Embedded newform 33.6.e.b.31.3

• Label: 33.6.e.b.31.3
• 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			31.3
Root			\(0.527694 + 1.62407i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.31
Dual form			33.6.e.b.16.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.218677 - 0.673018i) q^{2} +(7.28115 - 5.29007i) q^{3} +(25.4834 + 18.5148i) q^{4} +(-21.0138 - 64.6739i) q^{5} +(-1.96809 - 6.05716i) q^{6} +(98.3565 + 71.4602i) q^{7} +(36.3535 - 26.4124i) q^{8} +(25.0304 - 77.0356i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{2} + 45 q^{3} + 8 q^{4} - 11 q^{5} - 54 q^{6} - 139 q^{7} - 76 q^{8} - 405 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.218677	−	0.673018i	0.0386569	−	0.118974i	−0.929866	−	0.367899i	\(-0.880077\pi\)
							0.968523	+	0.248925i	\(0.0800773\pi\)
\(3\)	7.28115	−	5.29007i	0.467086	−	0.339358i
							\(5\)	−21.0138	−	64.6739i	−0.375907	−	1.15692i	−0.942865	−	0.333175i	\(-0.891880\pi\)
0.566958	−	0.823747i	\(-0.308120\pi\)
\(7\)	98.3565	+	71.4602i	0.758679	+	0.551213i	0.898505	−	0.438963i	\(-0.144654\pi\)
							−0.139826	+	0.990176i	\(0.544654\pi\)
\(11\)	251.233	−	312.943i	0.626029	−	0.779800i
							\(13\)	235.729	−	725.500i	0.386861	−	1.19064i	−0.548260	−	0.836308i	\(-0.684710\pi\)
0.935121	−	0.354328i	\(-0.115290\pi\)
\(17\)	468.035	+	1440.46i	0.392786	+	1.20887i	0.930673	+	0.365853i	\(0.119223\pi\)
							−0.537887	+	0.843017i	\(0.680777\pi\)
\(19\)	−2013.29	+	1462.74i	−1.27945	+	0.929573i	−0.999536	−	0.0304534i	\(-0.990305\pi\)
							−0.279911	+	0.960026i	\(0.590305\pi\)
\(23\)	−3942.75			−1.55410			−0.777052	−	0.629437i	\(-0.783286\pi\)
							−0.777052	+	0.629437i	\(0.783286\pi\)
\(29\)	178.184	+	129.459i	0.0393437	+	0.0285848i	0.607283	−	0.794485i	\(-0.292259\pi\)
							−0.567940	+	0.823070i	\(0.692259\pi\)
\(31\)	−2884.27	+	8876.86i	−0.539052	+	1.65903i	0.195674	+	0.980669i	\(0.437311\pi\)
							−0.734726	+	0.678364i	\(0.762689\pi\)
\(37\)	−1615.79	−	1173.94i	−0.194035	−	0.140975i	0.486526	−	0.873666i	\(-0.338264\pi\)
							−0.680561	+	0.732691i	\(0.738264\pi\)
\(41\)	5162.61	−	3750.86i	0.479634	−	0.348474i	−0.321550	−	0.946893i	\(-0.604204\pi\)
							0.801184	+	0.598418i	\(0.204204\pi\)
\(43\)	8057.77			0.664575			0.332287	−	0.943178i	\(-0.392180\pi\)
							0.332287	+	0.943178i	\(0.392180\pi\)
\(47\)	−8202.70	+	5959.61i	−0.541642	+	0.393526i	−0.824694	−	0.565579i	\(-0.808653\pi\)
							0.283053	+	0.959104i	\(0.408653\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.e.b.31.3	yes	20
3.2	odd	2		99.6.f.b.64.3		20
11.4	even	5		363.6.a.r.1.6		10
11.5	even	5	inner	33.6.e.b.16.3	✓	20
11.7	odd	10		363.6.a.t.1.5		10
33.5	odd	10		99.6.f.b.82.3		20
33.26	odd	10		1089.6.a.bk.1.5		10
33.29	even	10		1089.6.a.bi.1.6		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.e.b.16.3	✓	20	11.5	even	5	inner
33.6.e.b.31.3	yes	20	1.1	even	1	trivial
99.6.f.b.64.3		20	3.2	odd	2
99.6.f.b.82.3		20	33.5	odd	10
363.6.a.r.1.6		10	11.4	even	5
363.6.a.t.1.5		10	11.7	odd	10
1089.6.a.bi.1.6		10	33.29	even	10
1089.6.a.bk.1.5		10	33.26	odd	10