Embedded newform 667.2.s.a.16.12

• Label: 667.2.s.a.16.12
• 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			16.12
Character	\(\chi\)	\(=\)	667.16
Dual form			667.2.s.a.542.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.66649 + 0.720349i) q^{2} +(-0.320321 + 3.12952i) q^{3} +(0.888013 - 0.944093i) q^{4} +(0.0140387 - 0.688078i) q^{5} +(-1.72053 - 5.44606i) q^{6} +(0.237956 - 0.582866i) q^{7} +(0.434358 - 1.20186i) q^{8} +(-6.75347 - 1.39714i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3480 q - 49 q^{2} - 45 q^{3} + 7 q^{4} - 49 q^{5} - 39 q^{6} - 49 q^{7} - 50 q^{8} + q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.66649	+	0.720349i	−1.17839	+	0.509364i	−0.893441	−	0.449180i	\(-0.851716\pi\)
							−0.284947	+	0.958543i	\(0.591976\pi\)
\(3\)	−0.320321	+	3.12952i	−0.184938	+	1.80683i	0.324036	+	0.946045i	\(0.394960\pi\)
							−0.508974	+	0.860782i	\(0.669975\pi\)
\(5\)	0.0140387	−	0.688078i	0.00627830	−	0.307718i	−0.984743	−	0.174014i	\(-0.944326\pi\)
							0.991021	−	0.133703i	\(-0.0426869\pi\)
\(7\)	0.237956	−	0.582866i	0.0899389	−	0.220303i	−0.874274	−	0.485433i	\(-0.838662\pi\)
							0.964213	+	0.265131i	\(0.0854151\pi\)
\(11\)	3.55804	−	1.89599i	1.07279	−	0.571661i	0.162956	−	0.986633i	\(-0.447897\pi\)
							0.909834	+	0.414972i	\(0.136209\pi\)
\(13\)	1.37858	−	2.46457i	0.382348	−	0.683549i	−0.611866	−	0.790961i	\(-0.709581\pi\)
							0.994215	+	0.107412i	\(0.0342563\pi\)
\(17\)	0.0163344	−	0.113608i	0.00396167	−	0.0275540i	−0.987745	−	0.156078i	\(-0.950115\pi\)
							0.991706	+	0.128524i	\(0.0410239\pi\)
\(19\)	3.23798	−	3.44246i	0.742843	−	0.789755i	−0.240667	−	0.970608i	\(-0.577366\pi\)
							0.983510	+	0.180853i	\(0.0578857\pi\)
\(23\)	4.62128	−	1.28209i	0.963604	−	0.267334i
							\(29\)	−3.69191	−	3.92044i	−0.685570	−	0.728007i
\(31\)	−2.76666	−	4.50447i	−0.496908	−	0.809026i								0.501516	−	0.865148i	\(-0.332776\pi\)
							−0.998424	+	0.0561221i	\(0.982126\pi\)
\(37\)	−2.31045	−	0.477979i	−0.379835	−	0.0785792i	0.00882366	−	0.999961i	\(-0.497191\pi\)
							−0.388659	+	0.921382i	\(0.627061\pi\)
\(41\)	−2.88004	−	6.30642i	−0.449787	−	0.984897i	−0.989697	−	0.143174i	\(-0.954269\pi\)
							0.539910	−	0.841723i	\(-0.318458\pi\)
\(43\)	−1.60893	+	2.61954i	−0.245360	+	0.399476i	−0.951046	−	0.309048i	\(-0.899989\pi\)
							0.705686	+	0.708524i	\(0.250639\pi\)
\(47\)	1.51853	−	6.65310i	0.221500	−	0.970454i	−0.734850	−	0.678230i	\(-0.762748\pi\)
							0.956350	−	0.292224i	\(-0.0943953\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	667.2.s.a.16.12	✓	3480
23.13	even	11	inner	667.2.s.a.335.12	yes	3480
29.20	even	7	inner	667.2.s.a.223.12	yes	3480
667.542	even	77	inner	667.2.s.a.542.12	yes	3480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.s.a.16.12	✓	3480	1.1	even	1	trivial
667.2.s.a.223.12	yes	3480	29.20	even	7	inner
667.2.s.a.335.12	yes	3480	23.13	even	11	inner
667.2.s.a.542.12	yes	3480	667.542	even	77	inner