Embedded newform 140.3.j.a.27.14

• Label: 140.3.j.a.27.14
• 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			27.14
Character	\(\chi\)	\(=\)	140.27
Dual form			140.3.j.a.83.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.29551 - 1.52370i) q^{2} +(1.72993 - 1.72993i) q^{3} +(-0.643329 + 3.94793i) q^{4} +(-1.92397 + 4.61501i) q^{5} +(-4.87703 - 0.394761i) q^{6} +(2.78648 + 6.42149i) q^{7} +(6.84890 - 4.13432i) q^{8} +3.01469i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 4 q^{2} - 16 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.29551	−	1.52370i	−0.647753	−	0.761850i
							\(3\)	1.72993	−	1.72993i	0.576643	−	0.576643i	−0.357334	−	0.933977i	\(-0.616314\pi\)
0.933977	+	0.357334i	\(0.116314\pi\)
\(5\)	−1.92397	+	4.61501i	−0.384794	+	0.923002i
							\(7\)	2.78648	+	6.42149i	0.398069	+	0.917355i
\(11\)			18.3854i			1.67140i								0.549188	+	0.835699i	\(0.314937\pi\)
							−0.549188	+	0.835699i	\(0.685063\pi\)
\(13\)	−7.89590	−	7.89590i	−0.607377	−	0.607377i	0.334883	−	0.942260i	\(-0.391303\pi\)
							−0.942260	+	0.334883i	\(0.891303\pi\)
\(17\)	8.00506	−	8.00506i	0.470886	−	0.470886i	−0.431315	−	0.902201i	\(-0.641950\pi\)
							0.902201	+	0.431315i	\(0.141950\pi\)
\(19\)		−	16.9427i		−	0.891719i	−0.895103	−	0.445859i	\(-0.852898\pi\)
							0.895103	−	0.445859i	\(-0.147102\pi\)
\(23\)	23.0876	+	23.0876i	1.00381	+	1.00381i	0.999993	+	0.00381673i	\(0.00121491\pi\)
							0.00381673	+	0.999993i	\(0.498785\pi\)
\(29\)			0.0974941i			0.00336187i	0.999999	+	0.00168093i	\(0.000535058\pi\)
							−0.999999	+	0.00168093i	\(0.999465\pi\)
\(31\)	−1.24115			−0.0400371			−0.0200185	−	0.999800i	\(-0.506373\pi\)
							−0.0200185	+	0.999800i	\(0.506373\pi\)
\(37\)	8.74675	+	8.74675i	0.236399	+	0.236399i	0.815357	−	0.578958i	\(-0.196541\pi\)
							−0.578958	+	0.815357i	\(0.696541\pi\)
\(41\)		−	56.5964i		−	1.38040i	−0.723618	−	0.690201i	\(-0.757522\pi\)
							0.723618	−	0.690201i	\(-0.242478\pi\)
\(43\)	−34.3533	−	34.3533i	−0.798914	−	0.798914i	0.184011	−	0.982924i	\(-0.441092\pi\)
							−0.982924	+	0.184011i	\(0.941092\pi\)
\(47\)	46.5189	+	46.5189i	0.989763	+	0.989763i	0.999948	−	0.0101848i	\(-0.00324197\pi\)
							−0.0101848	+	0.999948i	\(0.503242\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.3.j.a.27.14	yes	88
4.3	odd	2	inner	140.3.j.a.27.9	✓	88
5.3	odd	4	inner	140.3.j.a.83.10	yes	88
7.6	odd	2	inner	140.3.j.a.27.13	yes	88
20.3	even	4	inner	140.3.j.a.83.13	yes	88
28.27	even	2	inner	140.3.j.a.27.10	yes	88
35.13	even	4	inner	140.3.j.a.83.9	yes	88
140.83	odd	4	inner	140.3.j.a.83.14	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.3.j.a.27.9	✓	88	4.3	odd	2	inner
140.3.j.a.27.10	yes	88	28.27	even	2	inner
140.3.j.a.27.13	yes	88	7.6	odd	2	inner
140.3.j.a.27.14	yes	88	1.1	even	1	trivial
140.3.j.a.83.9	yes	88	35.13	even	4	inner
140.3.j.a.83.10	yes	88	5.3	odd	4	inner
140.3.j.a.83.13	yes	88	20.3	even	4	inner
140.3.j.a.83.14	yes	88	140.83	odd	4	inner