Embedded newform 105.4.j.a.8.20

• Label: 105.4.j.a.8.20
• 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			8.20
Character	\(\chi\)	\(=\)	105.8
Dual form			105.4.j.a.92.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.554582 + 0.554582i) q^{2} +(-1.79005 - 4.87809i) q^{3} -7.38488i q^{4} +(7.42269 + 8.36084i) q^{5} +(1.71257 - 3.69803i) q^{6} +(4.94975 - 4.94975i) q^{7} +(8.53218 - 8.53218i) q^{8} +(-20.5915 + 17.4640i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 8 q^{3}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.554582	+	0.554582i	0.196074	+	0.196074i	0.798315	−	0.602240i	\(-0.205725\pi\)
							−0.602240	+	0.798315i	\(0.705725\pi\)
\(3\)	−1.79005	−	4.87809i	−0.344495	−	0.938788i
							\(5\)	7.42269	+	8.36084i	0.663906	+	0.747816i
\(7\)	4.94975	−	4.94975i	0.267261	−	0.267261i
							\(11\)		−	53.3076i		−	1.46117i	−0.682823	−	0.730584i	\(-0.739248\pi\)
0.682823	−	0.730584i	\(-0.260752\pi\)
\(13\)	−41.2527	−	41.2527i	−0.880110	−	0.880110i	0.113436	−	0.993545i	\(-0.463814\pi\)
							−0.993545	+	0.113436i	\(0.963814\pi\)
\(17\)	−11.1241	−	11.1241i	−0.158705	−	0.158705i	0.623288	−	0.781993i	\(-0.285797\pi\)
							−0.781993	+	0.623288i	\(0.785797\pi\)
\(19\)			15.9997i			0.193189i	0.995324	+	0.0965944i	\(0.0307950\pi\)
							−0.995324	+	0.0965944i	\(0.969205\pi\)
\(23\)	4.96019	−	4.96019i	0.0449683	−	0.0449683i	−0.684265	−	0.729233i	\(-0.739877\pi\)
							0.729233	+	0.684265i	\(0.239877\pi\)
\(29\)	203.170			1.30096			0.650479	−	0.759524i	\(-0.274568\pi\)
							0.650479	+	0.759524i	\(0.274568\pi\)
\(31\)	295.138			1.70995			0.854975	−	0.518670i	\(-0.173573\pi\)
							0.854975	+	0.518670i	\(0.173573\pi\)
\(37\)	−96.1954	+	96.1954i	−0.427417	+	0.427417i	−0.887747	−	0.460331i	\(-0.847731\pi\)
							0.460331	+	0.887747i	\(0.347731\pi\)
\(41\)		−	51.9791i		−	0.197995i	−0.995088	−	0.0989973i	\(-0.968436\pi\)
							0.995088	−	0.0989973i	\(-0.0315635\pi\)
\(43\)	−292.943	−	292.943i	−1.03892	−	1.03892i	−0.999211	−	0.0397055i	\(-0.987358\pi\)
							−0.0397055	−	0.999211i	\(-0.512642\pi\)
\(47\)	418.981	+	418.981i	1.30031	+	1.30031i	0.928180	+	0.372131i	\(0.121373\pi\)
							0.372131	+	0.928180i	\(0.378627\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.j.a.8.20	yes	72
3.2	odd	2	inner	105.4.j.a.8.17	✓	72
5.2	odd	4	inner	105.4.j.a.92.17	yes	72
15.2	even	4	inner	105.4.j.a.92.20	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.j.a.8.17	✓	72	3.2	odd	2	inner
105.4.j.a.8.20	yes	72	1.1	even	1	trivial
105.4.j.a.92.17	yes	72	5.2	odd	4	inner
105.4.j.a.92.20	yes	72	15.2	even	4	inner