Embedded newform 402.2.n.b.11.5

• Label: 402.2.n.b.11.5
• Level: $402$
• Weight: $2$
• Character: 402.11
• Analytic conductor: $3.210$
• Analytic rank: $0$
• Dimension: $220$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 402 = 2 \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	402.n (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.20998616126\)
Analytic rank:	\(0\)
Dimension:	\(220\)
Relative dimension:	\(11\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			11.5
Character	\(\chi\)	\(=\)	402.11
Dual form			402.2.n.b.329.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.327068 - 0.945001i) q^{2} +(-0.187659 + 1.72185i) q^{3} +(-0.786053 + 0.618159i) q^{4} +(2.83825 - 0.833385i) q^{5} +(1.68853 - 0.385826i) q^{6} +(0.584073 - 3.03046i) q^{7} +(0.841254 + 0.540641i) q^{8} +(-2.92957 - 0.646243i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q + 11 q^{2} - q^{3} + 11 q^{4} - q^{6} + 6 q^{7} - 22 q^{8} - q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/402\mathbb{Z}\right)^\times\).

\(n\)	\(269\)	\(337\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{59}{66}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.327068	−	0.945001i	−0.231272	−	0.668216i
							\(3\)	−0.187659	+	1.72185i	−0.108345	+	0.994113i
\(5\)	2.83825	−	0.833385i	1.26930	−	0.372701i								0.423354	−	0.905964i	\(-0.360853\pi\)
							0.845949	+	0.533263i	\(0.179035\pi\)
\(7\)	0.584073	−	3.03046i	0.220759	−	1.14541i	−0.687239	−	0.726432i	\(-0.741177\pi\)
							0.907998	−	0.418975i	\(-0.137610\pi\)
\(11\)	−3.19459	−	3.04603i	−0.963204	−	0.918413i	0.0335080	−	0.999438i	\(-0.489332\pi\)
							−0.996712	+	0.0810253i	\(0.974181\pi\)
\(13\)	5.12228	−	0.244004i	1.42066	−	0.0676746i	0.676879	−	0.736094i	\(-0.263332\pi\)
							0.743785	+	0.668419i	\(0.233029\pi\)
\(17\)	−1.32511	+	1.68501i	−0.321385	+	0.408675i	−0.919722	−	0.392570i	\(-0.871586\pi\)
							0.598337	+	0.801245i	\(0.295829\pi\)
\(19\)	6.88762	−	1.32748i	1.58013	−	0.304545i	0.677685	−	0.735352i	\(-0.262983\pi\)
							0.902444	+	0.430808i	\(0.141771\pi\)
\(23\)	−0.377170	+	3.94990i	−0.0786454	+	0.823612i	0.866978	+	0.498347i	\(0.166060\pi\)
							−0.945623	+	0.325265i	\(0.894547\pi\)
\(29\)	4.91800	−	2.83941i	0.913250	−	0.527265i	0.0317749	−	0.999495i	\(-0.489884\pi\)
							0.881476	+	0.472230i	\(0.156551\pi\)
\(31\)	−6.56898	−	0.312919i	−1.17982	−	0.0562019i	−0.551514	−	0.834166i	\(-0.685950\pi\)
							−0.628309	+	0.777964i	\(0.716253\pi\)
\(37\)	−0.596438	+	1.03306i	−0.0980538	+	0.169834i	−0.910879	−	0.412673i	\(-0.864595\pi\)
							0.812825	+	0.582508i	\(0.197928\pi\)
\(41\)	−1.51638	+	0.607065i	−0.236818	+	0.0948076i	−0.487036	−	0.873382i	\(-0.661922\pi\)
							0.250218	+	0.968190i	\(0.419498\pi\)
\(43\)	11.6833	+	1.67981i	1.78169	+	0.256169i	0.952878	−	0.303355i	\(-0.0981069\pi\)
							0.828814	+	0.559524i	\(0.189016\pi\)
\(47\)	−6.64797	−	4.73400i	−0.969706	−	0.690524i	−0.0187412	−	0.999824i	\(-0.505966\pi\)
							−0.950965	+	0.309300i	\(0.899905\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	402.2.n.b.11.5	yes	220
3.2	odd	2		402.2.n.a.11.3	✓	220
67.61	odd	66		402.2.n.a.329.3	yes	220
201.128	even	66	inner	402.2.n.b.329.5	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
402.2.n.a.11.3	✓	220	3.2	odd	2
402.2.n.a.329.3	yes	220	67.61	odd	66
402.2.n.b.11.5	yes	220	1.1	even	1	trivial
402.2.n.b.329.5	yes	220	201.128	even	66	inner