Embedded newform 140.3.x.a.103.17

• Label: 140.3.x.a.103.17
• 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			103.17
Character	\(\chi\)	\(=\)	140.103
Dual form			140.3.x.a.87.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.772090 + 1.84496i) q^{2} +(1.07759 - 4.02162i) q^{3} +(-2.80775 - 2.84895i) q^{4} +(-4.88643 + 1.05963i) q^{5} +(6.58773 + 5.09316i) q^{6} +(-4.65720 + 5.22595i) q^{7} +(7.42404 - 2.98055i) q^{8} +(-7.21801 - 4.16732i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 2 q^{2} - 12 q^{5} + 4 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.772090	+	1.84496i	−0.386045	+	0.922480i
							\(3\)	1.07759	−	4.02162i	0.359197	−	1.34054i	−0.515923	−	0.856635i	\(-0.672551\pi\)
0.875120	−	0.483906i	\(-0.160782\pi\)
\(5\)	−4.88643	+	1.05963i	−0.977286	+	0.211927i
							\(7\)	−4.65720	+	5.22595i	−0.665314	+	0.746564i
\(11\)	−8.45949	+	4.88409i	−0.769044	+	0.444008i								−0.832534	−	0.553975i	\(-0.813110\pi\)
							0.0634892	+	0.997983i	\(0.479777\pi\)
\(13\)	−11.7318	+	11.7318i	−0.902447	+	0.902447i	−0.995647	−	0.0932003i	\(-0.970290\pi\)
							0.0932003	+	0.995647i	\(0.470290\pi\)
\(17\)	0.767072	−	2.86275i	0.0451219	−	0.168397i	−0.939688	−	0.342032i	\(-0.888885\pi\)
							0.984810	+	0.173635i	\(0.0555514\pi\)
\(19\)	−19.1066	−	11.0312i	−1.00561	−	0.580591i	−0.0957087	−	0.995409i	\(-0.530512\pi\)
							−0.909904	+	0.414819i	\(0.863845\pi\)
\(23\)	−4.05608	−	15.1375i	−0.176351	−	0.658152i	−0.996318	−	0.0857399i	\(-0.972675\pi\)
							0.819966	−	0.572412i	\(-0.193992\pi\)
\(29\)		−	15.6040i		−	0.538068i	−0.963131	−	0.269034i	\(-0.913296\pi\)
							0.963131	−	0.269034i	\(-0.0867044\pi\)
\(31\)	−11.7767	−	20.3979i	−0.379895	−	0.657998i	0.611152	−	0.791514i	\(-0.290707\pi\)
							−0.991047	+	0.133516i	\(0.957373\pi\)
\(37\)	17.9991	+	67.1737i	0.486463	+	1.81550i	0.573381	+	0.819289i	\(0.305632\pi\)
							−0.0869177	+	0.996215i	\(0.527702\pi\)
\(41\)		−	65.7226i		−	1.60299i	−0.598001	−	0.801496i	\(-0.704038\pi\)
							0.598001	−	0.801496i	\(-0.295962\pi\)
\(43\)	−50.2247	+	50.2247i	−1.16802	+	1.16802i	−0.185342	+	0.982674i	\(0.559339\pi\)
							−0.982674	+	0.185342i	\(0.940661\pi\)
\(47\)	1.85307	+	6.91574i	0.0394269	+	0.147143i	0.982833	−	0.184496i	\(-0.0590651\pi\)
							−0.943406	+	0.331639i	\(0.892398\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.3.x.a.103.17	yes	176
4.3	odd	2	inner	140.3.x.a.103.35	yes	176
5.2	odd	4	inner	140.3.x.a.47.5	yes	176
7.3	odd	6	inner	140.3.x.a.3.13	yes	176
20.7	even	4	inner	140.3.x.a.47.13	yes	176
28.3	even	6	inner	140.3.x.a.3.5	✓	176
35.17	even	12	inner	140.3.x.a.87.35	yes	176
140.87	odd	12	inner	140.3.x.a.87.17	yes	176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.3.x.a.3.5	✓	176	28.3	even	6	inner
140.3.x.a.3.13	yes	176	7.3	odd	6	inner
140.3.x.a.47.5	yes	176	5.2	odd	4	inner
140.3.x.a.47.13	yes	176	20.7	even	4	inner
140.3.x.a.87.17	yes	176	140.87	odd	12	inner
140.3.x.a.87.35	yes	176	35.17	even	12	inner
140.3.x.a.103.17	yes	176	1.1	even	1	trivial
140.3.x.a.103.35	yes	176	4.3	odd	2	inner