Embedded newform 33.8.e.a.16.5

• Label: 33.8.e.a.16.5
• 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			16.5
Character	\(\chi\)	\(=\)	33.16
Dual form			33.8.e.a.31.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.98016 + 9.17200i) q^{2} +(21.8435 + 15.8702i) q^{3} +(28.3100 - 20.5684i) q^{4} +(33.3401 - 102.610i) q^{5} +(-80.4644 + 247.644i) q^{6} +(1247.17 - 906.120i) q^{7} +(1271.70 + 923.944i) q^{8} +(225.273 + 693.320i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 22 q^{2} + 189 q^{3} - 692 q^{4} - 777 q^{5} + 594 q^{6} - 83 q^{7} + 3568 q^{8} - 5103 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.98016	+	9.17200i	0.263412	+	0.810698i	0.992055	+	0.125805i	\(0.0401512\pi\)
							−0.728643	+	0.684893i	\(0.759849\pi\)
\(3\)	21.8435	+	15.8702i	0.467086	+	0.339358i
							\(5\)	33.3401	−	102.610i	0.119281	−	0.367110i	−0.873535	−	0.486762i	\(-0.838178\pi\)
0.992816	+	0.119652i	\(0.0381779\pi\)
\(7\)	1247.17	−	906.120i	1.37430	−	0.998487i	0.376912	−	0.926249i	\(-0.376986\pi\)
							0.997387	−	0.0722384i	\(-0.0230142\pi\)
\(11\)	−3089.86	−	3152.77i	−0.699945	−	0.714197i
							\(13\)	1308.11	+	4025.96i	0.165137	+	0.508239i	0.999046	−	0.0436624i	\(-0.0139026\pi\)
−0.833910	+	0.551901i	\(0.813903\pi\)
\(17\)	−5183.97	+	15954.6i	−0.255912	+	0.787617i	0.737736	+	0.675089i	\(0.235895\pi\)
							−0.993649	+	0.112528i	\(0.964105\pi\)
\(19\)	20268.2	+	14725.7i	0.677919	+	0.492537i	0.872667	−	0.488316i	\(-0.162389\pi\)
							−0.194748	+	0.980853i	\(0.562389\pi\)
\(23\)	−67974.3			−1.16492			−0.582461	−	0.812858i	\(-0.697910\pi\)
							−0.582461	+	0.812858i	\(0.697910\pi\)
\(29\)	−3536.46	+	2569.39i	−0.0269263	+	0.0195631i	−0.601167	−	0.799123i	\(-0.705297\pi\)
							0.574241	+	0.818687i	\(0.305297\pi\)
\(31\)	−89093.9	−	274203.i	−0.537133	−	1.65313i	−0.738994	−	0.673713i	\(-0.764699\pi\)
							0.201860	−	0.979414i	\(-0.435301\pi\)
\(37\)	−25520.5	+	18541.8i	−0.0828293	+	0.0601790i	−0.628429	−	0.777867i	\(-0.716302\pi\)
							0.545600	+	0.838046i	\(0.316302\pi\)
\(41\)	−396638.	−	288174.i	−0.898774	−	0.652998i	0.0393765	−	0.999224i	\(-0.487463\pi\)
							−0.938151	+	0.346227i	\(0.887463\pi\)
\(43\)	−611530.			−1.17295			−0.586473	−	0.809969i	\(-0.699484\pi\)
							−0.586473	+	0.809969i	\(0.699484\pi\)
\(47\)	562895.	+	408967.i	0.790834	+	0.574574i	0.908211	−	0.418513i	\(-0.137449\pi\)
							−0.117377	+	0.993087i	\(0.537449\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.8.e.a.16.5	✓	28
3.2	odd	2		99.8.f.c.82.3		28
11.3	even	5		363.8.a.q.1.9		14
11.8	odd	10		363.8.a.p.1.6		14
11.9	even	5	inner	33.8.e.a.31.5	yes	28
33.20	odd	10		99.8.f.c.64.3		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.e.a.16.5	✓	28	1.1	even	1	trivial
33.8.e.a.31.5	yes	28	11.9	even	5	inner
99.8.f.c.64.3		28	33.20	odd	10
99.8.f.c.82.3		28	3.2	odd	2
363.8.a.p.1.6		14	11.8	odd	10
363.8.a.q.1.9		14	11.3	even	5