Embedded newform 403.2.bl.b.16.2

• Label: 403.2.bl.b.16.2
• 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			16.2
Character	\(\chi\)	\(=\)	403.16
Dual form			403.2.bl.b.126.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.39574 - 1.06665i) q^{2} +(-0.206334 + 1.96314i) q^{3} +(3.26354 + 3.62453i) q^{4} +1.57328 q^{5} +(2.58831 - 4.48308i) q^{6} +(-2.63168 - 2.92278i) q^{7} +(-2.33171 - 7.17627i) q^{8} +(-0.876907 - 0.186392i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q + q^{2} + 39 q^{4} - 8 q^{5} - 2 q^{7} + 12 q^{8} + 29 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.39574	−	1.06665i	−1.69404	−	0.754235i	−0.999391	−	0.0348870i	\(-0.988893\pi\)
							−0.694649	−	0.719348i	\(-0.744440\pi\)
\(3\)	−0.206334	+	1.96314i	−0.119127	+	1.13342i	0.757696	+	0.652608i	\(0.226325\pi\)
							−0.876823	+	0.480813i	\(0.840342\pi\)
\(5\)	1.57328			0.703593			0.351796	−	0.936077i	\(-0.385571\pi\)
							0.351796	+	0.936077i	\(0.385571\pi\)
\(7\)	−2.63168	−	2.92278i	−0.994682	−	1.10471i	−0.994503	−	0.104707i	\(-0.966610\pi\)
							−0.000179220	−	1.00000i	\(-0.500057\pi\)
\(11\)	−1.71673	+	0.364902i	−0.517614	+	0.110022i	−0.459309	−	0.888277i	\(-0.651903\pi\)
							−0.0583049	+	0.998299i	\(0.518570\pi\)
\(13\)	−2.82309	−	2.24280i	−0.782984	−	0.622042i
							\(17\)	0.480691	+	0.102174i	0.116585	+	0.0247809i	0.265835	−	0.964019i	\(-0.414353\pi\)
−0.149250	+	0.988800i	\(0.547686\pi\)
\(19\)	−0.652164	−	6.20493i	−0.149617	−	1.42351i	−0.769416	−	0.638747i	\(-0.779453\pi\)
							0.619800	−	0.784760i	\(-0.287214\pi\)
\(23\)	5.24738	−	5.82781i	1.09416	−	1.21518i	0.119179	−	0.992873i	\(-0.461974\pi\)
							0.974976	−	0.222310i	\(-0.0713596\pi\)
\(29\)	3.02882	+	1.34852i	0.562437	+	0.250413i	0.668204	−	0.743978i	\(-0.267063\pi\)
							−0.105767	+	0.994391i	\(0.533730\pi\)
\(31\)	−5.54975	+	0.447564i	−0.996764	+	0.0803848i
							\(37\)	−1.36210	−	2.35923i	−0.223928	−	0.387854i	0.732069	−	0.681230i	\(-0.238555\pi\)
−0.955997	+	0.293376i	\(0.905221\pi\)
\(41\)	7.76129	+	3.45555i	1.21211	+	0.539666i	0.910398	−	0.413734i	\(-0.135775\pi\)
							0.301711	+	0.953399i	\(0.402442\pi\)
\(43\)	−0.708033	−	6.73648i	−0.107974	−	1.02730i	−0.905595	−	0.424145i	\(-0.860575\pi\)
							0.797620	−	0.603160i	\(-0.206092\pi\)
\(47\)	−3.01126	−	2.18781i	−0.439237	−	0.319124i	0.346095	−	0.938200i	\(-0.387508\pi\)
							−0.785332	+	0.619075i	\(0.787508\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bl.b.16.2	✓	280
13.9	even	3	inner	403.2.bl.b.295.34	yes	280
31.2	even	5	inner	403.2.bl.b.250.34	yes	280
403.126	even	15	inner	403.2.bl.b.126.2	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bl.b.16.2	✓	280	1.1	even	1	trivial
403.2.bl.b.126.2	yes	280	403.126	even	15	inner
403.2.bl.b.250.34	yes	280	31.2	even	5	inner
403.2.bl.b.295.34	yes	280	13.9	even	3	inner