Embedded newform 403.2.e.a.191.20

• Label: 403.2.e.a.191.20
• Level: $403$
• Weight: $2$
• Character: 403.191
• 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.e (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			191.20
Character	\(\chi\)	\(=\)	403.191
Dual form			403.2.e.a.211.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.170223 - 0.294835i) q^{2} +(-1.12750 - 1.95288i) q^{3} +(0.942048 + 1.63168i) q^{4} +(0.624468 + 1.08161i) q^{5} -0.767702 q^{6} -0.903522 q^{7} +1.32232 q^{8} +(-1.04250 + 1.80566i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 2 q^{2} + 4 q^{3} - 34 q^{4} - 2 q^{5} + 8 q^{7} - 6 q^{8} - 29 q^{9}+O(q^{10}) \)

Character values

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

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

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	\(0.128260\pi\)
\(3\)	−1.12750	−	1.95288i	−0.650960	−	1.12750i	−0.982890	−	0.184192i	\(-0.941033\pi\)
							0.331930	−	0.943304i	\(-0.392300\pi\)
\(5\)	0.624468	+	1.08161i	0.279271	+	0.483711i	0.971204	−	0.238251i	\(-0.0765740\pi\)
							−0.691933	+	0.721962i	\(0.743241\pi\)
\(7\)	−0.903522			−0.341499			−0.170750	−	0.985314i	\(-0.554619\pi\)
							−0.170750	+	0.985314i	\(0.554619\pi\)
\(11\)	2.02268			0.609862			0.304931	−	0.952374i	\(-0.401367\pi\)
							0.304931	+	0.952374i	\(0.401367\pi\)
\(13\)	3.60431	+	0.0947479i	0.999655	+	0.0262783i
							\(17\)	4.36763			1.05931			0.529653	−	0.848214i	\(-0.322322\pi\)
0.529653	+	0.848214i	\(0.322322\pi\)
\(19\)	0.834588			0.191468			0.0957338	−	0.995407i	\(-0.469480\pi\)
							0.0957338	+	0.995407i	\(0.469480\pi\)
\(23\)	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
							\(37\)	1.37650	+	2.38417i	0.226295	+	0.391955i	0.956707	−	0.291052i	\(-0.0940053\pi\)
−0.730412	+	0.683007i	\(0.760672\pi\)
\(41\)	4.10604			0.641256			0.320628	−	0.947205i	\(-0.396106\pi\)
							0.320628	+	0.947205i	\(0.396106\pi\)
\(43\)	4.70002			0.716747			0.358373	−	0.933578i	\(-0.383331\pi\)
							0.358373	+	0.933578i	\(0.383331\pi\)
\(47\)	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.e.a.191.20	✓	70
13.3	even	3		403.2.g.a.315.20	yes	70
31.25	even	3		403.2.g.a.87.20	yes	70
403.211	even	3	inner	403.2.e.a.211.20	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.e.a.191.20	✓	70	1.1	even	1	trivial
403.2.e.a.211.20	yes	70	403.211	even	3	inner
403.2.g.a.87.20	yes	70	31.25	even	3
403.2.g.a.315.20	yes	70	13.3	even	3