Embedded newform 403.2.bs.a.4.4

• Label: 403.2.bs.a.4.4
• 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			4.4
Character	\(\chi\)	\(=\)	403.4
Dual form			403.2.bs.a.101.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.502235 + 2.36283i) q^{2} +(0.181860 + 0.201976i) q^{3} +(-3.50363 - 1.55992i) q^{4} -2.70808i q^{5} +(-0.568571 + 0.328265i) q^{6} +(-0.369305 + 0.829472i) q^{7} +(2.60573 - 3.58648i) q^{8} +(0.305864 - 2.91010i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 9 q^{2} - q^{3} - 39 q^{4} - 42 q^{6} - 15 q^{7} + 31 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.502235	+	2.36283i	−0.355134	+	1.67077i	0.331258	+	0.943540i	\(0.392527\pi\)
							−0.686392	+	0.727232i	\(0.740806\pi\)
\(3\)	0.181860	+	0.201976i	0.104997	+	0.116611i	0.793353	−	0.608763i	\(-0.208334\pi\)
							−0.688356	+	0.725373i	\(0.741667\pi\)
\(5\)		−	2.70808i		−	1.21109i	−0.795810	−	0.605546i	\(-0.792955\pi\)
							0.795810	−	0.605546i	\(-0.207045\pi\)
\(7\)	−0.369305	+	0.829472i	−0.139584	+	0.313511i	−0.969788	−	0.243949i	\(-0.921557\pi\)
							0.830204	+	0.557459i	\(0.188224\pi\)
\(11\)	3.33873	−	0.350915i	1.00667	−	0.105805i	0.413180	−	0.910649i	\(-0.364418\pi\)
							0.593486	+	0.804844i	\(0.297751\pi\)
\(13\)	0.564544	−	3.56108i	0.156576	−	0.987666i
							\(17\)	−0.346673	+	3.29837i	−0.0840805	+	0.799973i	0.868501	+	0.495688i	\(0.165084\pi\)
−0.952581	+	0.304285i	\(0.901583\pi\)
\(19\)	2.06233	+	1.85693i	0.473131	+	0.426009i	0.870888	−	0.491482i	\(-0.163545\pi\)
							−0.397757	+	0.917491i	\(0.630211\pi\)
\(23\)	6.12777	−	2.72826i	1.27773	−	0.568881i	0.348127	−	0.937447i	\(-0.386818\pi\)
							0.929602	+	0.368566i	\(0.120151\pi\)
\(29\)	−5.27619	−	1.12149i	−0.979764	−	0.208255i	−0.309924	−	0.950761i	\(-0.600304\pi\)
							−0.669839	+	0.742506i	\(0.733637\pi\)
\(31\)	−0.522466	−	5.54320i	−0.0938377	−	0.995588i
							\(37\)	0.883690	+	0.510198i	0.145278	+	0.0838761i	0.570877	−	0.821036i	\(-0.306603\pi\)
−0.425599	+	0.904912i	\(0.639937\pi\)
\(41\)	−0.104186	+	0.490157i	−0.0162711	+	0.0765496i	−0.985534	−	0.169478i	\(-0.945792\pi\)
							0.969263	+	0.246027i	\(0.0791253\pi\)
\(43\)	−1.24301	+	1.38051i	−0.189558	+	0.210525i	−0.830431	−	0.557122i	\(-0.811906\pi\)
							0.640873	+	0.767647i	\(0.278572\pi\)
\(47\)	−0.594623	−	0.193205i	−0.0867347	−	0.0281818i	0.265328	−	0.964158i	\(-0.414520\pi\)
							−0.352063	+	0.935976i	\(0.614520\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bs.a.4.4	✓	288
13.10	even	6	inner	403.2.bs.a.283.33	yes	288
31.8	even	5	inner	403.2.bs.a.225.33	yes	288
403.101	even	30	inner	403.2.bs.a.101.4	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bs.a.4.4	✓	288	1.1	even	1	trivial
403.2.bs.a.101.4	yes	288	403.101	even	30	inner
403.2.bs.a.225.33	yes	288	31.8	even	5	inner
403.2.bs.a.283.33	yes	288	13.10	even	6	inner