Embedded newform 201.4.m.a.4.12

• Label: 201.4.m.a.4.12
• 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			4.12
Character	\(\chi\)	\(=\)	201.4
Dual form			201.4.m.a.151.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.82303 + 0.269566i) q^{2} +(2.52376 + 1.62192i) q^{3} +(0.0413796 + 0.00797527i) q^{4} +(-2.49944 - 17.3840i) q^{5} +(6.68743 + 5.25905i) q^{6} +(-2.70946 + 3.80491i) q^{7} +(-21.6533 - 6.35798i) q^{8} +(3.73874 + 8.18669i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 2 q^{2} - 96 q^{3} + 66 q^{4} - 4 q^{5} + 27 q^{6} + 14 q^{7} + 123 q^{8} - 288 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.82303	+	0.269566i	0.998090	+	0.0953061i	0.581294	−	0.813694i	\(-0.302547\pi\)
							0.416797	+	0.909000i	\(0.363153\pi\)
\(3\)	2.52376	+	1.62192i	0.485698	+	0.312139i
							\(5\)	−2.49944	−	17.3840i	−0.223556	−	1.55487i	−0.724430	−	0.689348i	\(-0.757897\pi\)
0.500874	−	0.865520i	\(-0.333012\pi\)
\(7\)	−2.70946	+	3.80491i	−0.146297	+	0.205446i	−0.881222	−	0.472703i	\(-0.843278\pi\)
							0.734924	+	0.678149i	\(0.237218\pi\)
\(11\)	27.8911	−	21.9338i	0.764500	−	0.601209i	−0.157896	−	0.987456i	\(-0.550471\pi\)
							0.922396	+	0.386247i	\(0.126229\pi\)
\(13\)	14.0660	−	57.9807i	0.300092	−	1.23700i	−0.599538	−	0.800346i	\(-0.704649\pi\)
							0.899631	−	0.436651i	\(-0.143836\pi\)
\(17\)	45.1908	−	8.70981i	0.644728	−	0.124261i	0.143602	−	0.989636i	\(-0.454131\pi\)
							0.501126	+	0.865374i	\(0.332919\pi\)
\(19\)	−50.2715	−	70.5965i	−0.607004	−	0.852418i	0.390674	−	0.920529i	\(-0.372242\pi\)
							−0.997678	+	0.0681114i	\(0.978303\pi\)
\(23\)	−41.5943	+	21.4434i	−0.377088	+	0.194402i	−0.636342	−	0.771407i	\(-0.719553\pi\)
							0.259255	+	0.965809i	\(0.416523\pi\)
\(29\)	78.1888	+	135.427i	0.500665	+	0.867178i	1.00000		0.000768383i	\(0.000244584\pi\)
							−0.499334	+	0.866409i	\(0.666422\pi\)
\(31\)	11.1716	+	46.0501i	0.0647253	+	0.266802i	0.994975	−	0.100126i	\(-0.0319247\pi\)
							−0.930249	+	0.366928i	\(0.880410\pi\)
\(37\)	81.6175	−	141.366i	0.362644	−	0.628118i	−0.625751	−	0.780023i	\(-0.715207\pi\)
							0.988395	+	0.151905i	\(0.0485407\pi\)
\(41\)	85.3875	−	246.711i	0.325251	−	0.939751i	−0.657279	−	0.753647i	\(-0.728293\pi\)
							0.982530	−	0.186104i	\(-0.0595860\pi\)
\(43\)	205.993	+	237.729i	0.730550	+	0.843100i	0.992534	−	0.121972i	\(-0.0389219\pi\)
							−0.261983	+	0.965072i	\(0.584376\pi\)
\(47\)	−16.0182	+	336.264i	−0.0497127	+	1.04360i	0.826884	+	0.562372i	\(0.190111\pi\)
							−0.876597	+	0.481226i	\(0.840192\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.m.a.4.12	✓	320
67.17	even	33	inner	201.4.m.a.151.12	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.m.a.4.12	✓	320	1.1	even	1	trivial
201.4.m.a.151.12	yes	320	67.17	even	33	inner