Embedded newform 403.2.v.a.36.10

• Label: 403.2.v.a.36.10
• Level: $403$
• Weight: $2$
• Character: 403.36
• 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.v (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(70\)
Relative dimension:	\(35\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			36.10
Character	\(\chi\)	\(=\)	403.36
Dual form			403.2.v.a.56.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.06311 + 0.613789i) q^{2} +0.887230 q^{3} +(-0.246526 + 0.426995i) q^{4} +(-0.237568 + 0.137160i) q^{5} +(-0.943226 + 0.544572i) q^{6} +(-1.34961 + 0.779199i) q^{7} -3.06042i q^{8} -2.21282 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 6 q^{2} + 4 q^{3} + 30 q^{4} + 6 q^{6} - 12 q^{7} + 58 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.06311	+	0.613789i	−0.751735	+	0.434015i	−0.826321	−	0.563200i	\(-0.809570\pi\)
							0.0745853	+	0.997215i	\(0.476237\pi\)
\(3\)	0.887230			0.512242			0.256121	−	0.966645i	\(-0.417555\pi\)
							0.256121	+	0.966645i	\(0.417555\pi\)
\(5\)	−0.237568	+	0.137160i	−0.106244	+	0.0613397i	−0.552180	−	0.833725i	\(-0.686204\pi\)
							0.445937	+	0.895064i	\(0.352871\pi\)
\(7\)	−1.34961	+	0.779199i	−0.510106	+	0.294510i	−0.732877	−	0.680361i	\(-0.761823\pi\)
							0.222771	+	0.974871i	\(0.428490\pi\)
\(11\)	−4.65713	−	2.68880i	−1.40418	−	0.810703i	−0.409360	−	0.912373i	\(-0.634248\pi\)
							−0.994818	+	0.101670i	\(0.967581\pi\)
\(13\)	−0.395334	−	3.58381i	−0.109646	−	0.993971i
							\(17\)	1.84514	+	3.19588i	0.447513	+	0.775115i	0.998223	−	0.0595810i	\(-0.0189765\pi\)
−0.550710	+	0.834696i	\(0.685643\pi\)
\(19\)	1.56033	−	0.900858i	0.357965	−	0.206671i	−0.310223	−	0.950664i	\(-0.600404\pi\)
							0.668188	+	0.743993i	\(0.267070\pi\)
\(23\)	−2.40111	−	4.15885i	−0.500667	−	0.867180i	−1.00000		0.000770114i	\(-0.999755\pi\)
							0.499333	−	0.866410i	\(-0.333578\pi\)
\(29\)	−0.431591	−	0.747537i	−0.0801444	−	0.138814i	0.823167	−	0.567799i	\(-0.192205\pi\)
							−0.903312	+	0.428985i	\(0.858871\pi\)
\(31\)	1.09696	−	5.45863i	0.197019	−	0.980400i
							\(37\)			3.05644i			0.502476i	0.967925	+	0.251238i	\(0.0808376\pi\)
−0.967925	+	0.251238i	\(0.919162\pi\)
\(41\)	−9.49675	−	5.48295i	−1.48314	−	0.856293i	−0.483326	−	0.875440i	\(-0.660571\pi\)
							−0.999817	+	0.0191477i	\(0.993905\pi\)
\(43\)	1.99664	+	3.45827i	0.304484	+	0.527382i	0.977146	−	0.212568i	\(-0.0681827\pi\)
							−0.672662	+	0.739950i	\(0.734849\pi\)
\(47\)			3.07294i			0.448235i	0.974562	+	0.224118i	\(0.0719499\pi\)
							−0.974562	+	0.224118i	\(0.928050\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.v.a.36.10	yes	70
13.4	even	6		403.2.s.a.160.10	✓	70
31.25	even	3		403.2.s.a.335.10	yes	70
403.56	even	6	inner	403.2.v.a.56.10	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.s.a.160.10	✓	70	13.4	even	6
403.2.s.a.335.10	yes	70	31.25	even	3
403.2.v.a.36.10	yes	70	1.1	even	1	trivial
403.2.v.a.56.10	yes	70	403.56	even	6	inner