Embedded newform 33.6.e.a.4.5

• Label: 33.6.e.a.4.5
• Level: $33$
• Weight: $6$
• Character: 33.4
• 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			4.5
Root			\(8.61948 - 6.26242i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.4
Dual form			33.6.e.a.25.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.81047 + 5.67464i) q^{2} +(2.78115 - 8.55951i) q^{3} +(18.9134 + 58.2093i) q^{4} +(25.1385 - 18.2642i) q^{5} +(70.2942 - 51.0717i) q^{6} +(56.7190 + 174.563i) q^{7} +(-87.1280 + 268.152i) q^{8} +(-65.5304 - 47.6106i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} - 45 q^{3} - 124 q^{4} - 33 q^{5} - 18 q^{6} - 335 q^{7} - 472 q^{8} - 405 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.81047	+	5.67464i	1.38071	+	1.00314i	0.996814	+	0.0797669i	\(0.0254176\pi\)
							0.383895	+	0.923377i	\(0.374582\pi\)
\(3\)	2.78115	−	8.55951i	0.178411	−	0.549093i
							\(5\)	25.1385	−	18.2642i	0.449692	−	0.326720i	−0.339782	−	0.940504i	\(-0.610353\pi\)
0.789474	+	0.613784i	\(0.210353\pi\)
\(7\)	56.7190	+	174.563i	0.437506	+	1.34650i	0.890497	+	0.454989i	\(0.150357\pi\)
							−0.452991	+	0.891515i	\(0.649643\pi\)
\(11\)	−30.5455	−	400.147i	−0.0761141	−	0.997099i
							\(13\)	−707.066	−	513.713i	−1.16038	−	0.843068i	−0.170557	−	0.985348i	\(-0.554557\pi\)
−0.989826	+	0.142280i	\(0.954557\pi\)
\(17\)	−627.084	+	455.604i	−0.526264	+	0.382353i	−0.818958	−	0.573853i	\(-0.805448\pi\)
							0.292694	+	0.956206i	\(0.405448\pi\)
\(19\)	462.848	−	1424.50i	0.294140	−	0.905271i	−0.689368	−	0.724411i	\(-0.742112\pi\)
							0.983509	−	0.180860i	\(-0.0578881\pi\)
\(23\)	1303.68			0.513867			0.256933	−	0.966429i	\(-0.417288\pi\)
							0.256933	+	0.966429i	\(0.417288\pi\)
\(29\)	−150.045	−	461.791i	−0.0331304	−	0.101965i	0.933124	−	0.359555i	\(-0.117072\pi\)
							−0.966254	+	0.257590i	\(0.917072\pi\)
\(31\)	−6111.03	−	4439.92i	−1.14212	−	0.829795i	−0.154703	−	0.987961i	\(-0.549442\pi\)
							−0.987413	+	0.158166i	\(0.949442\pi\)
\(37\)	3612.44	+	11118.0i	0.433807	+	1.33512i	0.894304	+	0.447460i	\(0.147671\pi\)
							−0.460497	+	0.887661i	\(0.652329\pi\)
\(41\)	2555.51	−	7865.05i	0.237420	−	0.730705i	−0.759371	−	0.650658i	\(-0.774493\pi\)
							0.996791	−	0.0800467i	\(-0.0255069\pi\)
\(43\)	−1748.03			−0.144171			−0.0720854	−	0.997398i	\(-0.522965\pi\)
							−0.0720854	+	0.997398i	\(0.522965\pi\)
\(47\)	−8232.49	+	25337.0i	−0.543609	+	1.67306i	0.180666	+	0.983544i	\(0.442175\pi\)
							−0.724275	+	0.689511i	\(0.757825\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.e.a.4.5	✓	20
3.2	odd	2		99.6.f.c.37.1		20
11.3	even	5	inner	33.6.e.a.25.5	yes	20
11.5	even	5		363.6.a.u.1.1		10
11.6	odd	10		363.6.a.q.1.10		10
33.5	odd	10		1089.6.a.bh.1.10		10
33.14	odd	10		99.6.f.c.91.1		20
33.17	even	10		1089.6.a.bl.1.1		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.e.a.4.5	✓	20	1.1	even	1	trivial
33.6.e.a.25.5	yes	20	11.3	even	5	inner
99.6.f.c.37.1		20	3.2	odd	2
99.6.f.c.91.1		20	33.14	odd	10
363.6.a.q.1.10		10	11.6	odd	10
363.6.a.u.1.1		10	11.5	even	5
1089.6.a.bh.1.10		10	33.5	odd	10
1089.6.a.bl.1.1		10	33.17	even	10