Embedded newform 105.4.j.a.8.2

• Label: 105.4.j.a.8.2
• 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			8.2
Character	\(\chi\)	\(=\)	105.8
Dual form			105.4.j.a.92.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.77908 - 3.77908i) q^{2} +(3.12291 - 4.15300i) q^{3} +20.5629i q^{4} +(9.30912 + 6.19195i) q^{5} +(-27.4963 + 3.89279i) q^{6} +(4.94975 - 4.94975i) q^{7} +(47.4763 - 47.4763i) q^{8} +(-7.49484 - 25.9389i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 8 q^{3}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.77908	−	3.77908i	−1.33611	−	1.33611i	−0.899796	−	0.436312i	\(-0.856285\pi\)
							−0.436312	−	0.899796i	\(-0.643715\pi\)
\(3\)	3.12291	−	4.15300i	0.601005	−	0.799245i
							\(5\)	9.30912	+	6.19195i	0.832633	+	0.553825i
\(7\)	4.94975	−	4.94975i	0.267261	−	0.267261i
							\(11\)		−	52.2605i		−	1.43247i	−0.697861	−	0.716234i	\(-0.745865\pi\)
0.697861	−	0.716234i	\(-0.254135\pi\)
\(13\)	23.1889	+	23.1889i	0.494727	+	0.494727i	0.909792	−	0.415065i	\(-0.136241\pi\)
							−0.415065	+	0.909792i	\(0.636241\pi\)
\(17\)	−10.8738	−	10.8738i	−0.155134	−	0.155134i	0.625273	−	0.780406i	\(-0.284988\pi\)
							−0.780406	+	0.625273i	\(0.784988\pi\)
\(19\)		−	27.8872i		−	0.336725i	−0.985725	−	0.168362i	\(-0.946152\pi\)
							0.985725	−	0.168362i	\(-0.0538478\pi\)
\(23\)	100.394	−	100.394i	0.910158	−	0.910158i	−0.0861260	−	0.996284i	\(-0.527449\pi\)
							0.996284	+	0.0861260i	\(0.0274488\pi\)
\(29\)	−125.218			−0.801805			−0.400903	−	0.916121i	\(-0.631303\pi\)
							−0.400903	+	0.916121i	\(0.631303\pi\)
\(31\)	−95.1932			−0.551523			−0.275761	−	0.961226i	\(-0.588930\pi\)
							−0.275761	+	0.961226i	\(0.588930\pi\)
\(37\)	234.775	−	234.775i	1.04316	−	1.04316i	0.0441293	−	0.999026i	\(-0.485949\pi\)
							0.999026	−	0.0441293i	\(-0.0140513\pi\)
\(41\)			216.874i			0.826098i	0.910709	+	0.413049i	\(0.135536\pi\)
							−0.910709	+	0.413049i	\(0.864464\pi\)
\(43\)	239.406	+	239.406i	0.849048	+	0.849048i	0.990014	−	0.140967i	\(-0.0450210\pi\)
							−0.140967	+	0.990014i	\(0.545021\pi\)
\(47\)	−301.842	−	301.842i	−0.936770	−	0.936770i	0.0613467	−	0.998117i	\(-0.480460\pi\)
							−0.998117	+	0.0613467i	\(0.980460\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.j.a.8.2	✓	72
3.2	odd	2	inner	105.4.j.a.8.35	yes	72
5.2	odd	4	inner	105.4.j.a.92.35	yes	72
15.2	even	4	inner	105.4.j.a.92.2	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.j.a.8.2	✓	72	1.1	even	1	trivial
105.4.j.a.8.35	yes	72	3.2	odd	2	inner
105.4.j.a.92.2	yes	72	15.2	even	4	inner
105.4.j.a.92.35	yes	72	5.2	odd	4	inner