Embedded newform 403.2.bj.a.100.21

• Label: 403.2.bj.a.100.21
• 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			100.21
Character	\(\chi\)	\(=\)	403.100
Dual form			403.2.bj.a.133.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.241694 + 0.0513737i) q^{2} +(0.762457 - 2.34660i) q^{3} +(-1.77131 - 0.788640i) q^{4} +(-1.68082 + 2.91126i) q^{5} +(0.304835 - 0.527989i) q^{6} +(0.00261706 + 0.0248997i) q^{7} +(-0.787407 - 0.572085i) q^{8} +(-2.49815 - 1.81501i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 3 q^{2} + 3 q^{3} + 29 q^{4} - 8 q^{5} + 5 q^{6} - q^{7} - 29 q^{8} - 63 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.241694	+	0.0513737i	0.170904	+	0.0363267i	0.292569	−	0.956245i	\(-0.405490\pi\)
							−0.121665	+	0.992571i	\(0.538823\pi\)
\(3\)	0.762457	−	2.34660i	0.440205	−	1.35481i	−0.447453	−	0.894307i	\(-0.647669\pi\)
							0.887658	−	0.460504i	\(-0.152331\pi\)
\(5\)	−1.68082	+	2.91126i	−0.751684	+	1.30195i	0.195323	+	0.980739i	\(0.437425\pi\)
							−0.947006	+	0.321215i	\(0.895909\pi\)
\(7\)	0.00261706	+	0.0248997i	0.000989156	+	0.00941119i	0.995005	−	0.0998217i	\(-0.0318272\pi\)
							−0.994016	+	0.109233i	\(0.965161\pi\)
\(11\)	−5.85195	−	2.60546i	−1.76443	−	0.785575i	−0.987822	−	0.155591i	\(-0.950272\pi\)
							−0.776608	−	0.629984i	\(-0.783062\pi\)
\(13\)	−2.85539	+	2.20154i	−0.791942	+	0.610596i
							\(17\)	2.24609	−	1.00003i	0.544758	−	0.242542i	−0.115858	−	0.993266i	\(-0.536962\pi\)
0.660616	+	0.750724i	\(0.270295\pi\)
\(19\)	−4.67577	−	0.993865i	−1.07269	−	0.228008i	−0.362478	−	0.931992i	\(-0.618069\pi\)
							−0.710216	+	0.703984i	\(0.751403\pi\)
\(23\)	−2.04411	+	0.910095i	−0.426226	+	0.189768i	−0.608627	−	0.793456i	\(-0.708280\pi\)
							0.182402	+	0.983224i	\(0.441613\pi\)
\(29\)	7.08271	+	1.50548i	1.31523	+	0.279560i	0.811525	−	0.584318i	\(-0.198638\pi\)
							0.503701	+	0.863878i	\(0.331971\pi\)
\(31\)	−0.730640	−	5.51962i	−0.131227	−	0.991352i
							\(37\)	−2.09796			−0.344902			−0.172451	−	0.985018i	\(-0.555169\pi\)
−0.172451	+	0.985018i	\(0.555169\pi\)
\(41\)	−5.02384	+	5.57954i	−0.784593	+	0.871378i	−0.994326	−	0.106380i	\(-0.966074\pi\)
							0.209733	+	0.977759i	\(0.432741\pi\)
\(43\)	−10.3294	−	2.19558i	−1.57522	−	0.334822i	−0.664316	−	0.747452i	\(-0.731277\pi\)
							−0.910899	+	0.412629i	\(0.864610\pi\)
\(47\)	2.13004	−	6.55559i	0.310698	−	0.956231i	−0.666791	−	0.745245i	\(-0.732332\pi\)
							0.977489	−	0.210986i	\(-0.0676675\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bj.a.100.21	✓	280
13.3	even	3		403.2.bk.a.224.15	yes	280
31.9	even	15		403.2.bk.a.9.15	yes	280
403.133	even	15	inner	403.2.bj.a.133.21	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bj.a.100.21	✓	280	1.1	even	1	trivial
403.2.bj.a.133.21	yes	280	403.133	even	15	inner
403.2.bk.a.9.15	yes	280	31.9	even	15
403.2.bk.a.224.15	yes	280	13.3	even	3