Embedded newform 33.6.e.a.16.2

• Label: 33.6.e.a.16.2
• 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			16.2
Root			\(-2.05193 + 6.31518i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.16
Dual form			33.6.e.a.31.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.74291 - 5.36413i) q^{2} +(-7.28115 - 5.29007i) q^{3} +(0.152432 - 0.110748i) q^{4} +(21.0541 - 64.7978i) q^{5} +(-15.6862 + 48.2771i) q^{6} +(-106.538 + 77.4041i) q^{7} +(-146.876 - 106.711i) q^{8} +(25.0304 + 77.0356i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} - 45 q^{3} - 124 q^{4} - 33 q^{5} - 18 q^{6} - 335 q^{7} - 472 q^{8} - 405 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.74291	−	5.36413i	−0.308106	−	0.948252i	−0.978500	−	0.206247i	\(-0.933875\pi\)
							0.670394	−	0.742005i	\(-0.266125\pi\)
\(3\)	−7.28115	−	5.29007i	−0.467086	−	0.339358i
							\(5\)	21.0541	−	64.7978i	0.376627	−	1.15914i	−0.565747	−	0.824579i	\(-0.691412\pi\)
0.942374	−	0.334560i	\(-0.108588\pi\)
\(7\)	−106.538	+	77.4041i	−0.821784	+	0.597061i	−0.917223	−	0.398374i	\(-0.869575\pi\)
							0.0954388	+	0.995435i	\(0.469575\pi\)
\(11\)	−98.6986	+	388.985i	−0.245940	+	0.969285i
							\(13\)	−4.06908	−	12.5233i	−0.00667786	−	0.0205524i	0.947662	−	0.319275i	\(-0.103439\pi\)
−0.954340	+	0.298723i	\(0.903439\pi\)
\(17\)	618.609	−	1903.88i	0.519151	−	1.59778i	−0.256449	−	0.966558i	\(-0.582553\pi\)
							0.775600	−	0.631224i	\(-0.217447\pi\)
\(19\)	−2250.32	−	1634.96i	−1.43008	−	1.03902i	−0.990002	−	0.141056i	\(-0.954950\pi\)
							−0.440080	−	0.897959i	\(-0.645050\pi\)
\(23\)	2936.26			1.15738			0.578689	−	0.815548i	\(-0.303564\pi\)
							0.578689	+	0.815548i	\(0.303564\pi\)
\(29\)	1927.70	−	1400.56i	0.425643	−	0.309248i	−0.354261	−	0.935146i	\(-0.615268\pi\)
							0.779904	+	0.625899i	\(0.215268\pi\)
\(31\)	−603.783	−	1858.25i	−0.112843	−	0.347296i	0.878648	−	0.477471i	\(-0.158446\pi\)
							−0.991491	+	0.130174i	\(0.958446\pi\)
\(37\)	7396.17	−	5373.63i	0.888183	−	0.645303i	−0.0472204	−	0.998884i	\(-0.515036\pi\)
							0.935404	+	0.353582i	\(0.115036\pi\)
\(41\)	−7644.56	−	5554.10i	−0.710220	−	0.516005i	0.173025	−	0.984917i	\(-0.444646\pi\)
							−0.883245	+	0.468912i	\(0.844646\pi\)
\(43\)	−1942.32			−0.160195			−0.0800977	−	0.996787i	\(-0.525523\pi\)
							−0.0800977	+	0.996787i	\(0.525523\pi\)
\(47\)	−9324.43	−	6774.59i	−0.615712	−	0.447341i	0.235709	−	0.971824i	\(-0.424259\pi\)
							−0.851421	+	0.524483i	\(0.824259\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.e.a.16.2	✓	20
3.2	odd	2		99.6.f.c.82.4		20
11.3	even	5		363.6.a.u.1.3		10
11.8	odd	10		363.6.a.q.1.8		10
11.9	even	5	inner	33.6.e.a.31.2	yes	20
33.8	even	10		1089.6.a.bl.1.3		10
33.14	odd	10		1089.6.a.bh.1.8		10
33.20	odd	10		99.6.f.c.64.4		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.e.a.16.2	✓	20	1.1	even	1	trivial
33.6.e.a.31.2	yes	20	11.9	even	5	inner
99.6.f.c.64.4		20	33.20	odd	10
99.6.f.c.82.4		20	3.2	odd	2
363.6.a.q.1.8		10	11.8	odd	10
363.6.a.u.1.3		10	11.3	even	5
1089.6.a.bh.1.8		10	33.14	odd	10
1089.6.a.bl.1.3		10	33.8	even	10