Embedded newform 333.3.n.a.110.13

• Label: 333.3.n.a.110.13
• Level: $333$
• Weight: $3$
• Character: 333.110
• Analytic conductor: $9.074$
• Analytic rank: $0$
• Dimension: $148$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 333 = 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	333.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.07359280320\)
Analytic rank:	\(0\)
Dimension:	\(148\)
Relative dimension:	\(74\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			110.13
Character	\(\chi\)	\(=\)	333.110
Dual form			333.3.n.a.221.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.47315 + 2.55158i) q^{2} +(1.83443 + 2.37379i) q^{3} +(-2.34037 - 4.05363i) q^{4} +(-1.42559 - 2.46920i) q^{5} +(-8.75930 + 1.18375i) q^{6} +(0.889782 - 1.54115i) q^{7} +2.00564 q^{8} +(-2.26971 + 8.70910i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 148 q - 6 q^{3} - 146 q^{4} - 14 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/333\mathbb{Z}\right)^\times\).

\(n\)	\(38\)	\(298\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.47315	+	2.55158i	−0.736577	+	1.27579i	0.217451	+	0.976071i	\(0.430226\pi\)
							−0.954028	+	0.299718i	\(0.903108\pi\)
\(3\)	1.83443	+	2.37379i	0.611478	+	0.791262i
							\(5\)	−1.42559	−	2.46920i	−0.285118	−	0.493839i	0.687520	−	0.726166i	\(-0.258700\pi\)
−0.972638	+	0.232326i	\(0.925366\pi\)
\(7\)	0.889782	−	1.54115i	0.127112	−	0.220164i	−0.795445	−	0.606026i	\(-0.792763\pi\)
							0.922556	+	0.385862i	\(0.126096\pi\)
\(11\)	−4.68950	−	2.70749i	−0.426319	−	0.246135i	0.271458	−	0.962450i	\(-0.412494\pi\)
							−0.697777	+	0.716315i	\(0.745827\pi\)
\(13\)	−15.6193	+	9.01780i	−1.20148	+	0.693677i	−0.960885	−	0.276948i	\(-0.910677\pi\)
							−0.240598	+	0.970625i	\(0.577344\pi\)
\(17\)	−13.8013			−0.811843			−0.405921	−	0.913908i	\(-0.633049\pi\)
							−0.405921	+	0.913908i	\(0.633049\pi\)
\(19\)		−	24.0738i		−	1.26704i	−0.773725	−	0.633522i	\(-0.781609\pi\)
							0.773725	−	0.633522i	\(-0.218391\pi\)
\(23\)	−13.1948	−	22.8541i	−0.573688	−	0.993656i	−0.996183	−	0.0872910i	\(-0.972179\pi\)
							0.422495	−	0.906365i	\(-0.361154\pi\)
\(29\)	−21.3549	+	36.9877i	−0.736375	+	1.27544i	0.217743	+	0.976006i	\(0.430131\pi\)
							−0.954118	+	0.299432i	\(0.903203\pi\)
\(31\)	26.3236	−	15.1979i	0.849147	−	0.490255i	−0.0112159	−	0.999937i	\(-0.503570\pi\)
							0.860363	+	0.509682i	\(0.170237\pi\)
\(37\)	−36.2094	−	7.60767i	−0.978633	−	0.205613i
							\(41\)	−57.9544	+	33.4600i	−1.41352	+	0.816097i	−0.995718	−	0.0924408i	\(-0.970533\pi\)
−0.417803	+	0.908538i	\(0.637200\pi\)
\(43\)	49.8626	+	28.7882i	1.15960	+	0.669493i	0.951207	−	0.308553i	\(-0.0998446\pi\)
							0.208389	+	0.978046i	\(0.433178\pi\)
\(47\)	−2.39935	−	1.38526i	−0.0510499	−	0.0294737i	0.474258	−	0.880386i	\(-0.342716\pi\)
							−0.525308	+	0.850912i	\(0.676050\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	333.3.n.a.110.13	✓	148
9.5	odd	6	inner	333.3.n.a.221.62	yes	148
37.36	even	2	inner	333.3.n.a.110.62	yes	148
333.221	odd	6	inner	333.3.n.a.221.13	yes	148

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
333.3.n.a.110.13	✓	148	1.1	even	1	trivial
333.3.n.a.110.62	yes	148	37.36	even	2	inner
333.3.n.a.221.13	yes	148	333.221	odd	6	inner
333.3.n.a.221.62	yes	148	9.5	odd	6	inner