Embedded newform 65.2.o.a.2.2

• Label: 65.2.o.a.2.2
• Level: $65$
• Weight: $2$
• Character: 65.2
• Analytic conductor: $0.519$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 65 = 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	65.o (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.519027613138\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(5\) over \(\Q(\zeta_{12})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial:	x^20 + 26*x^18 + 279*x^16 + 1604*x^14 + 5353*x^12 + 10466*x^10 + 11441*x^8 + 6176*x^6 + 1263*x^4 + 78*x^2 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			2.2
Root			\(1.83163i\) of defining polynomial
Character	\(\chi\)	\(=\)	65.2
Dual form			65.2.o.a.33.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.915816 - 1.58624i) q^{2} +(1.91432 + 0.512942i) q^{3} +(-0.677439 + 1.17336i) q^{4} +(1.45480 + 1.69810i) q^{5} +(-0.939520 - 3.50634i) q^{6} +(-3.06478 - 1.76945i) q^{7} -1.18163 q^{8} +(0.803451 + 0.463873i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{2} - 2 q^{3} - 6 q^{4} - 6 q^{5} - 8 q^{6} - 6 q^{7} + 12 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/65\mathbb{Z}\right)^\times\).

\(n\)	\(27\)	\(41\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.915816	−	1.58624i	−0.647580	−	1.12164i	−0.983699	−	0.179822i	\(-0.942448\pi\)
							0.336119	−	0.941819i	\(-0.390885\pi\)
\(3\)	1.91432	+	0.512942i	1.10524	+	0.296147i	0.764895	−	0.644155i	\(-0.222791\pi\)
							0.340341	+	0.940302i	\(0.389458\pi\)
\(5\)	1.45480	+	1.69810i	0.650608	+	0.759414i
							\(7\)	−3.06478	−	1.76945i	−1.15838	−	0.668790i	−0.207464	−	0.978243i	\(-0.566521\pi\)
−0.950915	+	0.309453i	\(0.899854\pi\)
\(11\)	−1.00269	+	3.74209i	−0.302323	+	1.12828i	0.632903	+	0.774231i	\(0.281863\pi\)
							−0.935226	+	0.354053i	\(0.884803\pi\)
\(13\)	0.573112	−	3.55971i	0.158953	−	0.987286i
							\(17\)	0.524334	+	1.95684i	0.127170	+	0.474603i	0.999908	−	0.0135853i	\(-0.00432448\pi\)
−0.872738	+	0.488189i	\(0.837658\pi\)
\(19\)	−0.518968	+	0.139057i	−0.119059	+	0.0319019i	−0.317857	−	0.948139i	\(-0.602963\pi\)
							0.198797	+	0.980041i	\(0.436296\pi\)
\(23\)	0.0788026	−	0.294095i	0.0164315	−	0.0613231i	−0.957223	−	0.289350i	\(-0.906561\pi\)
							0.973655	+	0.228027i	\(0.0732275\pi\)
\(29\)	1.71273	−	0.988843i	0.318045	−	0.183624i	−0.332476	−	0.943112i	\(-0.607884\pi\)
							0.650521	+	0.759488i	\(0.274551\pi\)
\(31\)	4.13563	−	4.13563i	0.742781	−	0.742781i	−0.230331	−	0.973112i	\(-0.573981\pi\)
							0.973112	+	0.230331i	\(0.0739810\pi\)
\(37\)	4.69189	−	2.70887i	0.771342	−	0.445335i	−0.0620109	−	0.998075i	\(-0.519751\pi\)
							0.833353	+	0.552741i	\(0.186418\pi\)
\(41\)	0.649884	+	0.174136i	0.101495	+	0.0271955i	0.309209	−	0.950994i	\(-0.399936\pi\)
							−0.207714	+	0.978190i	\(0.566602\pi\)
\(43\)	−8.51164	+	2.28069i	−1.29801	+	0.347802i	−0.840698	−	0.541504i	\(-0.817855\pi\)
							−0.457314	+	0.889305i	\(0.651189\pi\)
\(47\)			9.75201i			1.42248i	0.702951	+	0.711238i	\(0.251865\pi\)
							−0.702951	+	0.711238i	\(0.748135\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	65.2.o.a.2.2	✓	20
3.2	odd	2		585.2.cf.a.262.4		20
5.2	odd	4		325.2.x.b.93.4		20
5.3	odd	4		65.2.t.a.28.2	yes	20
5.4	even	2		325.2.s.b.132.4		20
13.2	odd	12		845.2.f.d.437.3		20
13.3	even	3		845.2.k.e.577.8		20
13.4	even	6		845.2.o.f.357.4		20
13.5	odd	4		845.2.t.e.427.2		20
13.6	odd	12		845.2.t.g.657.4		20
13.7	odd	12		65.2.t.a.7.2	yes	20
13.8	odd	4		845.2.t.f.427.4		20
13.9	even	3		845.2.o.e.357.2		20
13.10	even	6		845.2.k.d.577.3		20
13.11	odd	12		845.2.f.e.437.8		20
13.12	even	2		845.2.o.g.587.4		20
15.8	even	4		585.2.dp.a.28.4		20
39.20	even	12		585.2.dp.a.397.4		20
65.3	odd	12		845.2.f.e.408.3		20
65.7	even	12		325.2.s.b.293.4		20
65.8	even	4		845.2.o.e.258.2		20
65.18	even	4		845.2.o.f.258.4		20
65.23	odd	12		845.2.f.d.408.8		20
65.28	even	12		845.2.k.d.268.3		20
65.33	even	12	inner	65.2.o.a.33.2	yes	20
65.38	odd	4		845.2.t.g.418.4		20
65.43	odd	12		845.2.t.e.188.2		20
65.48	odd	12		845.2.t.f.188.4		20
65.58	even	12		845.2.o.g.488.4		20
65.59	odd	12		325.2.x.b.7.4		20
65.63	even	12		845.2.k.e.268.8		20
195.98	odd	12		585.2.cf.a.163.4		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.o.a.2.2	✓	20	1.1	even	1	trivial
65.2.o.a.33.2	yes	20	65.33	even	12	inner
65.2.t.a.7.2	yes	20	13.7	odd	12
65.2.t.a.28.2	yes	20	5.3	odd	4
325.2.s.b.132.4		20	5.4	even	2
325.2.s.b.293.4		20	65.7	even	12
325.2.x.b.7.4		20	65.59	odd	12
325.2.x.b.93.4		20	5.2	odd	4
585.2.cf.a.163.4		20	195.98	odd	12
585.2.cf.a.262.4		20	3.2	odd	2
585.2.dp.a.28.4		20	15.8	even	4
585.2.dp.a.397.4		20	39.20	even	12
845.2.f.d.408.8		20	65.23	odd	12
845.2.f.d.437.3		20	13.2	odd	12
845.2.f.e.408.3		20	65.3	odd	12
845.2.f.e.437.8		20	13.11	odd	12
845.2.k.d.268.3		20	65.28	even	12
845.2.k.d.577.3		20	13.10	even	6
845.2.k.e.268.8		20	65.63	even	12
845.2.k.e.577.8		20	13.3	even	3
845.2.o.e.258.2		20	65.8	even	4
845.2.o.e.357.2		20	13.9	even	3
845.2.o.f.258.4		20	65.18	even	4
845.2.o.f.357.4		20	13.4	even	6
845.2.o.g.488.4		20	65.58	even	12
845.2.o.g.587.4		20	13.12	even	2
845.2.t.e.188.2		20	65.43	odd	12
845.2.t.e.427.2		20	13.5	odd	4
845.2.t.f.188.4		20	65.48	odd	12
845.2.t.f.427.4		20	13.8	odd	4
845.2.t.g.418.4		20	65.38	odd	4
845.2.t.g.657.4		20	13.6	odd	12