Embedded newform 668.2.h.a.15.15

• Label: 668.2.h.a.15.15
• 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			15.15
Character	\(\chi\)	\(=\)	668.15
Dual form			668.2.h.a.579.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.16508 - 0.801617i) q^{2} +(-1.64821 - 2.19780i) q^{3} +(0.714821 + 1.86789i) q^{4} +(-0.943687 + 4.08364i) q^{5} +(0.158496 + 3.88184i) q^{6} +(0.269428 + 0.0255713i) q^{7} +(0.664513 - 2.74926i) q^{8} +(-1.27351 + 4.36494i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6724 q - 81 q^{2} - 85 q^{4} - 166 q^{5} - 75 q^{6} - 75 q^{8} - 84 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.16508	−	0.801617i	−0.823836	−	0.566829i
							\(3\)	−1.64821	−	2.19780i	−0.951592	−	1.26890i	−0.963149	−	0.268969i	\(-0.913317\pi\)
0.0115566	−	0.999933i	\(-0.496321\pi\)
\(5\)	−0.943687	+	4.08364i	−0.422029	+	1.82626i	0.128838	+	0.991666i	\(0.458875\pi\)
							−0.550867	+	0.834593i	\(0.685703\pi\)
\(7\)	0.269428	+	0.0255713i	0.101834	+	0.00966506i	0.145279	−	0.989391i	\(-0.453592\pi\)
							−0.0434444	+	0.999056i	\(0.513833\pi\)
\(11\)	−0.306952	+	4.04705i	−0.0925496	+	1.22023i	0.742420	+	0.669935i	\(0.233678\pi\)
							−0.834970	+	0.550296i	\(0.814515\pi\)
\(13\)	−1.69217	+	0.291043i	−0.469324	+	0.0807208i	−0.399292	−	0.916824i	\(-0.630744\pi\)
							−0.0700322	+	0.997545i	\(0.522310\pi\)
\(17\)	−1.02474	−	1.74933i	−0.248537	−	0.424274i	0.712102	−	0.702076i	\(-0.247743\pi\)
							−0.960638	+	0.277802i	\(0.910394\pi\)
\(19\)	4.04499	−	5.84197i	0.927984	−	1.34024i	−0.0123273	−	0.999924i	\(-0.503924\pi\)
							0.940311	−	0.340315i	\(-0.110534\pi\)
\(23\)	−7.41146	−	3.97279i	−1.54540	−	0.828383i	−0.545444	−	0.838147i	\(-0.683639\pi\)
							−0.999952	+	0.00976360i	\(0.996892\pi\)
\(29\)	1.51623	−	0.812747i	0.281556	−	0.150923i	−0.325596	−	0.945509i	\(-0.605565\pi\)
							0.607152	+	0.794586i	\(0.292312\pi\)
\(31\)	−1.39063	−	0.0263212i	−0.249764	−	0.00472742i	−0.106106	−	0.994355i	\(-0.533838\pi\)
							−0.143658	+	0.989627i	\(0.545887\pi\)
\(37\)	11.4888	−	3.35195i	1.88874	−	0.551057i	0.894078	−	0.447911i	\(-0.147832\pi\)
							0.994666	−	0.103146i	\(-0.0328909\pi\)
\(41\)	−0.0627972	+	1.65829i	−0.00980728	+	0.258982i	0.985955	+	0.167013i	\(0.0534123\pi\)
							−0.995762	+	0.0919685i	\(0.970684\pi\)
\(43\)	−5.34384	−	6.85168i	−0.814928	−	1.04487i	−0.998089	−	0.0617880i	\(-0.980320\pi\)
							0.183161	−	0.983083i	\(-0.441367\pi\)
\(47\)	6.95108	+	0.926282i	1.01392	+	0.135112i	0.620467	−	0.784233i	\(-0.286943\pi\)
							0.393453	+	0.919345i	\(0.371280\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	668.2.h.a.15.15	✓	6724
4.3	odd	2	inner	668.2.h.a.15.34	yes	6724
167.78	odd	166	inner	668.2.h.a.579.34	yes	6724
668.579	even	166	inner	668.2.h.a.579.15	yes	6724

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.h.a.15.15	✓	6724	1.1	even	1	trivial
668.2.h.a.15.34	yes	6724	4.3	odd	2	inner
668.2.h.a.579.15	yes	6724	668.579	even	166	inner
668.2.h.a.579.34	yes	6724	167.78	odd	166	inner