Embedded newform 403.2.bl.b.16.3

• Label: 403.2.bl.b.16.3
• 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			16.3
Character	\(\chi\)	\(=\)	403.16
Dual form			403.2.bl.b.126.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.24423 - 0.999195i) q^{2} +(0.330998 - 3.14923i) q^{3} +(2.69991 + 2.99855i) q^{4} +2.52374 q^{5} +(-3.88953 + 6.73687i) q^{6} +(2.19826 + 2.44142i) q^{7} +(-1.54481 - 4.75442i) q^{8} +(-6.87367 - 1.46104i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q + q^{2} + 39 q^{4} - 8 q^{5} - 2 q^{7} + 12 q^{8} + 29 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.24423	−	0.999195i	−1.58691	−	0.706538i	−0.591871	−	0.806033i	\(-0.701611\pi\)
							−0.995038	+	0.0994950i	\(0.968277\pi\)
\(3\)	0.330998	−	3.14923i	0.191102	−	1.81821i	−0.307687	−	0.951488i	\(-0.599555\pi\)
							0.498789	−	0.866724i	\(-0.333778\pi\)
\(5\)	2.52374			1.12865			0.564325	−	0.825552i	\(-0.309136\pi\)
							0.564325	+	0.825552i	\(0.309136\pi\)
\(7\)	2.19826	+	2.44142i	0.830865	+	0.922769i	0.998003	−	0.0631644i	\(-0.0201192\pi\)
							−0.167138	+	0.985933i	\(0.553453\pi\)
\(11\)	3.84030	−	0.816280i	1.15789	−	0.246118i	0.411364	−	0.911471i	\(-0.365052\pi\)
							0.746528	+	0.665353i	\(0.231719\pi\)
\(13\)	−3.53198	+	0.724642i	−0.979595	+	0.200980i
							\(17\)	5.56681	+	1.18326i	1.35015	+	0.286983i	0.825532	−	0.564355i	\(-0.190875\pi\)
0.524618	+	0.851338i	\(0.324208\pi\)
\(19\)	−0.647823	−	6.16362i	−0.148621	−	1.41403i	−0.773741	−	0.633503i	\(-0.781617\pi\)
							0.625120	−	0.780529i	\(-0.285050\pi\)
\(23\)	2.09597	−	2.32781i	0.437040	−	0.485383i	−0.483879	−	0.875135i	\(-0.660773\pi\)
							0.920920	+	0.389752i	\(0.127439\pi\)
\(29\)	−4.89981	−	2.18154i	−0.909872	−	0.405101i	−0.102221	−	0.994762i	\(-0.532595\pi\)
							−0.807651	+	0.589661i	\(0.799262\pi\)
\(31\)	−0.816937	−	5.50751i	−0.146726	−	0.989177i
							\(37\)	−1.06714	−	1.84835i	−0.175437	−	0.303866i	0.764875	−	0.644178i	\(-0.222801\pi\)
−0.940313	+	0.340312i	\(0.889467\pi\)
\(41\)	4.14724	+	1.84647i	0.647690	+	0.288370i	0.704168	−	0.710034i	\(-0.251320\pi\)
							−0.0564775	+	0.998404i	\(0.517987\pi\)
\(43\)	0.898527	+	8.54892i	0.137024	+	1.30370i	0.819623	+	0.572904i	\(0.194183\pi\)
							−0.682598	+	0.730794i	\(0.739150\pi\)
\(47\)	−1.19236	−	0.866298i	−0.173923	−	0.126363i	0.497418	−	0.867511i	\(-0.334282\pi\)
							−0.671341	+	0.741148i	\(0.734282\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bl.b.16.3	✓	280
13.9	even	3	inner	403.2.bl.b.295.33	yes	280
31.2	even	5	inner	403.2.bl.b.250.33	yes	280
403.126	even	15	inner	403.2.bl.b.126.3	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bl.b.16.3	✓	280	1.1	even	1	trivial
403.2.bl.b.126.3	yes	280	403.126	even	15	inner
403.2.bl.b.250.33	yes	280	31.2	even	5	inner
403.2.bl.b.295.33	yes	280	13.9	even	3	inner