Embedded newform 668.2.h.a.15.13

• Label: 668.2.h.a.15.13
• 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			15.13
Character	\(\chi\)	\(=\)	668.15
Dual form			668.2.h.a.579.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22179 + 0.712204i) q^{2} +(-1.40535 - 1.87397i) q^{3} +(0.985531 - 1.74032i) q^{4} +(-0.352669 + 1.52611i) q^{5} +(3.05169 + 1.28870i) q^{6} +(3.78061 + 0.358817i) q^{7} +(0.0353553 + 2.82821i) q^{8} +(-0.696501 + 2.38725i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6724 q - 81 q^{2} - 85 q^{4} - 166 q^{5} - 75 q^{6} - 75 q^{8} - 84 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.22179	+	0.712204i	−0.863935	+	0.503604i
							\(3\)	−1.40535	−	1.87397i	−0.811381	−	1.08194i	−0.995092	−	0.0989567i	\(-0.968449\pi\)
0.183711	−	0.982980i	\(-0.441189\pi\)
\(5\)	−0.352669	+	1.52611i	−0.157718	+	0.682498i	0.833749	+	0.552144i	\(0.186190\pi\)
							−0.991467	+	0.130355i	\(0.958388\pi\)
\(7\)	3.78061	+	0.358817i	1.42894	+	0.135620i	0.780265	−	0.625449i	\(-0.215084\pi\)
							0.648671	+	0.761069i	\(0.275325\pi\)
\(11\)	−0.238072	+	3.13889i	−0.0717814	+	0.946410i	0.841809	+	0.539775i	\(0.181491\pi\)
							−0.913591	+	0.406635i	\(0.866702\pi\)
\(13\)	−5.72603	+	0.984841i	−1.58811	+	0.273146i	−0.894455	−	0.447157i	\(-0.852436\pi\)
							−0.693659	+	0.720303i	\(0.744003\pi\)
\(17\)	−2.41475	−	4.12219i	−0.585663	−	0.999778i	−0.996129	−	0.0879027i	\(-0.971984\pi\)
							0.410467	−	0.911876i	\(-0.365366\pi\)
\(19\)	−0.771638	+	1.11444i	−0.177026	+	0.255669i	−0.900670	−	0.434504i	\(-0.856924\pi\)
							0.723644	+	0.690173i	\(0.242466\pi\)
\(23\)	6.66663	+	3.57353i	1.39009	+	0.745133i	0.985550	−	0.169387i	\(-0.0541789\pi\)
							0.404539	+	0.914521i	\(0.367432\pi\)
\(29\)	−6.72453	+	3.60457i	−1.24871	+	0.669351i	−0.957810	−	0.287401i	\(-0.907209\pi\)
							−0.290903	+	0.956753i	\(0.593956\pi\)
\(31\)	−4.76032	−	0.0901009i	−0.854978	−	0.0161826i	−0.410382	−	0.911914i	\(-0.634605\pi\)
							−0.444596	+	0.895731i	\(0.646653\pi\)
\(37\)	−1.97265	+	0.575538i	−0.324302	+	0.0946179i	−0.438208	−	0.898874i	\(-0.644387\pi\)
							0.113906	+	0.993492i	\(0.463664\pi\)
\(41\)	−0.340976	+	9.00418i	−0.0532514	+	1.40622i	0.683450	+	0.729998i	\(0.260479\pi\)
							−0.736701	+	0.676219i	\(0.763617\pi\)
\(43\)	−3.12739	−	4.00983i	−0.476923	−	0.611493i	0.488378	−	0.872632i	\(-0.337589\pi\)
							−0.965301	+	0.261139i	\(0.915902\pi\)
\(47\)	−6.08809	−	0.811283i	−0.888040	−	0.118338i	−0.325924	−	0.945396i	\(-0.605675\pi\)
							−0.562116	+	0.827058i	\(0.690013\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	668.2.h.a.15.13	yes	6724
4.3	odd	2	inner	668.2.h.a.15.5	✓	6724
167.78	odd	166	inner	668.2.h.a.579.5	yes	6724
668.579	even	166	inner	668.2.h.a.579.13	yes	6724

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.h.a.15.5	✓	6724	4.3	odd	2	inner
668.2.h.a.15.13	yes	6724	1.1	even	1	trivial
668.2.h.a.579.5	yes	6724	167.78	odd	166	inner
668.2.h.a.579.13	yes	6724	668.579	even	166	inner