Embedded newform 105.4.j.a.8.6

• Label: 105.4.j.a.8.6
• 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			8.6
Character	\(\chi\)	\(=\)	105.8
Dual form			105.4.j.a.92.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.97398 - 2.97398i) q^{2} +(4.37077 + 2.81005i) q^{3} +9.68906i q^{4} +(11.0280 - 1.83931i) q^{5} +(-4.64156 - 21.3556i) q^{6} +(-4.94975 + 4.94975i) q^{7} +(5.02323 - 5.02323i) q^{8} +(11.2073 + 24.5641i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 8 q^{3}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.97398	−	2.97398i	−1.05146	−	1.05146i	−0.998602	−	0.0528571i	\(-0.983167\pi\)
							−0.0528571	−	0.998602i	\(-0.516833\pi\)
\(3\)	4.37077	+	2.81005i	0.841155	+	0.540794i
							\(5\)	11.0280	−	1.83931i	0.986375	−	0.164513i
\(7\)	−4.94975	+	4.94975i	−0.267261	+	0.267261i
							\(11\)			33.5205i			0.918801i	0.888229	+	0.459400i	\(0.151936\pi\)
−0.888229	+	0.459400i	\(0.848064\pi\)
\(13\)	59.5103	+	59.5103i	1.26963	+	1.26963i	0.946279	+	0.323351i	\(0.104809\pi\)
							0.323351	+	0.946279i	\(0.395191\pi\)
\(17\)	−56.2311	−	56.2311i	−0.802237	−	0.802237i	0.181208	−	0.983445i	\(-0.441999\pi\)
							−0.983445	+	0.181208i	\(0.941999\pi\)
\(19\)		−	87.1490i		−	1.05228i	−0.850397	−	0.526141i	\(-0.823638\pi\)
							0.850397	−	0.526141i	\(-0.176362\pi\)
\(23\)	−30.4546	+	30.4546i	−0.276097	+	0.276097i	−0.831549	−	0.555452i	\(-0.812545\pi\)
							0.555452	+	0.831549i	\(0.312545\pi\)
\(29\)	276.981			1.77359			0.886796	−	0.462162i	\(-0.152926\pi\)
							0.886796	+	0.462162i	\(0.152926\pi\)
\(31\)	129.690			0.751389			0.375695	−	0.926744i	\(-0.377404\pi\)
							0.375695	+	0.926744i	\(0.377404\pi\)
\(37\)	4.62065	−	4.62065i	0.0205306	−	0.0205306i	−0.696767	−	0.717298i	\(-0.745379\pi\)
							0.717298	+	0.696767i	\(0.245379\pi\)
\(41\)			134.834i			0.513600i	0.966465	+	0.256800i	\(0.0826682\pi\)
							−0.966465	+	0.256800i	\(0.917332\pi\)
\(43\)	−268.723	−	268.723i	−0.953020	−	0.953020i	0.0459246	−	0.998945i	\(-0.485377\pi\)
							−0.998945	+	0.0459246i	\(0.985377\pi\)
\(47\)	−191.190	−	191.190i	−0.593360	−	0.593360i	0.345177	−	0.938538i	\(-0.387819\pi\)
							−0.938538	+	0.345177i	\(0.887819\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.j.a.8.6	✓	72
3.2	odd	2	inner	105.4.j.a.8.31	yes	72
5.2	odd	4	inner	105.4.j.a.92.31	yes	72
15.2	even	4	inner	105.4.j.a.92.6	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.j.a.8.6	✓	72	1.1	even	1	trivial
105.4.j.a.8.31	yes	72	3.2	odd	2	inner
105.4.j.a.92.6	yes	72	15.2	even	4	inner
105.4.j.a.92.31	yes	72	5.2	odd	4	inner