Embedded newform 403.2.bl.b.16.8

• Label: 403.2.bl.b.16.8
• Level: $403$
• Weight: $2$
• Character: 403.16
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $280$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bl (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(35\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			16.8
Character	\(\chi\)	\(=\)	403.16
Dual form			403.2.bl.b.126.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.58448 - 0.705458i) q^{2} +(-0.208907 + 1.98762i) q^{3} +(0.674658 + 0.749284i) q^{4} -0.120273 q^{5} +(1.73319 - 3.00197i) q^{6} +(-1.90609 - 2.11693i) q^{7} +(0.531542 + 1.63592i) q^{8} +(-0.972535 - 0.206719i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q + q^{2} + 39 q^{4} - 8 q^{5} - 2 q^{7} + 12 q^{8} + 29 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.58448	−	0.705458i	−1.12040	−	0.498834i	−0.238912	−	0.971041i	\(-0.576791\pi\)
							−0.881488	+	0.472207i	\(0.843457\pi\)
\(3\)	−0.208907	+	1.98762i	−0.120612	+	1.14755i	0.752009	+	0.659153i	\(0.229085\pi\)
							−0.872621	+	0.488398i	\(0.837582\pi\)
\(5\)	−0.120273			−0.0537875			−0.0268938	−	0.999638i	\(-0.508562\pi\)
							−0.0268938	+	0.999638i	\(0.508562\pi\)
\(7\)	−1.90609	−	2.11693i	−0.720435	−	0.800125i	0.266053	−	0.963959i	\(-0.414281\pi\)
							−0.986488	+	0.163834i	\(0.947614\pi\)
\(11\)	3.72316	−	0.791383i	1.12258	−	0.238611i	0.391013	−	0.920385i	\(-0.372125\pi\)
							0.731563	+	0.681774i	\(0.238791\pi\)
\(13\)	−0.0969973	+	3.60425i	−0.0269022	+	0.999638i
							\(17\)	−0.652972	−	0.138793i	−0.158369	−	0.0336623i	0.128045	−	0.991768i	\(-0.459130\pi\)
−0.286414	+	0.958106i	\(0.592463\pi\)
\(19\)	0.558652	+	5.31522i	0.128164	+	1.21940i	0.849793	+	0.527116i	\(0.176727\pi\)
							−0.721629	+	0.692280i	\(0.756607\pi\)
\(23\)	−2.32170	+	2.57851i	−0.484109	+	0.537657i	−0.934871	−	0.354987i	\(-0.884485\pi\)
							0.450763	+	0.892644i	\(0.351152\pi\)
\(29\)	−9.22585	−	4.10761i	−1.71320	−	0.762765i	−0.997957	−	0.0638864i	\(-0.979650\pi\)
							−0.715241	−	0.698878i	\(-0.753683\pi\)
\(31\)	−0.777578	+	5.51320i	−0.139657	+	0.990200i
							\(37\)	4.59594	+	7.96041i	0.755569	+	1.30868i	0.945091	+	0.326807i	\(0.105973\pi\)
−0.189523	+	0.981876i	\(0.560694\pi\)
\(41\)	4.72552	+	2.10394i	0.738003	+	0.328580i	0.741081	−	0.671415i	\(-0.234313\pi\)
							−0.00307842	+	0.999995i	\(0.500980\pi\)
\(43\)	0.487451	+	4.63779i	0.0743356	+	0.707256i	0.966695	+	0.255931i	\(0.0823819\pi\)
							−0.892360	+	0.451325i	\(0.850951\pi\)
\(47\)	−7.77273	−	5.64722i	−1.13377	−	0.823731i	−0.147530	−	0.989058i	\(-0.547132\pi\)
							−0.986239	+	0.165326i	\(0.947132\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bl.b.16.8	✓	280
13.9	even	3	inner	403.2.bl.b.295.28	yes	280
31.2	even	5	inner	403.2.bl.b.250.28	yes	280
403.126	even	15	inner	403.2.bl.b.126.8	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bl.b.16.8	✓	280	1.1	even	1	trivial
403.2.bl.b.126.8	yes	280	403.126	even	15	inner
403.2.bl.b.250.28	yes	280	31.2	even	5	inner
403.2.bl.b.295.28	yes	280	13.9	even	3	inner