
56.27	even	2		576.2.i.a.385.1		2
63.4	even	3		882.2.e.g.373.1		2
63.5	even	6		2646.2.h.h.361.1		2
63.13	odd	6		18.2.c.a.13.1	yes	2
63.20	even	6		162.2.a.b.1.1		1
63.23	odd	6		2646.2.h.i.361.1		2
63.31	odd	6		882.2.e.i.373.1		2
63.32	odd	6		2646.2.e.c.1549.1		2
63.34	odd	6		162.2.a.c.1.1		1
63.40	odd	6		882.2.h.c.67.1		2
63.41	even	6		54.2.c.a.37.1		2
63.58	even	3		882.2.h.b.67.1		2
63.59	even	6		2646.2.e.b.1549.1		2
84.83	odd	2		432.2.i.b.289.1		2
105.62	odd	4		1350.2.j.a.1099.2		4
105.83	odd	4		1350.2.j.a.1099.1		4
105.104	even	2		1350.2.e.c.451.1		2
168.83	odd	2		1728.2.i.f.1153.1		2
168.125	even	2		1728.2.i.e.1153.1		2
252.83	odd	6		1296.2.a.f.1.1		1
252.139	even	6		144.2.i.c.49.1		2
252.167	odd	6		432.2.i.b.145.1		2
252.223	even	6		1296.2.a.g.1.1		1
315.13	even	12		450.2.j.e.49.1		4
315.34	odd	6		4050.2.a.c.1.1		1
315.83	odd	12		4050.2.c.r.649.2		2
315.97	even	12		4050.2.c.c.649.2		2
315.104	even	6		1350.2.e.c.901.1		2
315.139	odd	6		450.2.e.i.301.1		2
315.167	odd	12		1350.2.j.a.199.1		4
315.202	even	12		450.2.j.e.49.2		4
315.209	even	6		4050.2.a.v.1.1		1
315.223	even	12		4050.2.c.c.649.1		2
315.272	odd	12		4050.2.c.r.649.1		2
315.293	odd	12		1350.2.j.a.199.2		4
504.13	odd	6		576.2.i.g.193.1		2
504.83	odd	6		5184.2.a.p.1.1		1
504.139	even	6		576.2.i.a.193.1		2
504.293	even	6		1728.2.i.e.577.1		2
504.349	odd	6		5184.2.a.r.1.1		1
504.419	odd	6		1728.2.i.f.577.1		2
504.461	even	6		5184.2.a.q.1.1		1
504.475	even	6		5184.2.a.o.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.2.c.a.7.1	✓	2	7.6	odd	2
18.2.c.a.13.1	yes	2	63.13	odd	6
54.2.c.a.19.1		2	21.20	even	2
54.2.c.a.37.1		2	63.41	even	6
144.2.i.c.49.1		2	252.139	even	6
144.2.i.c.97.1		2	28.27	even	2
162.2.a.b.1.1		1	63.20	even	6
162.2.a.c.1.1		1	63.34	odd	6
432.2.i.b.145.1		2	252.167	odd	6
432.2.i.b.289.1		2	84.83	odd	2
450.2.e.i.151.1		2	35.34	odd	2
450.2.e.i.301.1		2	315.139	odd	6
450.2.j.e.49.1		4	315.13	even	12
450.2.j.e.49.2		4	315.202	even	12
450.2.j.e.349.1		4	35.27	even	4
450.2.j.e.349.2		4	35.13	even	4
576.2.i.a.193.1		2	504.139	even	6
576.2.i.a.385.1		2	56.27	even	2
576.2.i.g.193.1		2	504.13	odd	6
576.2.i.g.385.1		2	56.13	odd	2
882.2.e.g.373.1		2	63.4	even	3
882.2.e.g.655.1		2	7.2	even	3
882.2.e.i.373.1		2	63.31	odd	6
882.2.e.i.655.1		2	7.5	odd	6
882.2.f.d.295.1		2	1.1	even	1	trivial
882.2.f.d.589.1		2	9.4	even	3	inner
882.2.h.b.67.1		2	63.58	even	3
882.2.h.b.79.1		2	7.4	even	3
882.2.h.c.67.1		2	63.40	odd	6
882.2.h.c.79.1		2	7.3	odd	6
1296.2.a.f.1.1		1	252.83	odd	6
1296.2.a.g.1.1		1	252.223	even	6
1350.2.e.c.451.1		2	105.104	even	2
1350.2.e.c.901.1		2	315.104	even	6
1350.2.j.a.199.1		4	315.167	odd	12
1350.2.j.a.199.2		4	315.293	odd	12
1350.2.j.a.1099.1		4	105.83	odd	4
1350.2.j.a.1099.2		4	105.62	odd	4
1728.2.i.e.577.1		2	504.293	even	6
1728.2.i.e.1153.1		2	168.125	even	2
1728.2.i.f.577.1		2	504.419	odd	6
1728.2.i.f.1153.1		2	168.83	odd	2
2646.2.e.b.1549.1		2	63.59	even	6
2646.2.e.b.2125.1		2	21.5	even	6
2646.2.e.c.1549.1		2	63.32	odd	6
2646.2.e.c.2125.1		2	21.2	odd	6
2646.2.f.g.883.1		2	3.2	odd	2
2646.2.f.g.1765.1		2	9.5	odd	6
2646.2.h.h.361.1		2	63.5	even	6
2646.2.h.h.667.1		2	21.17	even	6
2646.2.h.i.361.1		2	63.23	odd	6
2646.2.h.i.667.1		2	21.11	odd	6
4050.2.a.c.1.1		1	315.34	odd	6
4050.2.a.v.1.1		1	315.209	even	6
4050.2.c.c.649.1		2	315.223	even	12
4050.2.c.c.649.2		2	315.97	even	12
4050.2.c.r.649.1		2	315.272	odd	12
4050.2.c.r.649.2		2	315.83	odd	12
5184.2.a.o.1.1		1	504.475	even	6
5184.2.a.p.1.1		1	504.83	odd	6
5184.2.a.q.1.1		1	504.461	even	6
5184.2.a.r.1.1		1	504.349	odd	6
7938.2.a.i.1.1		1	9.2	odd	6
7938.2.a.x.1.1		1	9.7	even	3