Embedded newform 201.4.m.b.4.15

• Label: 201.4.m.b.4.15
• Level: $201$
• Weight: $4$
• Character: 201.4
• Analytic conductor: $11.859$
• Analytic rank: $0$
• Dimension: $360$
• 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:	\(360\)
Relative dimension:	\(18\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			4.15
Character	\(\chi\)	\(=\)	201.4
Dual form			201.4.m.b.151.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.06355 + 0.388022i) q^{2} +(-2.52376 - 1.62192i) q^{3} +(8.50643 + 1.63948i) q^{4} +(1.57648 + 10.9647i) q^{5} +(-9.62608 - 7.57003i) q^{6} +(-13.4370 + 18.8697i) q^{7} +(2.59669 + 0.762458i) q^{8} +(3.73874 + 8.18669i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q - 2 q^{2} + 108 q^{3} + 90 q^{4} + 4 q^{5} - 27 q^{6} - 22 q^{7} + 183 q^{8} - 324 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.06355	+	0.388022i	1.43668	+	0.137186i	0.784148	−	0.620574i	\(-0.213100\pi\)
							0.652533	+	0.757760i	\(0.273706\pi\)
\(3\)	−2.52376	−	1.62192i	−0.485698	−	0.312139i
							\(5\)	1.57648	+	10.9647i	0.141005	+	0.980710i	0.930327	+	0.366731i	\(0.119523\pi\)
−0.789322	+	0.613979i	\(0.789568\pi\)
\(7\)	−13.4370	+	18.8697i	−0.725531	+	1.01887i	0.272854	+	0.962055i	\(0.412032\pi\)
							−0.998385	+	0.0568101i	\(0.981907\pi\)
\(11\)	−42.8562	+	33.7025i	−1.17469	+	0.923790i	−0.998168	−	0.0605063i	\(-0.980728\pi\)
							−0.176526	+	0.984296i	\(0.556486\pi\)
\(13\)	4.01294	−	16.5416i	0.0856145	−	0.352908i	−0.912850	−	0.408294i	\(-0.866124\pi\)
							0.998465	+	0.0553861i	\(0.0176390\pi\)
\(17\)	93.0398	−	17.9320i	1.32738	−	0.255832i	0.524214	−	0.851586i	\(-0.324359\pi\)
							0.803166	+	0.595755i	\(0.203147\pi\)
\(19\)	57.9596	+	81.3929i	0.699834	+	0.982780i	0.999586	+	0.0287627i	\(0.00915672\pi\)
							−0.299752	+	0.954017i	\(0.596904\pi\)
\(23\)	−41.0114	+	21.1428i	−0.371803	+	0.191678i	−0.633993	−	0.773338i	\(-0.718585\pi\)
							0.262191	+	0.965016i	\(0.415555\pi\)
\(29\)	22.3082	+	38.6390i	0.142846	+	0.247417i	0.928567	−	0.371164i	\(-0.121041\pi\)
							−0.785721	+	0.618581i	\(0.787708\pi\)
\(31\)	0.298030	+	1.22850i	0.00172670	+	0.00711757i	0.972668	−	0.232199i	\(-0.0745919\pi\)
							−0.970942	+	0.239316i	\(0.923077\pi\)
\(37\)	−95.4046	+	165.246i	−0.423903	+	0.734222i	−0.996317	−	0.0857429i	\(-0.972674\pi\)
							0.572414	+	0.819965i	\(0.306007\pi\)
\(41\)	46.1778	−	133.422i	0.175897	−	0.508220i	−0.822247	−	0.569131i	\(-0.807280\pi\)
							0.998143	+	0.0609116i	\(0.0194008\pi\)
\(43\)	102.481	+	118.270i	0.363447	+	0.419441i	0.907792	−	0.419421i	\(-0.137767\pi\)
							−0.544344	+	0.838862i	\(0.683221\pi\)
\(47\)	−7.92039	+	166.269i	−0.0245810	+	0.516019i	0.952680	+	0.303977i	\(0.0983145\pi\)
							−0.977261	+	0.212042i	\(0.931989\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.m.b.4.15	✓	360
67.17	even	33	inner	201.4.m.b.151.15	yes	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.m.b.4.15	✓	360	1.1	even	1	trivial
201.4.m.b.151.15	yes	360	67.17	even	33	inner