Embedded newform 403.2.bj.a.100.20

• Label: 403.2.bj.a.100.20
• Level: $403$
• Weight: $2$
• Character: 403.100
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $280$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bj (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			100.20
Character	\(\chi\)	\(=\)	403.100
Dual form			403.2.bj.a.133.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.208786 + 0.0443788i) q^{2} +(0.200559 - 0.617258i) q^{3} +(-1.78547 - 0.794942i) q^{4} +(1.97676 - 3.42385i) q^{5} +(0.0692671 - 0.119974i) q^{6} +(0.0421544 + 0.401072i) q^{7} +(-0.682872 - 0.496136i) q^{8} +(2.08627 + 1.51576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 3 q^{2} + 3 q^{3} + 29 q^{4} - 8 q^{5} + 5 q^{6} - q^{7} - 29 q^{8} - 63 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{14}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.208786	+	0.0443788i	0.147634	+	0.0313806i	0.281136	−	0.959668i	\(-0.409289\pi\)
							−0.133502	+	0.991049i	\(0.542622\pi\)
\(3\)	0.200559	−	0.617258i	0.115793	−	0.356374i	−0.876319	−	0.481732i	\(-0.840008\pi\)
							0.992112	+	0.125358i	\(0.0400079\pi\)
\(5\)	1.97676	−	3.42385i	0.884035	−	1.53119i	0.0372200	−	0.999307i	\(-0.488150\pi\)
							0.846815	−	0.531887i	\(-0.178517\pi\)
\(7\)	0.0421544	+	0.401072i	0.0159329	+	0.151591i	0.999596	−	0.0284289i	\(-0.00905042\pi\)
							−0.983663	+	0.180020i	\(0.942384\pi\)
\(11\)	−3.05741	−	1.36124i	−0.921843	−	0.410431i	−0.109750	−	0.993959i	\(-0.535005\pi\)
							−0.812093	+	0.583528i	\(0.801672\pi\)
\(13\)	−3.30763	−	1.43513i	−0.917371	−	0.398033i
							\(17\)	1.22542	−	0.545590i	0.297207	−	0.132325i	−0.252716	−	0.967541i	\(-0.581324\pi\)
0.549923	+	0.835215i	\(0.314657\pi\)
\(19\)	−5.09640	−	1.08327i	−1.16919	−	0.248520i	−0.417903	−	0.908492i	\(-0.637235\pi\)
							−0.751290	+	0.659972i	\(0.770568\pi\)
\(23\)	7.29216	−	3.24668i	1.52052	−	0.676979i	0.534739	−	0.845017i	\(-0.320410\pi\)
							0.985780	+	0.168038i	\(0.0537432\pi\)
\(29\)	−4.03334	−	0.857312i	−0.748972	−	0.159199i	−0.182415	−	0.983222i	\(-0.558391\pi\)
							−0.566557	+	0.824023i	\(0.691725\pi\)
\(31\)	−2.06869	+	5.16919i	−0.371548	+	0.928414i
							\(37\)	2.08901			0.343431			0.171715	−	0.985147i	\(-0.445069\pi\)
0.171715	+	0.985147i	\(0.445069\pi\)
\(41\)	7.30845	−	8.11686i	1.14139	−	1.26764i	0.182704	−	0.983168i	\(-0.441515\pi\)
							0.958684	−	0.284472i	\(-0.0918184\pi\)
\(43\)	1.30748	+	0.277913i	0.199389	+	0.0423813i	0.306523	−	0.951863i	\(-0.400834\pi\)
							−0.107135	+	0.994245i	\(0.534168\pi\)
\(47\)	−1.26564	+	3.89525i	−0.184613	+	0.568180i	−0.999941	−	0.0108185i	\(-0.996556\pi\)
							0.815329	+	0.578998i	\(0.196556\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bj.a.100.20	✓	280
13.3	even	3		403.2.bk.a.224.16	yes	280
31.9	even	15		403.2.bk.a.9.16	yes	280
403.133	even	15	inner	403.2.bj.a.133.20	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bj.a.100.20	✓	280	1.1	even	1	trivial
403.2.bj.a.133.20	yes	280	403.133	even	15	inner
403.2.bk.a.9.16	yes	280	31.9	even	15
403.2.bk.a.224.16	yes	280	13.3	even	3