Embedded newform 403.2.e.a.191.8

• Label: 403.2.e.a.191.8
• 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			191.8
Character	\(\chi\)	\(=\)	403.191
Dual form			403.2.e.a.211.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.937381 + 1.62359i) q^{2} +(1.15361 + 1.99811i) q^{3} +(-0.757366 - 1.31180i) q^{4} +(-1.23736 - 2.14316i) q^{5} -4.32548 q^{6} -4.49839 q^{7} -0.909763 q^{8} +(-1.16162 + 2.01199i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 2 q^{2} + 4 q^{3} - 34 q^{4} - 2 q^{5} + 8 q^{7} - 6 q^{8} - 29 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.937381	+	1.62359i	−0.662828	+	1.14805i	0.317041	+	0.948412i	\(0.397311\pi\)
							−0.979869	+	0.199641i	\(0.936023\pi\)
\(3\)	1.15361	+	1.99811i	0.666036	+	1.15361i	0.979003	+	0.203844i	\(0.0653436\pi\)
							−0.312967	+	0.949764i	\(0.601323\pi\)
\(5\)	−1.23736	−	2.14316i	−0.553363	−	0.958452i	−0.998029	−	0.0627557i	\(-0.980011\pi\)
							0.444666	−	0.895696i	\(-0.353322\pi\)
\(7\)	−4.49839			−1.70023			−0.850115	−	0.526597i	\(-0.823468\pi\)
							−0.850115	+	0.526597i	\(0.823468\pi\)
\(11\)	3.24645			0.978841			0.489421	−	0.872048i	\(-0.337208\pi\)
							0.489421	+	0.872048i	\(0.337208\pi\)
\(13\)	−3.58415	−	0.392281i	−0.994064	−	0.108799i
							\(17\)	−5.31438			−1.28893			−0.644463	−	0.764635i	\(-0.722919\pi\)
−0.644463	+	0.764635i	\(0.722919\pi\)
\(19\)	0.271300			0.0622405			0.0311203	−	0.999516i	\(-0.490093\pi\)
							0.0311203	+	0.999516i	\(0.490093\pi\)
\(23\)	−0.343309	+	0.594628i	−0.0715849	+	0.123989i	−0.899596	−	0.436723i	\(-0.856139\pi\)
							0.828011	+	0.560712i	\(0.189472\pi\)
\(29\)	−5.08985	+	8.81587i	−0.945161	+	1.63707i	−0.189731	+	0.981836i	\(0.560762\pi\)
							−0.755429	+	0.655230i	\(0.772572\pi\)
\(31\)	−0.921559	+	5.49097i	−0.165517	+	0.986207i
							\(37\)	−5.46488	−	9.46546i	−0.898421	−	1.55611i	−0.829512	−	0.558489i	\(-0.811381\pi\)
−0.0689092	−	0.997623i	\(-0.521952\pi\)
\(41\)	−4.48599			−0.700594			−0.350297	−	0.936639i	\(-0.613919\pi\)
							−0.350297	+	0.936639i	\(0.613919\pi\)
\(43\)	−1.26729			−0.193260			−0.0966301	−	0.995320i	\(-0.530806\pi\)
							−0.0966301	+	0.995320i	\(0.530806\pi\)
\(47\)	−0.0778603			−0.0113571			−0.00567855	−	0.999984i	\(-0.501808\pi\)
							−0.00567855	+	0.999984i	\(0.501808\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.e.a.191.8	✓	70
13.3	even	3		403.2.g.a.315.8	yes	70
31.25	even	3		403.2.g.a.87.8	yes	70
403.211	even	3	inner	403.2.e.a.211.8	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.e.a.191.8	✓	70	1.1	even	1	trivial
403.2.e.a.211.8	yes	70	403.211	even	3	inner
403.2.g.a.87.8	yes	70	31.25	even	3
403.2.g.a.315.8	yes	70	13.3	even	3