Embedded newform 33.6.f.a.8.1

• Label: 33.6.f.a.8.1
• Level: $33$
• Weight: $6$
• Character: 33.8
• Analytic conductor: $5.293$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	33.f (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.29266605383\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(18\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			8.1
Character	\(\chi\)	\(=\)	33.8
Dual form			33.6.f.a.29.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-8.92830 + 6.48679i) q^{2} +(-5.92302 + 14.4194i) q^{3} +(27.7475 - 85.3981i) q^{4} +(33.7792 - 46.4931i) q^{5} +(-40.6528 - 167.162i) q^{6} +(-11.4606 - 3.72378i) q^{7} +(197.091 + 606.584i) q^{8} +(-172.836 - 170.812i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 18 q^{3} - 262 q^{4} + 15 q^{6} - 10 q^{7} + 292 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{3}{10}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−8.92830	+	6.48679i	−1.57831	+	1.14671i	−0.659720	+	0.751512i	\(0.729325\pi\)
							−0.918595	+	0.395201i	\(0.870675\pi\)
\(3\)	−5.92302	+	14.4194i	−0.379962	+	0.925002i
							\(5\)	33.7792	−	46.4931i	0.604261	−	0.831694i	−0.391829	−	0.920038i	\(-0.628158\pi\)
0.996090	+	0.0883444i	\(0.0281576\pi\)
\(7\)	−11.4606	−	3.72378i	−0.0884021	−	0.0287236i	0.264482	−	0.964391i	\(-0.414799\pi\)
							−0.352884	+	0.935667i	\(0.614799\pi\)
\(11\)	234.344	−	325.782i	0.583945	−	0.811793i
							\(13\)	−17.9032	−	24.6416i	−0.0293813	−	0.0404399i	0.794074	−	0.607822i	\(-0.207956\pi\)
−0.823455	+	0.567382i	\(0.807956\pi\)
\(17\)	743.029	+	539.842i	0.623567	+	0.453048i	0.854166	−	0.520001i	\(-0.174068\pi\)
							−0.230598	+	0.973049i	\(0.574068\pi\)
\(19\)	1398.06	−	454.258i	0.888469	−	0.288681i	0.171000	−	0.985271i	\(-0.445300\pi\)
							0.717470	+	0.696590i	\(0.245300\pi\)
\(23\)		−	1182.79i		−	0.466215i	−0.972451	−	0.233108i	\(-0.925111\pi\)
							0.972451	−	0.233108i	\(-0.0748895\pi\)
\(29\)	335.569	−	1032.78i	0.0740947	−	0.228040i	−0.907150	−	0.420808i	\(-0.861747\pi\)
							0.981244	+	0.192768i	\(0.0617465\pi\)
\(31\)	6582.67	−	4782.59i	1.23026	−	0.893838i	0.233353	−	0.972392i	\(-0.425030\pi\)
							0.996910	+	0.0785537i	\(0.0250302\pi\)
\(37\)	−923.903	+	2843.48i	−0.110949	+	0.341465i	−0.991081	−	0.133264i	\(-0.957454\pi\)
							0.880132	+	0.474729i	\(0.157454\pi\)
\(41\)	3447.83	+	10611.3i	0.320322	+	0.985848i	0.973508	+	0.228652i	\(0.0734316\pi\)
							−0.653187	+	0.757197i	\(0.726568\pi\)
\(43\)		−	18700.4i		−	1.54234i	−0.636632	−	0.771168i	\(-0.719673\pi\)
							0.636632	−	0.771168i	\(-0.280327\pi\)
\(47\)	5459.56	−	1773.92i	0.360506	−	0.117136i	−0.123163	−	0.992386i	\(-0.539304\pi\)
							0.483669	+	0.875251i	\(0.339304\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.f.a.8.1	✓	72
3.2	odd	2	inner	33.6.f.a.8.18	yes	72
11.7	odd	10	inner	33.6.f.a.29.18	yes	72
33.29	even	10	inner	33.6.f.a.29.1	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.f.a.8.1	✓	72	1.1	even	1	trivial
33.6.f.a.8.18	yes	72	3.2	odd	2	inner
33.6.f.a.29.1	yes	72	33.29	even	10	inner
33.6.f.a.29.18	yes	72	11.7	odd	10	inner