Embedded newform 668.2.h.a.15.10

• Label: 668.2.h.a.15.10
• 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			15.10
Character	\(\chi\)	\(=\)	668.15
Dual form			668.2.h.a.579.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.35512 + 0.404546i) q^{2} +(-0.666088 - 0.888195i) q^{3} +(1.67269 - 1.09641i) q^{4} +(0.185473 - 0.802603i) q^{5} +(1.26194 + 0.934146i) q^{6} +(0.603158 + 0.0572456i) q^{7} +(-1.82314 + 2.16245i) q^{8} +(0.495026 - 1.69670i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6724 q - 81 q^{2} - 85 q^{4} - 166 q^{5} - 75 q^{6} - 75 q^{8} - 84 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.35512	+	0.404546i	−0.958213	+	0.286057i
							\(3\)	−0.666088	−	0.888195i	−0.384566	−	0.512800i	0.565543	−	0.824719i	\(-0.308667\pi\)
−0.950109	+	0.311919i	\(0.899028\pi\)
\(5\)	0.185473	−	0.802603i	0.0829461	−	0.358935i	−0.916178	−	0.400771i	\(-0.868742\pi\)
							0.999124	+	0.0418359i	\(0.0133207\pi\)
\(7\)	0.603158	+	0.0572456i	0.227972	+	0.0216368i	0.207850	−	0.978161i	\(-0.433353\pi\)
							0.0201224	+	0.999798i	\(0.493594\pi\)
\(11\)	0.274566	−	3.62005i	0.0827848	−	1.09149i	−0.793141	−	0.609039i	\(-0.791555\pi\)
							0.875925	−	0.482447i	\(-0.160252\pi\)
\(13\)	−1.87878	+	0.323139i	−0.521080	+	0.0896225i	−0.424014	−	0.905655i	\(-0.639379\pi\)
							−0.0970655	+	0.995278i	\(0.530946\pi\)
\(17\)	2.12973	+	3.63564i	0.516535	+	0.881771i	0.999980	+	0.00639649i	\(0.00203608\pi\)
							−0.483445	+	0.875375i	\(0.660615\pi\)
\(19\)	1.86775	−	2.69749i	0.428490	−	0.618846i	−0.547462	−	0.836830i	\(-0.684406\pi\)
							0.975952	+	0.217984i	\(0.0699481\pi\)
\(23\)	−6.02284	−	3.22844i	−1.25585	−	0.673176i	−0.296401	−	0.955064i	\(-0.595787\pi\)
							−0.959447	+	0.281887i	\(0.909040\pi\)
\(29\)	1.09595	−	0.587466i	0.203513	−	0.109090i	−0.367524	−	0.930014i	\(-0.619794\pi\)
							0.571037	+	0.820924i	\(0.306541\pi\)
\(31\)	−1.95312	−	0.0369677i	−0.350791	−	0.00663960i	−0.156765	−	0.987636i	\(-0.550106\pi\)
							−0.194026	+	0.980996i	\(0.562155\pi\)
\(37\)	5.54023	−	1.61641i	0.910808	−	0.265736i	0.208844	−	0.977949i	\(-0.433030\pi\)
							0.701964	+	0.712213i	\(0.252307\pi\)
\(41\)	−0.0691438	+	1.82589i	−0.0107985	+	0.285156i	0.983661	+	0.180032i	\(0.0576202\pi\)
							−0.994459	+	0.105124i	\(0.966476\pi\)
\(43\)	−2.88088	−	3.69376i	−0.439330	−	0.563293i	0.516849	−	0.856077i	\(-0.327105\pi\)
							−0.956179	+	0.292784i	\(0.905418\pi\)
\(47\)	−11.5023	−	1.53276i	−1.67778	−	0.223577i	−0.768528	−	0.639817i	\(-0.779010\pi\)
							−0.909255	+	0.416240i	\(0.863348\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	668.2.h.a.15.10	✓	6724
4.3	odd	2	inner	668.2.h.a.15.11	yes	6724
167.78	odd	166	inner	668.2.h.a.579.11	yes	6724
668.579	even	166	inner	668.2.h.a.579.10	yes	6724

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.h.a.15.10	✓	6724	1.1	even	1	trivial
668.2.h.a.15.11	yes	6724	4.3	odd	2	inner
668.2.h.a.579.10	yes	6724	668.579	even	166	inner
668.2.h.a.579.11	yes	6724	167.78	odd	166	inner