Embedded newform 403.2.bs.a.4.13

• Label: 403.2.bs.a.4.13
• 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			4.13
Character	\(\chi\)	\(=\)	403.4
Dual form			403.2.bs.a.101.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.230096 + 1.08252i) q^{2} +(0.418891 + 0.465225i) q^{3} +(0.708195 + 0.315309i) q^{4} -1.86982i q^{5} +(-0.599999 + 0.346409i) q^{6} +(-2.06505 + 4.63818i) q^{7} +(-1.80528 + 2.48476i) q^{8} +(0.272620 - 2.59381i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 9 q^{2} - q^{3} - 39 q^{4} - 42 q^{6} - 15 q^{7} + 31 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.230096	+	1.08252i	−0.162702	+	0.765454i	0.818811	+	0.574063i	\(0.194633\pi\)
							−0.981514	+	0.191392i	\(0.938700\pi\)
\(3\)	0.418891	+	0.465225i	0.241847	+	0.268598i	0.851832	−	0.523814i	\(-0.175491\pi\)
							−0.609986	+	0.792412i	\(0.708825\pi\)
\(5\)		−	1.86982i		−	0.836208i	−0.908399	−	0.418104i	\(-0.862695\pi\)
							0.908399	−	0.418104i	\(-0.137305\pi\)
\(7\)	−2.06505	+	4.63818i	−0.780516	+	1.75307i	−0.132860	+	0.991135i	\(0.542416\pi\)
							−0.647656	+	0.761933i	\(0.724251\pi\)
\(11\)	−3.75668	+	0.394843i	−1.13268	+	0.119050i	−0.652266	−	0.757990i	\(-0.726181\pi\)
							−0.480417	+	0.877040i	\(0.659515\pi\)
\(13\)	1.50612	+	3.27591i	0.417723	+	0.908574i
							\(17\)	−0.577548	+	5.49500i	−0.140076	+	1.33273i	0.668223	+	0.743961i	\(0.267055\pi\)
−0.808299	+	0.588772i	\(0.799611\pi\)
\(19\)	2.66944	+	2.40357i	0.612410	+	0.551417i	0.915894	−	0.401420i	\(-0.131483\pi\)
							−0.303484	+	0.952837i	\(0.598150\pi\)
\(23\)	3.75781	−	1.67308i	0.783557	−	0.348862i	0.0243434	−	0.999704i	\(-0.492250\pi\)
							0.759214	+	0.650842i	\(0.225584\pi\)
\(29\)	−2.32166	−	0.493485i	−0.431122	−	0.0916378i	−0.0127609	−	0.999919i	\(-0.504062\pi\)
							−0.418361	+	0.908281i	\(0.637395\pi\)
\(31\)	5.00354	+	2.44224i	0.898663	+	0.438640i
							\(37\)	−1.43140	−	0.826420i	−0.235321	−	0.135863i	0.377703	−	0.925927i	\(-0.376714\pi\)
−0.613024	+	0.790064i	\(0.710047\pi\)
\(41\)	2.52485	−	11.8785i	0.394316	−	1.85511i	−0.113349	−	0.993555i	\(-0.536158\pi\)
							0.507664	−	0.861555i	\(-0.330509\pi\)
\(43\)	3.17631	−	3.52765i	0.484383	−	0.537961i	−0.450567	−	0.892743i	\(-0.648778\pi\)
							0.934949	+	0.354781i	\(0.115445\pi\)
\(47\)	−9.22439	−	2.99718i	−1.34551	−	0.437184i	−0.454334	−	0.890832i	\(-0.650123\pi\)
							−0.891181	+	0.453647i	\(0.850123\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bs.a.4.13	✓	288
13.10	even	6	inner	403.2.bs.a.283.24	yes	288
31.8	even	5	inner	403.2.bs.a.225.24	yes	288
403.101	even	30	inner	403.2.bs.a.101.13	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bs.a.4.13	✓	288	1.1	even	1	trivial
403.2.bs.a.101.13	yes	288	403.101	even	30	inner
403.2.bs.a.225.24	yes	288	31.8	even	5	inner
403.2.bs.a.283.24	yes	288	13.10	even	6	inner