Embedded newform 403.2.bs.a.101.5

• Label: 403.2.bs.a.101.5
• Level: $403$
• Weight: $2$
• Character: 403.101
• 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			101.5
Character	\(\chi\)	\(=\)	403.101
Dual form			403.2.bs.a.4.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.490829 - 2.30917i) q^{2} +(2.17363 - 2.41407i) q^{3} +(-3.26425 + 1.45334i) q^{4} +1.85570i q^{5} +(-6.64136 - 3.83439i) q^{6} +(-1.05002 - 2.35838i) q^{7} +(2.18296 + 3.00459i) q^{8} +(-0.789440 - 7.51102i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 9 q^{2} - q^{3} - 39 q^{4} - 42 q^{6} - 15 q^{7} + 31 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.490829	−	2.30917i	−0.347068	−	1.63283i	−0.712274	−	0.701902i	\(-0.752335\pi\)
							0.365206	−	0.930927i	\(-0.380999\pi\)
\(3\)	2.17363	−	2.41407i	1.25495	−	1.39376i	0.369363	−	0.929285i	\(-0.379576\pi\)
							0.885586	−	0.464476i	\(-0.153757\pi\)
\(5\)			1.85570i			0.829892i	0.909846	+	0.414946i	\(0.136200\pi\)
							−0.909846	+	0.414946i	\(0.863800\pi\)
\(7\)	−1.05002	−	2.35838i	−0.396869	−	0.891382i	−0.995883	−	0.0906441i	\(-0.971107\pi\)
							0.599014	−	0.800738i	\(-0.295559\pi\)
\(11\)	0.824110	+	0.0866174i	0.248478	+	0.0261161i	0.227949	−	0.973673i	\(-0.426798\pi\)
							0.0205296	+	0.999789i	\(0.493465\pi\)
\(13\)	1.84810	−	3.09589i	0.512571	−	0.858645i
							\(17\)	0.396669	+	3.77405i	0.0962064	+	0.915342i	0.931059	+	0.364869i	\(0.118886\pi\)
−0.834853	+	0.550474i	\(0.814447\pi\)
\(19\)	−3.65639	+	3.29223i	−0.838834	+	0.755289i	−0.971794	−	0.235832i	\(-0.924218\pi\)
							0.132960	+	0.991121i	\(0.457552\pi\)
\(23\)	1.70234	+	0.757931i	0.354963	+	0.158040i	0.576468	−	0.817119i	\(-0.304430\pi\)
							−0.221506	+	0.975159i	\(0.571097\pi\)
\(29\)	3.58482	−	0.761977i	0.665684	−	0.141495i	0.137337	−	0.990524i	\(-0.456146\pi\)
							0.528347	+	0.849029i	\(0.322812\pi\)
\(31\)	4.20136	+	3.65357i	0.754587	+	0.656200i
							\(37\)	−7.17635	+	4.14327i	−1.17978	+	0.681149i	−0.955965	−	0.293482i	\(-0.905186\pi\)
−0.223820	+	0.974631i	\(0.571853\pi\)
\(41\)	−2.17141	−	10.2157i	−0.339118	−	1.59542i	−0.735623	−	0.677391i	\(-0.763111\pi\)
							0.396506	−	0.918032i	\(-0.370223\pi\)
\(43\)	0.525350	+	0.583460i	0.0801151	+	0.0889769i	0.781870	−	0.623442i	\(-0.214266\pi\)
							−0.701754	+	0.712419i	\(0.747600\pi\)
\(47\)	4.31066	−	1.40062i	0.628774	−	0.204301i	0.0227422	−	0.999741i	\(-0.492760\pi\)
							0.606032	+	0.795440i	\(0.292760\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bs.a.101.5	yes	288
13.4	even	6	inner	403.2.bs.a.225.32	yes	288
31.4	even	5	inner	403.2.bs.a.283.32	yes	288
403.4	even	30	inner	403.2.bs.a.4.5	✓	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bs.a.4.5	✓	288	403.4	even	30	inner
403.2.bs.a.101.5	yes	288	1.1	even	1	trivial
403.2.bs.a.225.32	yes	288	13.4	even	6	inner
403.2.bs.a.283.32	yes	288	31.4	even	5	inner