Embedded newform 882.2.y.b.457.4

• Label: 882.2.y.b.457.4
• Level: $882$
• Weight: $2$
• Character: 882.457
• Analytic conductor: $7.043$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	882.y (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.04280545828\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(28\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			457.4
Character	\(\chi\)	\(=\)	882.457
Dual form			882.2.y.b.193.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.988831 - 0.149042i) q^{2} +(-1.34126 - 1.09591i) q^{3} +(0.955573 + 0.294755i) q^{4} +(-3.04871 - 1.46818i) q^{5} +(1.16294 + 1.28358i) q^{6} +(2.45568 + 0.984711i) q^{7} +(-0.900969 - 0.433884i) q^{8} +(0.597949 + 2.93981i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 28 q^{2} + 13 q^{3} + 28 q^{4} - 4 q^{5} - 5 q^{6} + 3 q^{7} - 56 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(785\)
\(\chi(n)\)	\(e\left(\frac{10}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.988831	−	0.149042i	−0.699209	−	0.105389i
							\(3\)	−1.34126	−	1.09591i	−0.774376	−	0.632726i
\(5\)	−3.04871	−	1.46818i	−1.36343	−	0.656591i								−0.398028	−	0.917373i	\(-0.630305\pi\)
							−0.965398	+	0.260782i	\(0.916020\pi\)
\(7\)	2.45568	+	0.984711i	0.928158	+	0.372186i
							\(11\)	3.56633	−	4.47204i	1.07529	−	1.34837i	0.141748	−	0.989903i	\(-0.454728\pi\)
0.933542	−	0.358468i	\(-0.116701\pi\)
\(13\)	−6.23350	−	0.939548i	−1.72886	−	0.260584i	−0.792059	−	0.610445i	\(-0.790991\pi\)
							−0.936802	+	0.349861i	\(0.886229\pi\)
\(17\)	−4.11992	+	1.27083i	−0.999226	+	0.308221i	−0.750871	−	0.660449i	\(-0.770366\pi\)
							−0.248355	+	0.968669i	\(0.579890\pi\)
\(19\)	3.59473	−	6.22625i	0.824688	−	1.42840i	−0.0774703	−	0.996995i	\(-0.524684\pi\)
							0.902158	−	0.431406i	\(-0.141982\pi\)
\(23\)	−0.593690	−	2.60113i	−0.123793	−	0.542372i	−0.998349	−	0.0574470i	\(-0.981704\pi\)
							0.874556	−	0.484925i	\(-0.161153\pi\)
\(29\)	−3.57762	−	3.31955i	−0.664348	−	0.616425i	0.274159	−	0.961684i	\(-0.411601\pi\)
							−0.938507	+	0.345259i	\(0.887791\pi\)
\(31\)	−2.50159	+	4.33289i	−0.449299	+	0.778209i	−0.998341	−	0.0575859i	\(-0.981660\pi\)
							0.549041	+	0.835795i	\(0.314993\pi\)
\(37\)	1.47081	+	1.36471i	0.241800	+	0.224357i	0.791758	−	0.610835i	\(-0.209166\pi\)
							−0.549959	+	0.835192i	\(0.685356\pi\)
\(41\)	0.162715	+	2.17128i	0.0254118	+	0.339097i	0.995390	+	0.0959066i	\(0.0305750\pi\)
							−0.969979	+	0.243191i	\(0.921806\pi\)
\(43\)	−0.805302	+	10.7460i	−0.122807	+	1.63875i	0.506961	+	0.861969i	\(0.330769\pi\)
							−0.629769	+	0.776783i	\(0.716850\pi\)
\(47\)	−0.805743	−	0.121446i	−0.117530	−	0.0177148i	0.0900140	−	0.995941i	\(-0.471309\pi\)
							−0.207544	+	0.978226i	\(0.566547\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.y.b.457.4	yes	336
9.4	even	3		882.2.bb.a.751.23	yes	336
49.46	even	21		882.2.bb.a.781.23	yes	336
441.193	even	21	inner	882.2.y.b.193.4	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.y.b.193.4	✓	336	441.193	even	21	inner
882.2.y.b.457.4	yes	336	1.1	even	1	trivial
882.2.bb.a.751.23	yes	336	9.4	even	3
882.2.bb.a.781.23	yes	336	49.46	even	21