Embedded newform 105.4.j.a.8.1

• Label: 105.4.j.a.8.1
• 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			8.1
Character	\(\chi\)	\(=\)	105.8
Dual form			105.4.j.a.92.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.89881 - 3.89881i) q^{2} +(-3.58895 - 3.75759i) q^{3} +22.4014i q^{4} +(-10.6015 - 3.55092i) q^{5} +(-0.657498 + 28.6427i) q^{6} +(-4.94975 + 4.94975i) q^{7} +(56.1481 - 56.1481i) q^{8} +(-1.23893 + 26.9716i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 8 q^{3}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.89881	−	3.89881i	−1.37844	−	1.37844i	−0.847264	−	0.531171i	\(-0.821752\pi\)
							−0.531171	−	0.847264i	\(-0.678248\pi\)
\(3\)	−3.58895	−	3.75759i	−0.690693	−	0.723148i
							\(5\)	−10.6015	−	3.55092i	−0.948223	−	0.317604i
\(7\)	−4.94975	+	4.94975i	−0.267261	+	0.267261i
							\(11\)			13.7309i			0.376367i	0.982134	+	0.188183i	\(0.0602599\pi\)
−0.982134	+	0.188183i	\(0.939740\pi\)
\(13\)	7.65816	+	7.65816i	0.163384	+	0.163384i	0.784064	−	0.620680i	\(-0.213143\pi\)
							−0.620680	+	0.784064i	\(0.713143\pi\)
\(17\)	−41.0096	−	41.0096i	−0.585076	−	0.585076i	0.351218	−	0.936294i	\(-0.385768\pi\)
							−0.936294	+	0.351218i	\(0.885768\pi\)
\(19\)		−	87.9327i		−	1.06174i	−0.847452	−	0.530872i	\(-0.821865\pi\)
							0.847452	−	0.530872i	\(-0.178135\pi\)
\(23\)	−82.6280	+	82.6280i	−0.749093	+	0.749093i	−0.974309	−	0.225216i	\(-0.927691\pi\)
							0.225216	+	0.974309i	\(0.427691\pi\)
\(29\)	44.5168			0.285054			0.142527	−	0.989791i	\(-0.454477\pi\)
							0.142527	+	0.989791i	\(0.454477\pi\)
\(31\)	230.858			1.33753			0.668763	−	0.743475i	\(-0.266824\pi\)
							0.668763	+	0.743475i	\(0.266824\pi\)
\(37\)	96.5877	−	96.5877i	0.429160	−	0.429160i	−0.459182	−	0.888342i	\(-0.651857\pi\)
							0.888342	+	0.459182i	\(0.151857\pi\)
\(41\)			322.058i			1.22676i	0.789789	+	0.613379i	\(0.210190\pi\)
							−0.789789	+	0.613379i	\(0.789810\pi\)
\(43\)	216.030	+	216.030i	0.766146	+	0.766146i	0.977426	−	0.211280i	\(-0.0677631\pi\)
							−0.211280	+	0.977426i	\(0.567763\pi\)
\(47\)	86.9832	+	86.9832i	0.269953	+	0.269953i	0.829081	−	0.559128i	\(-0.188864\pi\)
							−0.559128	+	0.829081i	\(0.688864\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.j.a.8.1	✓	72
3.2	odd	2	inner	105.4.j.a.8.36	yes	72
5.2	odd	4	inner	105.4.j.a.92.36	yes	72
15.2	even	4	inner	105.4.j.a.92.1	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.j.a.8.1	✓	72	1.1	even	1	trivial
105.4.j.a.8.36	yes	72	3.2	odd	2	inner
105.4.j.a.92.1	yes	72	15.2	even	4	inner
105.4.j.a.92.36	yes	72	5.2	odd	4	inner