Embedded newform 140.2.w.b.123.18

• Label: 140.2.w.b.123.18
• Level: $140$
• Weight: $2$
• Character: 140.123
• 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			123.18
Character	\(\chi\)	\(=\)	140.123
Dual form			140.2.w.b.107.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41349 + 0.0450897i) q^{2} +(-0.543458 - 2.02821i) q^{3} +(1.99593 + 0.127468i) q^{4} +(-1.62425 - 1.53682i) q^{5} +(-0.676724 - 2.89137i) q^{6} +(-0.0742486 + 2.64471i) q^{7} +(2.81549 + 0.270171i) q^{8} +(-1.22023 + 0.704499i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 2 q^{2} - 8 q^{5} - 16 q^{6} - 4 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41349	+	0.0450897i	0.999492	+	0.0318832i
							\(3\)	−0.543458	−	2.02821i	−0.313766	−	1.17099i	−0.925133	−	0.379643i	\(-0.876047\pi\)
0.611367	−	0.791347i	\(-0.290620\pi\)
\(5\)	−1.62425	−	1.53682i	−0.726386	−	0.687287i
							\(7\)	−0.0742486	+	2.64471i	−0.0280633	+	0.999606i
\(11\)	0.366133	+	0.211387i	0.110393	+	0.0637356i								0.554180	−	0.832397i	\(-0.313032\pi\)
							−0.443787	+	0.896132i	\(0.646365\pi\)
\(13\)	−1.56422	−	1.56422i	−0.433837	−	0.433837i	0.456095	−	0.889931i	\(-0.349248\pi\)
							−0.889931	+	0.456095i	\(0.849248\pi\)
\(17\)	1.18219	+	4.41198i	0.286722	+	1.07006i	0.947572	+	0.319544i	\(0.103530\pi\)
							−0.660849	+	0.750519i	\(0.729804\pi\)
\(19\)	3.66683	+	6.35114i	0.841228	+	1.45705i	0.888857	+	0.458185i	\(0.151500\pi\)
							−0.0476283	+	0.998865i	\(0.515166\pi\)
\(23\)	−6.44210	−	1.72616i	−1.34327	−	0.359928i	−0.485624	−	0.874168i	\(-0.661408\pi\)
							−0.857647	+	0.514239i	\(0.828074\pi\)
\(29\)		−	4.72835i		−	0.878033i	−0.898479	−	0.439016i	\(-0.855327\pi\)
							0.898479	−	0.439016i	\(-0.144673\pi\)
\(31\)	1.70170	+	0.982474i	0.305634	+	0.176458i	0.644971	−	0.764207i	\(-0.276869\pi\)
							−0.339337	+	0.940665i	\(0.610203\pi\)
\(37\)	−7.20177	−	1.92971i	−1.18396	−	0.317242i	−0.387467	−	0.921884i	\(-0.626650\pi\)
							−0.796498	+	0.604641i	\(0.793316\pi\)
\(41\)	−4.01882			−0.627634			−0.313817	−	0.949483i	\(-0.601608\pi\)
							−0.313817	+	0.949483i	\(0.601608\pi\)
\(43\)	1.88664	−	1.88664i	0.287710	−	0.287710i	−0.548464	−	0.836174i	\(-0.684787\pi\)
							0.836174	+	0.548464i	\(0.184787\pi\)
\(47\)	1.70358	−	6.35786i	0.248493	−	0.927389i	−0.723102	−	0.690741i	\(-0.757284\pi\)
							0.971595	−	0.236648i	\(-0.0760488\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.2.w.b.123.18	yes	72
4.3	odd	2	inner	140.2.w.b.123.16	yes	72
5.2	odd	4	inner	140.2.w.b.67.10	yes	72
5.3	odd	4		700.2.be.e.207.9		72
5.4	even	2		700.2.be.e.543.1		72
7.2	even	3	inner	140.2.w.b.23.8	✓	72
7.3	odd	6		980.2.k.j.883.6		36
7.4	even	3		980.2.k.k.883.6		36
7.5	odd	6		980.2.x.m.863.8		72
7.6	odd	2		980.2.x.m.263.18		72
20.3	even	4		700.2.be.e.207.11		72
20.7	even	4	inner	140.2.w.b.67.8	yes	72
20.19	odd	2		700.2.be.e.543.3		72
28.3	even	6		980.2.k.j.883.5		36
28.11	odd	6		980.2.k.k.883.5		36
28.19	even	6		980.2.x.m.863.10		72
28.23	odd	6	inner	140.2.w.b.23.10	yes	72
28.27	even	2		980.2.x.m.263.16		72
35.2	odd	12	inner	140.2.w.b.107.16	yes	72
35.9	even	6		700.2.be.e.443.11		72
35.12	even	12		980.2.x.m.667.16		72
35.17	even	12		980.2.k.j.687.5		36
35.23	odd	12		700.2.be.e.107.3		72
35.27	even	4		980.2.x.m.67.10		72
35.32	odd	12		980.2.k.k.687.5		36
140.23	even	12		700.2.be.e.107.1		72
140.27	odd	4		980.2.x.m.67.8		72
140.47	odd	12		980.2.x.m.667.18		72
140.67	even	12		980.2.k.k.687.6		36
140.79	odd	6		700.2.be.e.443.9		72
140.87	odd	12		980.2.k.j.687.6		36
140.107	even	12	inner	140.2.w.b.107.18	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.w.b.23.8	✓	72	7.2	even	3	inner
140.2.w.b.23.10	yes	72	28.23	odd	6	inner
140.2.w.b.67.8	yes	72	20.7	even	4	inner
140.2.w.b.67.10	yes	72	5.2	odd	4	inner
140.2.w.b.107.16	yes	72	35.2	odd	12	inner
140.2.w.b.107.18	yes	72	140.107	even	12	inner
140.2.w.b.123.16	yes	72	4.3	odd	2	inner
140.2.w.b.123.18	yes	72	1.1	even	1	trivial
700.2.be.e.107.1		72	140.23	even	12
700.2.be.e.107.3		72	35.23	odd	12
700.2.be.e.207.9		72	5.3	odd	4
700.2.be.e.207.11		72	20.3	even	4
700.2.be.e.443.9		72	140.79	odd	6
700.2.be.e.443.11		72	35.9	even	6
700.2.be.e.543.1		72	5.4	even	2
700.2.be.e.543.3		72	20.19	odd	2
980.2.k.j.687.5		36	35.17	even	12
980.2.k.j.687.6		36	140.87	odd	12
980.2.k.j.883.5		36	28.3	even	6
980.2.k.j.883.6		36	7.3	odd	6
980.2.k.k.687.5		36	35.32	odd	12
980.2.k.k.687.6		36	140.67	even	12
980.2.k.k.883.5		36	28.11	odd	6
980.2.k.k.883.6		36	7.4	even	3
980.2.x.m.67.8		72	140.27	odd	4
980.2.x.m.67.10		72	35.27	even	4
980.2.x.m.263.16		72	28.27	even	2
980.2.x.m.263.18		72	7.6	odd	2
980.2.x.m.667.16		72	35.12	even	12
980.2.x.m.667.18		72	140.47	odd	12
980.2.x.m.863.8		72	7.5	odd	6
980.2.x.m.863.10		72	28.19	even	6