Embedded newform 105.3.t.b.11.16

• Label: 105.3.t.b.11.16
• Level: $105$
• Weight: $3$
• Character: 105.11
• Analytic conductor: $2.861$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	105.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.86104277578\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(18\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.16
Character	\(\chi\)	\(=\)	105.11
Dual form			105.3.t.b.86.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.60397 - 1.50340i) q^{2} +(-0.740030 - 2.90729i) q^{3} +(2.52044 - 4.36553i) q^{4} +(-1.93649 + 1.11803i) q^{5} +(-6.29785 - 6.45794i) q^{6} +(-0.494553 - 6.98251i) q^{7} -3.12973i q^{8} +(-7.90471 + 4.30297i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 4 q^{3} + 36 q^{4} - 24 q^{6} - 58 q^{7} - 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.60397	−	1.50340i	1.30199	−	0.751701i	0.321241	−	0.946998i	\(-0.395900\pi\)
							0.980744	+	0.195296i	\(0.0625668\pi\)
\(3\)	−0.740030	−	2.90729i	−0.246677	−	0.969098i
							\(5\)	−1.93649	+	1.11803i	−0.387298	+	0.223607i
\(7\)	−0.494553	−	6.98251i	−0.0706504	−	0.997501i
							\(11\)	16.3666	+	9.44928i	1.48788	+	0.859025i	0.999904	−	0.0138346i	\(-0.00440384\pi\)
0.487971	+	0.872860i	\(0.337737\pi\)
\(13\)	17.2837			1.32951			0.664757	−	0.747059i	\(-0.268535\pi\)
							0.664757	+	0.747059i	\(0.268535\pi\)
\(17\)	−21.7883	−	12.5795i	−1.28166	−	0.739969i	−0.304511	−	0.952509i	\(-0.598493\pi\)
							−0.977152	+	0.212540i	\(0.931826\pi\)
\(19\)	−1.28629	−	2.22791i	−0.0676993	−	0.117259i	0.830189	−	0.557482i	\(-0.188233\pi\)
							−0.897888	+	0.440224i	\(0.854899\pi\)
\(23\)	−0.196138	+	0.113240i	−0.00852774	+	0.00492349i	−0.504258	−	0.863553i	\(-0.668234\pi\)
							0.495730	+	0.868477i	\(0.334901\pi\)
\(29\)			11.7618i			0.405580i	0.979222	+	0.202790i	\(0.0650009\pi\)
							−0.979222	+	0.202790i	\(0.934999\pi\)
\(31\)	−26.5324	+	45.9555i	−0.855885	+	1.48244i	0.0199372	+	0.999801i	\(0.493653\pi\)
							−0.875822	+	0.482635i	\(0.839680\pi\)
\(37\)	−5.41981	−	9.38738i	−0.146481	−	0.253713i	0.783443	−	0.621463i	\(-0.213461\pi\)
							−0.929925	+	0.367750i	\(0.880128\pi\)
\(41\)		−	4.80764i		−	0.117260i	−0.998280	−	0.0586298i	\(-0.981327\pi\)
							0.998280	−	0.0586298i	\(-0.0186731\pi\)
\(43\)	−1.49099			−0.0346742			−0.0173371	−	0.999850i	\(-0.505519\pi\)
							−0.0173371	+	0.999850i	\(0.505519\pi\)
\(47\)	−0.114843	+	0.0663044i	−0.00244346	+	0.00141073i	−0.501221	−	0.865319i	\(-0.667116\pi\)
							0.498778	+	0.866730i	\(0.333782\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.t.b.11.16	yes	36
3.2	odd	2	inner	105.3.t.b.11.3	✓	36
7.2	even	3	inner	105.3.t.b.86.3	yes	36
21.2	odd	6	inner	105.3.t.b.86.16	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.t.b.11.3	✓	36	3.2	odd	2	inner
105.3.t.b.11.16	yes	36	1.1	even	1	trivial
105.3.t.b.86.3	yes	36	7.2	even	3	inner
105.3.t.b.86.16	yes	36	21.2	odd	6	inner