Embedded newform 402.2.n.a.101.10

• Label: 402.2.n.a.101.10
• Level: $402$
• Weight: $2$
• Character: 402.101
• 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			101.10
Character	\(\chi\)	\(=\)	402.101
Dual form			402.2.n.a.203.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0475819 - 0.998867i) q^{2} +(1.62810 - 0.591001i) q^{3} +(-0.995472 + 0.0950560i) q^{4} +(1.71146 + 1.97513i) q^{5} +(-0.667800 - 1.59814i) q^{6} +(-1.85056 + 3.58958i) q^{7} +(0.142315 + 0.989821i) q^{8} +(2.30143 - 1.92442i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 11 q^{2} + q^{3} + 11 q^{4} + 2 q^{6} + 6 q^{7} + 22 q^{8} - q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(269\)	\(337\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{65}{66}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0475819	−	0.998867i	−0.0336455	−	0.706306i
							\(3\)	1.62810	−	0.591001i	0.939985	−	0.341215i
\(5\)	1.71146	+	1.97513i	0.765389	+	0.883306i								0.995964	−	0.0897482i	\(-0.0286062\pi\)
							−0.230575	+	0.973055i	\(0.574061\pi\)
\(7\)	−1.85056	+	3.58958i	−0.699444	+	1.35673i	0.224601	+	0.974451i	\(0.427892\pi\)
							−0.924046	+	0.382282i	\(0.875138\pi\)
\(11\)	0.423306	−	1.22306i	0.127631	−	0.368767i	−0.863023	−	0.505165i	\(-0.831432\pi\)
							0.990654	+	0.136398i	\(0.0435527\pi\)
\(13\)	3.07564	−	3.91099i	0.853029	−	1.08471i	−0.142456	−	0.989801i	\(-0.545500\pi\)
							0.995486	−	0.0949136i	\(-0.0302575\pi\)
\(17\)	−0.597732	+	6.25973i	−0.144971	+	1.51821i	0.571506	+	0.820598i	\(0.306359\pi\)
							−0.716478	+	0.697610i	\(0.754247\pi\)
\(19\)	4.89497	−	2.52353i	1.12298	−	0.578938i	0.206274	−	0.978494i	\(-0.433866\pi\)
							0.916709	+	0.399556i	\(0.130836\pi\)
\(23\)	0.746398	+	0.181074i	0.155635	+	0.0377566i	0.312819	−	0.949813i	\(-0.398727\pi\)
							−0.157185	+	0.987569i	\(0.550242\pi\)
\(29\)	−5.69215	+	3.28636i	−1.05701	+	0.610262i	−0.924602	−	0.380934i	\(-0.875603\pi\)
							−0.132403	+	0.991196i	\(0.542269\pi\)
\(31\)	−6.56181	−	8.34402i	−1.17854	−	1.49863i	−0.826768	−	0.562543i	\(-0.809823\pi\)
							−0.351767	−	0.936087i	\(-0.614419\pi\)
\(37\)	−4.78275	+	8.28396i	−0.786279	+	1.36187i	0.141954	+	0.989873i	\(0.454662\pi\)
							−0.928232	+	0.372001i	\(0.878672\pi\)
\(41\)	5.46199	−	7.67029i	0.853019	−	1.19790i	−0.125572	−	0.992084i	\(-0.540077\pi\)
							0.978591	−	0.205813i	\(-0.0659838\pi\)
\(43\)	−0.580066	+	0.264907i	−0.0884593	+	0.0403980i	−0.459153	−	0.888357i	\(-0.651847\pi\)
							0.370694	+	0.928755i	\(0.379120\pi\)
\(47\)	−3.37397	+	3.53851i	−0.492144	+	0.516145i	−0.922230	−	0.386642i	\(-0.873635\pi\)
							0.430086	+	0.902788i	\(0.358483\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	402.2.n.a.101.10	✓	220
3.2	odd	2		402.2.n.b.101.2	yes	220
67.2	odd	66		402.2.n.b.203.2	yes	220
201.2	even	66	inner	402.2.n.a.203.10	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
402.2.n.a.101.10	✓	220	1.1	even	1	trivial
402.2.n.a.203.10	yes	220	201.2	even	66	inner
402.2.n.b.101.2	yes	220	3.2	odd	2
402.2.n.b.203.2	yes	220	67.2	odd	66