Embedded newform 201.4.m.a.4.4

• Label: 201.4.m.a.4.4
• Level: $201$
• Weight: $4$
• Character: 201.4
• Analytic conductor: $11.859$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	201.m (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8593839112\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(16\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			4.4
Character	\(\chi\)	\(=\)	201.4
Dual form			201.4.m.a.151.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.06689 - 0.292852i) q^{2} +(2.52376 + 1.62192i) q^{3} +(1.46462 + 0.282282i) q^{4} +(1.97722 + 13.7518i) q^{5} +(-7.26511 - 5.71335i) q^{6} +(4.23013 - 5.94039i) q^{7} +(19.2392 + 5.64914i) q^{8} +(3.73874 + 8.18669i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 2 q^{2} - 96 q^{3} + 66 q^{4} - 4 q^{5} + 27 q^{6} + 14 q^{7} + 123 q^{8} - 288 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/201\mathbb{Z}\right)^\times\).

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{33}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.06689	−	0.292852i	−1.08431	−	0.103539i	−0.462433	−	0.886654i	\(-0.653023\pi\)
							−0.621876	+	0.783115i	\(0.713629\pi\)
\(3\)	2.52376	+	1.62192i	0.485698	+	0.312139i
							\(5\)	1.97722	+	13.7518i	0.176848	+	1.23000i	0.864001	+	0.503490i	\(0.167951\pi\)
−0.687153	+	0.726512i	\(0.741140\pi\)
\(7\)	4.23013	−	5.94039i	0.228406	−	0.320751i	−0.684430	−	0.729079i	\(-0.739949\pi\)
							0.912835	+	0.408328i	\(0.133888\pi\)
\(11\)	36.6280	−	28.8046i	1.00398	−	0.789537i	0.0263019	−	0.999654i	\(-0.491627\pi\)
							0.977676	+	0.210117i	\(0.0673845\pi\)
\(13\)	−9.73918	+	40.1454i	−0.207782	+	0.856488i	0.768477	+	0.639878i	\(0.221015\pi\)
							−0.976258	+	0.216610i	\(0.930500\pi\)
\(17\)	−12.3522	+	2.38070i	−0.176227	+	0.0339650i	−0.276601	−	0.960985i	\(-0.589208\pi\)
							0.100374	+	0.994950i	\(0.467996\pi\)
\(19\)	21.9079	+	30.7654i	0.264527	+	0.371477i	0.925381	−	0.379038i	\(-0.123745\pi\)
							−0.660854	+	0.750515i	\(0.729806\pi\)
\(23\)	13.4399	−	6.92873i	0.121844	−	0.0628148i	−0.396227	−	0.918153i	\(-0.629681\pi\)
							0.518071	+	0.855338i	\(0.326650\pi\)
\(29\)	−19.4819	−	33.7436i	−0.124748	−	0.216070i	0.796886	−	0.604129i	\(-0.206479\pi\)
							−0.921634	+	0.388059i	\(0.873146\pi\)
\(31\)	46.3870	+	191.210i	0.268753	+	1.10782i	0.932877	+	0.360194i	\(0.117289\pi\)
							−0.664124	+	0.747622i	\(0.731195\pi\)
\(37\)	−200.783	+	347.766i	−0.892121	+	1.54520i	−0.0547936	+	0.998498i	\(0.517450\pi\)
							−0.837328	+	0.546701i	\(0.815883\pi\)
\(41\)	−58.2105	+	168.188i	−0.221730	+	0.640648i	0.778190	+	0.628029i	\(0.216138\pi\)
							−0.999920	+	0.0126189i	\(0.995983\pi\)
\(43\)	144.168	+	166.379i	0.511289	+	0.590059i	0.951428	−	0.307871i	\(-0.0996165\pi\)
							−0.440139	+	0.897929i	\(0.645071\pi\)
\(47\)	3.01802	−	63.3561i	0.00936647	−	0.196626i	−0.989334	−	0.145664i	\(-0.953468\pi\)
							0.998701	−	0.0509623i	\(-0.0162288\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.m.a.4.4	✓	320
67.17	even	33	inner	201.4.m.a.151.4	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.m.a.4.4	✓	320	1.1	even	1	trivial
201.4.m.a.151.4	yes	320	67.17	even	33	inner