Embedded newform 140.3.t.a.11.2

• Label: 140.3.t.a.11.2
• 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			11.2
Character	\(\chi\)	\(=\)	140.11
Dual form			140.3.t.a.51.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.97671 + 0.304335i) q^{2} +(0.0515150 + 0.0297422i) q^{3} +(3.81476 - 1.20316i) q^{4} +(-1.11803 - 1.93649i) q^{5} +(-0.110882 - 0.0431139i) q^{6} +(6.36655 + 2.90982i) q^{7} +(-7.17451 + 3.53927i) q^{8} +(-4.49823 - 7.79116i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 2 q^{2} + 6 q^{4} + 20 q^{8} + 96 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.97671	+	0.304335i	−0.988355	+	0.152167i
							\(3\)	0.0515150	+	0.0297422i	0.0171717	+	0.00991406i	0.508561	−	0.861026i	\(-0.330178\pi\)
−0.491390	+	0.870940i	\(0.663511\pi\)
\(5\)	−1.11803	−	1.93649i	−0.223607	−	0.387298i
							\(7\)	6.36655	+	2.90982i	0.909507	+	0.415689i
\(11\)	−7.49057	−	4.32468i	−0.680961	−	0.393153i								0.119256	−	0.992864i	\(-0.461949\pi\)
							−0.800217	+	0.599710i	\(0.795282\pi\)
\(13\)	19.6476			1.51135			0.755676	−	0.654945i	\(-0.227308\pi\)
							0.755676	+	0.654945i	\(0.227308\pi\)
\(17\)	10.5918	−	18.3455i	0.623044	−	1.07914i	−0.365871	−	0.930666i	\(-0.619229\pi\)
							0.988916	−	0.148479i	\(-0.0474377\pi\)
\(19\)	4.61859	−	2.66655i	0.243084	−	0.140345i	−0.373509	−	0.927626i	\(-0.621846\pi\)
							0.616593	+	0.787282i	\(0.288512\pi\)
\(23\)	24.5095	−	14.1506i	1.06563	−	0.615242i	0.138645	−	0.990342i	\(-0.455725\pi\)
							0.926984	+	0.375101i	\(0.122392\pi\)
\(29\)	44.2933			1.52736			0.763678	−	0.645597i	\(-0.223391\pi\)
							0.763678	+	0.645597i	\(0.223391\pi\)
\(31\)	−38.7553	−	22.3754i	−1.25017	−	0.721787i	−0.279028	−	0.960283i	\(-0.590012\pi\)
							−0.971144	+	0.238496i	\(0.923346\pi\)
\(37\)	−9.12571	−	15.8062i	−0.246641	−	0.427194i	0.715951	−	0.698151i	\(-0.245993\pi\)
							−0.962592	+	0.270956i	\(0.912660\pi\)
\(41\)	−13.3841			−0.326442			−0.163221	−	0.986590i	\(-0.552188\pi\)
							−0.163221	+	0.986590i	\(0.552188\pi\)
\(43\)			41.6789i			0.969276i	0.874715	+	0.484638i	\(0.161049\pi\)
							−0.874715	+	0.484638i	\(0.838951\pi\)
\(47\)	−74.1733	+	42.8239i	−1.57815	+	0.911148i	−0.583037	+	0.812445i	\(0.698136\pi\)
							−0.995117	+	0.0987025i	\(0.968531\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.3.t.a.11.2	✓	64
4.3	odd	2	inner	140.3.t.a.11.23	yes	64
7.2	even	3	inner	140.3.t.a.51.23	yes	64
28.23	odd	6	inner	140.3.t.a.51.2	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.3.t.a.11.2	✓	64	1.1	even	1	trivial
140.3.t.a.11.23	yes	64	4.3	odd	2	inner
140.3.t.a.51.2	yes	64	28.23	odd	6	inner
140.3.t.a.51.23	yes	64	7.2	even	3	inner