Embedded newform 33.6.e.a.31.3

• Label: 33.6.e.a.31.3
• Level: $33$
• Weight: $6$
• Character: 33.31
• 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			31.3
Root			\(0.992840 + 3.05565i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.31
Dual form			33.6.e.a.16.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.30186 - 4.00670i) q^{2} +(-7.28115 + 5.29007i) q^{3} +(11.5297 + 8.37681i) q^{4} +(-26.9724 - 83.0125i) q^{5} +(11.7167 + 36.0603i) q^{6} +(-114.007 - 82.8310i) q^{7} +(157.639 - 114.532i) q^{8} +(25.0304 - 77.0356i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} - 45 q^{3} - 124 q^{4} - 33 q^{5} - 18 q^{6} - 335 q^{7} - 472 q^{8} - 405 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{3}{5}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.30186	−	4.00670i	0.230138	−	0.708292i	−0.767591	−	0.640940i	\(-0.778545\pi\)
							0.997729	−	0.0673523i	\(-0.0214551\pi\)
\(3\)	−7.28115	+	5.29007i	−0.467086	+	0.339358i
							\(5\)	−26.9724	−	83.0125i	−0.482497	−	1.48497i	−0.835574	−	0.549379i	\(-0.814864\pi\)
0.353077	−	0.935594i	\(-0.385136\pi\)
\(7\)	−114.007	−	82.8310i	−0.879401	−	0.638922i	0.0536919	−	0.998558i	\(-0.482901\pi\)
							−0.933093	+	0.359635i	\(0.882901\pi\)
\(11\)	−279.344	−	288.128i	−0.696078	−	0.717966i
							\(13\)	−100.022	+	307.837i	−0.164149	+	0.505199i	−0.998973	−	0.0453185i	\(-0.985570\pi\)
0.834823	+	0.550518i	\(0.185570\pi\)
\(17\)	59.5926	+	183.407i	0.0500115	+	0.153920i	0.972943	−	0.231044i	\(-0.0742141\pi\)
							−0.922932	+	0.384963i	\(0.874214\pi\)
\(19\)	1356.66	−	985.668i	0.862155	−	0.626392i	−0.0663152	−	0.997799i	\(-0.521124\pi\)
							0.928470	+	0.371406i	\(0.121124\pi\)
\(23\)	3665.89			1.44497			0.722487	−	0.691385i	\(-0.242999\pi\)
							0.722487	+	0.691385i	\(0.242999\pi\)
\(29\)	1173.00	+	852.234i	0.259002	+	0.188176i	0.709707	−	0.704497i	\(-0.248827\pi\)
							−0.450705	+	0.892673i	\(0.648827\pi\)
\(31\)	−1294.26	+	3983.32i	−0.241890	+	0.744459i	0.754243	+	0.656595i	\(0.228004\pi\)
							−0.996132	+	0.0878640i	\(0.971996\pi\)
\(37\)	−10765.7	−	7821.77i	−1.29282	−	0.939292i	−0.292966	−	0.956123i	\(-0.594642\pi\)
							−0.999858	+	0.0168306i	\(0.994642\pi\)
\(41\)	−9386.66	+	6819.81i	−0.872070	+	0.633596i	−0.931142	−	0.364658i	\(-0.881186\pi\)
							0.0590714	+	0.998254i	\(0.481186\pi\)
\(43\)	13362.8			1.10211			0.551057	−	0.834468i	\(-0.314225\pi\)
							0.551057	+	0.834468i	\(0.314225\pi\)
\(47\)	14908.1	−	10831.3i	0.984411	−	0.715217i	0.0257212	−	0.999669i	\(-0.491812\pi\)
							0.958690	+	0.284452i	\(0.0918118\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.e.a.31.3	yes	20
3.2	odd	2		99.6.f.c.64.3		20
11.4	even	5		363.6.a.u.1.6		10
11.5	even	5	inner	33.6.e.a.16.3	✓	20
11.7	odd	10		363.6.a.q.1.5		10
33.5	odd	10		99.6.f.c.82.3		20
33.26	odd	10		1089.6.a.bh.1.5		10
33.29	even	10		1089.6.a.bl.1.6		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.e.a.16.3	✓	20	11.5	even	5	inner
33.6.e.a.31.3	yes	20	1.1	even	1	trivial
99.6.f.c.64.3		20	3.2	odd	2
99.6.f.c.82.3		20	33.5	odd	10
363.6.a.q.1.5		10	11.7	odd	10
363.6.a.u.1.6		10	11.4	even	5
1089.6.a.bh.1.5		10	33.26	odd	10
1089.6.a.bl.1.6		10	33.29	even	10