Embedded newform 403.2.bl.b.16.6

• Label: 403.2.bl.b.16.6
• 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			16.6
Character	\(\chi\)	\(=\)	403.16
Dual form			403.2.bl.b.126.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.74019 - 0.774780i) q^{2} +(-0.239952 + 2.28299i) q^{3} +(1.08970 + 1.21023i) q^{4} +3.51072 q^{5} +(2.18638 - 3.78692i) q^{6} +(1.97153 + 2.18961i) q^{7} +(0.218662 + 0.672972i) q^{8} +(-2.22004 - 0.471884i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q + q^{2} + 39 q^{4} - 8 q^{5} - 2 q^{7} + 12 q^{8} + 29 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.74019	−	0.774780i	−1.23050	−	0.547852i	−0.314584	−	0.949230i	\(-0.601865\pi\)
							−0.915913	+	0.401377i	\(0.868532\pi\)
\(3\)	−0.239952	+	2.28299i	−0.138537	+	1.31809i	0.675537	+	0.737326i	\(0.263912\pi\)
							−0.814074	+	0.580761i	\(0.802755\pi\)
\(5\)	3.51072			1.57004			0.785021	−	0.619469i	\(-0.212652\pi\)
							0.785021	+	0.619469i	\(0.212652\pi\)
\(7\)	1.97153	+	2.18961i	0.745170	+	0.827595i	0.989866	−	0.142005i	\(-0.0453548\pi\)
							−0.244696	+	0.969600i	\(0.578688\pi\)
\(11\)	3.45353	−	0.734070i	1.04128	−	0.221331i	0.344626	−	0.938740i	\(-0.388006\pi\)
							0.696652	+	0.717409i	\(0.254672\pi\)
\(13\)	2.27365	−	2.79831i	0.630596	−	0.776111i
							\(17\)	−6.08742	−	1.29392i	−1.47642	−	0.313822i	−0.601803	−	0.798644i	\(-0.705551\pi\)
−0.874613	+	0.484823i	\(0.838884\pi\)
\(19\)	−0.154003	−	1.46524i	−0.0353307	−	0.336149i	−0.997882	−	0.0650495i	\(-0.979279\pi\)
							0.962551	−	0.271099i	\(-0.0873872\pi\)
\(23\)	0.858693	−	0.953675i	0.179050	−	0.198855i	−0.646939	−	0.762542i	\(-0.723951\pi\)
							0.825988	+	0.563687i	\(0.190618\pi\)
\(29\)	3.45688	+	1.53910i	0.641927	+	0.285804i	0.701770	−	0.712403i	\(-0.252393\pi\)
							−0.0598431	+	0.998208i	\(0.519060\pi\)
\(31\)	2.87974	−	4.76520i	0.517216	−	0.855855i
							\(37\)	1.83185	+	3.17285i	0.301154	+	0.521614i	0.976398	−	0.215981i	\(-0.0692949\pi\)
−0.675244	+	0.737595i	\(0.735962\pi\)
\(41\)	−10.7968	−	4.80704i	−1.68618	−	0.750734i	−0.999722	−	0.0235608i	\(-0.992500\pi\)
							−0.686454	−	0.727173i	\(-0.740834\pi\)
\(43\)	1.32672	+	12.6229i	0.202323	+	1.92497i	0.351546	+	0.936170i	\(0.385656\pi\)
							−0.149224	+	0.988803i	\(0.547677\pi\)
\(47\)	−1.14888	−	0.834707i	−0.167581	−	0.121754i	0.500834	−	0.865543i	\(-0.333027\pi\)
							−0.668415	+	0.743789i	\(0.733027\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bl.b.16.6	✓	280
13.9	even	3	inner	403.2.bl.b.295.30	yes	280
31.2	even	5	inner	403.2.bl.b.250.30	yes	280
403.126	even	15	inner	403.2.bl.b.126.6	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bl.b.16.6	✓	280	1.1	even	1	trivial
403.2.bl.b.126.6	yes	280	403.126	even	15	inner
403.2.bl.b.250.30	yes	280	31.2	even	5	inner
403.2.bl.b.295.30	yes	280	13.9	even	3	inner