Embedded newform 668.2.h.a.15.3

• Label: 668.2.h.a.15.3
• Level: $668$
• Weight: $2$
• Character: 668.15
• Analytic conductor: $5.334$
• Analytic rank: $0$
• Dimension: $6724$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 668 = 2^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	668.h (of order \(166\), degree \(82\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.33400685502\)
Analytic rank:	\(0\)
Dimension:	\(6724\)
Relative dimension:	\(82\) over \(\Q(\zeta_{166})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{166}]$

Embedding invariants

Embedding label			15.3
Character	\(\chi\)	\(=\)	668.15
Dual form			668.2.h.a.579.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40266 - 0.180378i) q^{2} +(-1.32817 - 1.77105i) q^{3} +(1.93493 + 0.506019i) q^{4} +(0.679950 - 2.94236i) q^{5} +(1.54351 + 2.72375i) q^{6} +(-1.65201 - 0.156792i) q^{7} +(-2.62278 - 1.05879i) q^{8} +(-0.532332 + 1.82456i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6724 q - 81 q^{2} - 85 q^{4} - 166 q^{5} - 75 q^{6} - 75 q^{8} - 84 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/668\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(335\)
\(\chi(n)\)	\(e\left(\frac{95}{166}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40266	−	0.180378i	−0.991833	−	0.127546i
							\(3\)	−1.32817	−	1.77105i	−0.766818	−	1.02251i	−0.998760	−	0.0497844i	\(-0.984147\pi\)
0.231942	−	0.972730i	\(-0.425492\pi\)
\(5\)	0.679950	−	2.94236i	0.304083	−	1.31586i	−0.566589	−	0.824000i	\(-0.691737\pi\)
							0.870672	−	0.491864i	\(-0.163684\pi\)
\(7\)	−1.65201	−	0.156792i	−0.624402	−	0.0592618i	−0.222482	−	0.974937i	\(-0.571416\pi\)
							−0.401920	+	0.915675i	\(0.631657\pi\)
\(11\)	−0.248906	+	3.28173i	−0.0750481	+	0.989480i	0.828220	+	0.560403i	\(0.189354\pi\)
							−0.903268	+	0.429077i	\(0.858839\pi\)
\(13\)	−3.48401	+	0.599227i	−0.966289	+	0.166196i	−0.630883	−	0.775878i	\(-0.717307\pi\)
							−0.335406	+	0.942074i	\(0.608874\pi\)
\(17\)	−0.994117	−	1.69705i	−0.241109	−	0.411594i	0.717398	−	0.696663i	\(-0.245333\pi\)
							−0.958507	+	0.285069i	\(0.907983\pi\)
\(19\)	−2.63437	+	3.80468i	−0.604366	+	0.872854i	−0.998941	−	0.0460094i	\(-0.985350\pi\)
							0.394575	+	0.918864i	\(0.370892\pi\)
\(23\)	−1.83752	−	0.984971i	−0.383149	−	0.205381i	0.269569	−	0.962981i	\(-0.413119\pi\)
							−0.652719	+	0.757600i	\(0.726372\pi\)
\(29\)	5.47363	−	2.93405i	1.01643	−	0.544839i	0.122226	−	0.992502i	\(-0.460997\pi\)
							0.894202	+	0.447663i	\(0.147744\pi\)
\(31\)	−1.29728	−	0.0245542i	−0.232998	−	0.00441006i	−0.0977035	−	0.995216i	\(-0.531150\pi\)
							−0.135294	+	0.990805i	\(0.543198\pi\)
\(37\)	2.32546	−	0.678473i	0.382303	−	0.111540i	−0.0833211	−	0.996523i	\(-0.526553\pi\)
							0.465624	+	0.884982i	\(0.345830\pi\)
\(41\)	−0.00730654	+	0.192945i	−0.00114109	+	0.0301329i	−0.999741	−	0.0227707i	\(-0.992751\pi\)
							0.998600	+	0.0529036i	\(0.0168476\pi\)
\(43\)	−2.54336	−	3.26100i	−0.387858	−	0.497297i	0.554370	−	0.832271i	\(-0.312959\pi\)
							−0.942228	+	0.334973i	\(0.891273\pi\)
\(47\)	5.31637	+	0.708445i	0.775472	+	0.103337i	0.509301	−	0.860588i	\(-0.329904\pi\)
							0.266171	+	0.963926i	\(0.414241\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	668.2.h.a.15.3	✓	6724
4.3	odd	2	inner	668.2.h.a.15.22	yes	6724
167.78	odd	166	inner	668.2.h.a.579.22	yes	6724
668.579	even	166	inner	668.2.h.a.579.3	yes	6724

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.h.a.15.3	✓	6724	1.1	even	1	trivial
668.2.h.a.15.22	yes	6724	4.3	odd	2	inner
668.2.h.a.579.3	yes	6724	668.579	even	166	inner
668.2.h.a.579.22	yes	6724	167.78	odd	166	inner