Embedded newform 105.4.s.a.26.7

• Label: 105.4.s.a.26.7
• Level: $105$
• Weight: $4$
• Character: 105.26
• 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			26.7
Character	\(\chi\)	\(=\)	105.26
Dual form			105.4.s.a.101.7

$q$-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 \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)

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	0.272180	−	0.962246i	\(-0.412255\pi\)
							−0.697240	+	0.716838i	\(0.745589\pi\)
\(3\)	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	−0.649858	−	0.760056i	\(-0.725172\pi\)
0.333299	+	0.942821i	\(0.391838\pi\)
\(13\)			27.3633i			0.583786i	0.956451	+	0.291893i	\(0.0942852\pi\)
							−0.956451	+	0.291893i	\(0.905715\pi\)
\(17\)	−8.19960	−	14.2021i	−0.116982	−	0.202619i	0.801588	−	0.597876i	\(-0.203989\pi\)
							−0.918570	+	0.395258i	\(0.870655\pi\)
\(19\)	3.26555	+	1.88537i	0.0394300	+	0.0227649i	0.519585	−	0.854419i	\(-0.326086\pi\)
							−0.480155	+	0.877183i	\(0.659420\pi\)
\(23\)	−171.457	−	98.9905i	−1.55440	−	0.897433i	−0.997775	−	0.0666676i	\(-0.978763\pi\)
							−0.556623	−	0.830765i	\(-0.687903\pi\)
\(29\)		−	68.2057i		−	0.436741i	−0.975866	−	0.218370i	\(-0.929926\pi\)
							0.975866	−	0.218370i	\(-0.0700741\pi\)
\(31\)	−4.82091	+	2.78335i	−0.0279310	+	0.0161260i	−0.513900	−	0.857850i	\(-0.671800\pi\)
							0.485969	+	0.873976i	\(0.338467\pi\)
\(37\)	125.414	−	217.224i	0.557243	−	0.965173i	−0.440483	−	0.897761i	\(-0.645193\pi\)
							0.997725	−	0.0674114i	\(-0.0214740\pi\)
\(41\)	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	−0.369722	−	0.929142i	\(-0.620547\pi\)
							0.989522	−	0.144382i	\(-0.0461195\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.s.a.26.7	✓	32	1.1	even	1	trivial
105.4.s.a.101.7	yes	32	21.17	even	6	inner
105.4.s.b.26.10	yes	32	3.2	odd	2
105.4.s.b.101.10	yes	32	7.3	odd	6