Embedded newform 105.4.j.a.8.15

• Label: 105.4.j.a.8.15
• Level: $105$
• Weight: $4$
• Character: 105.8
• Analytic conductor: $6.195$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.19520055060\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(36\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			8.15
Character	\(\chi\)	\(=\)	105.8
Dual form			105.4.j.a.92.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.777009 - 0.777009i) q^{2} +(4.13454 - 3.14732i) q^{3} -6.79251i q^{4} +(10.0976 - 4.79984i) q^{5} +(-5.65807 - 0.767075i) q^{6} +(4.94975 - 4.94975i) q^{7} +(-11.4939 + 11.4939i) q^{8} +(7.18876 - 26.0254i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 8 q^{3}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.777009	−	0.777009i	−0.274714	−	0.274714i	0.556280	−	0.830995i	\(-0.312228\pi\)
							−0.830995	+	0.556280i	\(0.812228\pi\)
\(3\)	4.13454	−	3.14732i	0.795692	−	0.605702i
							\(5\)	10.0976	−	4.79984i	0.903157	−	0.429311i
\(7\)	4.94975	−	4.94975i	0.267261	−	0.267261i
							\(11\)			48.0020i			1.31574i	0.753131	+	0.657871i	\(0.228543\pi\)
−0.753131	+	0.657871i	\(0.771457\pi\)
\(13\)	−10.0823	−	10.0823i	−0.215102	−	0.215102i	0.591328	−	0.806431i	\(-0.298604\pi\)
							−0.806431	+	0.591328i	\(0.798604\pi\)
\(17\)	13.9867	+	13.9867i	0.199546	+	0.199546i	0.799805	−	0.600260i	\(-0.204936\pi\)
							−0.600260	+	0.799805i	\(0.704936\pi\)
\(19\)		−	32.4890i		−	0.392289i	−0.980575	−	0.196145i	\(-0.937158\pi\)
							0.980575	−	0.196145i	\(-0.0628422\pi\)
\(23\)	−15.5544	+	15.5544i	−0.141014	+	0.141014i	−0.774090	−	0.633076i	\(-0.781792\pi\)
							0.633076	+	0.774090i	\(0.281792\pi\)
\(29\)	177.804			1.13853			0.569265	−	0.822154i	\(-0.307228\pi\)
							0.569265	+	0.822154i	\(0.307228\pi\)
\(31\)	−198.524			−1.15019			−0.575097	−	0.818085i	\(-0.695036\pi\)
							−0.575097	+	0.818085i	\(0.695036\pi\)
\(37\)	−124.682	+	124.682i	−0.553990	+	0.553990i	−0.927590	−	0.373600i	\(-0.878123\pi\)
							0.373600	+	0.927590i	\(0.378123\pi\)
\(41\)		−	128.266i		−	0.488579i	−0.969702	−	0.244290i	\(-0.921445\pi\)
							0.969702	−	0.244290i	\(-0.0785547\pi\)
\(43\)	333.360	+	333.360i	1.18225	+	1.18225i	0.979159	+	0.203096i	\(0.0651003\pi\)
							0.203096	+	0.979159i	\(0.434900\pi\)
\(47\)	370.670	+	370.670i	1.15038	+	1.15038i	0.986477	+	0.163901i	\(0.0524077\pi\)
							0.163901	+	0.986477i	\(0.447592\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.j.a.8.15	✓	72
3.2	odd	2	inner	105.4.j.a.8.22	yes	72
5.2	odd	4	inner	105.4.j.a.92.22	yes	72
15.2	even	4	inner	105.4.j.a.92.15	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.j.a.8.15	✓	72	1.1	even	1	trivial
105.4.j.a.8.22	yes	72	3.2	odd	2	inner
105.4.j.a.92.15	yes	72	15.2	even	4	inner
105.4.j.a.92.22	yes	72	5.2	odd	4	inner