Embedded newform 668.2.h.a.15.1

• Label: 668.2.h.a.15.1
• 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			15.1
Character	\(\chi\)	\(=\)	668.15
Dual form			668.2.h.a.579.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41330 - 0.0508355i) q^{2} +(-1.89908 - 2.53233i) q^{3} +(1.99483 + 0.143692i) q^{4} +(0.0776745 - 0.336123i) q^{5} +(2.55524 + 3.67549i) q^{6} +(1.39922 + 0.132799i) q^{7} +(-2.81199 - 0.304488i) q^{8} +(-1.96595 + 6.73829i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6724 q - 81 q^{2} - 85 q^{4} - 166 q^{5} - 75 q^{6} - 75 q^{8} - 84 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41330	−	0.0508355i	−0.999354	−	0.0359461i
							\(3\)	−1.89908	−	2.53233i	−1.09644	−	1.46204i	−0.873253	−	0.487267i	\(-0.837994\pi\)
−0.223182	−	0.974777i	\(-0.571645\pi\)
\(5\)	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	0.355519	−	0.934669i	\(-0.384304\pi\)
							0.173336	+	0.984863i	\(0.444545\pi\)
\(11\)	0.169203	−	2.23087i	0.0510165	−	0.672633i	−0.913199	−	0.407514i	\(-0.866396\pi\)
							0.964216	−	0.265119i	\(-0.0854114\pi\)
\(13\)	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	\(-0.527881\pi\)
							0.963238	−	0.268649i	\(-0.0865772\pi\)
\(23\)	6.67337	+	3.57715i	1.39149	+	0.745886i	0.985768	−	0.168109i	\(-0.0537659\pi\)
							0.405725	+	0.913995i	\(0.367019\pi\)
\(29\)	−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	−0.388731	−	0.921351i	\(-0.627087\pi\)
							−0.423318	+	0.905981i	\(0.639135\pi\)
\(37\)	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	0.689940	−	0.723867i	\(-0.257637\pi\)
							−0.870107	+	0.492863i	\(0.835950\pi\)
\(47\)	7.92743	+	1.05639i	1.15633	+	0.154090i	0.685460	−	0.728110i	\(-0.259601\pi\)
							0.470874	+	0.882200i	\(0.343939\pi\)

(See \(a_n\) instead)

Twists

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

    

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