Embedded newform 403.2.bs.a.4.8

• Label: 403.2.bs.a.4.8
• Level: $403$
• Weight: $2$
• Character: 403.4
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $288$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bs (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(36\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			4.8
Character	\(\chi\)	\(=\)	403.4
Dual form			403.2.bs.a.101.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.360928 + 1.69803i) q^{2} +(0.922989 + 1.02508i) q^{3} +(-0.925954 - 0.412261i) q^{4} -3.21206i q^{5} +(-2.07376 + 1.19728i) q^{6} +(2.07463 - 4.65970i) q^{7} +(-1.00652 + 1.38535i) q^{8} +(0.114699 - 1.09128i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 9 q^{2} - q^{3} - 39 q^{4} - 42 q^{6} - 15 q^{7} + 31 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.360928	+	1.69803i	−0.255215	+	1.20069i	0.644635	+	0.764490i	\(0.277009\pi\)
							−0.899850	+	0.436200i	\(0.856324\pi\)
\(3\)	0.922989	+	1.02508i	0.532888	+	0.591832i	0.948132	−	0.317878i	\(-0.102970\pi\)
							−0.415244	+	0.909710i	\(0.636304\pi\)
\(5\)		−	3.21206i		−	1.43648i	−0.695798	−	0.718238i	\(-0.744949\pi\)
							0.695798	−	0.718238i	\(-0.255051\pi\)
\(7\)	2.07463	−	4.65970i	0.784137	−	1.76120i	0.148946	−	0.988845i	\(-0.452412\pi\)
							0.635191	−	0.772355i	\(-0.280922\pi\)
\(11\)	0.851973	−	0.0895460i	0.256880	−	0.0269991i	0.0247867	−	0.999693i	\(-0.492109\pi\)
							0.232093	+	0.972694i	\(0.425443\pi\)
\(13\)	0.333053	+	3.59014i	0.0923722	+	0.995725i
							\(17\)	0.564410	−	5.37000i	0.136890	−	1.30242i	−0.683221	−	0.730212i	\(-0.739421\pi\)
0.820110	−	0.572205i	\(-0.193912\pi\)
\(19\)	5.79368	+	5.21666i	1.32916	+	1.19678i	0.964019	+	0.265832i	\(0.0856465\pi\)
							0.365143	+	0.930952i	\(0.381020\pi\)
\(23\)	−4.83762	+	2.15385i	−1.00871	+	0.449108i	−0.843488	−	0.537148i	\(-0.819502\pi\)
							−0.165225	+	0.986256i	\(0.552835\pi\)
\(29\)	−4.21286	−	0.895471i	−0.782308	−	0.166285i	−0.200590	−	0.979675i	\(-0.564286\pi\)
							−0.581718	+	0.813391i	\(0.697619\pi\)
\(31\)	−1.08645	+	5.46074i	−0.195131	+	0.980777i
							\(37\)	−0.511548	−	0.295342i	−0.0840980	−	0.0485540i	0.457361	−	0.889281i	\(-0.348795\pi\)
−0.541459	+	0.840727i	\(0.682128\pi\)
\(41\)	0.0643393	−	0.302692i	0.0100481	−	0.0472726i	−0.972838	−	0.231487i	\(-0.925641\pi\)
							0.982886	+	0.184215i	\(0.0589742\pi\)
\(43\)	−3.87945	+	4.30856i	−0.591610	+	0.657050i	−0.962390	−	0.271671i	\(-0.912424\pi\)
							0.370780	+	0.928721i	\(0.379090\pi\)
\(47\)	3.70113	+	1.20257i	0.539866	+	0.175413i	0.566242	−	0.824239i	\(-0.308397\pi\)
							−0.0263762	+	0.999652i	\(0.508397\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bs.a.4.8	✓	288
13.10	even	6	inner	403.2.bs.a.283.29	yes	288
31.8	even	5	inner	403.2.bs.a.225.29	yes	288
403.101	even	30	inner	403.2.bs.a.101.8	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bs.a.4.8	✓	288	1.1	even	1	trivial
403.2.bs.a.101.8	yes	288	403.101	even	30	inner
403.2.bs.a.225.29	yes	288	31.8	even	5	inner
403.2.bs.a.283.29	yes	288	13.10	even	6	inner