Embedded newform 105.4.j.a.8.19

• Label: 105.4.j.a.8.19
• 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			8.19
Character	\(\chi\)	\(=\)	105.8
Dual form			105.4.j.a.92.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.152180 + 0.152180i) q^{2} +(-5.00615 - 1.39229i) q^{3} -7.95368i q^{4} +(-3.10522 + 10.7405i) q^{5} +(-0.549957 - 0.973712i) q^{6} +(-4.94975 + 4.94975i) q^{7} +(2.42783 - 2.42783i) q^{8} +(23.1231 + 13.9400i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 8 q^{3}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.152180	+	0.152180i	0.0538037	+	0.0538037i	0.733497	−	0.679693i	\(-0.237887\pi\)
							−0.679693	+	0.733497i	\(0.737887\pi\)
\(3\)	−5.00615	−	1.39229i	−0.963434	−	0.267946i
							\(5\)	−3.10522	+	10.7405i	−0.277739	+	0.960656i
\(7\)	−4.94975	+	4.94975i	−0.267261	+	0.267261i
							\(11\)			6.51935i			0.178696i	0.996000	+	0.0893481i	\(0.0284784\pi\)
−0.996000	+	0.0893481i	\(0.971522\pi\)
\(13\)	51.9925	+	51.9925i	1.10924	+	1.10924i	0.993250	+	0.115990i	\(0.0370040\pi\)
							0.115990	+	0.993250i	\(0.462996\pi\)
\(17\)	70.3745	+	70.3745i	1.00402	+	1.00402i	0.999992	+	0.00402733i	\(0.00128194\pi\)
							0.00402733	+	0.999992i	\(0.498718\pi\)
\(19\)			13.9776i			0.168773i	0.996433	+	0.0843864i	\(0.0268930\pi\)
							−0.996433	+	0.0843864i	\(0.973107\pi\)
\(23\)	−139.848	+	139.848i	−1.26784	+	1.26784i	−0.320643	+	0.947200i	\(0.603899\pi\)
							−0.947200	+	0.320643i	\(0.896101\pi\)
\(29\)	79.7898			0.510917			0.255458	−	0.966820i	\(-0.417774\pi\)
							0.255458	+	0.966820i	\(0.417774\pi\)
\(31\)	−72.9627			−0.422725			−0.211363	−	0.977408i	\(-0.567790\pi\)
							−0.211363	+	0.977408i	\(0.567790\pi\)
\(37\)	137.037	−	137.037i	0.608885	−	0.608885i	−0.333769	−	0.942655i	\(-0.608321\pi\)
							0.942655	+	0.333769i	\(0.108321\pi\)
\(41\)		−	31.6201i		−	0.120445i	−0.998185	−	0.0602224i	\(-0.980819\pi\)
							0.998185	−	0.0602224i	\(-0.0191810\pi\)
\(43\)	126.071	+	126.071i	0.447109	+	0.447109i	0.894392	−	0.447283i	\(-0.147608\pi\)
							−0.447283	+	0.894392i	\(0.647608\pi\)
\(47\)	174.101	+	174.101i	0.540326	+	0.540326i	0.923624	−	0.383299i	\(-0.125212\pi\)
							−0.383299	+	0.923624i	\(0.625212\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.j.a.8.19	yes	72
3.2	odd	2	inner	105.4.j.a.8.18	✓	72
5.2	odd	4	inner	105.4.j.a.92.18	yes	72
15.2	even	4	inner	105.4.j.a.92.19	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.j.a.8.18	✓	72	3.2	odd	2	inner
105.4.j.a.8.19	yes	72	1.1	even	1	trivial
105.4.j.a.92.18	yes	72	5.2	odd	4	inner
105.4.j.a.92.19	yes	72	15.2	even	4	inner