Embedded newform 33.8.e.b.4.2

• Label: 33.8.e.b.4.2
• Level: $33$
• Weight: $8$
• Character: 33.4
• Analytic conductor: $10.309$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	33.e (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.3087058410\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(7\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			4.2
Character	\(\chi\)	\(=\)	33.4
Dual form			33.8.e.b.25.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-12.1293 - 8.81245i) q^{2} +(8.34346 - 25.6785i) q^{3} +(29.9064 + 92.0423i) q^{4} +(-3.10808 + 2.25815i) q^{5} +(-327.491 + 237.936i) q^{6} +(300.631 + 925.246i) q^{7} +(-144.646 + 445.175i) q^{8} +(-589.773 - 428.495i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 6 q^{2} - 189 q^{3} + 160 q^{4} + 773 q^{5} - 162 q^{6} + 1289 q^{7} + 2956 q^{8} - 5103 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−12.1293	−	8.81245i	−1.07209	−	0.778918i	−0.0958018	−	0.995400i	\(-0.530542\pi\)
							−0.976286	+	0.216483i	\(0.930542\pi\)
\(3\)	8.34346	−	25.6785i	0.178411	−	0.549093i
							\(5\)	−3.10808	+	2.25815i	−0.0111198	+	0.00807901i	−0.593331	−	0.804958i	\(-0.702188\pi\)
0.582211	+	0.813037i	\(0.302188\pi\)
\(7\)	300.631	+	925.246i	0.331276	+	1.01956i	0.968527	+	0.248907i	\(0.0800713\pi\)
							−0.637251	+	0.770656i	\(0.719929\pi\)
\(11\)	−3928.96	+	2012.57i	−0.890027	+	0.455908i
							\(13\)	7881.27	+	5726.08i	0.994935	+	0.722862i	0.960996	−	0.276562i	\(-0.0891950\pi\)
0.0339384	+	0.999424i	\(0.489195\pi\)
\(17\)	29814.1	−	21661.2i	1.47180	−	1.06933i	0.491719	−	0.870754i	\(-0.336369\pi\)
							0.980086	−	0.198575i	\(-0.0636312\pi\)
\(19\)	−3261.53	+	10037.9i	−0.109090	+	0.335743i	−0.990669	−	0.136293i	\(-0.956481\pi\)
							0.881579	+	0.472037i	\(0.156481\pi\)
\(23\)	−13400.6			−0.229655			−0.114828	−	0.993385i	\(-0.536632\pi\)
							−0.114828	+	0.993385i	\(0.536632\pi\)
\(29\)	−12789.3	−	39361.5i	−0.0973766	−	0.299694i	0.890489	−	0.455004i	\(-0.150362\pi\)
							−0.987866	+	0.155310i	\(0.950362\pi\)
\(31\)	76358.0	+	55477.3i	0.460350	+	0.334464i	0.793669	−	0.608350i	\(-0.208168\pi\)
							−0.333318	+	0.942814i	\(0.608168\pi\)
\(37\)	145533.	+	447903.i	0.472340	+	1.45371i	0.849512	+	0.527569i	\(0.176896\pi\)
							−0.377173	+	0.926143i	\(0.623104\pi\)
\(41\)	13634.9	−	41964.0i	0.0308965	−	0.0950898i	−0.934419	−	0.356176i	\(-0.884081\pi\)
							0.965316	+	0.261086i	\(0.0840805\pi\)
\(43\)	321422.			0.616504			0.308252	−	0.951305i	\(-0.400256\pi\)
							0.308252	+	0.951305i	\(0.400256\pi\)
\(47\)	−59337.5	+	182622.i	−0.0833656	+	0.256573i	−0.984047	−	0.177906i	\(-0.943068\pi\)
							0.900682	+	0.434479i	\(0.143068\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.8.e.b.4.2	✓	28
3.2	odd	2		99.8.f.b.37.6		28
11.3	even	5	inner	33.8.e.b.25.2	yes	28
11.5	even	5		363.8.a.o.1.12		14
11.6	odd	10		363.8.a.r.1.3		14
33.14	odd	10		99.8.f.b.91.6		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.e.b.4.2	✓	28	1.1	even	1	trivial
33.8.e.b.25.2	yes	28	11.3	even	5	inner
99.8.f.b.37.6		28	3.2	odd	2
99.8.f.b.91.6		28	33.14	odd	10
363.8.a.o.1.12		14	11.5	even	5
363.8.a.r.1.3		14	11.6	odd	10