Embedded newform 403.2.bl.b.126.19

• Label: 403.2.bl.b.126.19
• Level: $403$
• Weight: $2$
• Character: 403.126
• 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			126.19
Character	\(\chi\)	\(=\)	403.126
Dual form			403.2.bl.b.16.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.192109 - 0.0855325i) q^{2} +(0.298908 + 2.84392i) q^{3} +(-1.30867 + 1.45343i) q^{4} -1.10354 q^{5} +(0.300670 + 0.520777i) q^{6} +(0.556681 - 0.618257i) q^{7} +(-0.257059 + 0.791146i) q^{8} +(-5.06408 + 1.07640i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q + q^{2} + 39 q^{4} - 8 q^{5} - 2 q^{7} + 12 q^{8} + 29 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.192109	−	0.0855325i	0.135842	−	0.0604806i	−0.337690	−	0.941257i	\(-0.609646\pi\)
							0.473532	+	0.880777i	\(0.342979\pi\)
\(3\)	0.298908	+	2.84392i	0.172574	+	1.64194i	0.647611	+	0.761971i	\(0.275768\pi\)
							−0.475037	+	0.879966i	\(0.657565\pi\)
\(5\)	−1.10354			−0.493518			−0.246759	−	0.969077i	\(-0.579366\pi\)
							−0.246759	+	0.969077i	\(0.579366\pi\)
\(7\)	0.556681	−	0.618257i	0.210406	−	0.233679i	−0.628700	−	0.777648i	\(-0.716412\pi\)
							0.839105	+	0.543969i	\(0.183079\pi\)
\(11\)	−4.31024	−	0.916169i	−1.29959	−	0.276235i	−0.494387	−	0.869242i	\(-0.664608\pi\)
							−0.805198	+	0.593006i	\(0.797941\pi\)
\(13\)	0.218340	+	3.59893i	0.0605567	+	0.998165i
							\(17\)	7.87533	−	1.67395i	1.91005	−	0.405993i	0.910083	−	0.414425i	\(-0.136017\pi\)
0.999964	+	0.00843182i	\(0.00268396\pi\)
\(19\)	0.249921	−	2.37784i	0.0573358	−	0.545514i	−0.927720	−	0.373277i	\(-0.878234\pi\)
							0.985056	−	0.172237i	\(-0.0550994\pi\)
\(23\)	3.02162	+	3.35585i	0.630051	+	0.699743i	0.970658	−	0.240463i	\(-0.0772992\pi\)
							−0.340607	+	0.940206i	\(0.610633\pi\)
\(29\)	−4.50948	+	2.00775i	−0.837389	+	0.372830i	−0.780197	−	0.625534i	\(-0.784881\pi\)
							−0.0571917	+	0.998363i	\(0.518215\pi\)
\(31\)	−5.03931	+	2.36755i	−0.905088	+	0.425225i
							\(37\)	−2.54264	+	4.40398i	−0.418007	+	0.724009i	−0.995739	−	0.0922174i	\(-0.970605\pi\)
0.577732	+	0.816227i	\(0.303938\pi\)
\(41\)	−5.21095	+	2.32006i	−0.813813	+	0.362333i	−0.771068	−	0.636753i	\(-0.780277\pi\)
							−0.0427453	+	0.999086i	\(0.513610\pi\)
\(43\)	−1.14508	+	10.8947i	−0.174622	+	1.66142i	0.459493	+	0.888182i	\(0.348031\pi\)
							−0.634115	+	0.773239i	\(0.718635\pi\)
\(47\)	8.60198	−	6.24970i	1.25473	−	0.911613i	0.256241	−	0.966613i	\(-0.417516\pi\)
							0.998486	+	0.0549998i	\(0.0175158\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bl.b.126.19	yes	280
13.3	even	3	inner	403.2.bl.b.250.17	yes	280
31.16	even	5	inner	403.2.bl.b.295.17	yes	280
403.16	even	15	inner	403.2.bl.b.16.19	✓	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bl.b.16.19	✓	280	403.16	even	15	inner
403.2.bl.b.126.19	yes	280	1.1	even	1	trivial
403.2.bl.b.250.17	yes	280	13.3	even	3	inner
403.2.bl.b.295.17	yes	280	31.16	even	5	inner