Embedded newform 105.4.p.a.59.9

• Label: 105.4.p.a.59.9
• Level: $105$
• Weight: $4$
• Character: 105.59
• Analytic conductor: $6.195$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.19520055060\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(44\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			59.9
Character	\(\chi\)	\(=\)	105.59
Dual form			105.4.p.a.89.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.82858 - 3.16720i) q^{2} +(-2.45762 + 4.57822i) q^{3} +(-2.68744 + 4.65479i) q^{4} +(-6.67931 - 8.96587i) q^{5} +(18.9941 - 0.587877i) q^{6} +(2.93091 + 18.2869i) q^{7} -9.60048 q^{8} +(-14.9202 - 22.5031i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 164 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.82858	−	3.16720i	−0.646502	−	1.11977i	−0.983952	−	0.178431i	\(-0.942898\pi\)
							0.337450	−	0.941343i	\(-0.390436\pi\)
\(3\)	−2.45762	+	4.57822i	−0.472970	+	0.881079i
							\(5\)	−6.67931	−	8.96587i	−0.597416	−	0.801932i
\(7\)	2.93091	+	18.2869i	0.158255	+	0.987398i
							\(11\)	49.5536	+	28.6098i	1.35827	+	0.784197i	0.989391	−	0.145280i	\(-0.0464084\pi\)
0.368879	+	0.929477i	\(0.379742\pi\)
\(13\)	28.2640			0.603001			0.301500	−	0.953466i	\(-0.402513\pi\)
							0.301500	+	0.953466i	\(0.402513\pi\)
\(17\)	50.7431	+	29.2965i	0.723941	+	0.417968i	0.816202	−	0.577767i	\(-0.196076\pi\)
							−0.0922602	+	0.995735i	\(0.529409\pi\)
\(19\)	57.2243	−	33.0385i	0.690956	−	0.398924i	−0.113014	−	0.993593i	\(-0.536051\pi\)
							0.803970	+	0.594670i	\(0.202717\pi\)
\(23\)	−49.5701	−	85.8580i	−0.449395	−	0.778375i	0.548952	−	0.835854i	\(-0.315027\pi\)
							−0.998347	+	0.0574792i	\(0.981694\pi\)
\(29\)			146.774i			0.939834i	0.882710	+	0.469917i	\(0.155716\pi\)
							−0.882710	+	0.469917i	\(0.844284\pi\)
\(31\)	219.604	+	126.789i	1.27233	+	0.734577i	0.975425	−	0.220331i	\(-0.0707138\pi\)
							0.296900	+	0.954909i	\(0.404047\pi\)
\(37\)	−44.6243	+	25.7639i	−0.198275	+	0.114474i	−0.595851	−	0.803095i	\(-0.703185\pi\)
							0.397575	+	0.917570i	\(0.369852\pi\)
\(41\)	−178.366			−0.679416			−0.339708	−	0.940531i	\(-0.610328\pi\)
							−0.339708	+	0.940531i	\(0.610328\pi\)
\(43\)			435.043i			1.54287i	0.636308	+	0.771435i	\(0.280461\pi\)
							−0.636308	+	0.771435i	\(0.719539\pi\)
\(47\)	80.0881	−	46.2389i	0.248554	−	0.143503i	−0.370548	−	0.928813i	\(-0.620830\pi\)
							0.619102	+	0.785311i	\(0.287497\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.p.a.59.9	✓	88
3.2	odd	2	inner	105.4.p.a.59.35	yes	88
5.4	even	2	inner	105.4.p.a.59.36	yes	88
7.5	odd	6	inner	105.4.p.a.89.10	yes	88
15.14	odd	2	inner	105.4.p.a.59.10	yes	88
21.5	even	6	inner	105.4.p.a.89.36	yes	88
35.19	odd	6	inner	105.4.p.a.89.35	yes	88
105.89	even	6	inner	105.4.p.a.89.9	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.p.a.59.9	✓	88	1.1	even	1	trivial
105.4.p.a.59.10	yes	88	15.14	odd	2	inner
105.4.p.a.59.35	yes	88	3.2	odd	2	inner
105.4.p.a.59.36	yes	88	5.4	even	2	inner
105.4.p.a.89.9	yes	88	105.89	even	6	inner
105.4.p.a.89.10	yes	88	7.5	odd	6	inner
105.4.p.a.89.35	yes	88	35.19	odd	6	inner
105.4.p.a.89.36	yes	88	21.5	even	6	inner