Embedded newform 105.4.j.a.8.14

• Label: 105.4.j.a.8.14
• 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			8.14
Character	\(\chi\)	\(=\)	105.8
Dual form			105.4.j.a.92.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.948593 - 0.948593i) q^{2} +(-4.45711 + 2.67099i) q^{3} -6.20034i q^{4} +(11.1478 + 0.851917i) q^{5} +(6.76166 + 1.69430i) q^{6} +(-4.94975 + 4.94975i) q^{7} +(-13.4703 + 13.4703i) q^{8} +(12.7317 - 23.8098i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 8 q^{3}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.948593	−	0.948593i	−0.335378	−	0.335378i	0.519246	−	0.854625i	\(-0.326213\pi\)
							−0.854625	+	0.519246i	\(0.826213\pi\)
\(3\)	−4.45711	+	2.67099i	−0.857771	+	0.514031i
							\(5\)	11.1478	+	0.851917i	0.997093	+	0.0761977i
\(7\)	−4.94975	+	4.94975i	−0.267261	+	0.267261i
							\(11\)		−	4.85038i		−	0.132950i	−0.997788	−	0.0664748i	\(-0.978825\pi\)
0.997788	−	0.0664748i	\(-0.0211752\pi\)
\(13\)	−45.5058	−	45.5058i	−0.970850	−	0.970850i	0.0287372	−	0.999587i	\(-0.490851\pi\)
							−0.999587	+	0.0287372i	\(0.990851\pi\)
\(17\)	−76.6056	−	76.6056i	−1.09292	−	1.09292i	−0.995216	−	0.0977018i	\(-0.968851\pi\)
							−0.0977018	−	0.995216i	\(-0.531149\pi\)
\(19\)		−	109.846i		−	1.32634i	−0.748468	−	0.663171i	\(-0.769210\pi\)
							0.748468	−	0.663171i	\(-0.230790\pi\)
\(23\)	−62.6719	+	62.6719i	−0.568174	+	0.568174i	−0.931617	−	0.363443i	\(-0.881601\pi\)
							0.363443	+	0.931617i	\(0.381601\pi\)
\(29\)	−192.727			−1.23409			−0.617043	−	0.786930i	\(-0.711669\pi\)
							−0.617043	+	0.786930i	\(0.711669\pi\)
\(31\)	83.9583			0.486431			0.243215	−	0.969972i	\(-0.421798\pi\)
							0.243215	+	0.969972i	\(0.421798\pi\)
\(37\)	108.049	−	108.049i	0.480084	−	0.480084i	−0.425075	−	0.905158i	\(-0.639752\pi\)
							0.905158	+	0.425075i	\(0.139752\pi\)
\(41\)		−	311.084i		−	1.18495i	−0.805588	−	0.592477i	\(-0.798150\pi\)
							0.805588	−	0.592477i	\(-0.201850\pi\)
\(43\)	137.727	+	137.727i	0.488446	+	0.488446i	0.907816	−	0.419369i	\(-0.137749\pi\)
							−0.419369	+	0.907816i	\(0.637749\pi\)
\(47\)	−42.0121	−	42.0121i	−0.130385	−	0.130385i	0.638903	−	0.769288i	\(-0.279389\pi\)
							−0.769288	+	0.638903i	\(0.779389\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.j.a.8.14	✓	72
3.2	odd	2	inner	105.4.j.a.8.23	yes	72
5.2	odd	4	inner	105.4.j.a.92.23	yes	72
15.2	even	4	inner	105.4.j.a.92.14	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.j.a.8.14	✓	72	1.1	even	1	trivial
105.4.j.a.8.23	yes	72	3.2	odd	2	inner
105.4.j.a.92.14	yes	72	15.2	even	4	inner
105.4.j.a.92.23	yes	72	5.2	odd	4	inner