Embedded newform 33.6.e.b.25.2

• Label: 33.6.e.b.25.2
• Level: $33$
• Weight: $6$
• Character: 33.25
• 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			25.2
Root			\(-3.89647 - 2.83095i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.25
Dual form			33.6.e.b.4.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.08746 + 2.24317i) q^{2} +(-2.78115 - 8.55951i) q^{3} +(-5.38796 + 16.5824i) q^{4} +(3.77724 + 2.74432i) q^{5} +(27.7871 + 20.1885i) q^{6} +(49.4196 - 152.098i) q^{7} +(-58.2998 - 179.428i) q^{8} +(-65.5304 + 47.6106i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{2} + 45 q^{3} + 8 q^{4} - 11 q^{5} - 54 q^{6} - 139 q^{7} - 76 q^{8} - 405 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.08746	+	2.24317i	−0.545791	+	0.396540i	−0.826231	−	0.563331i	\(-0.809519\pi\)
							0.280441	+	0.959871i	\(0.409519\pi\)
\(3\)	−2.78115	−	8.55951i	−0.178411	−	0.549093i
							\(5\)	3.77724	+	2.74432i	0.0675692	+	0.0490919i	0.621057	−	0.783765i	\(-0.286703\pi\)
−0.553488	+	0.832857i	\(0.686703\pi\)
\(7\)	49.4196	−	152.098i	0.381201	−	1.17322i	−0.557998	−	0.829842i	\(-0.688430\pi\)
							0.939199	−	0.343374i	\(-0.111570\pi\)
\(11\)	85.6495	−	392.065i	0.213424	−	0.976960i
							\(13\)	821.799	−	597.072i	1.34867	−	0.979869i	0.349598	−	0.936900i	\(-0.386318\pi\)
0.999076	−	0.0429698i	\(-0.0136819\pi\)
\(17\)	−1647.76	−	1197.17i	−1.38284	−	1.00469i	−0.996609	−	0.0822865i	\(-0.973778\pi\)
							−0.386228	−	0.922403i	\(-0.626222\pi\)
\(19\)	919.594	+	2830.22i	0.584402	+	1.79861i	0.601657	+	0.798754i	\(0.294507\pi\)
							−0.0172548	+	0.999851i	\(0.505493\pi\)
\(23\)	−1906.44			−0.751456			−0.375728	−	0.926730i	\(-0.622607\pi\)
							−0.375728	+	0.926730i	\(0.622607\pi\)
\(29\)	−487.060	+	1499.02i	−0.107544	+	0.330988i	−0.990319	−	0.138809i	\(-0.955673\pi\)
							0.882775	+	0.469796i	\(0.155673\pi\)
\(31\)	1910.76	−	1388.25i	0.357109	−	0.259455i	−0.394736	−	0.918794i	\(-0.629164\pi\)
							0.751845	+	0.659340i	\(0.229164\pi\)
\(37\)	1230.94	−	3788.44i	0.147819	−	0.454942i	−0.849543	−	0.527519i	\(-0.823122\pi\)
							0.997363	+	0.0725773i	\(0.0231224\pi\)
\(41\)	1063.73	+	3273.84i	0.0988265	+	0.304157i	0.988232	−	0.152962i	\(-0.0488813\pi\)
							−0.889406	+	0.457119i	\(0.848881\pi\)
\(43\)	−2635.68			−0.217381			−0.108691	−	0.994076i	\(-0.534666\pi\)
							−0.108691	+	0.994076i	\(0.534666\pi\)
\(47\)	−5004.26	−	15401.5i	−0.330442	−	1.01700i	−0.968924	−	0.247359i	\(-0.920437\pi\)
							0.638482	−	0.769637i	\(-0.279563\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.e.b.25.2	yes	20
3.2	odd	2		99.6.f.b.91.4		20
11.2	odd	10		363.6.a.t.1.3		10
11.4	even	5	inner	33.6.e.b.4.2	✓	20
11.9	even	5		363.6.a.r.1.8		10
33.2	even	10		1089.6.a.bi.1.8		10
33.20	odd	10		1089.6.a.bk.1.3		10
33.26	odd	10		99.6.f.b.37.4		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.e.b.4.2	✓	20	11.4	even	5	inner
33.6.e.b.25.2	yes	20	1.1	even	1	trivial
99.6.f.b.37.4		20	33.26	odd	10
99.6.f.b.91.4		20	3.2	odd	2
363.6.a.r.1.8		10	11.9	even	5
363.6.a.t.1.3		10	11.2	odd	10
1089.6.a.bi.1.8		10	33.2	even	10
1089.6.a.bk.1.3		10	33.20	odd	10