Embedded newform 33.6.e.a.16.5

• Label: 33.6.e.a.16.5
• 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			16.5
Root			\(3.07809 - 9.47338i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.16
Dual form			33.6.e.a.31.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.38711 + 10.4244i) q^{2} +(-7.28115 - 5.29007i) q^{3} +(-71.3079 + 51.8082i) q^{4} +(8.39633 - 25.8412i) q^{5} +(30.4839 - 93.8199i) q^{6} +(-110.374 + 80.1917i) q^{7} +(-497.837 - 361.700i) q^{8} +(25.0304 + 77.0356i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} - 45 q^{3} - 124 q^{4} - 33 q^{5} - 18 q^{6} - 335 q^{7} - 472 q^{8} - 405 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.38711	+	10.4244i	0.598761	+	1.84280i	0.535034	+	0.844830i	\(0.320299\pi\)
							0.0637272	+	0.997967i	\(0.479701\pi\)
\(3\)	−7.28115	−	5.29007i	−0.467086	−	0.339358i
							\(5\)	8.39633	−	25.8412i	0.150198	−	0.462262i	−0.847445	−	0.530884i	\(-0.821860\pi\)
0.997643	+	0.0686216i	\(0.0218601\pi\)
\(7\)	−110.374	+	80.1917i	−0.851380	+	0.618564i	−0.925526	−	0.378684i	\(-0.876377\pi\)
							0.0741464	+	0.997247i	\(0.476377\pi\)
\(11\)	272.319	+	294.777i	0.678571	+	0.734534i
							\(13\)	32.7197	+	100.701i	0.0536972	+	0.165263i	0.974309	−	0.225217i	\(-0.0723091\pi\)
−0.920611	+	0.390480i	\(0.872309\pi\)
\(17\)	−583.925	+	1797.14i	−0.490044	+	1.50820i	0.334497	+	0.942397i	\(0.391434\pi\)
							−0.824541	+	0.565803i	\(0.808566\pi\)
\(19\)	−8.60651	−	6.25300i	−0.00546944	−	0.00397378i	0.585047	−	0.810999i	\(-0.301076\pi\)
							−0.590517	+	0.807025i	\(0.701076\pi\)
\(23\)	−3878.53			−1.52879			−0.764395	−	0.644749i	\(-0.776962\pi\)
							−0.764395	+	0.644749i	\(0.776962\pi\)
\(29\)	3413.30	−	2479.91i	0.753668	−	0.547572i	−0.143294	−	0.989680i	\(-0.545769\pi\)
							0.896962	+	0.442108i	\(0.145769\pi\)
\(31\)	1250.63	+	3849.03i	0.233735	+	0.719362i	0.997287	+	0.0736152i	\(0.0234537\pi\)
							−0.763552	+	0.645746i	\(0.776546\pi\)
\(37\)	5133.73	−	3729.87i	0.616494	−	0.447909i	−0.235201	−	0.971947i	\(-0.575575\pi\)
							0.851695	+	0.524038i	\(0.175575\pi\)
\(41\)	6896.60	+	5010.67i	0.640730	+	0.465518i	0.860101	−	0.510124i	\(-0.170400\pi\)
							−0.219371	+	0.975642i	\(0.570400\pi\)
\(43\)	14570.2			1.20170			0.600848	−	0.799363i	\(-0.294830\pi\)
							0.600848	+	0.799363i	\(0.294830\pi\)
\(47\)	−7378.76	−	5360.98i	−0.487235	−	0.353997i	0.316885	−	0.948464i	\(-0.397363\pi\)
							−0.804120	+	0.594467i	\(0.797363\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.e.a.16.5	✓	20
3.2	odd	2		99.6.f.c.82.1		20
11.3	even	5		363.6.a.u.1.10		10
11.8	odd	10		363.6.a.q.1.1		10
11.9	even	5	inner	33.6.e.a.31.5	yes	20
33.8	even	10		1089.6.a.bl.1.10		10
33.14	odd	10		1089.6.a.bh.1.1		10
33.20	odd	10		99.6.f.c.64.1		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.e.a.16.5	✓	20	1.1	even	1	trivial
33.6.e.a.31.5	yes	20	11.9	even	5	inner
99.6.f.c.64.1		20	33.20	odd	10
99.6.f.c.82.1		20	3.2	odd	2
363.6.a.q.1.1		10	11.8	odd	10
363.6.a.u.1.10		10	11.3	even	5
1089.6.a.bh.1.1		10	33.14	odd	10
1089.6.a.bl.1.10		10	33.8	even	10