Embedded newform 403.2.bl.b.16.1

• Label: 403.2.bl.b.16.1
• 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			16.1
Character	\(\chi\)	\(=\)	403.16
Dual form			403.2.bl.b.126.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.44344 - 1.08789i) q^{2} +(-0.216774 + 2.06246i) q^{3} +(3.44862 + 3.83009i) q^{4} -2.87099 q^{5} +(2.77340 - 4.80367i) q^{6} +(2.21936 + 2.46485i) q^{7} +(-2.60675 - 8.02276i) q^{8} +(-1.27232 - 0.270439i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q + q^{2} + 39 q^{4} - 8 q^{5} - 2 q^{7} + 12 q^{8} + 29 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.44344	−	1.08789i	−1.72777	−	0.769253i	−0.996162	−	0.0875291i	\(-0.972103\pi\)
							−0.731609	−	0.681724i	\(-0.761230\pi\)
\(3\)	−0.216774	+	2.06246i	−0.125154	+	1.19076i	0.734037	+	0.679109i	\(0.237634\pi\)
							−0.859192	+	0.511654i	\(0.829033\pi\)
\(5\)	−2.87099			−1.28395			−0.641974	−	0.766727i	\(-0.721884\pi\)
							−0.641974	+	0.766727i	\(0.721884\pi\)
\(7\)	2.21936	+	2.46485i	0.838839	+	0.931625i	0.998456	−	0.0555537i	\(-0.0176924\pi\)
							−0.159616	+	0.987179i	\(0.551026\pi\)
\(11\)	3.29131	−	0.699590i	0.992369	−	0.210934i	0.317009	−	0.948423i	\(-0.397322\pi\)
							0.675360	+	0.737488i	\(0.263988\pi\)
\(13\)	3.57419	+	0.474524i	0.991302	+	0.131609i
							\(17\)	5.58366	+	1.18684i	1.35424	+	0.287852i	0.827160	−	0.561967i	\(-0.189955\pi\)
0.527076	+	0.849818i	\(0.323288\pi\)
\(19\)	−0.0720284	−	0.685305i	−0.0165245	−	0.157220i	0.983148	−	0.182811i	\(-0.0585197\pi\)
							−0.999672	+	0.0255916i	\(0.991853\pi\)
\(23\)	−5.71945	+	6.35209i	−1.19259	+	1.32450i	−0.259118	+	0.965846i	\(0.583432\pi\)
							−0.933469	+	0.358657i	\(0.883235\pi\)
\(29\)	−0.973563	−	0.433458i	−0.180786	−	0.0804912i	0.314348	−	0.949308i	\(-0.398214\pi\)
							−0.495134	+	0.868817i	\(0.664881\pi\)
\(31\)	−5.49165	−	0.917494i	−0.986329	−	0.164787i
							\(37\)	−3.42469	−	5.93174i	−0.563016	−	0.975172i	−0.997231	−	0.0743631i	\(-0.976308\pi\)
0.434215	−	0.900809i	\(-0.357026\pi\)
\(41\)	6.37189	+	2.83695i	0.995121	+	0.443057i	0.838676	−	0.544630i	\(-0.183330\pi\)
							0.156445	+	0.987687i	\(0.449997\pi\)
\(43\)	0.641770	+	6.10603i	0.0978690	+	0.931161i	0.927745	+	0.373214i	\(0.121744\pi\)
							−0.829876	+	0.557947i	\(0.811589\pi\)
\(47\)	4.86584	+	3.53524i	0.709756	+	0.515668i	0.883095	−	0.469194i	\(-0.155456\pi\)
							−0.173339	+	0.984862i	\(0.555456\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bl.b.16.1	✓	280
13.9	even	3	inner	403.2.bl.b.295.35	yes	280
31.2	even	5	inner	403.2.bl.b.250.35	yes	280
403.126	even	15	inner	403.2.bl.b.126.1	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bl.b.16.1	✓	280	1.1	even	1	trivial
403.2.bl.b.126.1	yes	280	403.126	even	15	inner
403.2.bl.b.250.35	yes	280	31.2	even	5	inner
403.2.bl.b.295.35	yes	280	13.9	even	3	inner