Embedded newform 882.2.e.i.373.1

• Label: 882.2.e.i.373.1
• Level: $882$
• Weight: $2$
• Character: 882.373
• 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.e (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:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			373.1
Root			\(0.500000 - 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	882.373
Dual form			882.2.e.i.655.1

$q$-expansion

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

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.00000			0.707107
							\(3\)	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	\(-0.922790\pi\)
							0.693375	+	0.720577i	\(0.256123\pi\)
\(17\)	1.50000	+	2.59808i	0.363803	+	0.630126i	0.988583	−	0.150675i	\(-0.0481447\pi\)
							−0.624780	+	0.780801i	\(0.714811\pi\)
\(19\)	0.500000	−	0.866025i	0.114708	−	0.198680i	−0.802955	−	0.596040i	\(-0.796740\pi\)
							0.917663	+	0.397360i	\(0.130073\pi\)
\(23\)	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	−0.997738	−	0.0672232i	\(-0.978586\pi\)
							0.440652	−	0.897678i	\(-0.354747\pi\)
\(31\)	−4.00000			−0.718421			−0.359211	−	0.933257i	\(-0.616954\pi\)
							−0.359211	+	0.933257i	\(0.616954\pi\)
\(37\)	2.00000	−	3.46410i	0.328798	−	0.569495i	−0.653476	−	0.756948i	\(-0.726690\pi\)
							0.982274	+	0.187453i	\(0.0600231\pi\)
\(41\)	−4.50000	+	7.79423i	−0.702782	+	1.21725i	0.264704	+	0.964330i	\(0.414726\pi\)
							−0.967486	+	0.252924i	\(0.918608\pi\)
\(43\)	0.500000	+	0.866025i	0.0762493	+	0.132068i	0.901629	−	0.432511i	\(-0.142372\pi\)
							−0.825380	+	0.564578i	\(0.809039\pi\)
\(47\)	−6.00000			−0.875190			−0.437595	−	0.899172i	\(-0.644170\pi\)
							−0.437595	+	0.899172i	\(0.644170\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.e.i.373.1		2
3.2	odd	2		2646.2.e.b.1549.1		2
7.2	even	3		18.2.c.a.13.1	yes	2
7.3	odd	6		882.2.h.b.67.1		2
7.4	even	3		882.2.h.c.67.1		2
7.5	odd	6		882.2.f.d.589.1		2
7.6	odd	2		882.2.e.g.373.1		2
9.2	odd	6		2646.2.h.h.667.1		2
9.7	even	3		882.2.h.c.79.1		2
21.2	odd	6		54.2.c.a.37.1		2
21.5	even	6		2646.2.f.g.1765.1		2
21.11	odd	6		2646.2.h.h.361.1		2
21.17	even	6		2646.2.h.i.361.1		2
21.20	even	2		2646.2.e.c.1549.1		2
28.23	odd	6		144.2.i.c.49.1		2
35.2	odd	12		450.2.j.e.49.2		4
35.9	even	6		450.2.e.i.301.1		2
35.23	odd	12		450.2.j.e.49.1		4
56.37	even	6		576.2.i.g.193.1		2
56.51	odd	6		576.2.i.a.193.1		2
63.2	odd	6		54.2.c.a.19.1		2
63.5	even	6		7938.2.a.i.1.1		1
63.11	odd	6		2646.2.e.b.2125.1		2
63.16	even	3		18.2.c.a.7.1	✓	2
63.20	even	6		2646.2.h.i.667.1		2
63.23	odd	6		162.2.a.b.1.1		1
63.25	even	3	inner	882.2.e.i.655.1		2
63.34	odd	6		882.2.h.b.79.1		2
63.38	even	6		2646.2.e.c.2125.1		2
63.40	odd	6		7938.2.a.x.1.1		1
63.47	even	6		2646.2.f.g.883.1		2
63.52	odd	6		882.2.e.g.655.1		2
63.58	even	3		162.2.a.c.1.1		1
63.61	odd	6		882.2.f.d.295.1		2
84.23	even	6		432.2.i.b.145.1		2
105.2	even	12		1350.2.j.a.199.1		4
105.23	even	12		1350.2.j.a.199.2		4
105.44	odd	6		1350.2.e.c.901.1		2
168.107	even	6		1728.2.i.f.577.1		2
168.149	odd	6		1728.2.i.e.577.1		2
252.23	even	6		1296.2.a.f.1.1		1
252.79	odd	6		144.2.i.c.97.1		2
252.191	even	6		432.2.i.b.289.1		2
252.247	odd	6		1296.2.a.g.1.1		1
315.2	even	12		1350.2.j.a.1099.2		4
315.23	even	12		4050.2.c.r.649.2		2
315.58	odd	12		4050.2.c.c.649.1		2
315.79	even	6		450.2.e.i.151.1		2
315.128	even	12		1350.2.j.a.1099.1		4
315.142	odd	12		450.2.j.e.349.1		4
315.149	odd	6		4050.2.a.v.1.1		1
315.184	even	6		4050.2.a.c.1.1		1
315.212	even	12		4050.2.c.r.649.1		2
315.247	odd	12		4050.2.c.c.649.2		2
315.254	odd	6		1350.2.e.c.451.1		2
315.268	odd	12		450.2.j.e.349.2		4
504.149	odd	6		5184.2.a.q.1.1		1
504.205	even	6		576.2.i.g.385.1		2
504.275	even	6		5184.2.a.p.1.1		1
504.317	odd	6		1728.2.i.e.1153.1		2
504.331	odd	6		576.2.i.a.385.1		2
504.373	even	6		5184.2.a.r.1.1		1
504.443	even	6		1728.2.i.f.1153.1		2
504.499	odd	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	63.16	even	3
18.2.c.a.13.1	yes	2	7.2	even	3
54.2.c.a.19.1		2	63.2	odd	6
54.2.c.a.37.1		2	21.2	odd	6
144.2.i.c.49.1		2	28.23	odd	6
144.2.i.c.97.1		2	252.79	odd	6
162.2.a.b.1.1		1	63.23	odd	6
162.2.a.c.1.1		1	63.58	even	3
432.2.i.b.145.1		2	84.23	even	6
432.2.i.b.289.1		2	252.191	even	6
450.2.e.i.151.1		2	315.79	even	6
450.2.e.i.301.1		2	35.9	even	6
450.2.j.e.49.1		4	35.23	odd	12
450.2.j.e.49.2		4	35.2	odd	12
450.2.j.e.349.1		4	315.142	odd	12
450.2.j.e.349.2		4	315.268	odd	12
576.2.i.a.193.1		2	56.51	odd	6
576.2.i.a.385.1		2	504.331	odd	6
576.2.i.g.193.1		2	56.37	even	6
576.2.i.g.385.1		2	504.205	even	6
882.2.e.g.373.1		2	7.6	odd	2
882.2.e.g.655.1		2	63.52	odd	6
882.2.e.i.373.1		2	1.1	even	1	trivial
882.2.e.i.655.1		2	63.25	even	3	inner
882.2.f.d.295.1		2	63.61	odd	6
882.2.f.d.589.1		2	7.5	odd	6
882.2.h.b.67.1		2	7.3	odd	6
882.2.h.b.79.1		2	63.34	odd	6
882.2.h.c.67.1		2	7.4	even	3
882.2.h.c.79.1		2	9.7	even	3
1296.2.a.f.1.1		1	252.23	even	6
1296.2.a.g.1.1		1	252.247	odd	6
1350.2.e.c.451.1		2	315.254	odd	6
1350.2.e.c.901.1		2	105.44	odd	6
1350.2.j.a.199.1		4	105.2	even	12
1350.2.j.a.199.2		4	105.23	even	12
1350.2.j.a.1099.1		4	315.128	even	12
1350.2.j.a.1099.2		4	315.2	even	12
1728.2.i.e.577.1		2	168.149	odd	6
1728.2.i.e.1153.1		2	504.317	odd	6
1728.2.i.f.577.1		2	168.107	even	6
1728.2.i.f.1153.1		2	504.443	even	6
2646.2.e.b.1549.1		2	3.2	odd	2
2646.2.e.b.2125.1		2	63.11	odd	6
2646.2.e.c.1549.1		2	21.20	even	2
2646.2.e.c.2125.1		2	63.38	even	6
2646.2.f.g.883.1		2	63.47	even	6
2646.2.f.g.1765.1		2	21.5	even	6
2646.2.h.h.361.1		2	21.11	odd	6
2646.2.h.h.667.1		2	9.2	odd	6
2646.2.h.i.361.1		2	21.17	even	6
2646.2.h.i.667.1		2	63.20	even	6
4050.2.a.c.1.1		1	315.184	even	6
4050.2.a.v.1.1		1	315.149	odd	6
4050.2.c.c.649.1		2	315.58	odd	12
4050.2.c.c.649.2		2	315.247	odd	12
4050.2.c.r.649.1		2	315.212	even	12
4050.2.c.r.649.2		2	315.23	even	12
5184.2.a.o.1.1		1	504.499	odd	6
5184.2.a.p.1.1		1	504.275	even	6
5184.2.a.q.1.1		1	504.149	odd	6
5184.2.a.r.1.1		1	504.373	even	6
7938.2.a.i.1.1		1	63.5	even	6
7938.2.a.x.1.1		1	63.40	odd	6