Embedded newform 35.4.j.a.4.7

• Label: 35.4.j.a.4.7
• Level: $35$
• Weight: $4$
• Character: 35.4
• Analytic conductor: $2.065$
• Analytic rank: $0$
• Dimension: $20$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 35 = 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	35.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.06506685020\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(10\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial:	x^20 - 55*x^18 + 2018*x^16 - 42095*x^14 + 639938*x^12 - 5744691*x^10 + 35287093*x^8 - 51070316*x^6 + 53741776*x^4 - 27082496*x^2 + 9834496
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{10}\cdot 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			4.7
Root			\(-0.793194 + 0.457951i\) of defining polynomial
Character	\(\chi\)	\(=\)	35.4
Dual form			35.4.j.a.9.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.793194 - 0.457951i) q^{2} +(-7.59866 - 4.38709i) q^{3} +(-3.58056 + 6.20172i) q^{4} +(-1.99348 - 11.0012i) q^{5} -8.03628 q^{6} +(-13.8805 - 12.2610i) q^{7} +13.8861i q^{8} +(24.9931 + 43.2892i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 30 q^{4} + 3 q^{5} - 96 q^{6} + 82 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\)	\(22\)	\(31\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.793194	−	0.457951i	0.280436	−	0.161910i	−0.353185	−	0.935554i	\(-0.614901\pi\)
							0.633621	+	0.773644i	\(0.281568\pi\)
\(3\)	−7.59866	−	4.38709i	−1.46236	−	0.844295i	−0.463242	−	0.886232i	\(-0.653314\pi\)
							−0.999120	+	0.0419367i	\(0.986647\pi\)
\(5\)	−1.99348	−	11.0012i	−0.178302	−	0.983976i
							\(7\)	−13.8805	−	12.2610i	−0.749475	−	0.662032i
\(11\)	22.8271	−	39.5377i	0.625693	−	1.08373i								−0.362713	−	0.931901i	\(-0.618149\pi\)
							0.988406	−	0.151832i	\(-0.0485172\pi\)
\(13\)			16.0789i			0.343037i	0.985181	+	0.171518i	\(0.0548673\pi\)
							−0.985181	+	0.171518i	\(0.945133\pi\)
\(17\)	−6.30744	−	3.64160i	−0.0899869	−	0.0519540i	0.454331	−	0.890833i	\(-0.349878\pi\)
							−0.544318	+	0.838879i	\(0.683212\pi\)
\(19\)	−37.8128	−	65.4936i	−0.456571	−	0.790803i	0.542206	−	0.840245i	\(-0.317589\pi\)
							−0.998777	+	0.0494419i	\(0.984256\pi\)
\(23\)	−13.7451	+	7.93576i	−0.124611	+	0.0719444i	−0.561010	−	0.827809i	\(-0.689587\pi\)
							0.436399	+	0.899753i	\(0.356254\pi\)
\(29\)	−119.356			−0.764271			−0.382135	−	0.924106i	\(-0.624811\pi\)
							−0.382135	+	0.924106i	\(0.624811\pi\)
\(31\)	29.7834	−	51.5863i	0.172556	−	0.298876i	−0.766757	−	0.641938i	\(-0.778131\pi\)
							0.939313	+	0.343062i	\(0.111464\pi\)
\(37\)	263.284	−	152.007i	1.16983	−	0.675401i	0.216190	−	0.976351i	\(-0.430637\pi\)
							0.953640	+	0.300950i	\(0.0973037\pi\)
\(41\)	238.104			0.906966			0.453483	−	0.891265i	\(-0.350181\pi\)
							0.453483	+	0.891265i	\(0.350181\pi\)
\(43\)		−	365.491i		−	1.29621i	−0.761552	−	0.648104i	\(-0.775562\pi\)
							0.761552	−	0.648104i	\(-0.224438\pi\)
\(47\)	−87.0587	+	50.2633i	−0.270187	+	0.155993i	−0.628973	−	0.777427i	\(-0.716524\pi\)
							0.358785	+	0.933420i	\(0.383191\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	35.4.j.a.4.7	yes	20
5.2	odd	4		175.4.e.g.151.7		20
5.3	odd	4		175.4.e.g.151.4		20
5.4	even	2	inner	35.4.j.a.4.4	✓	20
7.2	even	3	inner	35.4.j.a.9.4	yes	20
7.3	odd	6		245.4.b.f.99.7		10
7.4	even	3		245.4.b.e.99.7		10
7.5	odd	6		245.4.j.d.79.4		20
7.6	odd	2		245.4.j.d.214.7		20
35.2	odd	12		175.4.e.g.51.7		20
35.3	even	12		1225.4.a.bq.1.7		10
35.4	even	6		245.4.b.e.99.4		10
35.9	even	6	inner	35.4.j.a.9.7	yes	20
35.17	even	12		1225.4.a.bq.1.4		10
35.18	odd	12		1225.4.a.bp.1.7		10
35.19	odd	6		245.4.j.d.79.7		20
35.23	odd	12		175.4.e.g.51.4		20
35.24	odd	6		245.4.b.f.99.4		10
35.32	odd	12		1225.4.a.bp.1.4		10
35.34	odd	2		245.4.j.d.214.4		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.j.a.4.4	✓	20	5.4	even	2	inner
35.4.j.a.4.7	yes	20	1.1	even	1	trivial
35.4.j.a.9.4	yes	20	7.2	even	3	inner
35.4.j.a.9.7	yes	20	35.9	even	6	inner
175.4.e.g.51.4		20	35.23	odd	12
175.4.e.g.51.7		20	35.2	odd	12
175.4.e.g.151.4		20	5.3	odd	4
175.4.e.g.151.7		20	5.2	odd	4
245.4.b.e.99.4		10	35.4	even	6
245.4.b.e.99.7		10	7.4	even	3
245.4.b.f.99.4		10	35.24	odd	6
245.4.b.f.99.7		10	7.3	odd	6
245.4.j.d.79.4		20	7.5	odd	6
245.4.j.d.79.7		20	35.19	odd	6
245.4.j.d.214.4		20	35.34	odd	2
245.4.j.d.214.7		20	7.6	odd	2
1225.4.a.bp.1.4		10	35.32	odd	12
1225.4.a.bp.1.7		10	35.18	odd	12
1225.4.a.bq.1.4		10	35.17	even	12
1225.4.a.bq.1.7		10	35.3	even	12