Embedded newform 33.6.e.b.31.4

• Label: 33.6.e.b.31.4
• 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 - 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			31.4
Root			\(1.34436 + 4.13753i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.31
Dual form			33.6.e.b.16.4

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

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

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)

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

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.e.b.31.4	yes	20
3.2	odd	2		99.6.f.b.64.2		20
11.4	even	5		363.6.a.r.1.7		10
11.5	even	5	inner	33.6.e.b.16.4	✓	20
11.7	odd	10		363.6.a.t.1.4		10
33.5	odd	10		99.6.f.b.82.2		20
33.26	odd	10		1089.6.a.bk.1.4		10
33.29	even	10		1089.6.a.bi.1.7		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.e.b.16.4	✓	20	11.5	even	5	inner
33.6.e.b.31.4	yes	20	1.1	even	1	trivial
99.6.f.b.64.2		20	3.2	odd	2
99.6.f.b.82.2		20	33.5	odd	10
363.6.a.r.1.7		10	11.4	even	5
363.6.a.t.1.4		10	11.7	odd	10
1089.6.a.bi.1.7		10	33.29	even	10
1089.6.a.bk.1.4		10	33.26	odd	10