Embedded newform 403.2.e.a.191.4

• Label: 403.2.e.a.191.4
• 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			191.4
Character	\(\chi\)	\(=\)	403.191
Dual form			403.2.e.a.211.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18452 + 2.05166i) q^{2} +(-0.300552 - 0.520571i) q^{3} +(-1.80620 - 3.12843i) q^{4} +(2.14607 + 3.71711i) q^{5} +1.42404 q^{6} +1.99772 q^{7} +3.81984 q^{8} +(1.31934 - 2.28516i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 2 q^{2} + 4 q^{3} - 34 q^{4} - 2 q^{5} + 8 q^{7} - 6 q^{8} - 29 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18452	+	2.05166i	−0.837585	+	1.45074i	0.0543224	+	0.998523i	\(0.482700\pi\)
							−0.891908	+	0.452217i	\(0.850633\pi\)
\(3\)	−0.300552	−	0.520571i	−0.173524	−	0.300552i	0.766126	−	0.642691i	\(-0.222182\pi\)
							−0.939649	+	0.342139i	\(0.888849\pi\)
\(5\)	2.14607	+	3.71711i	0.959753	+	1.66234i	0.723096	+	0.690748i	\(0.242719\pi\)
							0.236657	+	0.971593i	\(0.423948\pi\)
\(7\)	1.99772			0.755066			0.377533	−	0.925996i	\(-0.376772\pi\)
							0.377533	+	0.925996i	\(0.376772\pi\)
\(11\)	4.97362			1.49960			0.749802	−	0.661662i	\(-0.230149\pi\)
							0.749802	+	0.661662i	\(0.230149\pi\)
\(13\)	2.33926	−	2.74370i	0.648793	−	0.760965i
							\(17\)	0.00513090			0.00124443			0.000622213	−	1.00000i	\(-0.499802\pi\)
0.000622213		1.00000i	\(0.499802\pi\)
\(19\)	−0.329608			−0.0756172			−0.0378086	−	0.999285i	\(-0.512038\pi\)
							−0.0378086	+	0.999285i	\(0.512038\pi\)
\(23\)	3.12651	−	5.41527i	0.651922	−	1.12916i	−0.330734	−	0.943724i	\(-0.607296\pi\)
							0.982656	−	0.185438i	\(-0.0593705\pi\)
\(29\)	−3.13159	+	5.42407i	−0.581521	+	1.00722i	0.413778	+	0.910378i	\(0.364209\pi\)
							−0.995299	+	0.0968466i	\(0.969124\pi\)
\(31\)	−4.25706	+	3.58852i	−0.764590	+	0.644517i
							\(37\)	−2.80149	−	4.85232i	−0.460562	−	0.797717i	0.538427	−	0.842672i	\(-0.319019\pi\)
−0.998989	+	0.0449555i	\(0.985685\pi\)
\(41\)	−6.24104			−0.974686			−0.487343	−	0.873210i	\(-0.662034\pi\)
							−0.487343	+	0.873210i	\(0.662034\pi\)
\(43\)	−0.779650			−0.118895			−0.0594477	−	0.998231i	\(-0.518934\pi\)
							−0.0594477	+	0.998231i	\(0.518934\pi\)
\(47\)	10.6914			1.55950			0.779752	−	0.626089i	\(-0.215345\pi\)
							0.779752	+	0.626089i	\(0.215345\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.e.a.191.4	✓	70
13.3	even	3		403.2.g.a.315.4	yes	70
31.25	even	3		403.2.g.a.87.4	yes	70
403.211	even	3	inner	403.2.e.a.211.4	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.e.a.191.4	✓	70	1.1	even	1	trivial
403.2.e.a.211.4	yes	70	403.211	even	3	inner
403.2.g.a.87.4	yes	70	31.25	even	3
403.2.g.a.315.4	yes	70	13.3	even	3