Embedded newform 105.4.j.a.8.5

• Label: 105.4.j.a.8.5
• 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			8.5
Character	\(\chi\)	\(=\)	105.8
Dual form			105.4.j.a.92.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.31161 - 3.31161i) q^{2} +(5.02457 + 1.32428i) q^{3} +13.9335i q^{4} +(-11.0620 + 1.62271i) q^{5} +(-12.2539 - 21.0249i) q^{6} +(4.94975 - 4.94975i) q^{7} +(19.6493 - 19.6493i) q^{8} +(23.4926 + 13.3078i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 8 q^{3}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.31161	−	3.31161i	−1.17083	−	1.17083i	−0.982012	−	0.188817i	\(-0.939535\pi\)
							−0.188817	−	0.982012i	\(-0.560465\pi\)
\(3\)	5.02457	+	1.32428i	0.966979	+	0.254857i
							\(5\)	−11.0620	+	1.62271i	−0.989411	+	0.145140i
\(7\)	4.94975	−	4.94975i	0.267261	−	0.267261i
							\(11\)			52.5395i			1.44011i	0.693915	+	0.720057i	\(0.255884\pi\)
−0.693915	+	0.720057i	\(0.744116\pi\)
\(13\)	−4.28258	−	4.28258i	−0.0913672	−	0.0913672i	0.659946	−	0.751313i	\(-0.270579\pi\)
							−0.751313	+	0.659946i	\(0.770579\pi\)
\(17\)	86.2289	+	86.2289i	1.23021	+	1.23021i	0.963882	+	0.266329i	\(0.0858108\pi\)
							0.266329	+	0.963882i	\(0.414189\pi\)
\(19\)			37.7800i			0.456176i	0.973641	+	0.228088i	\(0.0732473\pi\)
							−0.973641	+	0.228088i	\(0.926753\pi\)
\(23\)	84.2799	−	84.2799i	0.764068	−	0.764068i	−0.212987	−	0.977055i	\(-0.568319\pi\)
							0.977055	+	0.212987i	\(0.0683191\pi\)
\(29\)	−166.906			−1.06875			−0.534375	−	0.845248i	\(-0.679453\pi\)
							−0.534375	+	0.845248i	\(0.679453\pi\)
\(31\)	−17.3409			−0.100469			−0.0502343	−	0.998737i	\(-0.515997\pi\)
							−0.0502343	+	0.998737i	\(0.515997\pi\)
\(37\)	−202.605	+	202.605i	−0.900220	+	0.900220i	−0.995455	−	0.0952350i	\(-0.969640\pi\)
							0.0952350	+	0.995455i	\(0.469640\pi\)
\(41\)		−	55.6056i		−	0.211808i	−0.994376	−	0.105904i	\(-0.966226\pi\)
							0.994376	−	0.105904i	\(-0.0337736\pi\)
\(43\)	339.073	+	339.073i	1.20252	+	1.20252i	0.973399	+	0.229117i	\(0.0735838\pi\)
							0.229117	+	0.973399i	\(0.426416\pi\)
\(47\)	166.870	+	166.870i	0.517883	+	0.517883i	0.916930	−	0.399048i	\(-0.130659\pi\)
							−0.399048	+	0.916930i	\(0.630659\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.j.a.8.5	✓	72
3.2	odd	2	inner	105.4.j.a.8.32	yes	72
5.2	odd	4	inner	105.4.j.a.92.32	yes	72
15.2	even	4	inner	105.4.j.a.92.5	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.j.a.8.5	✓	72	1.1	even	1	trivial
105.4.j.a.8.32	yes	72	3.2	odd	2	inner
105.4.j.a.92.5	yes	72	15.2	even	4	inner
105.4.j.a.92.32	yes	72	5.2	odd	4	inner