Embedded newform 33.6.e.a.31.2

• Label: 33.6.e.a.31.2
• 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			31.2
Root			\(-2.05193 - 6.31518i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.31
Dual form			33.6.e.a.16.2

$q$-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 \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.e.a.31.2	yes	20
3.2	odd	2		99.6.f.c.64.4		20
11.4	even	5		363.6.a.u.1.3		10
11.5	even	5	inner	33.6.e.a.16.2	✓	20
11.7	odd	10		363.6.a.q.1.8		10
33.5	odd	10		99.6.f.c.82.4		20
33.26	odd	10		1089.6.a.bh.1.8		10
33.29	even	10		1089.6.a.bl.1.3		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.e.a.16.2	✓	20	11.5	even	5	inner
33.6.e.a.31.2	yes	20	1.1	even	1	trivial
99.6.f.c.64.4		20	3.2	odd	2
99.6.f.c.82.4		20	33.5	odd	10
363.6.a.q.1.8		10	11.7	odd	10
363.6.a.u.1.3		10	11.4	even	5
1089.6.a.bh.1.8		10	33.26	odd	10
1089.6.a.bl.1.3		10	33.29	even	10