Embedded newform 667.2.s.a.16.8

• Label: 667.2.s.a.16.8
• Level: $667$
• Weight: $2$
• Character: 667.16
• Analytic conductor: $5.326$
• Analytic rank: $0$
• Dimension: $3480$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 667 = 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	667.s (of order \(77\), degree \(60\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.32602181482\)
Analytic rank:	\(0\)
Dimension:	\(3480\)
Relative dimension:	\(58\) over \(\Q(\zeta_{77})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{77}]$

Embedding invariants

Embedding label			16.8
Character	\(\chi\)	\(=\)	667.16
Dual form			667.2.s.a.542.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.01330 + 0.870257i) q^{2} +(0.169590 - 1.65688i) q^{3} +(1.92574 - 2.04736i) q^{4} +(-0.0877619 + 4.30147i) q^{5} +(1.10048 + 3.48338i) q^{6} +(0.176453 - 0.432216i) q^{7} +(-0.604387 + 1.67232i) q^{8} +(0.221308 + 0.0457837i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3480 q - 49 q^{2} - 45 q^{3} + 7 q^{4} - 49 q^{5} - 39 q^{6} - 49 q^{7} - 50 q^{8} + q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/667\mathbb{Z}\right)^\times\).

\(n\)	\(465\)	\(553\)
\(\chi(n)\)	\(e\left(\frac{4}{11}\right)\)	\(e\left(\frac{1}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.01330	+	0.870257i	−1.42362	+	0.615365i	−0.962330	−	0.271885i	\(-0.912353\pi\)
							−0.461288	+	0.887250i	\(0.652613\pi\)
\(3\)	0.169590	−	1.65688i	0.0979127	−	0.956599i	−0.823318	−	0.567581i	\(-0.807879\pi\)
							0.921230	−	0.389018i	\(-0.127186\pi\)
\(5\)	−0.0877619	+	4.30147i	−0.0392483	+	1.92367i	0.253238	+	0.967404i	\(0.418504\pi\)
							−0.292486	+	0.956270i	\(0.594483\pi\)
\(7\)	0.176453	−	0.432216i	0.0666928	−	0.163362i	−0.888863	−	0.458174i	\(-0.848504\pi\)
							0.955555	+	0.294812i	\(0.0952569\pi\)
\(11\)	4.82705	−	2.57220i	1.45541	−	0.775548i	0.461646	−	0.887064i	\(-0.347259\pi\)
							0.993763	+	0.111516i	\(0.0355706\pi\)
\(13\)	−1.64171	+	2.93499i	−0.455328	+	0.814020i	−0.999692	−	0.0248292i	\(-0.992096\pi\)
							0.544364	+	0.838849i	\(0.316771\pi\)
\(17\)	0.768863	−	5.34756i	0.186477	−	1.29697i	−0.654566	−	0.756005i	\(-0.727149\pi\)
							0.841043	−	0.540969i	\(-0.181942\pi\)
\(19\)	−0.671759	+	0.714182i	−0.154112	+	0.163845i	−0.800844	−	0.598873i	\(-0.795616\pi\)
							0.646732	+	0.762717i	\(0.276135\pi\)
\(23\)	3.63191	+	3.13197i	0.757306	+	0.653060i
							\(29\)	−5.15075	+	1.57156i	−0.956470	+	0.291831i
\(31\)	1.96229	+	3.19485i	0.352438	+	0.573812i								0.978572	−	0.205903i	\(-0.0660133\pi\)
							−0.626134	+	0.779715i	\(0.715364\pi\)
\(37\)	5.83640	+	1.20742i	0.959498	+	0.198498i	0.656360	−	0.754448i	\(-0.272095\pi\)
							0.303138	+	0.952947i	\(0.401966\pi\)
\(41\)	1.77179	+	3.87969i	0.276708	+	0.605905i	0.996054	−	0.0887462i	\(-0.0282860\pi\)
							−0.719347	+	0.694651i	\(0.755559\pi\)
\(43\)	−1.12829	+	1.83699i	−0.172062	+	0.280139i	−0.926547	−	0.376179i	\(-0.877238\pi\)
							0.754484	+	0.656318i	\(0.227887\pi\)
\(47\)	1.00269	−	4.39308i	0.146258	−	0.640796i	−0.847648	−	0.530559i	\(-0.821982\pi\)
							0.993905	−	0.110237i	\(-0.0351609\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	667.2.s.a.16.8	✓	3480
23.13	even	11	inner	667.2.s.a.335.8	yes	3480
29.20	even	7	inner	667.2.s.a.223.8	yes	3480
667.542	even	77	inner	667.2.s.a.542.8	yes	3480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.s.a.16.8	✓	3480	1.1	even	1	trivial
667.2.s.a.223.8	yes	3480	29.20	even	7	inner
667.2.s.a.335.8	yes	3480	23.13	even	11	inner
667.2.s.a.542.8	yes	3480	667.542	even	77	inner