Embedded newform 403.2.g.a.87.20

• Label: 403.2.g.a.87.20
• Level: $403$
• Weight: $2$
• Character: 403.87
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $70$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(70\)
Relative dimension:	\(35\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			87.20
Character	\(\chi\)	\(=\)	403.87
Dual form			403.2.g.a.315.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.170223 - 0.294835i) q^{2} +2.25499 q^{3} +(0.942048 + 1.63168i) q^{4} +(0.624468 - 1.08161i) q^{5} +(0.383851 - 0.664850i) q^{6} +(0.451761 - 0.782473i) q^{7} +1.32232 q^{8} +2.08499 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 2 q^{2} - 8 q^{3} - 34 q^{4} - 2 q^{5} - 4 q^{7} - 6 q^{8} + 58 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/403\mathbb{Z}\right)^\times\).

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.170223	−	0.294835i	0.120366	−	0.208479i	−0.799546	−	0.600605i	\(-0.794927\pi\)
							0.919912	+	0.392125i	\(0.128260\pi\)
\(3\)	2.25499			1.30192			0.650960	−	0.759112i	\(-0.274366\pi\)
							0.650960	+	0.759112i	\(0.274366\pi\)
\(5\)	0.624468	−	1.08161i	0.279271	−	0.483711i	−0.691933	−	0.721962i	\(-0.743241\pi\)
							0.971204	+	0.238251i	\(0.0765740\pi\)
\(7\)	0.451761	−	0.782473i	0.170750	−	0.295747i	−0.767933	−	0.640531i	\(-0.778714\pi\)
							0.938682	+	0.344784i	\(0.112048\pi\)
\(11\)	−1.01134	−	1.75170i	−0.304931	−	0.528156i	0.672315	−	0.740265i	\(-0.265300\pi\)
							−0.977246	+	0.212109i	\(0.931967\pi\)
\(13\)	−1.72010	−	3.16879i	−0.477070	−	0.878865i
							\(17\)	−2.18382	+	3.78248i	−0.529653	+	0.917387i	0.469748	+	0.882800i	\(0.344345\pi\)
−0.999402	+	0.0345862i	\(0.988989\pi\)
\(19\)	−0.417294	+	0.722775i	−0.0957338	+	0.165816i	−0.909915	−	0.414796i	\(-0.863853\pi\)
							0.814181	+	0.580611i	\(0.197186\pi\)
\(23\)	0.275828	−	0.477748i	0.0575141	−	0.0996173i	−0.835835	−	0.548981i	\(-0.815016\pi\)
							0.893349	+	0.449364i	\(0.148349\pi\)
\(29\)	0.580088	−	1.00474i	0.107720	−	0.186576i	−0.807126	−	0.590379i	\(-0.798978\pi\)
							0.914846	+	0.403803i	\(0.132312\pi\)
\(31\)	−3.76590	+	4.10097i	−0.676376	+	0.736557i
							\(37\)	−2.75300			−0.452590			−0.226295	−	0.974059i	\(-0.572661\pi\)
−0.226295	+	0.974059i	\(0.572661\pi\)
\(41\)	−2.05302	−	3.55594i	−0.320628	−	0.555344i	0.659990	−	0.751275i	\(-0.270561\pi\)
							−0.980618	+	0.195931i	\(0.937227\pi\)
\(43\)	−2.35001	+	4.07034i	−0.358373	+	0.620721i	−0.987689	−	0.156429i	\(-0.950002\pi\)
							0.629316	+	0.777150i	\(0.283335\pi\)
\(47\)	1.34735			0.196531			0.0982655	−	0.995160i	\(-0.468671\pi\)
							0.0982655	+	0.995160i	\(0.468671\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.g.a.87.20	yes	70
13.3	even	3		403.2.e.a.211.20	yes	70
31.5	even	3		403.2.e.a.191.20	✓	70
403.315	even	3	inner	403.2.g.a.315.20	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.e.a.191.20	✓	70	31.5	even	3
403.2.e.a.211.20	yes	70	13.3	even	3
403.2.g.a.87.20	yes	70	1.1	even	1	trivial
403.2.g.a.315.20	yes	70	403.315	even	3	inner