Embedded newform 403.2.g.a.87.5

• Label: 403.2.g.a.87.5
• Level: $403$
• Weight: $2$
• Character: 403.87
• 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.g (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			87.5
Character	\(\chi\)	\(=\)	403.87
Dual form			403.2.g.a.315.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.06665 + 1.84749i) q^{2} -1.21169 q^{3} +(-1.27547 - 2.20918i) q^{4} +(-1.78766 + 3.09632i) q^{5} +(1.29245 - 2.23859i) q^{6} +(-0.455966 + 0.789757i) q^{7} +1.17532 q^{8} -1.53180 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 2 q^{2} - 8 q^{3} - 34 q^{4} - 2 q^{5} - 4 q^{7} - 6 q^{8} + 58 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.06665	+	1.84749i	−0.754233	+	1.30637i	0.191521	+	0.981488i	\(0.438658\pi\)
							−0.945755	+	0.324882i	\(0.894675\pi\)
\(3\)	−1.21169			−0.699571			−0.349786	−	0.936830i	\(-0.613746\pi\)
							−0.349786	+	0.936830i	\(0.613746\pi\)
\(5\)	−1.78766	+	3.09632i	−0.799465	+	1.38471i	0.120500	+	0.992713i	\(0.461550\pi\)
							−0.919965	+	0.392001i	\(0.871783\pi\)
\(7\)	−0.455966	+	0.789757i	−0.172339	+	0.298500i	−0.939237	−	0.343269i	\(-0.888466\pi\)
							0.766898	+	0.641769i	\(0.221799\pi\)
\(11\)	2.62090	+	4.53953i	0.790231	+	1.36872i	0.925824	+	0.377956i	\(0.123373\pi\)
							−0.135592	+	0.990765i	\(0.543294\pi\)
\(13\)	−1.49562	−	3.28072i	−0.414812	−	0.909907i
							\(17\)	−1.06096	+	1.83764i	−0.257321	+	0.445694i	−0.965523	−	0.260316i	\(-0.916173\pi\)
0.708202	+	0.706010i	\(0.249507\pi\)
\(19\)	−2.76924	+	4.79647i	−0.635308	+	1.10039i	0.351142	+	0.936322i	\(0.385793\pi\)
							−0.986450	+	0.164063i	\(0.947540\pi\)
\(23\)	4.21989	−	7.30906i	0.879907	−	1.52404i	0.0284660	−	0.999595i	\(-0.490938\pi\)
							0.851441	−	0.524450i	\(-0.175729\pi\)
\(29\)	2.11755	−	3.66770i	0.393219	−	0.681075i	−0.599653	−	0.800260i	\(-0.704695\pi\)
							0.992872	+	0.119185i	\(0.0380282\pi\)
\(31\)	−5.56533	−	0.164588i	−0.999563	−	0.0295608i
							\(37\)	5.91088			0.971743			0.485872	−	0.874030i	\(-0.338502\pi\)
0.485872	+	0.874030i	\(0.338502\pi\)
\(41\)	−0.278647	−	0.482631i	−0.0435174	−	0.0753743i	0.843446	−	0.537214i	\(-0.180523\pi\)
							−0.886964	+	0.461839i	\(0.847190\pi\)
\(43\)	2.21258	−	3.83230i	0.337415	−	0.584420i	−0.646531	−	0.762888i	\(-0.723781\pi\)
							0.983946	+	0.178468i	\(0.0571141\pi\)
\(47\)	−5.68283			−0.828926			−0.414463	−	0.910066i	\(-0.636031\pi\)
							−0.414463	+	0.910066i	\(0.636031\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.g.a.87.5	yes	70
13.3	even	3		403.2.e.a.211.5	yes	70
31.5	even	3		403.2.e.a.191.5	✓	70
403.315	even	3	inner	403.2.g.a.315.5	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.e.a.191.5	✓	70	31.5	even	3
403.2.e.a.211.5	yes	70	13.3	even	3
403.2.g.a.87.5	yes	70	1.1	even	1	trivial
403.2.g.a.315.5	yes	70	403.315	even	3	inner