Embedded newform 33.8.e.a.4.3

• Label: 33.8.e.a.4.3
• 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			4.3
Character	\(\chi\)	\(=\)	33.4
Dual form			33.8.e.a.25.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-8.43584 - 6.12900i) q^{2} +(-8.34346 + 25.6785i) q^{3} +(-5.95538 - 18.3288i) q^{4} +(-186.461 + 135.472i) q^{5} +(227.768 - 165.483i) q^{6} +(-43.6776 - 134.426i) q^{7} +(-474.540 + 1460.49i) q^{8} +(-589.773 - 428.495i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 22 q^{2} + 189 q^{3} - 692 q^{4} - 777 q^{5} + 594 q^{6} - 83 q^{7} + 3568 q^{8} - 5103 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−8.43584	−	6.12900i	−0.745630	−	0.541732i	0.148839	−	0.988861i	\(-0.452446\pi\)
							−0.894469	+	0.447130i	\(0.852446\pi\)
\(3\)	−8.34346	+	25.6785i	−0.178411	+	0.549093i
							\(5\)	−186.461	+	135.472i	−0.667103	+	0.484678i	−0.869054	−	0.494717i	\(-0.835272\pi\)
0.201951	+	0.979396i	\(0.435272\pi\)
\(7\)	−43.6776	−	134.426i	−0.0481300	−	0.148129i	0.924103	−	0.382143i	\(-0.124814\pi\)
							−0.972233	+	0.234014i	\(0.924814\pi\)
\(11\)	4405.31	+	283.647i	0.997934	+	0.0642545i
							\(13\)	−2402.15	−	1745.26i	−0.303248	−	0.220323i	0.425746	−	0.904843i	\(-0.360012\pi\)
−0.728994	+	0.684520i	\(0.760012\pi\)
\(17\)	32135.9	−	23348.1i	1.58643	−	1.15261i	0.677601	−	0.735430i	\(-0.263020\pi\)
							0.908826	−	0.417176i	\(-0.136980\pi\)
\(19\)	11338.9	−	34897.4i	0.379255	−	1.16723i	−0.561307	−	0.827607i	\(-0.689701\pi\)
							0.940563	−	0.339620i	\(-0.110299\pi\)
\(23\)	26232.0			0.449556			0.224778	−	0.974410i	\(-0.427834\pi\)
							0.224778	+	0.974410i	\(0.427834\pi\)
\(29\)	62224.8	+	191508.i	0.473773	+	1.45812i	0.847605	+	0.530628i	\(0.178044\pi\)
							−0.373832	+	0.927497i	\(0.621956\pi\)
\(31\)	34405.6	+	24997.1i	0.207426	+	0.150704i	0.686648	−	0.726990i	\(-0.259081\pi\)
							−0.479222	+	0.877694i	\(0.659081\pi\)
\(37\)	−93379.6	−	287393.i	−0.303072	−	0.932760i	−0.980390	−	0.197069i	\(-0.936858\pi\)
							0.677317	−	0.735691i	\(-0.263142\pi\)
\(41\)	−166492.	+	512408.i	−0.377267	+	1.16111i	0.564669	+	0.825317i	\(0.309004\pi\)
							−0.941936	+	0.335791i	\(0.890996\pi\)
\(43\)	282374.			0.541607			0.270804	−	0.962635i	\(-0.412711\pi\)
							0.270804	+	0.962635i	\(0.412711\pi\)
\(47\)	225672.	−	694546.i	0.317055	−	0.975795i	−0.657845	−	0.753153i	\(-0.728532\pi\)
							0.974900	−	0.222642i	\(-0.0714681\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.8.e.a.4.3	✓	28
3.2	odd	2		99.8.f.c.37.5		28
11.3	even	5	inner	33.8.e.a.25.3	yes	28
11.5	even	5		363.8.a.q.1.10		14
11.6	odd	10		363.8.a.p.1.5		14
33.14	odd	10		99.8.f.c.91.5		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.e.a.4.3	✓	28	1.1	even	1	trivial
33.8.e.a.25.3	yes	28	11.3	even	5	inner
99.8.f.c.37.5		28	3.2	odd	2
99.8.f.c.91.5		28	33.14	odd	10
363.8.a.p.1.5		14	11.6	odd	10
363.8.a.q.1.10		14	11.5	even	5