Embedded newform 140.2.w.b.67.16

• Label: 140.2.w.b.67.16
• Level: $140$
• Weight: $2$
• Character: 140.67
• Analytic conductor: $1.118$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	140.w (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.11790562830\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(18\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			67.16
Character	\(\chi\)	\(=\)	140.67
Dual form			140.2.w.b.23.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.20559 - 0.739299i) q^{2} +(-2.47915 + 0.664287i) q^{3} +(0.906874 - 1.78258i) q^{4} +(1.93364 - 1.12296i) q^{5} +(-2.49773 + 2.63369i) q^{6} +(1.41045 - 2.23844i) q^{7} +(-0.224544 - 2.81950i) q^{8} +(3.10685 - 1.79374i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 2 q^{2} - 8 q^{5} - 16 q^{6} - 4 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(57\)	\(71\)	\(101\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.20559	−	0.739299i	0.852478	−	0.522763i
							\(3\)	−2.47915	+	0.664287i	−1.43134	+	0.383527i	−0.889492	−	0.456950i	\(-0.848942\pi\)
−0.541848	+	0.840476i	\(0.682275\pi\)
\(5\)	1.93364	−	1.12296i	0.864751	−	0.502202i
							\(7\)	1.41045	−	2.23844i	0.533101	−	0.846051i
\(11\)	1.59653	+	0.921758i	0.481372	+	0.277920i								0.720988	−	0.692947i	\(-0.243688\pi\)
							−0.239616	+	0.970868i	\(0.577022\pi\)
\(13\)	−2.94578	+	2.94578i	−0.817012	+	0.817012i	−0.985674	−	0.168662i	\(-0.946055\pi\)
							0.168662	+	0.985674i	\(0.446055\pi\)
\(17\)	−2.96933	+	0.795631i	−0.720169	+	0.192969i	−0.600247	−	0.799815i	\(-0.704931\pi\)
							−0.119922	+	0.992783i	\(0.538264\pi\)
\(19\)	2.66374	+	4.61374i	0.611104	+	1.05846i	0.991055	+	0.133457i	\(0.0426077\pi\)
							−0.379950	+	0.925007i	\(0.624059\pi\)
\(23\)	−0.654945	+	2.44429i	−0.136565	+	0.509669i	0.863421	+	0.504484i	\(0.168317\pi\)
							−0.999987	+	0.00518527i	\(0.998349\pi\)
\(29\)			6.35796i			1.18064i	0.807168	+	0.590322i	\(0.200999\pi\)
							−0.807168	+	0.590322i	\(0.799001\pi\)
\(31\)	−3.78330	−	2.18429i	−0.679500	−	0.392310i	0.120166	−	0.992754i	\(-0.461657\pi\)
							−0.799667	+	0.600444i	\(0.794991\pi\)
\(37\)	−1.03655	+	3.86848i	−0.170409	+	0.635973i	0.826880	+	0.562379i	\(0.190114\pi\)
							−0.997288	+	0.0735946i	\(0.976553\pi\)
\(41\)	10.6462			1.66266			0.831331	−	0.555777i	\(-0.187579\pi\)
							0.831331	+	0.555777i	\(0.187579\pi\)
\(43\)	−1.02386	−	1.02386i	−0.156137	−	0.156137i	0.624715	−	0.780853i	\(-0.285215\pi\)
							−0.780853	+	0.624715i	\(0.785215\pi\)
\(47\)	−0.785472	−	0.210467i	−0.114573	−	0.0306997i	0.201077	−	0.979575i	\(-0.435556\pi\)
							−0.315650	+	0.948876i	\(0.602223\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.2.w.b.67.16	yes	72
4.3	odd	2	inner	140.2.w.b.67.7	yes	72
5.2	odd	4		700.2.be.e.543.14		72
5.3	odd	4	inner	140.2.w.b.123.5	yes	72
5.4	even	2		700.2.be.e.207.3		72
7.2	even	3	inner	140.2.w.b.107.2	yes	72
7.3	odd	6		980.2.k.j.687.10		36
7.4	even	3		980.2.k.k.687.10		36
7.5	odd	6		980.2.x.m.667.2		72
7.6	odd	2		980.2.x.m.67.16		72
20.3	even	4	inner	140.2.w.b.123.2	yes	72
20.7	even	4		700.2.be.e.543.17		72
20.19	odd	2		700.2.be.e.207.12		72
28.3	even	6		980.2.k.j.687.18		36
28.11	odd	6		980.2.k.k.687.18		36
28.19	even	6		980.2.x.m.667.5		72
28.23	odd	6	inner	140.2.w.b.107.5	yes	72
28.27	even	2		980.2.x.m.67.7		72
35.2	odd	12		700.2.be.e.443.12		72
35.3	even	12		980.2.k.j.883.18		36
35.9	even	6		700.2.be.e.107.17		72
35.13	even	4		980.2.x.m.263.5		72
35.18	odd	12		980.2.k.k.883.18		36
35.23	odd	12	inner	140.2.w.b.23.7	✓	72
35.33	even	12		980.2.x.m.863.7		72
140.3	odd	12		980.2.k.j.883.10		36
140.23	even	12	inner	140.2.w.b.23.16	yes	72
140.79	odd	6		700.2.be.e.107.14		72
140.83	odd	4		980.2.x.m.263.2		72
140.103	odd	12		980.2.x.m.863.16		72
140.107	even	12		700.2.be.e.443.3		72
140.123	even	12		980.2.k.k.883.10		36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.w.b.23.7	✓	72	35.23	odd	12	inner
140.2.w.b.23.16	yes	72	140.23	even	12	inner
140.2.w.b.67.7	yes	72	4.3	odd	2	inner
140.2.w.b.67.16	yes	72	1.1	even	1	trivial
140.2.w.b.107.2	yes	72	7.2	even	3	inner
140.2.w.b.107.5	yes	72	28.23	odd	6	inner
140.2.w.b.123.2	yes	72	20.3	even	4	inner
140.2.w.b.123.5	yes	72	5.3	odd	4	inner
700.2.be.e.107.14		72	140.79	odd	6
700.2.be.e.107.17		72	35.9	even	6
700.2.be.e.207.3		72	5.4	even	2
700.2.be.e.207.12		72	20.19	odd	2
700.2.be.e.443.3		72	140.107	even	12
700.2.be.e.443.12		72	35.2	odd	12
700.2.be.e.543.14		72	5.2	odd	4
700.2.be.e.543.17		72	20.7	even	4
980.2.k.j.687.10		36	7.3	odd	6
980.2.k.j.687.18		36	28.3	even	6
980.2.k.j.883.10		36	140.3	odd	12
980.2.k.j.883.18		36	35.3	even	12
980.2.k.k.687.10		36	7.4	even	3
980.2.k.k.687.18		36	28.11	odd	6
980.2.k.k.883.10		36	140.123	even	12
980.2.k.k.883.18		36	35.18	odd	12
980.2.x.m.67.7		72	28.27	even	2
980.2.x.m.67.16		72	7.6	odd	2
980.2.x.m.263.2		72	140.83	odd	4
980.2.x.m.263.5		72	35.13	even	4
980.2.x.m.667.2		72	7.5	odd	6
980.2.x.m.667.5		72	28.19	even	6
980.2.x.m.863.7		72	35.33	even	12
980.2.x.m.863.16		72	140.103	odd	12