Embedded newform 33.6.e.a.4.1

• Label: 33.6.e.a.4.1
• 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			4.1
Root			\(-7.12695 + 5.17803i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.4
Dual form			33.6.e.a.25.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.93597 - 5.76582i) q^{2} +(2.78115 - 8.55951i) q^{3} +(19.8464 + 61.0809i) q^{4} +(-37.2750 + 27.0819i) q^{5} +(-71.4237 + 51.8924i) q^{6} +(37.2644 + 114.688i) q^{7} +(97.6803 - 300.629i) q^{8} +(-65.5304 - 47.6106i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} - 45 q^{3} - 124 q^{4} - 33 q^{5} - 18 q^{6} - 335 q^{7} - 472 q^{8} - 405 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.93597	−	5.76582i	−1.40289	−	1.01926i	−0.994308	−	0.106542i	\(-0.966022\pi\)
							−0.408586	−	0.912720i	\(-0.633978\pi\)
\(3\)	2.78115	−	8.55951i	0.178411	−	0.549093i
							\(5\)	−37.2750	+	27.0819i	−0.666796	+	0.484456i	−0.868951	−	0.494898i	\(-0.835206\pi\)
0.202155	+	0.979354i	\(0.435206\pi\)
\(7\)	37.2644	+	114.688i	0.287441	+	0.884653i	0.985656	+	0.168765i	\(0.0539779\pi\)
							−0.698215	+	0.715888i	\(0.746022\pi\)
\(11\)	319.427	−	242.935i	0.795957	−	0.605354i
							\(13\)	488.591	+	354.982i	0.801839	+	0.582570i	0.911453	−	0.411404i	\(-0.134962\pi\)
−0.109614	+	0.993974i	\(0.534962\pi\)
\(17\)	−1125.77	+	817.922i	−0.944776	+	0.686420i	−0.949565	−	0.313569i	\(-0.898475\pi\)
							0.00478977	+	0.999989i	\(0.498475\pi\)
\(19\)	−258.169	+	794.562i	−0.164066	+	0.504944i	−0.998966	−	0.0454574i	\(-0.985525\pi\)
							0.834900	+	0.550402i	\(0.185525\pi\)
\(23\)	3264.14			1.28662			0.643309	−	0.765607i	\(-0.277561\pi\)
							0.643309	+	0.765607i	\(0.277561\pi\)
\(29\)	2614.37	+	8046.20i	0.577261	+	1.77663i	0.628349	+	0.777931i	\(0.283731\pi\)
							−0.0510888	+	0.998694i	\(0.516269\pi\)
\(31\)	5178.44	+	3762.36i	0.967821	+	0.703163i	0.954954	−	0.296754i	\(-0.0959042\pi\)
							0.0128669	+	0.999917i	\(0.495904\pi\)
\(37\)	−3631.02	−	11175.1i	−0.436038	−	1.34199i	−0.892019	−	0.451997i	\(-0.850712\pi\)
							0.455982	−	0.889989i	\(-0.349288\pi\)
\(41\)	−3290.02	+	10125.6i	−0.305660	+	0.940725i	0.673770	+	0.738941i	\(0.264674\pi\)
							−0.979430	+	0.201784i	\(0.935326\pi\)
\(43\)	−2640.34			−0.217765			−0.108883	−	0.994055i	\(-0.534727\pi\)
							−0.108883	+	0.994055i	\(0.534727\pi\)
\(47\)	−4454.65	+	13710.0i	−0.294150	+	0.905300i	0.689356	+	0.724423i	\(0.257894\pi\)
							−0.983506	+	0.180877i	\(0.942106\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.e.a.4.1	✓	20
3.2	odd	2		99.6.f.c.37.5		20
11.3	even	5	inner	33.6.e.a.25.1	yes	20
11.5	even	5		363.6.a.u.1.9		10
11.6	odd	10		363.6.a.q.1.2		10
33.5	odd	10		1089.6.a.bh.1.2		10
33.14	odd	10		99.6.f.c.91.5		20
33.17	even	10		1089.6.a.bl.1.9		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.e.a.4.1	✓	20	1.1	even	1	trivial
33.6.e.a.25.1	yes	20	11.3	even	5	inner
99.6.f.c.37.5		20	3.2	odd	2
99.6.f.c.91.5		20	33.14	odd	10
363.6.a.q.1.2		10	11.6	odd	10
363.6.a.u.1.9		10	11.5	even	5
1089.6.a.bh.1.2		10	33.5	odd	10
1089.6.a.bl.1.9		10	33.17	even	10