Embedded newform 403.2.v.a.36.17

• Label: 403.2.v.a.36.17
• 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			36.17
Character	\(\chi\)	\(=\)	403.36
Dual form			403.2.v.a.56.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.135075 + 0.0779857i) q^{2} +3.20789 q^{3} +(-0.987836 + 1.71098i) q^{4} +(1.02567 - 0.592170i) q^{5} +(-0.433307 + 0.250170i) q^{6} +(3.25567 - 1.87966i) q^{7} -0.620091i q^{8} +7.29058 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 6 q^{2} + 4 q^{3} + 30 q^{4} + 6 q^{6} - 12 q^{7} + 58 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.135075	+	0.0779857i	−0.0955126	+	0.0551442i	−0.546996	−	0.837136i	\(-0.684229\pi\)
							0.451483	+	0.892280i	\(0.350895\pi\)
\(3\)	3.20789			1.85208			0.926039	−	0.377428i	\(-0.123191\pi\)
							0.926039	+	0.377428i	\(0.123191\pi\)
\(5\)	1.02567	−	0.592170i	0.458693	−	0.264826i	−0.252802	−	0.967518i	\(-0.581352\pi\)
							0.711494	+	0.702692i	\(0.248019\pi\)
\(7\)	3.25567	−	1.87966i	1.23053	−	0.710446i	0.263388	−	0.964690i	\(-0.415160\pi\)
							0.967140	+	0.254244i	\(0.0818268\pi\)
\(11\)	−3.77898	−	2.18179i	−1.13940	−	0.657835i	−0.193121	−	0.981175i	\(-0.561861\pi\)
							−0.946283	+	0.323340i	\(0.895194\pi\)
\(13\)	−3.60197	+	0.160654i	−0.999007	+	0.0445574i
							\(17\)	−1.10474	−	1.91347i	−0.267940	−	0.464085i	0.700390	−	0.713760i	\(-0.253009\pi\)
−0.968330	+	0.249675i	\(0.919676\pi\)
\(19\)	−5.99050	+	3.45862i	−1.37432	+	0.793462i	−0.991468	−	0.130350i	\(-0.958390\pi\)
							−0.382848	+	0.923811i	\(0.625057\pi\)
\(23\)	3.79985	+	6.58153i	0.792323	+	1.37234i	0.924525	+	0.381122i	\(0.124462\pi\)
							−0.132202	+	0.991223i	\(0.542205\pi\)
\(29\)	0.918868	+	1.59153i	0.170630	+	0.295539i	0.938640	−	0.344898i	\(-0.112087\pi\)
							−0.768011	+	0.640437i	\(0.778753\pi\)
\(31\)	−2.96048	−	4.71546i	−0.531718	−	0.846922i
							\(37\)			2.77697i			0.456532i	0.973599	+	0.228266i	\(0.0733055\pi\)
−0.973599	+	0.228266i	\(0.926694\pi\)
\(41\)	−1.71530	−	0.990328i	−0.267885	−	0.154663i	0.360041	−	0.932936i	\(-0.382763\pi\)
							−0.627926	+	0.778273i	\(0.716096\pi\)
\(43\)	1.18280	+	2.04867i	0.180376	+	0.312420i	0.942009	−	0.335589i	\(-0.108935\pi\)
							−0.761633	+	0.648009i	\(0.775602\pi\)
\(47\)			4.11241i			0.599857i	0.953962	+	0.299929i	\(0.0969629\pi\)
							−0.953962	+	0.299929i	\(0.903037\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.v.a.36.17	yes	70
13.4	even	6		403.2.s.a.160.17	✓	70
31.25	even	3		403.2.s.a.335.17	yes	70
403.56	even	6	inner	403.2.v.a.56.17	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.s.a.160.17	✓	70	13.4	even	6
403.2.s.a.335.17	yes	70	31.25	even	3
403.2.v.a.36.17	yes	70	1.1	even	1	trivial
403.2.v.a.56.17	yes	70	403.56	even	6	inner