Embedded newform 402.2.n.a.11.9

• Label: 402.2.n.a.11.9
• 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			11.9
Character	\(\chi\)	\(=\)	402.11
Dual form			402.2.n.a.329.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.327068 + 0.945001i) q^{2} +(1.35130 + 1.08350i) q^{3} +(-0.786053 + 0.618159i) q^{4} +(2.79086 - 0.819470i) q^{5} +(-0.581943 + 1.63136i) q^{6} +(-0.548725 + 2.84706i) q^{7} +(-0.841254 - 0.540641i) q^{8} +(0.652042 + 2.92828i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 11 q^{2} + q^{3} + 11 q^{4} + 2 q^{6} + 6 q^{7} + 22 q^{8} - q^{9}+O(q^{10}) \)

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
							\(3\)	1.35130	+	1.08350i	0.780175	+	0.625561i
\(5\)	2.79086	−	0.819470i	1.24811	−	0.366478i								0.410054	−	0.912061i	\(-0.365510\pi\)
							0.838057	+	0.545583i	\(0.183692\pi\)
\(7\)	−0.548725	+	2.84706i	−0.207399	+	1.07609i	0.717698	+	0.696354i	\(0.245196\pi\)
							−0.925097	+	0.379731i	\(0.876016\pi\)
\(11\)	−3.28214	−	3.12951i	−0.989601	−	0.943583i	0.00880807	−	0.999961i	\(-0.497196\pi\)
							−0.998409	+	0.0563782i	\(0.982045\pi\)
\(13\)	1.12186	−	0.0534408i	0.311148	−	0.0148218i	0.108573	−	0.994088i	\(-0.465372\pi\)
							0.202575	+	0.979267i	\(0.435069\pi\)
\(17\)	1.01883	−	1.29555i	0.247103	−	0.314217i	−0.646641	−	0.762795i	\(-0.723827\pi\)
							0.893744	+	0.448577i	\(0.148069\pi\)
\(19\)	0.814290	−	0.156941i	0.186811	−	0.0360048i	−0.0949876	−	0.995478i	\(-0.530281\pi\)
							0.281798	+	0.959474i	\(0.409069\pi\)
\(23\)	0.114234	−	1.19632i	0.0238195	−	0.249449i	−0.975718	−	0.219031i	\(-0.929710\pi\)
							0.999537	−	0.0304175i	\(-0.00968368\pi\)
\(29\)	6.32275	−	3.65044i	1.17410	−	0.677870i	0.219461	−	0.975621i	\(-0.429570\pi\)
							0.954643	+	0.297751i	\(0.0962367\pi\)
\(31\)	−2.97292	−	0.141618i	−0.533952	−	0.0254353i	−0.221125	−	0.975245i	\(-0.570973\pi\)
							−0.312827	+	0.949810i	\(0.601276\pi\)
\(37\)	−1.62091	+	2.80750i	−0.266476	+	0.461550i	−0.967949	−	0.251146i	\(-0.919193\pi\)
							0.701473	+	0.712696i	\(0.252526\pi\)
\(41\)	−2.49460	+	0.998685i	−0.389590	+	0.155968i	−0.558184	−	0.829717i	\(-0.688502\pi\)
							0.168594	+	0.985686i	\(0.446077\pi\)
\(43\)	−8.31698	−	1.19580i	−1.26833	−	0.182358i	−0.524894	−	0.851167i	\(-0.675895\pi\)
							−0.743434	+	0.668809i	\(0.766804\pi\)
\(47\)	−9.32003	−	6.63676i	−1.35947	−	0.968071i	−0.999382	−	0.0351512i	\(-0.988809\pi\)
							−0.360084	−	0.932920i	\(-0.617252\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	402.2.n.a.11.9	✓	220
3.2	odd	2		402.2.n.b.11.6	yes	220
67.61	odd	66		402.2.n.b.329.6	yes	220
201.128	even	66	inner	402.2.n.a.329.9	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
402.2.n.a.11.9	✓	220	1.1	even	1	trivial
402.2.n.a.329.9	yes	220	201.128	even	66	inner
402.2.n.b.11.6	yes	220	3.2	odd	2
402.2.n.b.329.6	yes	220	67.61	odd	66