Embedded newform 140.3.j.a.27.16

• Label: 140.3.j.a.27.16
• Level: $140$
• Weight: $3$
• Character: 140.27
• Analytic conductor: $3.815$
• Analytic rank: $0$
• Dimension: $88$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	140.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.81472370104\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(44\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			27.16
Character	\(\chi\)	\(=\)	140.27
Dual form			140.3.j.a.83.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18798 + 1.60895i) q^{2} +(0.419773 - 0.419773i) q^{3} +(-1.17743 - 3.82278i) q^{4} +(-3.27835 - 3.77523i) q^{5} +(0.176713 + 1.17407i) q^{6} +(1.57725 + 6.81999i) q^{7} +(7.54941 + 2.64695i) q^{8} +8.64758i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 4 q^{2} - 16 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/140\mathbb{Z}\right)^\times\).

\(n\)	\(57\)	\(71\)	\(101\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18798	+	1.60895i	−0.593988	+	0.804474i
							\(3\)	0.419773	−	0.419773i	0.139924	−	0.139924i	−0.633675	−	0.773599i	\(-0.718454\pi\)
0.773599	+	0.633675i	\(0.218454\pi\)
\(5\)	−3.27835	−	3.77523i	−0.655671	−	0.755047i
							\(7\)	1.57725	+	6.81999i	0.225321	+	0.974285i
\(11\)			13.2698i			1.20635i								0.797610	+	0.603174i	\(0.206098\pi\)
							−0.797610	+	0.603174i	\(0.793902\pi\)
\(13\)	12.7614	+	12.7614i	0.981644	+	0.981644i	0.999835	−	0.0181904i	\(-0.00579051\pi\)
							−0.0181904	+	0.999835i	\(0.505791\pi\)
\(17\)	12.4770	−	12.4770i	0.733940	−	0.733940i	−0.237458	−	0.971398i	\(-0.576314\pi\)
							0.971398	+	0.237458i	\(0.0763143\pi\)
\(19\)			0.277851i			0.0146238i	0.999973	+	0.00731188i	\(0.00232746\pi\)
							−0.999973	+	0.00731188i	\(0.997673\pi\)
\(23\)	−11.9500	−	11.9500i	−0.519565	−	0.519565i	0.397875	−	0.917440i	\(-0.369748\pi\)
							−0.917440	+	0.397875i	\(0.869748\pi\)
\(29\)			17.1941i			0.592900i	0.955048	+	0.296450i	\(0.0958028\pi\)
							−0.955048	+	0.296450i	\(0.904197\pi\)
\(31\)	−18.3615			−0.592307			−0.296153	−	0.955140i	\(-0.595704\pi\)
							−0.296153	+	0.955140i	\(0.595704\pi\)
\(37\)	18.6599	+	18.6599i	0.504322	+	0.504322i	0.912778	−	0.408456i	\(-0.133933\pi\)
							−0.408456	+	0.912778i	\(0.633933\pi\)
\(41\)		−	14.2914i		−	0.348571i	−0.984695	−	0.174286i	\(-0.944238\pi\)
							0.984695	−	0.174286i	\(-0.0557616\pi\)
\(43\)	23.8261	+	23.8261i	0.554096	+	0.554096i	0.927620	−	0.373524i	\(-0.121851\pi\)
							−0.373524	+	0.927620i	\(0.621851\pi\)
\(47\)	−62.0743	−	62.0743i	−1.32073	−	1.32073i	−0.913185	−	0.407546i	\(-0.866385\pi\)
							−0.407546	−	0.913185i	\(-0.633615\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.3.j.a.27.16	yes	88
4.3	odd	2	inner	140.3.j.a.27.37	yes	88
5.3	odd	4	inner	140.3.j.a.83.38	yes	88
7.6	odd	2	inner	140.3.j.a.27.15	✓	88
20.3	even	4	inner	140.3.j.a.83.15	yes	88
28.27	even	2	inner	140.3.j.a.27.38	yes	88
35.13	even	4	inner	140.3.j.a.83.37	yes	88
140.83	odd	4	inner	140.3.j.a.83.16	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.3.j.a.27.15	✓	88	7.6	odd	2	inner
140.3.j.a.27.16	yes	88	1.1	even	1	trivial
140.3.j.a.27.37	yes	88	4.3	odd	2	inner
140.3.j.a.27.38	yes	88	28.27	even	2	inner
140.3.j.a.83.15	yes	88	20.3	even	4	inner
140.3.j.a.83.16	yes	88	140.83	odd	4	inner
140.3.j.a.83.37	yes	88	35.13	even	4	inner
140.3.j.a.83.38	yes	88	5.3	odd	4	inner