Embedded newform 33.6.e.a.4.4

• Label: 33.6.e.a.4.4
• 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			4.4
Root			\(3.88934 - 2.82577i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.4
Dual form			33.6.e.a.25.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.08033 + 2.23799i) q^{2} +(2.78115 - 8.55951i) q^{3} +(-5.40872 - 16.6463i) q^{4} +(20.5684 - 14.9438i) q^{5} +(27.7229 - 20.1419i) q^{6} +(-26.9084 - 82.8155i) q^{7} +(58.2442 - 179.257i) q^{8} +(-65.5304 - 47.6106i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} - 45 q^{3} - 124 q^{4} - 33 q^{5} - 18 q^{6} - 335 q^{7} - 472 q^{8} - 405 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.08033	+	2.23799i	0.544530	+	0.395624i	0.825765	−	0.564015i	\(-0.190744\pi\)
							−0.281235	+	0.959639i	\(0.590744\pi\)
\(3\)	2.78115	−	8.55951i	0.178411	−	0.549093i
							\(5\)	20.5684	−	14.9438i	0.367938	−	0.267323i	−0.388417	−	0.921484i	\(-0.626978\pi\)
0.756356	+	0.654161i	\(0.226978\pi\)
\(7\)	−26.9084	−	82.8155i	−0.207560	−	0.638803i	−0.999599	−	0.0283322i	\(-0.990980\pi\)
							0.792039	−	0.610471i	\(-0.209020\pi\)
\(11\)	375.272	+	142.203i	0.935114	+	0.354346i
							\(13\)	270.435	+	196.482i	0.443817	+	0.322452i	0.787150	−	0.616762i	\(-0.211556\pi\)
−0.343333	+	0.939214i	\(0.611556\pi\)
\(17\)	−125.262	+	91.0081i	−0.105123	+	0.0763761i	−0.639105	−	0.769120i	\(-0.720695\pi\)
							0.533982	+	0.845496i	\(0.320695\pi\)
\(19\)	−153.046	+	471.028i	−0.0972609	+	0.299338i	−0.987836	−	0.155497i	\(-0.950302\pi\)
							0.890575	+	0.454836i	\(0.150302\pi\)
\(23\)	−1320.61			−0.520542			−0.260271	−	0.965536i	\(-0.583812\pi\)
							−0.260271	+	0.965536i	\(0.583812\pi\)
\(29\)	−329.136	−	1012.98i	−0.0726743	−	0.223668i	0.908121	−	0.418707i	\(-0.137517\pi\)
							−0.980796	+	0.195039i	\(0.937517\pi\)
\(31\)	8149.64	+	5921.06i	1.52312	+	1.10661i	0.959914	+	0.280295i	\(0.0904322\pi\)
							0.563206	+	0.826317i	\(0.309568\pi\)
\(37\)	1421.10	+	4373.70i	0.170655	+	0.525224i	0.999408	−	0.0343910i	\(-0.0109492\pi\)
							−0.828753	+	0.559615i	\(0.810949\pi\)
\(41\)	4828.95	−	14862.0i	0.448635	−	1.38076i	−0.429813	−	0.902918i	\(-0.641421\pi\)
							0.878448	−	0.477838i	\(-0.158579\pi\)
\(43\)	11364.2			0.937276			0.468638	−	0.883390i	\(-0.344745\pi\)
							0.468638	+	0.883390i	\(0.344745\pi\)
\(47\)	2162.73	−	6656.19i	0.142809	−	0.439522i	−0.853913	−	0.520415i	\(-0.825777\pi\)
							0.996723	+	0.0808930i	\(0.0257772\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.e.a.4.4	✓	20
3.2	odd	2		99.6.f.c.37.2		20
11.3	even	5	inner	33.6.e.a.25.4	yes	20
11.5	even	5		363.6.a.u.1.4		10
11.6	odd	10		363.6.a.q.1.7		10
33.5	odd	10		1089.6.a.bh.1.7		10
33.14	odd	10		99.6.f.c.91.2		20
33.17	even	10		1089.6.a.bl.1.4		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.e.a.4.4	✓	20	1.1	even	1	trivial
33.6.e.a.25.4	yes	20	11.3	even	5	inner
99.6.f.c.37.2		20	3.2	odd	2
99.6.f.c.91.2		20	33.14	odd	10
363.6.a.q.1.7		10	11.6	odd	10
363.6.a.u.1.4		10	11.5	even	5
1089.6.a.bh.1.7		10	33.5	odd	10
1089.6.a.bl.1.4		10	33.17	even	10