Embedded newform 201.4.m.a.4.11

• Label: 201.4.m.a.4.11
• 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			4.11
Character	\(\chi\)	\(=\)	201.4
Dual form			201.4.m.a.151.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.23941 + 0.213838i) q^{2} +(2.52376 + 1.62192i) q^{3} +(-2.88618 - 0.556265i) q^{4} +(0.118431 + 0.823704i) q^{5} +(5.30492 + 4.17183i) q^{6} +(-8.75517 + 12.2949i) q^{7} +(-23.6122 - 6.93317i) q^{8} +(3.73874 + 8.18669i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 2 q^{2} - 96 q^{3} + 66 q^{4} - 4 q^{5} + 27 q^{6} + 14 q^{7} + 123 q^{8} - 288 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.23941	+	0.213838i	0.791753	+	0.0756032i	0.483093	−	0.875569i	\(-0.339513\pi\)
							0.308660	+	0.951172i	\(0.400119\pi\)
\(3\)	2.52376	+	1.62192i	0.485698	+	0.312139i
							\(5\)	0.118431	+	0.823704i	0.0105928	+	0.0736743i	0.994432	−	0.105379i	\(-0.0336056\pi\)
−0.983839	+	0.179053i	\(0.942697\pi\)
\(7\)	−8.75517	+	12.2949i	−0.472735	+	0.663863i	−0.980246	−	0.197782i	\(-0.936626\pi\)
							0.507511	+	0.861645i	\(0.330566\pi\)
\(11\)	−30.2950	+	23.8242i	−0.830389	+	0.653025i	−0.940085	−	0.340940i	\(-0.889255\pi\)
							0.109696	+	0.993965i	\(0.465012\pi\)
\(13\)	−18.4058	+	75.8698i	−0.392681	+	1.61865i	0.341626	+	0.939836i	\(0.389022\pi\)
							−0.734307	+	0.678817i	\(0.762493\pi\)
\(17\)	104.776	−	20.1939i	1.49481	−	0.288102i	0.624666	−	0.780892i	\(-0.285235\pi\)
							0.870148	+	0.492790i	\(0.164023\pi\)
\(19\)	−20.5344	−	28.8365i	−0.247943	−	0.348187i	0.671783	−	0.740748i	\(-0.265529\pi\)
							−0.919726	+	0.392561i	\(0.871589\pi\)
\(23\)	−188.192	+	97.0198i	−1.70612	+	0.879567i	−0.724366	+	0.689415i	\(0.757868\pi\)
							−0.981755	+	0.190151i	\(0.939102\pi\)
\(29\)	−55.0250	−	95.3061i	−0.352341	−	0.610272i	0.634318	−	0.773072i	\(-0.281281\pi\)
							−0.986659	+	0.162800i	\(0.947948\pi\)
\(31\)	29.2425	+	120.539i	0.169423	+	0.698370i	0.991622	+	0.129171i	\(0.0412315\pi\)
							−0.822200	+	0.569199i	\(0.807253\pi\)
\(37\)	45.8776	−	79.4623i	0.203844	−	0.353068i	−0.745920	−	0.666036i	\(-0.767990\pi\)
							0.949764	+	0.312968i	\(0.101323\pi\)
\(41\)	−17.0173	+	49.1684i	−0.0648210	+	0.187288i	−0.972760	−	0.231814i	\(-0.925534\pi\)
							0.907939	+	0.419102i	\(0.137655\pi\)
\(43\)	−194.907	−	224.935i	−0.691234	−	0.797727i	0.296306	−	0.955093i	\(-0.404245\pi\)
							−0.987540	+	0.157366i	\(0.949700\pi\)
\(47\)	8.74116	−	183.500i	0.0271283	−	0.569493i	−0.943857	−	0.330354i	\(-0.892832\pi\)
							0.970985	−	0.239139i	\(-0.0768651\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.m.a.4.11	✓	320
67.17	even	33	inner	201.4.m.a.151.11	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.m.a.4.11	✓	320	1.1	even	1	trivial
201.4.m.a.151.11	yes	320	67.17	even	33	inner