Embedded newform 201.4.m.a.151.13

• Label: 201.4.m.a.151.13
• Level: $201$
• Weight: $4$
• Character: 201.151
• 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			151.13
Character	\(\chi\)	\(=\)	201.151
Dual form			201.4.m.a.4.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.84022 - 0.271208i) q^{2} +(2.52376 - 1.62192i) q^{3} +(0.137874 - 0.0265731i) q^{4} +(1.10231 - 7.66675i) q^{5} +(6.72816 - 5.29108i) q^{6} +(-17.6977 - 24.8529i) q^{7} +(-21.5162 + 6.31771i) q^{8} +(3.73874 - 8.18669i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 2 q^{2} - 96 q^{3} + 66 q^{4} - 4 q^{5} + 27 q^{6} + 14 q^{7} + 123 q^{8} - 288 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{32}{33}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.84022	−	0.271208i	1.00417	−	0.0958866i	0.420005	−	0.907522i	\(-0.362028\pi\)
							0.584165	+	0.811635i	\(0.301422\pi\)
\(3\)	2.52376	−	1.62192i	0.485698	−	0.312139i
							\(5\)	1.10231	−	7.66675i	0.0985938	−	0.685735i	−0.879244	−	0.476371i	\(-0.841952\pi\)
0.977838	−	0.209363i	\(-0.0671391\pi\)
\(7\)	−17.6977	−	24.8529i	−0.955585	−	1.34193i	−0.939679	−	0.342058i	\(-0.888876\pi\)
							−0.0159057	−	0.999873i	\(-0.505063\pi\)
\(11\)	11.3562	+	8.93063i	0.311275	+	0.244790i	0.761558	−	0.648096i	\(-0.224435\pi\)
							−0.450283	+	0.892886i	\(0.648677\pi\)
\(13\)	1.07011	+	4.41104i	0.0228303	+	0.0941079i	0.982087	−	0.188428i	\(-0.0603394\pi\)
							−0.959257	+	0.282536i	\(0.908824\pi\)
\(17\)	−124.276	−	23.9522i	−1.77302	−	0.341722i	−0.805130	−	0.593098i	\(-0.797905\pi\)
							−0.967889	+	0.251376i	\(0.919117\pi\)
\(19\)	63.5184	−	89.1991i	0.766954	−	1.07704i	−0.227696	−	0.973732i	\(-0.573119\pi\)
							0.994650	−	0.103304i	\(-0.0329414\pi\)
\(23\)	118.349	+	61.0131i	1.07293	+	0.553135i	0.901818	−	0.432116i	\(-0.142233\pi\)
							0.171114	+	0.985251i	\(0.445263\pi\)
\(29\)	143.225	−	248.073i	0.917110	−	1.58848i	0.113328	−	0.993558i	\(-0.463849\pi\)
							0.803782	−	0.594924i	\(-0.202818\pi\)
\(31\)	−24.6337	+	101.542i	−0.142721	+	0.588303i	0.854865	+	0.518851i	\(0.173640\pi\)
							−0.997585	+	0.0694518i	\(0.977875\pi\)
\(37\)	−1.20350	−	2.08452i	−0.00534740	−	0.00926196i	0.863339	−	0.504624i	\(-0.168369\pi\)
							−0.868687	+	0.495362i	\(0.835035\pi\)
\(41\)	−72.9462	−	210.764i	−0.277861	−	0.802826i	−0.994439	−	0.105318i	\(-0.966414\pi\)
							0.716578	−	0.697507i	\(-0.245707\pi\)
\(43\)	240.974	−	278.099i	0.854611	−	0.986273i	−0.145384	−	0.989375i	\(-0.546442\pi\)
							0.999995	+	0.00310189i	\(0.000987363\pi\)
\(47\)	14.9688	+	314.233i	0.0464558	+	0.975227i	0.894990	+	0.446086i	\(0.147183\pi\)
							−0.848534	+	0.529141i	\(0.822514\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.m.a.151.13	yes	320
67.4	even	33	inner	201.4.m.a.4.13	✓	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.m.a.4.13	✓	320	67.4	even	33	inner
201.4.m.a.151.13	yes	320	1.1	even	1	trivial