Embedded newform 140.3.x.a.103.44

• Label: 140.3.x.a.103.44
• Level: $140$
• Weight: $3$
• Character: 140.103
• Analytic conductor: $3.815$
• Analytic rank: $0$
• Dimension: $176$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	140.x (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.81472370104\)
Analytic rank:	\(0\)
Dimension:	\(176\)
Relative dimension:	\(44\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			103.44
Character	\(\chi\)	\(=\)	140.103
Dual form			140.3.x.a.87.44

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.99996 - 0.0127017i) q^{2} +(-1.22147 + 4.55859i) q^{3} +(3.99968 - 0.0508057i) q^{4} +(3.64464 + 3.42295i) q^{5} +(-2.38499 + 9.13251i) q^{6} +(-5.07374 - 4.82257i) q^{7} +(7.99855 - 0.152412i) q^{8} +(-11.4945 - 6.63636i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 2 q^{2} - 12 q^{5} + 4 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(57\)	\(71\)	\(101\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.99996	−	0.0127017i	0.999980	−	0.00635084i
							\(3\)	−1.22147	+	4.55859i	−0.407157	+	1.51953i	0.392886	+	0.919587i	\(0.371477\pi\)
−0.800043	+	0.599943i	\(0.795190\pi\)
\(5\)	3.64464	+	3.42295i	0.728929	+	0.684589i
							\(7\)	−5.07374	−	4.82257i	−0.724820	−	0.688938i
\(11\)	−9.58491	+	5.53385i	−0.871355	+	0.503077i								−0.867798	−	0.496916i	\(-0.834466\pi\)
							−0.00355698	+	0.999994i	\(0.501132\pi\)
\(13\)	8.35697	−	8.35697i	0.642844	−	0.642844i	−0.308410	−	0.951254i	\(-0.599797\pi\)
							0.951254	+	0.308410i	\(0.0997968\pi\)
\(17\)	0.550863	−	2.05585i	0.0324037	−	0.120932i	−0.947830	−	0.318778i	\(-0.896728\pi\)
							0.980233	+	0.197845i	\(0.0633944\pi\)
\(19\)	27.9980	+	16.1646i	1.47358	+	0.850771i	0.999558	−	0.0297435i	\(-0.00946904\pi\)
							0.474020	+	0.880514i	\(0.342802\pi\)
\(23\)	−8.01428	−	29.9097i	−0.348447	−	1.30042i	−0.888533	−	0.458812i	\(-0.848275\pi\)
							0.540086	−	0.841610i	\(-0.318392\pi\)
\(29\)		−	23.6996i		−	0.817228i	−0.912707	−	0.408614i	\(-0.866012\pi\)
							0.912707	−	0.408614i	\(-0.133988\pi\)
\(31\)	3.58243	+	6.20495i	0.115562	+	0.200160i	0.918004	−	0.396570i	\(-0.129800\pi\)
							−0.802442	+	0.596730i	\(0.796466\pi\)
\(37\)	−10.1003	−	37.6949i	−0.272982	−	1.01878i	−0.957182	−	0.289488i	\(-0.906515\pi\)
							0.684200	−	0.729295i	\(-0.260152\pi\)
\(41\)			2.34310i			0.0571487i	0.999592	+	0.0285743i	\(0.00909673\pi\)
							−0.999592	+	0.0285743i	\(0.990903\pi\)
\(43\)	27.8788	−	27.8788i	0.648345	−	0.648345i	−0.304248	−	0.952593i	\(-0.598405\pi\)
							0.952593	+	0.304248i	\(0.0984052\pi\)
\(47\)	−11.5158	−	42.9774i	−0.245016	−	0.914413i	−0.973375	−	0.229217i	\(-0.926383\pi\)
							0.728359	−	0.685196i	\(-0.240283\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.3.x.a.103.44	yes	176
4.3	odd	2	inner	140.3.x.a.103.8	yes	176
5.2	odd	4	inner	140.3.x.a.47.23	yes	176
7.3	odd	6	inner	140.3.x.a.3.16	✓	176
20.7	even	4	inner	140.3.x.a.47.16	yes	176
28.3	even	6	inner	140.3.x.a.3.23	yes	176
35.17	even	12	inner	140.3.x.a.87.8	yes	176
140.87	odd	12	inner	140.3.x.a.87.44	yes	176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.3.x.a.3.16	✓	176	7.3	odd	6	inner
140.3.x.a.3.23	yes	176	28.3	even	6	inner
140.3.x.a.47.16	yes	176	20.7	even	4	inner
140.3.x.a.47.23	yes	176	5.2	odd	4	inner
140.3.x.a.87.8	yes	176	35.17	even	12	inner
140.3.x.a.87.44	yes	176	140.87	odd	12	inner
140.3.x.a.103.8	yes	176	4.3	odd	2	inner
140.3.x.a.103.44	yes	176	1.1	even	1	trivial