Embedded newform 882.2.g.c.667.1

• Label: 882.2.g.c.667.1
• Level: $882$
• Weight: $2$
• Character: 882.667
• 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.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 14)
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.c.361.1

$q$-expansion

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

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			0									−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(7\)		0			0
							\(11\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
−0.866025	+	0.500000i	\(0.833333\pi\)
\(13\)	−4.00000			−1.10940			−0.554700	−	0.832050i	\(-0.687167\pi\)
							−0.554700	+	0.832050i	\(0.687167\pi\)
\(17\)	3.00000	+	5.19615i	0.727607	+	1.26025i	0.957892	+	0.287129i	\(0.0927008\pi\)
							−0.230285	+	0.973123i	\(0.573966\pi\)
\(19\)	−1.00000	+	1.73205i	−0.229416	+	0.397360i	−0.957635	−	0.287984i	\(-0.907015\pi\)
							0.728219	+	0.685344i	\(0.240348\pi\)
\(23\)		0			0		−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(29\)	6.00000			1.11417			0.557086	−	0.830455i	\(-0.311919\pi\)
							0.557086	+	0.830455i	\(0.311919\pi\)
\(31\)	2.00000	+	3.46410i	0.359211	+	0.622171i	0.987829	−	0.155543i	\(-0.0497126\pi\)
							−0.628619	+	0.777714i	\(0.716379\pi\)
\(37\)	−1.00000	+	1.73205i	−0.164399	+	0.284747i	−0.936442	−	0.350823i	\(-0.885902\pi\)
							0.772043	+	0.635571i	\(0.219235\pi\)
\(41\)	−6.00000			−0.937043			−0.468521	−	0.883452i	\(-0.655213\pi\)
							−0.468521	+	0.883452i	\(0.655213\pi\)
\(43\)	8.00000			1.21999			0.609994	−	0.792406i	\(-0.291172\pi\)
							0.609994	+	0.792406i	\(0.291172\pi\)
\(47\)	−6.00000	+	10.3923i	−0.875190	+	1.51587i	−0.0186297	+	0.999826i	\(0.505930\pi\)
							−0.856560	+	0.516047i	\(0.827403\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.g.c.667.1		2
3.2	odd	2		98.2.c.b.79.1		2
7.2	even	3		126.2.a.b.1.1		1
7.3	odd	6		882.2.g.d.361.1		2
7.4	even	3	inner	882.2.g.c.361.1		2
7.5	odd	6		882.2.a.i.1.1		1
7.6	odd	2		882.2.g.d.667.1		2
12.11	even	2		784.2.i.c.177.1		2
21.2	odd	6		14.2.a.a.1.1	✓	1
21.5	even	6		98.2.a.a.1.1		1
21.11	odd	6		98.2.c.b.67.1		2
21.17	even	6		98.2.c.a.67.1		2
21.20	even	2		98.2.c.a.79.1		2
28.19	even	6		7056.2.a.bd.1.1		1
28.23	odd	6		1008.2.a.h.1.1		1
35.2	odd	12		3150.2.g.j.2899.2		2
35.9	even	6		3150.2.a.i.1.1		1
35.23	odd	12		3150.2.g.j.2899.1		2
56.37	even	6		4032.2.a.w.1.1		1
56.51	odd	6		4032.2.a.r.1.1		1
63.2	odd	6		1134.2.f.l.757.1		2
63.16	even	3		1134.2.f.f.757.1		2
63.23	odd	6		1134.2.f.l.379.1		2
63.58	even	3		1134.2.f.f.379.1		2
84.11	even	6		784.2.i.c.753.1		2
84.23	even	6		112.2.a.c.1.1		1
84.47	odd	6		784.2.a.b.1.1		1
84.59	odd	6		784.2.i.i.753.1		2
84.83	odd	2		784.2.i.i.177.1		2
105.2	even	12		350.2.c.d.99.1		2
105.23	even	12		350.2.c.d.99.2		2
105.44	odd	6		350.2.a.f.1.1		1
105.47	odd	12		2450.2.c.c.99.1		2
105.68	odd	12		2450.2.c.c.99.2		2
105.89	even	6		2450.2.a.t.1.1		1
168.5	even	6		3136.2.a.e.1.1		1
168.107	even	6		448.2.a.a.1.1		1
168.131	odd	6		3136.2.a.z.1.1		1
168.149	odd	6		448.2.a.g.1.1		1
231.65	even	6		1694.2.a.e.1.1		1
273.44	even	12		2366.2.d.b.337.2		2
273.86	even	12		2366.2.d.b.337.1		2
273.233	odd	6		2366.2.a.j.1.1		1
336.107	even	12		1792.2.b.g.897.1		2
336.149	odd	12		1792.2.b.c.897.2		2
336.275	even	12		1792.2.b.g.897.2		2
336.317	odd	12		1792.2.b.c.897.1		2
357.254	odd	6		4046.2.a.f.1.1		1
399.170	even	6		5054.2.a.c.1.1		1
420.23	odd	12		2800.2.g.h.449.2		2
420.107	odd	12		2800.2.g.h.449.1		2
420.359	even	6		2800.2.a.g.1.1		1
483.275	even	6		7406.2.a.a.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.2.a.a.1.1	✓	1	21.2	odd	6
98.2.a.a.1.1		1	21.5	even	6
98.2.c.a.67.1		2	21.17	even	6
98.2.c.a.79.1		2	21.20	even	2
98.2.c.b.67.1		2	21.11	odd	6
98.2.c.b.79.1		2	3.2	odd	2
112.2.a.c.1.1		1	84.23	even	6
126.2.a.b.1.1		1	7.2	even	3
350.2.a.f.1.1		1	105.44	odd	6
350.2.c.d.99.1		2	105.2	even	12
350.2.c.d.99.2		2	105.23	even	12
448.2.a.a.1.1		1	168.107	even	6
448.2.a.g.1.1		1	168.149	odd	6
784.2.a.b.1.1		1	84.47	odd	6
784.2.i.c.177.1		2	12.11	even	2
784.2.i.c.753.1		2	84.11	even	6
784.2.i.i.177.1		2	84.83	odd	2
784.2.i.i.753.1		2	84.59	odd	6
882.2.a.i.1.1		1	7.5	odd	6
882.2.g.c.361.1		2	7.4	even	3	inner
882.2.g.c.667.1		2	1.1	even	1	trivial
882.2.g.d.361.1		2	7.3	odd	6
882.2.g.d.667.1		2	7.6	odd	2
1008.2.a.h.1.1		1	28.23	odd	6
1134.2.f.f.379.1		2	63.58	even	3
1134.2.f.f.757.1		2	63.16	even	3
1134.2.f.l.379.1		2	63.23	odd	6
1134.2.f.l.757.1		2	63.2	odd	6
1694.2.a.e.1.1		1	231.65	even	6
1792.2.b.c.897.1		2	336.317	odd	12
1792.2.b.c.897.2		2	336.149	odd	12
1792.2.b.g.897.1		2	336.107	even	12
1792.2.b.g.897.2		2	336.275	even	12
2366.2.a.j.1.1		1	273.233	odd	6
2366.2.d.b.337.1		2	273.86	even	12
2366.2.d.b.337.2		2	273.44	even	12
2450.2.a.t.1.1		1	105.89	even	6
2450.2.c.c.99.1		2	105.47	odd	12
2450.2.c.c.99.2		2	105.68	odd	12
2800.2.a.g.1.1		1	420.359	even	6
2800.2.g.h.449.1		2	420.107	odd	12
2800.2.g.h.449.2		2	420.23	odd	12
3136.2.a.e.1.1		1	168.5	even	6
3136.2.a.z.1.1		1	168.131	odd	6
3150.2.a.i.1.1		1	35.9	even	6
3150.2.g.j.2899.1		2	35.23	odd	12
3150.2.g.j.2899.2		2	35.2	odd	12
4032.2.a.r.1.1		1	56.51	odd	6
4032.2.a.w.1.1		1	56.37	even	6
4046.2.a.f.1.1		1	357.254	odd	6
5054.2.a.c.1.1		1	399.170	even	6
7056.2.a.bd.1.1		1	28.19	even	6
7406.2.a.a.1.1		1	483.275	even	6