Embedded newform 668.2.h.a.15.6

• Label: 668.2.h.a.15.6
• 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			15.6
Character	\(\chi\)	\(=\)	668.15
Dual form			668.2.h.a.579.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37664 - 0.323827i) q^{2} +(1.92126 + 2.56191i) q^{3} +(1.79027 + 0.891587i) q^{4} +(0.303580 - 1.31369i) q^{5} +(-1.81527 - 4.14898i) q^{6} +(1.06823 + 0.101385i) q^{7} +(-2.17584 - 1.80713i) q^{8} +(-2.03189 + 6.96427i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6724 q - 81 q^{2} - 85 q^{4} - 166 q^{5} - 75 q^{6} - 75 q^{8} - 84 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37664	−	0.323827i	−0.973431	−	0.228981i
							\(3\)	1.92126	+	2.56191i	1.10924	+	1.47912i	0.859731	+	0.510747i	\(0.170631\pi\)
0.249511	+	0.968372i	\(0.419730\pi\)
\(5\)	0.303580	−	1.31369i	0.135765	−	0.587500i	−0.861104	−	0.508429i	\(-0.830226\pi\)
							0.996869	−	0.0790708i	\(-0.0251953\pi\)
\(7\)	1.06823	+	0.101385i	0.403752	+	0.0383200i	0.294398	−	0.955683i	\(-0.404881\pi\)
							0.109354	+	0.994003i	\(0.465122\pi\)
\(11\)	0.401848	−	5.29821i	0.121162	−	1.59747i	−0.536330	−	0.844008i	\(-0.680190\pi\)
							0.657492	−	0.753461i	\(-0.271617\pi\)
\(13\)	−0.111679	+	0.0192081i	−0.0309742	+	0.00532738i	−0.184971	−	0.982744i	\(-0.559219\pi\)
							0.153997	+	0.988071i	\(0.450785\pi\)
\(17\)	3.91561	+	6.68430i	0.949676	+	1.62118i	0.770574	+	0.637351i	\(0.219970\pi\)
							0.179102	+	0.983831i	\(0.442681\pi\)
\(19\)	−2.93899	+	4.24464i	−0.674252	+	0.973787i	0.325351	+	0.945593i	\(0.394518\pi\)
							−0.999602	+	0.0281934i	\(0.991025\pi\)
\(23\)	5.43028	+	2.91081i	1.13229	+	0.606946i	0.928228	−	0.372012i	\(-0.121332\pi\)
							0.204064	+	0.978958i	\(0.434585\pi\)
\(29\)	7.24283	−	3.88240i	1.34496	−	0.720943i	0.367096	−	0.930183i	\(-0.380352\pi\)
							0.977864	+	0.209240i	\(0.0670990\pi\)
\(31\)	−3.17452	−	0.0600857i	−0.570160	−	0.0107917i	−0.266941	−	0.963713i	\(-0.586013\pi\)
							−0.303219	+	0.952921i	\(0.598061\pi\)
\(37\)	−8.99568	+	2.62457i	−1.47888	+	0.431476i	−0.918054	−	0.396456i	\(-0.870240\pi\)
							−0.560827	+	0.827933i	\(0.689517\pi\)
\(41\)	−0.146609	+	3.87152i	−0.0228965	+	0.604629i	0.941009	+	0.338383i	\(0.109880\pi\)
							−0.963905	+	0.266246i	\(0.914217\pi\)
\(43\)	−1.92414	−	2.46706i	−0.293429	−	0.376223i	0.619047	−	0.785354i	\(-0.287519\pi\)
							−0.912476	+	0.409130i	\(0.865832\pi\)
\(47\)	6.97270	+	0.929164i	1.01707	+	0.135532i	0.621918	−	0.783082i	\(-0.286354\pi\)
							0.395155	+	0.918614i	\(0.370691\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	668.2.h.a.15.6	✓	6724
4.3	odd	2	inner	668.2.h.a.15.26	yes	6724
167.78	odd	166	inner	668.2.h.a.579.26	yes	6724
668.579	even	166	inner	668.2.h.a.579.6	yes	6724

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.h.a.15.6	✓	6724	1.1	even	1	trivial
668.2.h.a.15.26	yes	6724	4.3	odd	2	inner
668.2.h.a.579.6	yes	6724	668.579	even	166	inner
668.2.h.a.579.26	yes	6724	167.78	odd	166	inner