Embedded newform 403.2.bl.b.16.11

• Label: 403.2.bl.b.16.11
• Level: $403$
• Weight: $2$
• Character: 403.16
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $280$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bl (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			16.11
Character	\(\chi\)	\(=\)	403.16
Dual form			403.2.bl.b.126.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18678 - 0.528387i) q^{2} +(0.262692 - 2.49935i) q^{3} +(-0.209017 - 0.232137i) q^{4} -0.0704053 q^{5} +(-1.63238 + 2.82736i) q^{6} +(-3.21641 - 3.57218i) q^{7} +(0.928279 + 2.85695i) q^{8} +(-3.24328 - 0.689381i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q + q^{2} + 39 q^{4} - 8 q^{5} - 2 q^{7} + 12 q^{8} + 29 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18678	−	0.528387i	−0.839177	−	0.373626i	−0.0582914	−	0.998300i	\(-0.518565\pi\)
							−0.780886	+	0.624674i	\(0.785232\pi\)
\(3\)	0.262692	−	2.49935i	0.151665	−	1.44300i	−0.608647	−	0.793441i	\(-0.708288\pi\)
							0.760313	−	0.649557i	\(-0.225046\pi\)
\(5\)	−0.0704053			−0.0314862			−0.0157431	−	0.999876i	\(-0.505011\pi\)
							−0.0157431	+	0.999876i	\(0.505011\pi\)
\(7\)	−3.21641	−	3.57218i	−1.21569	−	1.35016i	−0.918541	−	0.395326i	\(-0.870632\pi\)
							−0.297147	−	0.954832i	\(-0.596035\pi\)
\(11\)	−1.13650	+	0.241571i	−0.342669	+	0.0728365i	−0.376032	−	0.926607i	\(-0.622712\pi\)
							0.0333630	+	0.999443i	\(0.489378\pi\)
\(13\)	−1.16983	+	3.41050i	−0.324453	+	0.945902i
							\(17\)	3.33824	+	0.709564i	0.809641	+	0.172095i	0.594091	−	0.804398i	\(-0.297512\pi\)
0.215551	+	0.976493i	\(0.430845\pi\)
\(19\)	−0.450778	−	4.28887i	−0.103416	−	0.983934i	−0.916024	−	0.401124i	\(-0.868620\pi\)
							0.812608	−	0.582810i	\(-0.198047\pi\)
\(23\)	3.06679	−	3.40602i	0.639471	−	0.710204i	−0.333079	−	0.942899i	\(-0.608088\pi\)
							0.972550	+	0.232695i	\(0.0747543\pi\)
\(29\)	1.26800	+	0.564551i	0.235462	+	0.104835i	0.521076	−	0.853510i	\(-0.325531\pi\)
							−0.285614	+	0.958345i	\(0.592197\pi\)
\(31\)	1.97723	+	5.20486i	0.355120	+	0.934821i
							\(37\)	1.88949	+	3.27269i	0.310630	+	0.538027i	0.978499	−	0.206252i	\(-0.0661266\pi\)
−0.667869	+	0.744279i	\(0.732793\pi\)
\(41\)	−10.9559	−	4.87790i	−1.71103	−	0.761800i	−0.998168	−	0.0604951i	\(-0.980732\pi\)
							−0.712862	−	0.701305i	\(-0.752601\pi\)
\(43\)	0.765593	+	7.28413i	0.116752	+	1.11082i	0.883357	+	0.468700i	\(0.155278\pi\)
							−0.766605	+	0.642119i	\(0.778056\pi\)
\(47\)	−1.44472	−	1.04965i	−0.210734	−	0.153107i	0.477412	−	0.878680i	\(-0.341575\pi\)
							−0.688146	+	0.725572i	\(0.741575\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bl.b.16.11	✓	280
13.9	even	3	inner	403.2.bl.b.295.25	yes	280
31.2	even	5	inner	403.2.bl.b.250.25	yes	280
403.126	even	15	inner	403.2.bl.b.126.11	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bl.b.16.11	✓	280	1.1	even	1	trivial
403.2.bl.b.126.11	yes	280	403.126	even	15	inner
403.2.bl.b.250.25	yes	280	31.2	even	5	inner
403.2.bl.b.295.25	yes	280	13.9	even	3	inner