Embedded newform 201.4.m.a.4.5

• Label: 201.4.m.a.4.5
• 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			4.5
Character	\(\chi\)	\(=\)	201.4
Dual form			201.4.m.a.151.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.05732 - 0.291939i) q^{2} +(2.52376 + 1.62192i) q^{3} +(1.40655 + 0.271090i) q^{4} +(-1.09359 - 7.60610i) q^{5} +(-7.24244 - 5.69552i) q^{6} +(20.0509 - 28.1576i) q^{7} +(19.3534 + 5.68268i) q^{8} +(3.73874 + 8.18669i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 2 q^{2} - 96 q^{3} + 66 q^{4} - 4 q^{5} + 27 q^{6} + 14 q^{7} + 123 q^{8} - 288 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.05732	−	0.291939i	−1.08093	−	0.103216i	−0.460636	−	0.887589i	\(-0.652379\pi\)
							−0.620289	+	0.784373i	\(0.712985\pi\)
\(3\)	2.52376	+	1.62192i	0.485698	+	0.312139i
							\(5\)	−1.09359	−	7.60610i	−0.0978139	−	0.680310i	−0.978445	−	0.206508i	\(-0.933790\pi\)
0.880631	−	0.473803i	\(-0.157119\pi\)
\(7\)	20.0509	−	28.1576i	1.08265	−	1.52036i	0.248652	−	0.968593i	\(-0.420013\pi\)
							0.833995	−	0.551772i	\(-0.186048\pi\)
\(11\)	−19.3123	+	15.1873i	−0.529352	+	0.416287i	−0.846738	−	0.532009i	\(-0.821437\pi\)
							0.317387	+	0.948296i	\(0.397195\pi\)
\(13\)	−8.37113	+	34.5063i	−0.178595	+	0.736178i	0.810107	+	0.586282i	\(0.199409\pi\)
							−0.988702	+	0.149896i	\(0.952106\pi\)
\(17\)	6.37477	−	1.22864i	0.0909475	−	0.0175287i	−0.143574	−	0.989640i	\(-0.545860\pi\)
							0.234522	+	0.972111i	\(0.424648\pi\)
\(19\)	−75.5545	−	106.101i	−0.912284	−	1.28112i	−0.959346	−	0.282231i	\(-0.908925\pi\)
							0.0470626	−	0.998892i	\(-0.485014\pi\)
\(23\)	9.84251	−	5.07417i	0.0892307	−	0.0460016i	−0.413034	−	0.910716i	\(-0.635531\pi\)
							0.502264	+	0.864714i	\(0.332500\pi\)
\(29\)	16.4662	+	28.5202i	0.105438	+	0.182623i	0.913917	−	0.405901i	\(-0.133042\pi\)
							−0.808479	+	0.588525i	\(0.799709\pi\)
\(31\)	−46.4132	−	191.318i	−0.268905	−	1.10844i	−0.932736	−	0.360559i	\(-0.882586\pi\)
							0.663831	−	0.747882i	\(-0.268929\pi\)
\(37\)	121.539	−	210.512i	0.540024	−	0.935350i	−0.458878	−	0.888500i	\(-0.651748\pi\)
							0.998902	−	0.0468502i	\(-0.0149183\pi\)
\(41\)	95.5236	−	275.997i	0.363860	−	1.05131i	−0.603382	−	0.797452i	\(-0.706181\pi\)
							0.967243	−	0.253853i	\(-0.0816981\pi\)
\(43\)	−139.933	−	161.491i	−0.496270	−	0.572726i	0.451260	−	0.892392i	\(-0.350975\pi\)
							−0.947530	+	0.319666i	\(0.896429\pi\)
\(47\)	−7.25421	+	152.285i	−0.0225135	+	0.472617i	0.959256	+	0.282540i	\(0.0911769\pi\)
							−0.981769	+	0.190077i	\(0.939126\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.m.a.4.5	✓	320
67.17	even	33	inner	201.4.m.a.151.5	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.m.a.4.5	✓	320	1.1	even	1	trivial
201.4.m.a.151.5	yes	320	67.17	even	33	inner