Embedded newform 201.2.m.b.10.4

• Label: 201.2.m.b.10.4
• Level: $201$
• Weight: $2$
• Character: 201.10
• Analytic conductor: $1.605$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	201.m (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.60499308063\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(6\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			10.4
Character	\(\chi\)	\(=\)	201.10
Dual form			201.2.m.b.181.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.453484 + 0.432396i) q^{2} +(-0.142315 - 0.989821i) q^{3} +(-0.0764825 - 1.60557i) q^{4} +(-0.976375 + 2.13796i) q^{5} +(0.363457 - 0.510404i) q^{6} +(1.16821 - 4.81544i) q^{7} +(1.48021 - 1.70826i) q^{8} +(-0.959493 + 0.281733i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + 2 q^{2} - 12 q^{3} + 8 q^{4} + 2 q^{5} - 9 q^{6} + 5 q^{7} - 33 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{8}{33}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.453484	+	0.432396i	0.320661	+	0.305750i	0.833263	−	0.552876i	\(-0.186470\pi\)
							−0.512602	+	0.858626i	\(0.671318\pi\)
\(3\)	−0.142315	−	0.989821i	−0.0821655	−	0.571474i
							\(5\)	−0.976375	+	2.13796i	−0.436648	+	0.956126i	0.555553	+	0.831481i	\(0.312506\pi\)
−0.992201	+	0.124645i	\(0.960221\pi\)
\(7\)	1.16821	−	4.81544i	0.441543	−	1.82007i	−0.120194	−	0.992750i	\(-0.538352\pi\)
							0.561737	−	0.827316i	\(-0.310133\pi\)
\(11\)	1.39687	+	1.96163i	0.421173	+	0.591455i	0.969585	−	0.244753i	\(-0.0787070\pi\)
							−0.548412	+	0.836208i	\(0.684768\pi\)
\(13\)	0.634429	−	1.83306i	0.175959	−	0.508400i	−0.822190	−	0.569213i	\(-0.807248\pi\)
							0.998149	+	0.0608130i	\(0.0193693\pi\)
\(17\)	0.243145	−	5.10424i	0.0589713	−	1.23796i	−0.754479	−	0.656325i	\(-0.772110\pi\)
							0.813450	−	0.581635i	\(-0.197587\pi\)
\(19\)	1.66419	+	6.85988i	0.381791	+	1.57376i	0.761188	+	0.648531i	\(0.224616\pi\)
							−0.379397	+	0.925234i	\(0.623869\pi\)
\(23\)	−0.328883	+	0.258637i	−0.0685769	+	0.0539294i	−0.651859	−	0.758340i	\(-0.726011\pi\)
							0.583282	+	0.812270i	\(0.301768\pi\)
\(29\)	−3.67810	+	6.37065i	−0.683005	+	1.18300i	0.291054	+	0.956707i	\(0.405994\pi\)
							−0.974059	+	0.226293i	\(0.927339\pi\)
\(31\)	1.07295	+	3.10008i	0.192707	+	0.556791i	0.999416	−	0.0341578i	\(-0.0108749\pi\)
							−0.806709	+	0.590948i	\(0.798754\pi\)
\(37\)	1.46536	+	2.53809i	0.240904	+	0.417259i	0.960972	−	0.276645i	\(-0.0892226\pi\)
							−0.720068	+	0.693904i	\(0.755889\pi\)
\(41\)	5.21191	−	2.68692i	0.813963	−	0.419627i	−0.000387858	−	1.00000i	\(-0.500123\pi\)
							0.814351	+	0.580373i	\(0.197093\pi\)
\(43\)	4.17289	+	2.68176i	0.636360	+	0.408964i	0.818660	−	0.574279i	\(-0.194718\pi\)
							−0.182299	+	0.983243i	\(0.558354\pi\)
\(47\)	6.95496	+	2.78435i	1.01448	+	0.406139i	0.818523	−	0.574473i	\(-0.194793\pi\)
							0.195961	+	0.980612i	\(0.437217\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.m.b.10.4	✓	120
3.2	odd	2		603.2.z.d.10.3		120
67.47	even	33	inner	201.2.m.b.181.4	yes	120
201.47	odd	66		603.2.z.d.181.3		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.m.b.10.4	✓	120	1.1	even	1	trivial
201.2.m.b.181.4	yes	120	67.47	even	33	inner
603.2.z.d.10.3		120	3.2	odd	2
603.2.z.d.181.3		120	201.47	odd	66