Embedded newform 403.2.e.a.191.16

• Label: 403.2.e.a.191.16
• 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			191.16
Character	\(\chi\)	\(=\)	403.191
Dual form			403.2.e.a.211.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.333798 + 0.578156i) q^{2} +(-1.40074 - 2.42616i) q^{3} +(0.777157 + 1.34608i) q^{4} +(-1.97182 - 3.41529i) q^{5} +1.87026 q^{6} -1.53640 q^{7} -2.37285 q^{8} +(-2.42416 + 4.19877i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 2 q^{2} + 4 q^{3} - 34 q^{4} - 2 q^{5} + 8 q^{7} - 6 q^{8} - 29 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.333798	+	0.578156i	−0.236031	+	0.408818i	−0.959572	−	0.281464i	\(-0.909180\pi\)
							0.723541	+	0.690282i	\(0.242513\pi\)
\(3\)	−1.40074	−	2.42616i	−0.808719	−	1.40074i	−0.913751	−	0.406274i	\(-0.866828\pi\)
							0.105032	−	0.994469i	\(-0.466505\pi\)
\(5\)	−1.97182	−	3.41529i	−0.881825	−	1.52736i	−0.849310	−	0.527894i	\(-0.822982\pi\)
							−0.0325141	−	0.999471i	\(-0.510351\pi\)
\(7\)	−1.53640			−0.580705			−0.290353	−	0.956920i	\(-0.593773\pi\)
							−0.290353	+	0.956920i	\(0.593773\pi\)
\(11\)	0.213405			0.0643442			0.0321721	−	0.999482i	\(-0.489758\pi\)
							0.0321721	+	0.999482i	\(0.489758\pi\)
\(13\)	2.37566	+	2.71224i	0.658889	+	0.752240i
							\(17\)	−4.88801			−1.18552			−0.592759	−	0.805380i	\(-0.701961\pi\)
−0.592759	+	0.805380i	\(0.701961\pi\)
\(19\)	3.88168			0.890518			0.445259	−	0.895402i	\(-0.353112\pi\)
							0.445259	+	0.895402i	\(0.353112\pi\)
\(23\)	−2.93168	+	5.07782i	−0.611298	+	1.05880i	0.379724	+	0.925100i	\(0.376019\pi\)
							−0.991022	+	0.133699i	\(0.957315\pi\)
\(29\)	1.13809	−	1.97122i	0.211337	−	0.366047i	−0.740796	−	0.671730i	\(-0.765552\pi\)
							0.952133	+	0.305683i	\(0.0988849\pi\)
\(31\)	5.38989	+	1.39611i	0.968052	+	0.250750i
							\(37\)	−1.64485	−	2.84897i	−0.270412	−	0.468367i	0.698555	−	0.715556i	\(-0.253827\pi\)
−0.968967	+	0.247189i	\(0.920493\pi\)
\(41\)	−5.47466			−0.854999			−0.427499	−	0.904016i	\(-0.640605\pi\)
							−0.427499	+	0.904016i	\(0.640605\pi\)
\(43\)	−12.5069			−1.90729			−0.953646	−	0.300932i	\(-0.902702\pi\)
							−0.953646	+	0.300932i	\(0.902702\pi\)
\(47\)	−7.15586			−1.04379			−0.521895	−	0.853010i	\(-0.674775\pi\)
							−0.521895	+	0.853010i	\(0.674775\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.e.a.191.16	✓	70
13.3	even	3		403.2.g.a.315.16	yes	70
31.25	even	3		403.2.g.a.87.16	yes	70
403.211	even	3	inner	403.2.e.a.211.16	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.e.a.191.16	✓	70	1.1	even	1	trivial
403.2.e.a.211.16	yes	70	403.211	even	3	inner
403.2.g.a.87.16	yes	70	31.25	even	3
403.2.g.a.315.16	yes	70	13.3	even	3