Embedded newform 33.6.e.b.4.4

• Label: 33.6.e.b.4.4
• 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			4.4
Root			\(4.70404 - 3.41769i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.4
Dual form			33.6.e.b.25.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.51306 + 4.00547i) q^{2} +(-2.78115 + 8.55951i) q^{3} +(4.46148 + 13.7310i) q^{4} +(-62.5986 + 45.4806i) q^{5} +(-49.6176 + 36.0493i) q^{6} +(59.3828 + 182.761i) q^{7} +(36.9828 - 113.821i) q^{8} +(-65.5304 - 47.6106i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{2} + 45 q^{3} + 8 q^{4} - 11 q^{5} - 54 q^{6} - 139 q^{7} - 76 q^{8} - 405 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.51306	+	4.00547i	0.974581	+	0.708074i	0.956491	−	0.291762i	\(-0.0942415\pi\)
							0.0180898	+	0.999836i	\(0.494242\pi\)
\(3\)	−2.78115	+	8.55951i	−0.178411	+	0.549093i
							\(5\)	−62.5986	+	45.4806i	−1.11980	+	0.813581i	−0.984178	−	0.177180i	\(-0.943303\pi\)
−0.135620	+	0.990761i	\(0.543303\pi\)
\(7\)	59.3828	+	182.761i	0.458053	+	1.40974i	0.867513	+	0.497414i	\(0.165717\pi\)
							−0.409461	+	0.912328i	\(0.634283\pi\)
\(11\)	385.600	+	111.193i	0.960848	+	0.277075i
							\(13\)	−216.388	−	157.215i	−0.355120	−	0.258010i	0.395894	−	0.918296i	\(-0.370435\pi\)
−0.751014	+	0.660286i	\(0.770435\pi\)
\(17\)	1189.63	−	864.319i	0.998368	−	0.725357i	0.0366302	−	0.999329i	\(-0.488338\pi\)
							0.961738	+	0.273972i	\(0.0883376\pi\)
\(19\)	−727.309	+	2238.43i	−0.462205	+	1.42252i	0.400259	+	0.916402i	\(0.368920\pi\)
							−0.862464	+	0.506119i	\(0.831080\pi\)
\(23\)	2228.76			0.878503			0.439251	−	0.898364i	\(-0.355244\pi\)
							0.439251	+	0.898364i	\(0.355244\pi\)
\(29\)	−2118.06	−	6518.72i	−0.467674	−	1.43935i	−0.855588	−	0.517657i	\(-0.826804\pi\)
							0.387915	−	0.921695i	\(-0.373196\pi\)
\(31\)	4802.37	+	3489.12i	0.897534	+	0.652097i	0.937832	−	0.347091i	\(-0.112830\pi\)
							−0.0402971	+	0.999188i	\(0.512830\pi\)
\(37\)	165.708	+	509.998i	0.0198994	+	0.0612441i	0.960513	−	0.278235i	\(-0.0897493\pi\)
							−0.940614	+	0.339479i	\(0.889749\pi\)
\(41\)	−3609.97	+	11110.3i	−0.335385	+	1.03221i	0.631147	+	0.775664i	\(0.282585\pi\)
							−0.966532	+	0.256546i	\(0.917415\pi\)
\(43\)	−12622.3			−1.04104			−0.520521	−	0.853849i	\(-0.674262\pi\)
							−0.520521	+	0.853849i	\(0.674262\pi\)
\(47\)	143.345	−	441.170i	0.00946536	−	0.0291314i	−0.946212	−	0.323546i	\(-0.895125\pi\)
							0.955678	+	0.294415i	\(0.0951248\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.e.b.4.4	✓	20
3.2	odd	2		99.6.f.b.37.2		20
11.3	even	5	inner	33.6.e.b.25.4	yes	20
11.5	even	5		363.6.a.r.1.3		10
11.6	odd	10		363.6.a.t.1.8		10
33.5	odd	10		1089.6.a.bk.1.8		10
33.14	odd	10		99.6.f.b.91.2		20
33.17	even	10		1089.6.a.bi.1.3		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.e.b.4.4	✓	20	1.1	even	1	trivial
33.6.e.b.25.4	yes	20	11.3	even	5	inner
99.6.f.b.37.2		20	3.2	odd	2
99.6.f.b.91.2		20	33.14	odd	10
363.6.a.r.1.3		10	11.5	even	5
363.6.a.t.1.8		10	11.6	odd	10
1089.6.a.bi.1.3		10	33.17	even	10
1089.6.a.bk.1.8		10	33.5	odd	10