Embedded newform 33.6.e.b.4.3

• Label: 33.6.e.b.4.3
• 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			4.3
Root			\(-0.490010 + 0.356013i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.4
Dual form			33.6.e.b.25.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.319007 + 0.231772i) q^{2} +(-2.78115 + 8.55951i) q^{3} +(-9.84050 - 30.2859i) q^{4} +(-17.3054 + 12.5731i) q^{5} +(-2.87106 + 2.08595i) q^{6} +(-52.3175 - 161.017i) q^{7} +(7.77944 - 23.9427i) q^{8} +(-65.5304 - 47.6106i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{2} + 45 q^{3} + 8 q^{4} - 11 q^{5} - 54 q^{6} - 139 q^{7} - 76 q^{8} - 405 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.319007	+	0.231772i	0.0563930	+	0.0409719i	0.615625	−	0.788040i	\(-0.288904\pi\)
							−0.559232	+	0.829011i	\(0.688904\pi\)
\(3\)	−2.78115	+	8.55951i	−0.178411	+	0.549093i
							\(5\)	−17.3054	+	12.5731i	−0.309569	+	0.224915i	−0.731712	−	0.681614i	\(-0.761278\pi\)
0.422143	+	0.906529i	\(0.361278\pi\)
\(7\)	−52.3175	−	161.017i	−0.403554	−	1.24201i	−0.922097	−	0.386960i	\(-0.873525\pi\)
							0.518543	−	0.855052i	\(-0.326475\pi\)
\(11\)	−52.4927	−	397.864i	−0.130803	−	0.991408i
							\(13\)	−256.745	−	186.536i	−0.421351	−	0.306130i	0.356830	−	0.934169i	\(-0.383858\pi\)
−0.778181	+	0.628040i	\(0.783858\pi\)
\(17\)	−406.637	+	295.439i	−0.341260	+	0.247940i	−0.745193	−	0.666849i	\(-0.767643\pi\)
							0.403934	+	0.914788i	\(0.367643\pi\)
\(19\)	−53.6197	+	165.024i	−0.0340753	+	0.104873i	−0.966648	−	0.256110i	\(-0.917559\pi\)
							0.932572	+	0.360984i	\(0.117559\pi\)
\(23\)	3677.58			1.44958			0.724791	−	0.688969i	\(-0.241936\pi\)
							0.724791	+	0.688969i	\(0.241936\pi\)
\(29\)	810.645	+	2494.91i	0.178993	+	0.550884i	0.999793	−	0.0203316i	\(-0.00647219\pi\)
							−0.820800	+	0.571215i	\(0.806472\pi\)
\(31\)	−2054.46	−	1492.65i	−0.383966	−	0.278968i	0.379012	−	0.925392i	\(-0.376264\pi\)
							−0.762978	+	0.646424i	\(0.776264\pi\)
\(37\)	−3879.42	−	11939.6i	−0.465868	−	1.43379i	−0.857888	−	0.513837i	\(-0.828223\pi\)
							0.392019	−	0.919957i	\(-0.371777\pi\)
\(41\)	5963.90	−	18355.0i	0.554078	−	1.70528i	−0.144289	−	0.989536i	\(-0.546089\pi\)
							0.698366	−	0.715740i	\(-0.253911\pi\)
\(43\)	−6550.25			−0.540240			−0.270120	−	0.962827i	\(-0.587063\pi\)
							−0.270120	+	0.962827i	\(0.587063\pi\)
\(47\)	2398.65	−	7382.30i	0.158388	−	0.487469i	−0.840100	−	0.542431i	\(-0.817504\pi\)
							0.998488	+	0.0549623i	\(0.0175039\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.e.b.4.3	✓	20
3.2	odd	2		99.6.f.b.37.3		20
11.3	even	5	inner	33.6.e.b.25.3	yes	20
11.5	even	5		363.6.a.r.1.5		10
11.6	odd	10		363.6.a.t.1.6		10
33.5	odd	10		1089.6.a.bk.1.6		10
33.14	odd	10		99.6.f.b.91.3		20
33.17	even	10		1089.6.a.bi.1.5		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.e.b.4.3	✓	20	1.1	even	1	trivial
33.6.e.b.25.3	yes	20	11.3	even	5	inner
99.6.f.b.37.3		20	3.2	odd	2
99.6.f.b.91.3		20	33.14	odd	10
363.6.a.r.1.5		10	11.5	even	5
363.6.a.t.1.6		10	11.6	odd	10
1089.6.a.bi.1.5		10	33.17	even	10
1089.6.a.bk.1.6		10	33.5	odd	10