Embedded newform 201.2.m.b.4.1

• Label: 201.2.m.b.4.1
• Level: $201$
• Weight: $2$
• Character: 201.4
• 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			4.1
Character	\(\chi\)	\(=\)	201.4
Dual form			201.2.m.b.151.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.46190 - 0.235083i) q^{2} +(0.841254 + 0.540641i) q^{3} +(4.04182 + 0.778997i) q^{4} +(-0.0802800 - 0.558360i) q^{5} +(-1.94398 - 1.52877i) q^{6} +(1.37223 - 1.92702i) q^{7} +(-5.02158 - 1.47447i) q^{8} +(0.415415 + 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + 2 q^{2} - 12 q^{3} + 8 q^{4} + 2 q^{5} - 9 q^{6} + 5 q^{7} - 33 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.46190	−	0.235083i	−1.74082	−	0.166229i	−0.824286	−	0.566174i	\(-0.808423\pi\)
							−0.916539	+	0.399946i	\(0.869029\pi\)
\(3\)	0.841254	+	0.540641i	0.485698	+	0.312139i
							\(5\)	−0.0802800	−	0.558360i	−0.0359023	−	0.249706i	0.963964	−	0.266031i	\(-0.0857124\pi\)
−0.999867	+	0.0163251i	\(0.994803\pi\)
\(7\)	1.37223	−	1.92702i	0.518653	−	0.728347i	−0.469310	−	0.883033i	\(-0.655497\pi\)
							0.987964	+	0.154687i	\(0.0494368\pi\)
\(11\)	−0.332151	+	0.261206i	−0.100147	+	0.0787567i	−0.666977	−	0.745078i	\(-0.732412\pi\)
							0.566830	+	0.823835i	\(0.308170\pi\)
\(13\)	0.109203	−	0.450141i	0.0302875	−	0.124847i	−0.954661	−	0.297695i	\(-0.903782\pi\)
							0.984948	+	0.172849i	\(0.0552972\pi\)
\(17\)	3.44853	−	0.664649i	0.836391	−	0.161201i	0.246966	−	0.969024i	\(-0.420566\pi\)
							0.589425	+	0.807823i	\(0.299354\pi\)
\(19\)	0.695047	+	0.976057i	0.159455	+	0.223923i	0.886572	−	0.462590i	\(-0.153080\pi\)
							−0.727118	+	0.686513i	\(0.759140\pi\)
\(23\)	8.42181	−	4.34175i	1.75607	−	0.905316i	0.806816	−	0.590803i	\(-0.201189\pi\)
							0.949253	−	0.314514i	\(-0.101841\pi\)
\(29\)	−2.51886	−	4.36279i	−0.467741	−	0.810151i	0.531580	−	0.847008i	\(-0.321599\pi\)
							−0.999321	+	0.0368576i	\(0.988265\pi\)
\(31\)	0.720427	+	2.96964i	0.129393	+	0.533363i	0.999201	+	0.0399733i	\(0.0127273\pi\)
							−0.869808	+	0.493390i	\(0.835758\pi\)
\(37\)	−1.50819	+	2.61227i	−0.247946	+	0.429455i	−0.962956	−	0.269660i	\(-0.913089\pi\)
							0.715010	+	0.699114i	\(0.246422\pi\)
\(41\)	−3.64342	+	10.5270i	−0.569007	+	1.64404i	0.181664	+	0.983361i	\(0.441852\pi\)
							−0.750671	+	0.660677i	\(0.770269\pi\)
\(43\)	−5.51385	−	6.36332i	−0.840854	−	0.970397i	0.159004	−	0.987278i	\(-0.449172\pi\)
							−0.999858	+	0.0168811i	\(0.994626\pi\)
\(47\)	−0.301670	+	6.33283i	−0.0440031	+	0.923738i	0.863669	+	0.504059i	\(0.168161\pi\)
							−0.907672	+	0.419679i	\(0.862143\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.m.b.4.1	✓	120
3.2	odd	2		603.2.z.d.406.6		120
67.17	even	33	inner	201.2.m.b.151.1	yes	120
201.17	odd	66		603.2.z.d.352.6		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.m.b.4.1	✓	120	1.1	even	1	trivial
201.2.m.b.151.1	yes	120	67.17	even	33	inner
603.2.z.d.352.6		120	201.17	odd	66
603.2.z.d.406.6		120	3.2	odd	2