Embedded newform 140.3.t.a.11.14

• Label: 140.3.t.a.11.14
• Level: $140$
• Weight: $3$
• Character: 140.11
• Analytic conductor: $3.815$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	140.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.81472370104\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.14
Character	\(\chi\)	\(=\)	140.11
Dual form			140.3.t.a.51.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.422124 + 1.95495i) q^{2} +(-3.79087 - 2.18866i) q^{3} +(-3.64362 - 1.65046i) q^{4} +(-1.11803 - 1.93649i) q^{5} +(5.87893 - 6.48706i) q^{6} +(2.80797 + 6.41212i) q^{7} +(4.76462 - 6.42638i) q^{8} +(5.08046 + 8.79962i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 2 q^{2} + 6 q^{4} + 20 q^{8} + 96 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.422124	+	1.95495i	−0.211062	+	0.977473i
							\(3\)	−3.79087	−	2.18866i	−1.26362	−	0.729553i	−0.289850	−	0.957072i	\(-0.593605\pi\)
−0.973774	+	0.227519i	\(0.926939\pi\)
\(5\)	−1.11803	−	1.93649i	−0.223607	−	0.387298i
							\(7\)	2.80797	+	6.41212i	0.401139	+	0.916017i
\(11\)	14.4025	+	8.31528i	1.30932	+	0.755935i								0.981982	−	0.188974i	\(-0.0605161\pi\)
							0.327335	+	0.944908i	\(0.393849\pi\)
\(13\)	0.222899			0.0171461			0.00857305	−	0.999963i	\(-0.497271\pi\)
							0.00857305	+	0.999963i	\(0.497271\pi\)
\(17\)	−2.05476	+	3.55894i	−0.120868	+	0.209349i	−0.920110	−	0.391660i	\(-0.871901\pi\)
							0.799242	+	0.601009i	\(0.205234\pi\)
\(19\)	26.0576	−	15.0443i	1.37145	−	0.791808i	0.380341	−	0.924846i	\(-0.375807\pi\)
							0.991111	+	0.133039i	\(0.0424734\pi\)
\(23\)	−8.06000	+	4.65344i	−0.350435	+	0.202324i	−0.664877	−	0.746953i	\(-0.731516\pi\)
							0.314442	+	0.949277i	\(0.398183\pi\)
\(29\)	45.7265			1.57678			0.788389	−	0.615178i	\(-0.210916\pi\)
							0.788389	+	0.615178i	\(0.210916\pi\)
\(31\)	−23.1602	−	13.3716i	−0.747104	−	0.431341i	0.0775424	−	0.996989i	\(-0.475293\pi\)
							−0.824647	+	0.565648i	\(0.808626\pi\)
\(37\)	15.2459	+	26.4066i	0.412051	+	0.713693i	0.995114	−	0.0987338i	\(-0.0314792\pi\)
							−0.583063	+	0.812427i	\(0.698146\pi\)
\(41\)	24.4295			0.595841			0.297920	−	0.954591i	\(-0.403707\pi\)
							0.297920	+	0.954591i	\(0.403707\pi\)
\(43\)			57.4160i			1.33526i	0.744495	+	0.667628i	\(0.232690\pi\)
							−0.744495	+	0.667628i	\(0.767310\pi\)
\(47\)	49.3478	−	28.4909i	1.04995	−	0.606190i	0.127316	−	0.991862i	\(-0.459364\pi\)
							0.922636	+	0.385672i	\(0.126030\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.3.t.a.11.14	✓	64
4.3	odd	2	inner	140.3.t.a.11.29	yes	64
7.2	even	3	inner	140.3.t.a.51.29	yes	64
28.23	odd	6	inner	140.3.t.a.51.14	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.3.t.a.11.14	✓	64	1.1	even	1	trivial
140.3.t.a.11.29	yes	64	4.3	odd	2	inner
140.3.t.a.51.14	yes	64	28.23	odd	6	inner
140.3.t.a.51.29	yes	64	7.2	even	3	inner