Embedded newform 201.4.m.a.4.8

• Label: 201.4.m.a.4.8
• 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			4.8
Character	\(\chi\)	\(=\)	201.4
Dual form			201.4.m.a.151.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0857546 - 0.00818857i) q^{2} +(2.52376 + 1.62192i) q^{3} +(-7.84814 - 1.51261i) q^{4} +(2.32615 + 16.1787i) q^{5} +(-0.203143 - 0.159753i) q^{6} +(3.76989 - 5.29407i) q^{7} +(1.32187 + 0.388136i) q^{8} +(3.73874 + 8.18669i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 2 q^{2} - 96 q^{3} + 66 q^{4} - 4 q^{5} + 27 q^{6} + 14 q^{7} + 123 q^{8} - 288 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0857546	−	0.00818857i	−0.0303188	−	0.00289510i	0.0798856	−	0.996804i	\(-0.474544\pi\)
							−0.110204	+	0.993909i	\(0.535151\pi\)
\(3\)	2.52376	+	1.62192i	0.485698	+	0.312139i
							\(5\)	2.32615	+	16.1787i	0.208057	+	1.44707i	0.779487	+	0.626418i	\(0.215480\pi\)
−0.571430	+	0.820651i	\(0.693611\pi\)
\(7\)	3.76989	−	5.29407i	0.203555	−	0.285853i	−0.700162	−	0.713985i	\(-0.746889\pi\)
							0.903716	+	0.428132i	\(0.140828\pi\)
\(11\)	−24.9979	+	19.6586i	−0.685195	+	0.538844i	−0.898954	−	0.438043i	\(-0.855672\pi\)
							0.213759	+	0.976886i	\(0.431429\pi\)
\(13\)	8.57388	−	35.3420i	0.182920	−	0.754008i	−0.804244	−	0.594300i	\(-0.797429\pi\)
							0.987164	−	0.159709i	\(-0.0510555\pi\)
\(17\)	−102.442	+	19.7441i	−1.46152	+	0.281685i	−0.857171	−	0.515032i	\(-0.827780\pi\)
							−0.604352	+	0.796718i	\(0.706568\pi\)
\(19\)	−48.7301	−	68.4318i	−0.588392	−	0.826281i	0.407836	−	0.913055i	\(-0.366283\pi\)
							−0.996228	+	0.0867744i	\(0.972344\pi\)
\(23\)	−165.861	+	85.5073i	−1.50367	+	0.775196i	−0.996252	−	0.0865029i	\(-0.972431\pi\)
							−0.507419	+	0.861699i	\(0.669401\pi\)
\(29\)	42.7473	+	74.0406i	0.273724	+	0.474103i	0.969812	−	0.243853i	\(-0.0784113\pi\)
							−0.696089	+	0.717956i	\(0.745078\pi\)
\(31\)	−7.72238	−	31.8321i	−0.0447413	−	0.184426i	0.945056	−	0.326908i	\(-0.106007\pi\)
							−0.989797	+	0.142482i	\(0.954492\pi\)
\(37\)	−52.2821	+	90.5553i	−0.232301	+	0.402357i	−0.958485	−	0.285144i	\(-0.907959\pi\)
							0.726184	+	0.687500i	\(0.241292\pi\)
\(41\)	5.82818	−	16.8394i	0.0222002	−	0.0641434i	−0.933356	−	0.358951i	\(-0.883134\pi\)
							0.955557	+	0.294808i	\(0.0952556\pi\)
\(43\)	−144.820	−	167.131i	−0.513600	−	0.592726i	0.438417	−	0.898772i	\(-0.355539\pi\)
							−0.952017	+	0.306046i	\(0.900994\pi\)
\(47\)	−23.1293	+	485.543i	−0.0717819	+	1.50689i	0.619898	+	0.784683i	\(0.287174\pi\)
							−0.691680	+	0.722204i	\(0.743129\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.m.a.4.8	✓	320
67.17	even	33	inner	201.4.m.a.151.8	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.m.a.4.8	✓	320	1.1	even	1	trivial
201.4.m.a.151.8	yes	320	67.17	even	33	inner