Embedded newform 403.2.e.a.191.6

• Label: 403.2.e.a.191.6
• 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			191.6
Character	\(\chi\)	\(=\)	403.191
Dual form			403.2.e.a.211.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.05012 + 1.81887i) q^{2} +(-1.37200 - 2.37637i) q^{3} +(-1.20552 - 2.08802i) q^{4} +(0.498066 + 0.862675i) q^{5} +5.76307 q^{6} -3.52844 q^{7} +0.863293 q^{8} +(-2.26475 + 3.92266i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 2 q^{2} + 4 q^{3} - 34 q^{4} - 2 q^{5} + 8 q^{7} - 6 q^{8} - 29 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.05012	+	1.81887i	−0.742550	+	1.28613i	0.208781	+	0.977962i	\(0.433050\pi\)
							−0.951331	+	0.308172i	\(0.900283\pi\)
\(3\)	−1.37200	−	2.37637i	−0.792122	−	1.37200i	−0.924650	−	0.380817i	\(-0.875643\pi\)
							0.132528	−	0.991179i	\(-0.457691\pi\)
\(5\)	0.498066	+	0.862675i	0.222742	+	0.385800i	0.955640	−	0.294539i	\(-0.0951660\pi\)
							−0.732898	+	0.680339i	\(0.761833\pi\)
\(7\)	−3.52844			−1.33363			−0.666813	−	0.745225i	\(-0.732342\pi\)
							−0.666813	+	0.745225i	\(0.732342\pi\)
\(11\)	1.96543			0.592598			0.296299	−	0.955095i	\(-0.404248\pi\)
							0.296299	+	0.955095i	\(0.404248\pi\)
\(13\)	−1.43533	+	3.30754i	−0.398089	+	0.917347i
							\(17\)	6.16311			1.49477			0.747387	−	0.664389i	\(-0.231308\pi\)
0.747387	+	0.664389i	\(0.231308\pi\)
\(19\)	4.69632			1.07741			0.538705	−	0.842494i	\(-0.318914\pi\)
							0.538705	+	0.842494i	\(0.318914\pi\)
\(23\)	4.08816	−	7.08091i	0.852441	−	1.47647i	−0.0265580	−	0.999647i	\(-0.508455\pi\)
							0.878999	−	0.476824i	\(-0.158212\pi\)
\(29\)	−3.21532	+	5.56909i	−0.597069	+	1.03415i	0.396182	+	0.918172i	\(0.370335\pi\)
							−0.993251	+	0.115982i	\(0.962998\pi\)
\(31\)	4.20362	+	3.65097i	0.754992	+	0.655734i
							\(37\)	2.66269	+	4.61191i	0.437743	+	0.758194i	0.997515	−	0.0704532i	\(-0.0224446\pi\)
−0.559772	+	0.828647i	\(0.689111\pi\)
\(41\)	4.01461			0.626976			0.313488	−	0.949592i	\(-0.398502\pi\)
							0.313488	+	0.949592i	\(0.398502\pi\)
\(43\)	2.63526			0.401874			0.200937	−	0.979604i	\(-0.435601\pi\)
							0.200937	+	0.979604i	\(0.435601\pi\)
\(47\)	−9.15867			−1.33593			−0.667965	−	0.744193i	\(-0.732834\pi\)
							−0.667965	+	0.744193i	\(0.732834\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.e.a.191.6	✓	70
13.3	even	3		403.2.g.a.315.6	yes	70
31.25	even	3		403.2.g.a.87.6	yes	70
403.211	even	3	inner	403.2.e.a.211.6	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.e.a.191.6	✓	70	1.1	even	1	trivial
403.2.e.a.211.6	yes	70	403.211	even	3	inner
403.2.g.a.87.6	yes	70	31.25	even	3
403.2.g.a.315.6	yes	70	13.3	even	3