Embedded newform 105.4.s.a.101.7

• Label: 105.4.s.a.101.7
• Level: $105$
• Weight: $4$
• Character: 105.101
• Analytic conductor: $6.195$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	105.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.19520055060\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			101.7
Character	\(\chi\)	\(=\)	105.101
Dual form			105.4.s.a.26.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.20225 + 0.694119i) q^{2} +(3.08234 + 4.18320i) q^{3} +(-3.03640 + 5.25919i) q^{4} +(-2.50000 - 4.33013i) q^{5} +(-6.60938 - 2.88975i) q^{6} +(-11.1211 + 14.8095i) q^{7} -19.5364i q^{8} +(-7.99841 + 25.7881i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 2 q^{3} + 64 q^{4} - 80 q^{5} - 28 q^{6} + 46 q^{7} + 100 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.20225	+	0.694119i	−0.425059	+	0.245408i	−0.697240	−	0.716838i	\(-0.745589\pi\)
							0.272180	+	0.962246i	\(0.412255\pi\)
\(3\)	3.08234	+	4.18320i	0.593196	+	0.805058i
							\(5\)	−2.50000	−	4.33013i	−0.223607	−	0.387298i
\(7\)	−11.1211	+	14.8095i	−0.600484	+	0.799637i
							\(11\)	−11.5490	−	6.66781i	−0.316559	−	0.182765i	0.333299	−	0.942821i	\(-0.391838\pi\)
−0.649858	+	0.760056i	\(0.725172\pi\)
\(13\)		−	27.3633i		−	0.583786i	−0.956451	−	0.291893i	\(-0.905715\pi\)
							0.956451	−	0.291893i	\(-0.0942852\pi\)
\(17\)	−8.19960	+	14.2021i	−0.116982	+	0.202619i	−0.918570	−	0.395258i	\(-0.870655\pi\)
							0.801588	+	0.597876i	\(0.203989\pi\)
\(19\)	3.26555	−	1.88537i	0.0394300	−	0.0227649i	−0.480155	−	0.877183i	\(-0.659420\pi\)
							0.519585	+	0.854419i	\(0.326086\pi\)
\(23\)	−171.457	+	98.9905i	−1.55440	+	0.897433i	−0.556623	+	0.830765i	\(0.687903\pi\)
							−0.997775	+	0.0666676i	\(0.978763\pi\)
\(29\)			68.2057i			0.436741i	0.975866	+	0.218370i	\(0.0700741\pi\)
							−0.975866	+	0.218370i	\(0.929926\pi\)
\(31\)	−4.82091	−	2.78335i	−0.0279310	−	0.0161260i	0.485969	−	0.873976i	\(-0.338467\pi\)
							−0.513900	+	0.857850i	\(0.671800\pi\)
\(37\)	125.414	+	217.224i	0.557243	+	0.965173i	0.997725	+	0.0674114i	\(0.0214740\pi\)
							−0.440483	+	0.897761i	\(0.645193\pi\)
\(41\)	113.675			0.433003			0.216501	−	0.976282i	\(-0.430535\pi\)
							0.216501	+	0.976282i	\(0.430535\pi\)
\(43\)	15.9752			0.0566556			0.0283278	−	0.999599i	\(-0.490982\pi\)
							0.0283278	+	0.999599i	\(0.490982\pi\)
\(47\)	199.709	+	345.907i	0.619800	+	1.07352i	0.989522	+	0.144382i	\(0.0461195\pi\)
							−0.369722	+	0.929142i	\(0.620547\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.s.a.101.7	yes	32
3.2	odd	2		105.4.s.b.101.10	yes	32
7.5	odd	6		105.4.s.b.26.10	yes	32
21.5	even	6	inner	105.4.s.a.26.7	✓	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.s.a.26.7	✓	32	21.5	even	6	inner
105.4.s.a.101.7	yes	32	1.1	even	1	trivial
105.4.s.b.26.10	yes	32	7.5	odd	6
105.4.s.b.101.10	yes	32	3.2	odd	2