Embedded newform 33.8.e.a.16.1

• Label: 33.8.e.a.16.1
• Level: $33$
• Weight: $8$
• Character: 33.16
• 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			16.1
Character	\(\chi\)	\(=\)	33.16
Dual form			33.8.e.a.31.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.06001 - 18.6508i) q^{2} +(21.8435 + 15.8702i) q^{3} +(-207.575 + 150.812i) q^{4} +(-96.8862 + 298.185i) q^{5} +(163.620 - 503.572i) q^{6} +(902.102 - 655.415i) q^{7} +(2039.90 + 1482.08i) q^{8} +(225.273 + 693.320i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 22 q^{2} + 189 q^{3} - 692 q^{4} - 777 q^{5} + 594 q^{6} - 83 q^{7} + 3568 q^{8} - 5103 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.06001	−	18.6508i	−0.535635	−	1.64851i	−0.742274	−	0.670097i	\(-0.766253\pi\)
							0.206639	−	0.978417i	\(-0.433747\pi\)
\(3\)	21.8435	+	15.8702i	0.467086	+	0.339358i
							\(5\)	−96.8862	+	298.185i	−0.346631	+	1.06682i	0.614074	+	0.789248i	\(0.289530\pi\)
−0.960705	+	0.277572i	\(0.910470\pi\)
\(7\)	902.102	−	655.415i	0.994059	−	0.722226i	0.0332531	−	0.999447i	\(-0.489413\pi\)
							0.960806	+	0.277220i	\(0.0894132\pi\)
\(11\)	3877.44	−	2110.12i	0.878356	−	0.478006i
							\(13\)	2274.05	+	6998.81i	0.287077	+	0.883533i	0.985768	+	0.168109i	\(0.0537662\pi\)
−0.698691	+	0.715424i	\(0.746234\pi\)
\(17\)	8745.95	−	26917.3i	0.431754	−	1.32880i	−0.464623	−	0.885508i	\(-0.653810\pi\)
							0.896377	−	0.443293i	\(-0.146190\pi\)
\(19\)	26348.2	+	19143.1i	0.881280	+	0.640287i	0.933590	−	0.358344i	\(-0.116658\pi\)
							−0.0523100	+	0.998631i	\(0.516658\pi\)
\(23\)	−13018.3			−0.223103			−0.111552	−	0.993759i	\(-0.535582\pi\)
							−0.111552	+	0.993759i	\(0.535582\pi\)
\(29\)	14100.7	−	10244.8i	0.107361	−	0.0780026i	−0.532809	−	0.846235i	\(-0.678864\pi\)
							0.640171	+	0.768233i	\(0.278864\pi\)
\(31\)	86326.1	+	265684.i	0.520446	+	1.60177i	0.773148	+	0.634225i	\(0.218681\pi\)
							−0.252702	+	0.967544i	\(0.581319\pi\)
\(37\)	496326.	−	360602.i	1.61087	−	1.17037i	0.750196	−	0.661216i	\(-0.229959\pi\)
							0.860677	−	0.509152i	\(-0.170041\pi\)
\(41\)	−303055.	−	220182.i	−0.686717	−	0.498929i	0.188862	−	0.982004i	\(-0.439520\pi\)
							−0.875579	+	0.483074i	\(0.839520\pi\)
\(43\)	261150.			0.500900			0.250450	−	0.968130i	\(-0.419421\pi\)
							0.250450	+	0.968130i	\(0.419421\pi\)
\(47\)	−256115.	−	186078.i	−0.359826	−	0.261429i	0.393154	−	0.919473i	\(-0.371384\pi\)
							−0.752980	+	0.658044i	\(0.771384\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.8.e.a.16.1	✓	28
3.2	odd	2		99.8.f.c.82.7		28
11.3	even	5		363.8.a.q.1.1		14
11.8	odd	10		363.8.a.p.1.14		14
11.9	even	5	inner	33.8.e.a.31.1	yes	28
33.20	odd	10		99.8.f.c.64.7		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.e.a.16.1	✓	28	1.1	even	1	trivial
33.8.e.a.31.1	yes	28	11.9	even	5	inner
99.8.f.c.64.7		28	33.20	odd	10
99.8.f.c.82.7		28	3.2	odd	2
363.8.a.p.1.14		14	11.8	odd	10
363.8.a.q.1.1		14	11.3	even	5