Embedded newform 201.4.m.b.4.5

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

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.31558 - 0.316600i) q^{2} +(-2.52376 - 1.62192i) q^{3} +(3.03741 + 0.585414i) q^{4} +(-2.57817 - 17.9316i) q^{5} +(7.85423 + 6.17664i) q^{6} +(-18.7844 + 26.3790i) q^{7} +(15.6805 + 4.60422i) q^{8} +(3.73874 + 8.18669i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q - 2 q^{2} + 108 q^{3} + 90 q^{4} + 4 q^{5} - 27 q^{6} - 22 q^{7} + 183 q^{8} - 324 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.31558	−	0.316600i	−1.17224	−	0.111935i	−0.509285	−	0.860598i	\(-0.670090\pi\)
							−0.662950	+	0.748663i	\(0.730696\pi\)
\(3\)	−2.52376	−	1.62192i	−0.485698	−	0.312139i
							\(5\)	−2.57817	−	17.9316i	−0.230598	−	1.60385i	−0.695527	−	0.718500i	\(-0.744829\pi\)
0.464929	−	0.885348i	\(-0.346080\pi\)
\(7\)	−18.7844	+	26.3790i	−1.01426	+	1.42433i	−0.111714	+	0.993740i	\(0.535634\pi\)
							−0.902547	+	0.430590i	\(0.858305\pi\)
\(11\)	−27.1358	+	21.3398i	−0.743795	+	0.584927i	−0.916494	−	0.400048i	\(-0.868994\pi\)
							0.172699	+	0.984975i	\(0.444751\pi\)
\(13\)	1.06852	−	4.40452i	0.0227966	−	0.0939687i	−0.959277	−	0.282468i	\(-0.908847\pi\)
							0.982073	+	0.188499i	\(0.0603622\pi\)
\(17\)	−47.2641	+	9.10940i	−0.674307	+	0.129962i	−0.514899	−	0.857251i	\(-0.672171\pi\)
							−0.159408	+	0.987213i	\(0.550958\pi\)
\(19\)	−12.6979	−	17.8318i	−0.153322	−	0.215310i	0.730770	−	0.682624i	\(-0.239161\pi\)
							−0.884092	+	0.467314i	\(0.845222\pi\)
\(23\)	112.480	−	57.9876i	1.01973	−	0.525707i	0.134525	−	0.990910i	\(-0.457049\pi\)
							0.885204	+	0.465203i	\(0.154019\pi\)
\(29\)	97.0565	+	168.107i	0.621481	+	1.07644i	0.989210	+	0.146504i	\(0.0468020\pi\)
							−0.367729	+	0.929933i	\(0.619865\pi\)
\(31\)	−5.66721	−	23.3606i	−0.0328343	−	0.135345i	0.953035	−	0.302859i	\(-0.0979411\pi\)
							−0.985870	+	0.167514i	\(0.946426\pi\)
\(37\)	100.104	−	173.385i	0.444782	−	0.770385i	−0.553255	−	0.833012i	\(-0.686614\pi\)
							0.998037	+	0.0626267i	\(0.0199478\pi\)
\(41\)	67.4468	−	194.875i	0.256913	−	0.742301i	−0.740609	−	0.671937i	\(-0.765463\pi\)
							0.997521	−	0.0703640i	\(-0.0224161\pi\)
\(43\)	122.455	+	141.320i	0.434282	+	0.501188i	0.930135	−	0.367218i	\(-0.119690\pi\)
							−0.495852	+	0.868407i	\(0.665144\pi\)
\(47\)	3.47038	−	72.8523i	0.0107704	−	0.226098i	−0.987064	−	0.160325i	\(-0.948746\pi\)
							0.997835	−	0.0657728i	\(-0.0209513\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.m.b.4.5	✓	360
67.17	even	33	inner	201.4.m.b.151.5	yes	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.m.b.4.5	✓	360	1.1	even	1	trivial
201.4.m.b.151.5	yes	360	67.17	even	33	inner