Embedded newform 201.3.n.a.13.9

• Label: 201.3.n.a.13.9
• Level: $201$
• Weight: $3$
• Character: 201.13
• Analytic conductor: $5.477$
• Analytic rank: $0$
• Dimension: $220$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	201.n (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.47685331364\)
Analytic rank:	\(0\)
Dimension:	\(220\)
Relative dimension:	\(11\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			13.9
Character	\(\chi\)	\(=\)	201.13
Dual form			201.3.n.a.31.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.03156 + 1.31174i) q^{2} +(1.30900 + 1.13425i) q^{3} +(0.286498 - 1.18096i) q^{4} +(0.446140 + 0.694208i) q^{5} +(-0.137530 + 2.88711i) q^{6} +(2.69847 - 6.74045i) q^{7} +(7.91651 - 3.61535i) q^{8} +(0.426945 + 2.96946i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 26 q^{4} + 33 q^{6} + 15 q^{7} - 33 q^{8} + 66 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{19}{66}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.03156	+	1.31174i	0.515782	+	0.655870i	0.972270	−	0.233860i	\(-0.0751356\pi\)
							−0.456489	+	0.889729i	\(0.650893\pi\)
\(3\)	1.30900	+	1.13425i	0.436332	+	0.378084i
							\(5\)	0.446140	+	0.694208i	0.0892281	+	0.138842i	0.882998	−	0.469376i	\(-0.155521\pi\)
−0.793770	+	0.608218i	\(0.791885\pi\)
\(7\)	2.69847	−	6.74045i	0.385495	−	0.962921i	−0.601017	−	0.799236i	\(-0.705238\pi\)
							0.986513	−	0.163685i	\(-0.0523381\pi\)
\(11\)	11.1096	−	0.529214i	1.00996	−	0.0481104i	0.463927	−	0.885873i	\(-0.346440\pi\)
							0.546034	+	0.837763i	\(0.316137\pi\)
\(13\)	1.45869	+	15.2760i	0.112207	+	1.17508i	0.859696	+	0.510807i	\(0.170653\pi\)
							−0.747489	+	0.664274i	\(0.768741\pi\)
\(17\)	−0.443202	−	1.82690i	−0.0260707	−	0.107465i	0.957290	−	0.289131i	\(-0.0933662\pi\)
							−0.983360	+	0.181666i	\(0.941851\pi\)
\(19\)	−30.1021	+	12.0511i	−1.58432	+	0.634267i	−0.985963	−	0.166962i	\(-0.946604\pi\)
							−0.598358	+	0.801229i	\(0.704180\pi\)
\(23\)	−9.30762	−	1.79390i	−0.404679	−	0.0779955i	−0.0171499	−	0.999853i	\(-0.505459\pi\)
							−0.387529	+	0.921857i	\(0.626671\pi\)
\(29\)	1.93971	−	3.35968i	0.0668866	−	0.115851i	−0.830643	−	0.556806i	\(-0.812027\pi\)
							0.897529	+	0.440955i	\(0.145360\pi\)
\(31\)	−2.19975	+	23.0368i	−0.0709595	+	0.743122i	0.888050	+	0.459747i	\(0.152060\pi\)
							−0.959009	+	0.283375i	\(0.908546\pi\)
\(37\)	−22.8344	−	39.5503i	−0.617145	−	1.06893i	−0.990004	−	0.141039i	\(-0.954956\pi\)
							0.372859	−	0.927888i	\(-0.378378\pi\)
\(41\)	7.51461	+	7.88109i	0.183283	+	0.192222i	0.808966	−	0.587856i	\(-0.200028\pi\)
							−0.625682	+	0.780078i	\(0.715179\pi\)
\(43\)	−16.4922	+	56.1674i	−0.383540	+	1.30622i	0.511135	+	0.859501i	\(0.329225\pi\)
							−0.894675	+	0.446718i	\(0.852593\pi\)
\(47\)	6.84098	+	19.7657i	0.145553	+	0.420547i	0.994086	−	0.108595i	\(-0.0346351\pi\)
							−0.848533	+	0.529142i	\(0.822514\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.n.a.13.9	✓	220
67.31	odd	66	inner	201.3.n.a.31.9	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.n.a.13.9	✓	220	1.1	even	1	trivial
201.3.n.a.31.9	yes	220	67.31	odd	66	inner