Embedded newform 33.6.e.b.16.4

• Label: 33.6.e.b.16.4
• 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 - x^19 + 78*x^18 + 79*x^17 + 10573*x^16 - 33409*x^15 + 1262953*x^14 - 1581925*x^13 + 89291182*x^12 - 100741271*x^11 + 2277268901*x^10 - 1144062486*x^9 + 70677624924*x^8 + 93641098134*x^7 + 1732079176577*x^6 + 1113900934906*x^5 + 27749747552381*x^4 + 4154923377746*x^3 + 57625758470132*x^2 + 60036938210920*x + 25599187870096
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{4}\cdot 11^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			16.4
Root			\(1.34436 - 4.13753i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.16
Dual form			33.6.e.b.31.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.03535 + 3.18647i) q^{2} +(7.28115 + 5.29007i) q^{3} +(16.8069 - 12.2109i) q^{4} +(32.9083 - 101.281i) q^{5} +(-9.31813 + 28.6783i) q^{6} +(-32.7853 + 23.8199i) q^{7} +(143.049 + 103.931i) q^{8} +(25.0304 + 77.0356i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{2} + 45 q^{3} + 8 q^{4} - 11 q^{5} - 54 q^{6} - 139 q^{7} - 76 q^{8} - 405 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.03535	+	3.18647i	0.183025	+	0.563294i	0.999909	−	0.0135087i	\(-0.00430008\pi\)
							−0.816883	+	0.576803i	\(0.804300\pi\)
\(3\)	7.28115	+	5.29007i	0.467086	+	0.339358i
							\(5\)	32.9083	−	101.281i	0.588681	−	1.81177i	0.00472315	−	0.999989i	\(-0.498497\pi\)
0.583958	−	0.811784i	\(-0.301503\pi\)
\(7\)	−32.7853	+	23.8199i	−0.252891	+	0.183736i	−0.707007	−	0.707206i	\(-0.749955\pi\)
							0.454116	+	0.890943i	\(0.349955\pi\)
\(11\)	55.2651	+	397.488i	0.137711	+	0.990472i
							\(13\)	−202.056	−	621.863i	−0.331598	−	1.02055i	−0.968373	−	0.249505i	\(-0.919732\pi\)
0.636775	−	0.771050i	\(-0.280268\pi\)
\(17\)	−601.087	+	1849.95i	−0.504446	+	1.55253i	0.297254	+	0.954798i	\(0.403929\pi\)
							−0.801700	+	0.597727i	\(0.796071\pi\)
\(19\)	394.744	+	286.798i	0.250860	+	0.182260i	0.706108	−	0.708104i	\(-0.250449\pi\)
							−0.455248	+	0.890365i	\(0.650449\pi\)
\(23\)	805.987			0.317694			0.158847	−	0.987303i	\(-0.449222\pi\)
							0.158847	+	0.987303i	\(0.449222\pi\)
\(29\)	−2949.77	+	2143.13i	−0.651318	+	0.473210i	−0.863720	−	0.503972i	\(-0.831871\pi\)
							0.212402	+	0.977182i	\(0.431871\pi\)
\(31\)	−457.166	−	1407.01i	−0.0854417	−	0.262963i	0.899203	−	0.437531i	\(-0.144147\pi\)
							−0.984645	+	0.174568i	\(0.944147\pi\)
\(37\)	590.547	−	429.058i	0.0709170	−	0.0515242i	−0.551762	−	0.834002i	\(-0.686044\pi\)
							0.622679	+	0.782477i	\(0.286044\pi\)
\(41\)	−129.943	−	94.4091i	−0.0120724	−	0.00877110i	0.581733	−	0.813380i	\(-0.302375\pi\)
							−0.593805	+	0.804609i	\(0.702375\pi\)
\(43\)	−9144.84			−0.754232			−0.377116	−	0.926166i	\(-0.623084\pi\)
							−0.377116	+	0.926166i	\(0.623084\pi\)
\(47\)	18217.2	+	13235.6i	1.20292	+	0.873973i	0.994569	−	0.104081i	\(-0.0331901\pi\)
							0.208352	+	0.978054i	\(0.433190\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.e.b.16.4	✓	20
3.2	odd	2		99.6.f.b.82.2		20
11.3	even	5		363.6.a.r.1.7		10
11.8	odd	10		363.6.a.t.1.4		10
11.9	even	5	inner	33.6.e.b.31.4	yes	20
33.8	even	10		1089.6.a.bi.1.7		10
33.14	odd	10		1089.6.a.bk.1.4		10
33.20	odd	10		99.6.f.b.64.2		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.e.b.16.4	✓	20	1.1	even	1	trivial
33.6.e.b.31.4	yes	20	11.9	even	5	inner
99.6.f.b.64.2		20	33.20	odd	10
99.6.f.b.82.2		20	3.2	odd	2
363.6.a.r.1.7		10	11.3	even	5
363.6.a.t.1.4		10	11.8	odd	10
1089.6.a.bi.1.7		10	33.8	even	10
1089.6.a.bk.1.4		10	33.14	odd	10