Embedded newform 403.2.bl.b.16.14

• Label: 403.2.bl.b.16.14
• 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			16.14
Character	\(\chi\)	\(=\)	403.16
Dual form			403.2.bl.b.126.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.654441 - 0.291376i) q^{2} +(0.0273063 - 0.259802i) q^{3} +(-0.994868 - 1.10491i) q^{4} +4.00393 q^{5} +(-0.0935704 + 0.162069i) q^{6} +(0.155298 + 0.172476i) q^{7} +(0.771881 + 2.37561i) q^{8} +(2.86769 + 0.609547i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q + q^{2} + 39 q^{4} - 8 q^{5} - 2 q^{7} + 12 q^{8} + 29 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.654441	−	0.291376i	−0.462760	−	0.206034i	0.162095	−	0.986775i	\(-0.448175\pi\)
							−0.624854	+	0.780741i	\(0.714842\pi\)
\(3\)	0.0273063	−	0.259802i	0.0157653	−	0.149997i	−0.983807	−	0.179229i	\(-0.942640\pi\)
							0.999573	+	0.0292324i	\(0.00930630\pi\)
\(5\)	4.00393			1.79061			0.895306	−	0.445451i	\(-0.146957\pi\)
							0.895306	+	0.445451i	\(0.146957\pi\)
\(7\)	0.155298	+	0.172476i	0.0586971	+	0.0651897i	0.771778	−	0.635892i	\(-0.219368\pi\)
							−0.713081	+	0.701082i	\(0.752701\pi\)
\(11\)	1.14900	−	0.244229i	0.346438	−	0.0736377i	−0.0314066	−	0.999507i	\(-0.509999\pi\)
							0.377845	+	0.925869i	\(0.376665\pi\)
\(13\)	−3.10385	+	1.83469i	−0.860854	+	0.508851i
							\(17\)	−0.0529109	−	0.0112466i	−0.0128328	−	0.00272769i	0.201491	−	0.979490i	\(-0.435421\pi\)
−0.214324	+	0.976763i	\(0.568755\pi\)
\(19\)	0.263694	+	2.50888i	0.0604955	+	0.575576i	0.982220	+	0.187732i	\(0.0601135\pi\)
							−0.921725	+	0.387844i	\(0.873220\pi\)
\(23\)	5.76608	−	6.40388i	1.20231	−	1.33530i	0.274802	−	0.961501i	\(-0.411388\pi\)
							0.927509	−	0.373801i	\(-0.121946\pi\)
\(29\)	−3.71469	−	1.65389i	−0.689801	−	0.307119i	0.0317347	−	0.999496i	\(-0.489897\pi\)
							−0.721536	+	0.692377i	\(0.756564\pi\)
\(31\)	−5.56635	+	0.125521i	−0.999746	+	0.0225443i
							\(37\)	0.982029	+	1.70092i	0.161445	+	0.279630i	0.935387	−	0.353626i	\(-0.115051\pi\)
−0.773942	+	0.633256i	\(0.781718\pi\)
\(41\)	0.282052	+	0.125578i	0.0440491	+	0.0196119i	0.428643	−	0.903474i	\(-0.358992\pi\)
							−0.384594	+	0.923086i	\(0.625658\pi\)
\(43\)	−0.969141	−	9.22076i	−0.147793	−	1.40615i	−0.777286	−	0.629147i	\(-0.783404\pi\)
							0.629493	−	0.777006i	\(-0.283262\pi\)
\(47\)	3.22245	+	2.34125i	0.470043	+	0.341506i	0.797458	−	0.603374i	\(-0.206178\pi\)
							−0.327415	+	0.944881i	\(0.606178\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bl.b.16.14	✓	280
13.9	even	3	inner	403.2.bl.b.295.22	yes	280
31.2	even	5	inner	403.2.bl.b.250.22	yes	280
403.126	even	15	inner	403.2.bl.b.126.14	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bl.b.16.14	✓	280	1.1	even	1	trivial
403.2.bl.b.126.14	yes	280	403.126	even	15	inner
403.2.bl.b.250.22	yes	280	31.2	even	5	inner
403.2.bl.b.295.22	yes	280	13.9	even	3	inner