Embedded newform 668.2.h.a.15.18

• Label: 668.2.h.a.15.18
• 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			15.18
Character	\(\chi\)	\(=\)	668.15
Dual form			668.2.h.a.579.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.06576 + 0.929597i) q^{2} +(1.89908 + 2.53233i) q^{3} +(0.271697 - 1.98146i) q^{4} +(0.0776745 - 0.336123i) q^{5} +(-4.37802 - 0.933483i) q^{6} +(-1.39922 - 0.132799i) q^{7} +(1.55239 + 2.36433i) q^{8} +(-1.96595 + 6.73829i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6724 q - 81 q^{2} - 85 q^{4} - 166 q^{5} - 75 q^{6} - 75 q^{8} - 84 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.06576	+	0.929597i	−0.753608	+	0.657325i
							\(3\)	1.89908	+	2.53233i	1.09644	+	1.46204i	0.873253	+	0.487267i	\(0.162006\pi\)
0.223182	+	0.974777i	\(0.428355\pi\)
\(5\)	0.0776745	−	0.336123i	0.0347371	−	0.150319i	−0.954051	−	0.299644i	\(-0.903132\pi\)
							0.988788	+	0.149326i	\(0.0477103\pi\)
\(7\)	−1.39922	−	0.132799i	−0.528855	−	0.0501935i	−0.173336	−	0.984863i	\(-0.555455\pi\)
							−0.355519	+	0.934669i	\(0.615696\pi\)
\(11\)	−0.169203	+	2.23087i	−0.0510165	+	0.672633i	0.913199	+	0.407514i	\(0.133604\pi\)
							−0.964216	+	0.265119i	\(0.914589\pi\)
\(13\)	6.37344	−	1.09619i	1.76767	−	0.304029i	0.808847	−	0.588020i	\(-0.200092\pi\)
							0.958827	+	0.283991i	\(0.0916585\pi\)
\(17\)	0.733795	+	1.25265i	0.177971	+	0.303813i	0.938363	−	0.345652i	\(-0.112342\pi\)
							−0.760391	+	0.649465i	\(0.774993\pi\)
\(19\)	−3.81735	+	5.51320i	−0.875760	+	1.26482i	0.0874778	+	0.996166i	\(0.472119\pi\)
							−0.963238	+	0.268649i	\(0.913423\pi\)
\(23\)	−6.67337	−	3.57715i	−1.39149	−	0.745886i	−0.405725	−	0.913995i	\(-0.632981\pi\)
							−0.985768	+	0.168109i	\(0.946234\pi\)
\(29\)	−5.68146	+	3.04545i	−1.05502	+	0.565526i	−0.905987	−	0.423306i	\(-0.860870\pi\)
							−0.149034	+	0.988832i	\(0.547616\pi\)
\(31\)	4.52130	+	0.0855770i	0.812049	+	0.0153701i	0.423318	−	0.905981i	\(-0.360865\pi\)
							0.388731	+	0.921351i	\(0.372913\pi\)
\(37\)	2.24060	−	0.653714i	0.368352	−	0.107470i	−0.0907022	−	0.995878i	\(-0.528911\pi\)
							0.459055	+	0.888408i	\(0.348188\pi\)
\(41\)	0.286973	−	7.57813i	0.0448177	−	1.18351i	−0.782822	−	0.622245i	\(-0.786221\pi\)
							0.827640	−	0.561260i	\(-0.189683\pi\)
\(43\)	1.18143	+	1.51479i	0.180167	+	0.231003i	0.870107	−	0.492863i	\(-0.164050\pi\)
							−0.689940	+	0.723867i	\(0.742363\pi\)
\(47\)	−7.92743	−	1.05639i	−1.15633	−	0.154090i	−0.470874	−	0.882200i	\(-0.656061\pi\)
							−0.685460	+	0.728110i	\(0.740399\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	668.2.h.a.15.18	yes	6724
4.3	odd	2	inner	668.2.h.a.15.1	✓	6724
167.78	odd	166	inner	668.2.h.a.579.1	yes	6724
668.579	even	166	inner	668.2.h.a.579.18	yes	6724

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.h.a.15.1	✓	6724	4.3	odd	2	inner
668.2.h.a.15.18	yes	6724	1.1	even	1	trivial
668.2.h.a.579.1	yes	6724	167.78	odd	166	inner
668.2.h.a.579.18	yes	6724	668.579	even	166	inner