Embedded newform 403.2.s.a.160.8

• Label: 403.2.s.a.160.8
• 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			160.8
Character	\(\chi\)	\(=\)	403.160
Dual form			403.2.s.a.335.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.77854 - 1.02684i) q^{2} +(1.56768 + 2.71530i) q^{3} +(1.10881 + 1.92051i) q^{4} +(0.188573 - 0.108873i) q^{5} -6.43902i q^{6} +0.507266i q^{7} -0.446916i q^{8} +(-3.41522 + 5.91533i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 6 q^{2} - 2 q^{3} + 30 q^{4} - 29 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.77854	−	1.02684i	−1.25762	−	0.726087i	−0.285008	−	0.958525i	\(-0.591996\pi\)
							−0.972611	+	0.232438i	\(0.925330\pi\)
\(3\)	1.56768	+	2.71530i	0.905098	+	1.56768i	0.820785	+	0.571237i	\(0.193536\pi\)
							0.0843136	+	0.996439i	\(0.473130\pi\)
\(5\)	0.188573	−	0.108873i	0.0843324	−	0.0486894i	−0.457241	−	0.889343i	\(-0.651162\pi\)
							0.541573	+	0.840654i	\(0.317829\pi\)
\(7\)			0.507266i			0.191728i	0.995394	+	0.0958642i	\(0.0305615\pi\)
							−0.995394	+	0.0958642i	\(0.969439\pi\)
\(11\)			0.767453i			0.231396i	0.993284	+	0.115698i	\(0.0369104\pi\)
							−0.993284	+	0.115698i	\(0.963090\pi\)
\(13\)	2.60377	+	2.49407i	0.722155	+	0.691731i
							\(17\)	−1.71042			−0.414838			−0.207419	−	0.978252i	\(-0.566506\pi\)
−0.207419	+	0.978252i	\(0.566506\pi\)
\(19\)		−	6.42155i		−	1.47320i	−0.676326	−	0.736602i	\(-0.736429\pi\)
							0.676326	−	0.736602i	\(-0.263571\pi\)
\(23\)	−3.83089	+	6.63529i	−0.798795	+	1.38355i	0.121605	+	0.992579i	\(0.461196\pi\)
							−0.920401	+	0.390976i	\(0.872138\pi\)
\(29\)	−0.564228	+	0.977272i	−0.104775	+	0.181475i	−0.913646	−	0.406511i	\(-0.866745\pi\)
							0.808872	+	0.587985i	\(0.200079\pi\)
\(31\)	5.49748	−	0.881879i	0.987377	−	0.158390i
							\(37\)	−3.65242	+	2.10873i	−0.600455	+	0.346673i	−0.769221	−	0.638983i	\(-0.779355\pi\)
0.168766	+	0.985656i	\(0.446022\pi\)
\(41\)		−	0.999173i		−	0.156045i	−0.996952	−	0.0780223i	\(-0.975139\pi\)
							0.996952	−	0.0780223i	\(-0.0248605\pi\)
\(43\)	0.336841			0.0513678			0.0256839	−	0.999670i	\(-0.491824\pi\)
							0.0256839	+	0.999670i	\(0.491824\pi\)
\(47\)		−	5.20790i		−	0.759650i	−0.925058	−	0.379825i	\(-0.875984\pi\)
							0.925058	−	0.379825i	\(-0.124016\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.s.a.160.8	✓	70
13.10	even	6		403.2.v.a.36.8	yes	70
31.25	even	3		403.2.v.a.56.8	yes	70
403.335	even	6	inner	403.2.s.a.335.8	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.s.a.160.8	✓	70	1.1	even	1	trivial
403.2.s.a.335.8	yes	70	403.335	even	6	inner
403.2.v.a.36.8	yes	70	13.10	even	6
403.2.v.a.56.8	yes	70	31.25	even	3