Embedded newform 403.2.bj.a.100.4

• Label: 403.2.bj.a.100.4
• 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			100.4
Character	\(\chi\)	\(=\)	403.100
Dual form			403.2.bj.a.133.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.24686 - 0.477586i) q^{2} +(-0.679969 + 2.09273i) q^{3} +(2.99322 + 1.33267i) q^{4} +(0.449810 - 0.779094i) q^{5} +(2.52726 - 4.37734i) q^{6} +(0.399570 + 3.80166i) q^{7} +(-2.37217 - 1.72348i) q^{8} +(-1.49011 - 1.08263i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 3 q^{2} + 3 q^{3} + 29 q^{4} - 8 q^{5} + 5 q^{6} - q^{7} - 29 q^{8} - 63 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.24686	−	0.477586i	−1.58877	−	0.337704i	−0.673076	−	0.739573i	\(-0.735027\pi\)
							−0.915697	+	0.401869i	\(0.868361\pi\)
\(3\)	−0.679969	+	2.09273i	−0.392580	+	1.20824i	0.538250	+	0.842786i	\(0.319086\pi\)
							−0.930830	+	0.365453i	\(0.880914\pi\)
\(5\)	0.449810	−	0.779094i	0.201161	−	0.348422i	−0.747742	−	0.663990i	\(-0.768862\pi\)
							0.948903	+	0.315568i	\(0.102195\pi\)
\(7\)	0.399570	+	3.80166i	0.151023	+	1.43689i	0.763196	+	0.646167i	\(0.223629\pi\)
							−0.612173	+	0.790724i	\(0.709704\pi\)
\(11\)	1.57707	+	0.702157i	0.475505	+	0.211708i	0.630473	−	0.776211i	\(-0.282861\pi\)
							−0.154968	+	0.987919i	\(0.549528\pi\)
\(13\)	3.52549	+	0.755597i	0.977795	+	0.209565i
							\(17\)	4.22603	−	1.88155i	1.02496	−	0.456343i	0.175773	−	0.984431i	\(-0.443758\pi\)
0.849190	+	0.528088i	\(0.177091\pi\)
\(19\)	−2.92705	−	0.622163i	−0.671511	−	0.142734i	−0.140476	−	0.990084i	\(-0.544863\pi\)
							−0.531035	+	0.847350i	\(0.678197\pi\)
\(23\)	6.34751	−	2.82609i	1.32355	−	0.589281i	0.381378	−	0.924419i	\(-0.375450\pi\)
							0.942169	+	0.335138i	\(0.108783\pi\)
\(29\)	−5.85088	−	1.24364i	−1.08648	−	0.230939i	−0.370342	−	0.928896i	\(-0.620759\pi\)
							−0.716140	+	0.697957i	\(0.754093\pi\)
\(31\)	0.745505	+	5.51763i	0.133897	+	0.990995i
							\(37\)	−10.1531			−1.66916			−0.834582	−	0.550884i	\(-0.814290\pi\)
−0.834582	+	0.550884i	\(0.814290\pi\)
\(41\)	−3.06074	+	3.39930i	−0.478007	+	0.530881i	−0.933126	−	0.359550i	\(-0.882930\pi\)
							0.455118	+	0.890431i	\(0.349597\pi\)
\(43\)	7.53470	+	1.60155i	1.14903	+	0.244234i	0.742786	−	0.669529i	\(-0.233504\pi\)
							0.406246	+	0.913764i	\(0.366838\pi\)
\(47\)	−0.198192	+	0.609971i	−0.0289092	+	0.0889735i	−0.964470	−	0.264192i	\(-0.914895\pi\)
							0.935561	+	0.353165i	\(0.114895\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bj.a.100.4	✓	280
13.3	even	3		403.2.bk.a.224.32	yes	280
31.9	even	15		403.2.bk.a.9.32	yes	280
403.133	even	15	inner	403.2.bj.a.133.4	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bj.a.100.4	✓	280	1.1	even	1	trivial
403.2.bj.a.133.4	yes	280	403.133	even	15	inner
403.2.bk.a.9.32	yes	280	31.9	even	15
403.2.bk.a.224.32	yes	280	13.3	even	3