Embedded newform 33.6.e.b.4.1

• Label: 33.6.e.b.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 - 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			4.1
Root			\(-8.59311 + 6.24326i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.4
Dual form			33.6.e.b.25.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.78409 - 5.65548i) q^{2} +(-2.78115 + 8.55951i) q^{3} +(18.7192 + 57.6117i) q^{4} +(48.1408 - 34.9763i) q^{5} +(70.0568 - 50.8993i) q^{6} +(9.43666 + 29.0431i) q^{7} +(84.9654 - 261.497i) q^{8} +(-65.5304 - 47.6106i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{2} + 45 q^{3} + 8 q^{4} - 11 q^{5} - 54 q^{6} - 139 q^{7} - 76 q^{8} - 405 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.78409	−	5.65548i	−1.37605	−	0.999756i	−0.997237	−	0.0742813i	\(-0.976334\pi\)
							−0.378809	−	0.925475i	\(-0.623666\pi\)
\(3\)	−2.78115	+	8.55951i	−0.178411	+	0.549093i
							\(5\)	48.1408	−	34.9763i	0.861169	−	0.625676i	−0.0670339	−	0.997751i	\(-0.521354\pi\)
0.928203	+	0.372075i	\(0.121354\pi\)
\(7\)	9.43666	+	29.0431i	0.0727903	+	0.224025i	0.980832	−	0.194854i	\(-0.0624232\pi\)
							−0.908042	+	0.418879i	\(0.862423\pi\)
\(11\)	−70.0591	−	395.149i	−0.174575	−	0.984644i
							\(13\)	−460.773	−	334.771i	−0.756186	−	0.549401i	0.141552	−	0.989931i	\(-0.454791\pi\)
−0.897738	+	0.440529i	\(0.854791\pi\)
\(17\)	1619.20	−	1176.42i	1.35887	−	0.987280i	0.360359	−	0.932814i	\(-0.382654\pi\)
							0.998516	−	0.0544664i	\(-0.0173458\pi\)
\(19\)	497.601	−	1531.46i	0.316226	−	0.973243i	−0.659021	−	0.752124i	\(-0.729029\pi\)
							0.975247	−	0.221118i	\(-0.0709707\pi\)
\(23\)	−1214.78			−0.478827			−0.239413	−	0.970918i	\(-0.576955\pi\)
							−0.239413	+	0.970918i	\(0.576955\pi\)
\(29\)	−2348.47	−	7227.84i	−0.518549	−	1.59593i	−0.776731	−	0.629833i	\(-0.783123\pi\)
							0.258182	−	0.966096i	\(-0.416877\pi\)
\(31\)	5873.48	+	4267.33i	1.09772	+	0.797540i	0.980686	−	0.195588i	\(-0.0626615\pi\)
							0.117034	+	0.993128i	\(0.462662\pi\)
\(37\)	−406.300	−	1250.46i	−0.0487913	−	0.150164i	0.923693	−	0.383135i	\(-0.125155\pi\)
							−0.972484	+	0.232970i	\(0.925155\pi\)
\(41\)	1804.73	−	5554.40i	0.167669	−	0.516033i	−0.831554	−	0.555444i	\(-0.812548\pi\)
							0.999223	+	0.0394114i	\(0.0125483\pi\)
\(43\)	6357.22			0.524320			0.262160	−	0.965024i	\(-0.415565\pi\)
							0.262160	+	0.965024i	\(0.415565\pi\)
\(47\)	−9346.16	+	28764.5i	−0.617147	+	1.89938i	−0.257631	+	0.966243i	\(0.582942\pi\)
							−0.359516	+	0.933139i	\(0.617058\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.e.b.4.1	✓	20
3.2	odd	2		99.6.f.b.37.5		20
11.3	even	5	inner	33.6.e.b.25.1	yes	20
11.5	even	5		363.6.a.r.1.10		10
11.6	odd	10		363.6.a.t.1.1		10
33.5	odd	10		1089.6.a.bk.1.1		10
33.14	odd	10		99.6.f.b.91.5		20
33.17	even	10		1089.6.a.bi.1.10		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.e.b.4.1	✓	20	1.1	even	1	trivial
33.6.e.b.25.1	yes	20	11.3	even	5	inner
99.6.f.b.37.5		20	3.2	odd	2
99.6.f.b.91.5		20	33.14	odd	10
363.6.a.r.1.10		10	11.5	even	5
363.6.a.t.1.1		10	11.6	odd	10
1089.6.a.bi.1.10		10	33.17	even	10
1089.6.a.bk.1.1		10	33.5	odd	10