Embedded newform 105.3.t.b.86.8

• Label: 105.3.t.b.86.8
• Level: $105$
• Weight: $3$
• Character: 105.86
• 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			86.8
Character	\(\chi\)	\(=\)	105.86
Dual form			105.3.t.b.11.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.860118 - 0.496589i) q^{2} +(2.51213 - 1.63987i) q^{3} +(-1.50680 - 2.60985i) q^{4} +(1.93649 + 1.11803i) q^{5} +(-2.97507 + 0.162987i) q^{6} +(-1.66850 - 6.79824i) q^{7} +6.96575i q^{8} +(3.62163 - 8.23916i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 4 q^{3} + 36 q^{4} - 24 q^{6} - 58 q^{7} - 2 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.860118	−	0.496589i	−0.430059	−	0.248295i	0.269313	−	0.963053i	\(-0.413203\pi\)
							−0.699372	+	0.714758i	\(0.746537\pi\)
\(3\)	2.51213	−	1.63987i	0.837378	−	0.546625i
							\(5\)	1.93649	+	1.11803i	0.387298	+	0.223607i
\(7\)	−1.66850	−	6.79824i	−0.238358	−	0.971177i
							\(11\)	−0.568919	+	0.328465i	−0.0517199	+	0.0298605i	−0.525637	−	0.850709i	\(-0.676173\pi\)
0.473917	+	0.880569i	\(0.342840\pi\)
\(13\)	−10.1624			−0.781724			−0.390862	−	0.920449i	\(-0.627823\pi\)
							−0.390862	+	0.920449i	\(0.627823\pi\)
\(17\)	16.8362	−	9.72041i	0.990367	−	0.571789i	0.0849830	−	0.996382i	\(-0.472916\pi\)
							0.905384	+	0.424594i	\(0.139583\pi\)
\(19\)	9.11844	−	15.7936i	0.479918	−	0.831242i	−0.519817	−	0.854278i	\(-0.674000\pi\)
							0.999735	+	0.0230359i	\(0.00733320\pi\)
\(23\)	3.29154	+	1.90037i	0.143110	+	0.0826248i	0.569846	−	0.821752i	\(-0.307003\pi\)
							−0.426735	+	0.904377i	\(0.640336\pi\)
\(29\)			50.8888i			1.75478i	0.479774	+	0.877392i	\(0.340719\pi\)
							−0.479774	+	0.877392i	\(0.659281\pi\)
\(31\)	26.8895	+	46.5740i	0.867404	+	1.50239i	0.864640	+	0.502392i	\(0.167546\pi\)
							0.00276405	+	0.999996i	\(0.499120\pi\)
\(37\)	7.81834	−	13.5418i	0.211306	−	0.365993i	−0.740817	−	0.671707i	\(-0.765562\pi\)
							0.952124	+	0.305713i	\(0.0988949\pi\)
\(41\)			57.3936i			1.39985i	0.714219	+	0.699923i	\(0.246782\pi\)
							−0.714219	+	0.699923i	\(0.753218\pi\)
\(43\)	65.7755			1.52966			0.764831	−	0.644230i	\(-0.222822\pi\)
							0.764831	+	0.644230i	\(0.222822\pi\)
\(47\)	−22.4206	−	12.9445i	−0.477034	−	0.275416i	0.242146	−	0.970240i	\(-0.422149\pi\)
							−0.719180	+	0.694824i	\(0.755482\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.t.b.86.8	yes	36
3.2	odd	2	inner	105.3.t.b.86.11	yes	36
7.4	even	3	inner	105.3.t.b.11.11	yes	36
21.11	odd	6	inner	105.3.t.b.11.8	✓	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.t.b.11.8	✓	36	21.11	odd	6	inner
105.3.t.b.11.11	yes	36	7.4	even	3	inner
105.3.t.b.86.8	yes	36	1.1	even	1	trivial
105.3.t.b.86.11	yes	36	3.2	odd	2	inner