Embedded newform 33.6.e.a.16.4

• Label: 33.6.e.a.16.4
• Level: $33$
• Weight: $6$
• Character: 33.16
• Analytic conductor: $5.293$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	33.e (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.29266605383\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(5\) over \(\Q(\zeta_{5})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial:	x^20 - 3*x^19 + 142*x^18 - 389*x^17 + 14927*x^16 - 6599*x^15 + 1399353*x^14 + 1969563*x^13 + 129076450*x^12 + 14028509*x^11 + 4158720649*x^10 - 7877307064*x^9 + 107078602608*x^8 - 53496436514*x^7 + 2527225278875*x^6 - 6173773919508*x^5 + 63231264058241*x^4 - 118816044595930*x^3 + 398041877081860*x^2 - 161017247606600*x + 25735126080400
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{4}\cdot 11^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			16.4
Root			\(1.17535 - 3.61737i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.16
Dual form			33.6.e.a.31.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.48437 + 4.56842i) q^{2} +(-7.28115 - 5.29007i) q^{3} +(7.22140 - 5.24666i) q^{4} +(18.4760 - 56.8633i) q^{5} +(13.3593 - 41.1158i) q^{6} +(58.2735 - 42.3382i) q^{7} +(159.044 + 115.553i) q^{8} +(25.0304 + 77.0356i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} - 45 q^{3} - 124 q^{4} - 33 q^{5} - 18 q^{6} - 335 q^{7} - 472 q^{8} - 405 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.48437	+	4.56842i	0.262402	+	0.807591i	0.992280	+	0.124014i	\(0.0395767\pi\)
							−0.729878	+	0.683577i	\(0.760423\pi\)
\(3\)	−7.28115	−	5.29007i	−0.467086	−	0.339358i
							\(5\)	18.4760	−	56.8633i	0.330509	−	1.01720i	−0.638383	−	0.769719i	\(-0.720397\pi\)
0.968892	−	0.247483i	\(-0.0796035\pi\)
\(7\)	58.2735	−	42.3382i	0.449496	−	0.326578i	−0.339901	−	0.940461i	\(-0.610393\pi\)
							0.789397	+	0.613883i	\(0.210393\pi\)
\(11\)	64.9717	−	396.017i	0.161898	−	0.986807i
							\(13\)	167.114	+	514.323i	0.274254	+	0.844069i	0.989416	+	0.145109i	\(0.0463533\pi\)
−0.715161	+	0.698960i	\(0.753647\pi\)
\(17\)	466.835	−	1436.77i	0.391779	−	1.20577i	−0.539663	−	0.841881i	\(-0.681448\pi\)
							0.931442	−	0.363891i	\(-0.118552\pi\)
\(19\)	−255.388	−	185.550i	−0.162299	−	0.117917i	0.503671	−	0.863895i	\(-0.331982\pi\)
							−0.665970	+	0.745978i	\(0.731982\pi\)
\(23\)	−2610.94			−1.02914			−0.514572	−	0.857447i	\(-0.672049\pi\)
							−0.514572	+	0.857447i	\(0.672049\pi\)
\(29\)	−5364.44	+	3897.49i	−1.18448	+	0.860578i	−0.992670	−	0.120853i	\(-0.961437\pi\)
							−0.191814	+	0.981431i	\(0.561437\pi\)
\(31\)	2297.76	+	7071.77i	0.429438	+	1.32167i	0.898680	+	0.438604i	\(0.144527\pi\)
							−0.469243	+	0.883069i	\(0.655473\pi\)
\(37\)	−5210.22	+	3785.44i	−0.625679	+	0.454582i	−0.854900	−	0.518792i	\(-0.826382\pi\)
							0.229222	+	0.973374i	\(0.426382\pi\)
\(41\)	966.499	+	702.203i	0.0897929	+	0.0652383i	0.631776	−	0.775151i	\(-0.282326\pi\)
							−0.541983	+	0.840389i	\(0.682326\pi\)
\(43\)	21637.2			1.78456			0.892278	−	0.451487i	\(-0.149106\pi\)
							0.892278	+	0.451487i	\(0.149106\pi\)
\(47\)	−1094.51	−	795.206i	−0.0722727	−	0.0525092i	0.551062	−	0.834464i	\(-0.314223\pi\)
							−0.623335	+	0.781955i	\(0.714223\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.e.a.16.4	✓	20
3.2	odd	2		99.6.f.c.82.2		20
11.3	even	5		363.6.a.u.1.7		10
11.8	odd	10		363.6.a.q.1.4		10
11.9	even	5	inner	33.6.e.a.31.4	yes	20
33.8	even	10		1089.6.a.bl.1.7		10
33.14	odd	10		1089.6.a.bh.1.4		10
33.20	odd	10		99.6.f.c.64.2		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.e.a.16.4	✓	20	1.1	even	1	trivial
33.6.e.a.31.4	yes	20	11.9	even	5	inner
99.6.f.c.64.2		20	33.20	odd	10
99.6.f.c.82.2		20	3.2	odd	2
363.6.a.q.1.4		10	11.8	odd	10
363.6.a.u.1.7		10	11.3	even	5
1089.6.a.bh.1.4		10	33.14	odd	10
1089.6.a.bl.1.7		10	33.8	even	10