Embedded newform 403.2.s.a.160.2

• Label: 403.2.s.a.160.2
• Level: $403$
• Weight: $2$
• Character: 403.160
• 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.s (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			160.2
Character	\(\chi\)	\(=\)	403.160
Dual form			403.2.s.a.335.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.31373 - 1.33583i) q^{2} +(0.915310 + 1.58536i) q^{3} +(2.56889 + 4.44945i) q^{4} +(2.88237 - 1.66414i) q^{5} -4.89080i q^{6} +0.418923i q^{7} -8.38310i q^{8} +(-0.175583 + 0.304119i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 6 q^{2} - 2 q^{3} + 30 q^{4} - 29 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.31373	−	1.33583i	−1.63605	−	0.944576i	−0.982173	−	0.187977i	\(-0.939807\pi\)
							−0.653879	−	0.756599i	\(-0.726860\pi\)
\(3\)	0.915310	+	1.58536i	0.528454	+	0.915310i	0.999450	+	0.0331739i	\(0.0105615\pi\)
							−0.470995	+	0.882136i	\(0.656105\pi\)
\(5\)	2.88237	−	1.66414i	1.28903	−	0.744224i	0.310552	−	0.950556i	\(-0.399486\pi\)
							0.978482	+	0.206332i	\(0.0661528\pi\)
\(7\)			0.418923i			0.158338i	0.996861	+	0.0791691i	\(0.0252267\pi\)
							−0.996861	+	0.0791691i	\(0.974773\pi\)
\(11\)		−	5.32495i		−	1.60553i	−0.596293	−	0.802767i	\(-0.703360\pi\)
							0.596293	−	0.802767i	\(-0.296640\pi\)
\(13\)	−1.85762	−	3.09019i	−0.515211	−	0.857063i
							\(17\)	−0.402486			−0.0976172			−0.0488086	−	0.998808i	\(-0.515542\pi\)
−0.0488086	+	0.998808i	\(0.515542\pi\)
\(19\)			0.587709i			0.134830i	0.997725	+	0.0674148i	\(0.0214751\pi\)
							−0.997725	+	0.0674148i	\(0.978525\pi\)
\(23\)	2.77294	−	4.80288i	0.578199	−	1.00147i	−0.417487	−	0.908683i	\(-0.637089\pi\)
							0.995686	−	0.0927866i	\(-0.0295774\pi\)
\(29\)	−4.11226	+	7.12265i	−0.763628	+	1.32264i	0.177340	+	0.984150i	\(0.443251\pi\)
							−0.940969	+	0.338494i	\(0.890083\pi\)
\(31\)	4.01711	−	3.85524i	0.721494	−	0.692421i
							\(37\)	−6.56792	+	3.79199i	−1.07976	+	0.623399i	−0.930831	−	0.365449i	\(-0.880915\pi\)
−0.148927	+	0.988848i	\(0.547582\pi\)
\(41\)			5.81409i			0.908009i	0.890999	+	0.454004i	\(0.150005\pi\)
							−0.890999	+	0.454004i	\(0.849995\pi\)
\(43\)	−7.44982			−1.13609			−0.568044	−	0.822998i	\(-0.692299\pi\)
							−0.568044	+	0.822998i	\(0.692299\pi\)
\(47\)			2.22421i			0.324434i	0.986755	+	0.162217i	\(0.0518645\pi\)
							−0.986755	+	0.162217i	\(0.948135\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.s.a.160.2	✓	70
13.10	even	6		403.2.v.a.36.2	yes	70
31.25	even	3		403.2.v.a.56.2	yes	70
403.335	even	6	inner	403.2.s.a.335.2	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.s.a.160.2	✓	70	1.1	even	1	trivial
403.2.s.a.335.2	yes	70	403.335	even	6	inner
403.2.v.a.36.2	yes	70	13.10	even	6
403.2.v.a.56.2	yes	70	31.25	even	3