Embedded newform 105.4.p.a.59.17

• Label: 105.4.p.a.59.17
• Level: $105$
• Weight: $4$
• Character: 105.59
• Analytic conductor: $6.195$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	105.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.19520055060\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(44\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			59.17
Character	\(\chi\)	\(=\)	105.59
Dual form			105.4.p.a.89.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.850362 - 1.47287i) q^{2} +(-3.03689 + 4.21632i) q^{3} +(2.55377 - 4.42326i) q^{4} +(-6.34140 + 9.20796i) q^{5} +(8.79254 + 0.887546i) q^{6} +(17.5269 - 5.98407i) q^{7} -22.2923 q^{8} +(-8.55463 - 25.6089i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 164 q^{4}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.850362	−	1.47287i	−0.300648	−	0.520738i	0.675635	−	0.737237i	\(-0.263870\pi\)
							−0.976283	+	0.216498i	\(0.930536\pi\)
\(3\)	−3.03689	+	4.21632i	−0.584449	+	0.811430i
							\(5\)	−6.34140	+	9.20796i	−0.567192	+	0.823585i
\(7\)	17.5269	−	5.98407i	0.946362	−	0.323109i
							\(11\)	−31.7953	−	18.3570i	−0.871514	−	0.503169i	−0.00366263	−	0.999993i	\(-0.501166\pi\)
−0.867851	+	0.496825i	\(0.834499\pi\)
\(13\)	−39.4894			−0.842492			−0.421246	−	0.906946i	\(-0.638407\pi\)
							−0.421246	+	0.906946i	\(0.638407\pi\)
\(17\)	−45.7949	−	26.4397i	−0.653347	−	0.377210i	0.136390	−	0.990655i	\(-0.456450\pi\)
							−0.789737	+	0.613445i	\(0.789783\pi\)
\(19\)	−39.5063	+	22.8090i	−0.477019	+	0.275407i	−0.719173	−	0.694831i	\(-0.755479\pi\)
							0.242154	+	0.970238i	\(0.422146\pi\)
\(23\)	−82.7101	−	143.258i	−0.749836	−	1.29875i	−0.947901	−	0.318566i	\(-0.896799\pi\)
							0.198064	−	0.980189i	\(-0.436535\pi\)
\(29\)			152.884i			0.978961i	0.872014	+	0.489481i	\(0.162814\pi\)
							−0.872014	+	0.489481i	\(0.837186\pi\)
\(31\)	−76.0454	−	43.9048i	−0.440586	−	0.254372i	0.263260	−	0.964725i	\(-0.415202\pi\)
							−0.703846	+	0.710353i	\(0.748536\pi\)
\(37\)	52.1685	−	30.1195i	0.231796	−	0.133827i	−0.379604	−	0.925149i	\(-0.623940\pi\)
							0.611400	+	0.791321i	\(0.290607\pi\)
\(41\)	143.203			0.545478			0.272739	−	0.962088i	\(-0.412071\pi\)
							0.272739	+	0.962088i	\(0.412071\pi\)
\(43\)			144.671i			0.513071i	0.966535	+	0.256536i	\(0.0825811\pi\)
							−0.966535	+	0.256536i	\(0.917419\pi\)
\(47\)	408.016	−	235.568i	1.26628	−	0.731088i	0.292000	−	0.956418i	\(-0.405679\pi\)
							0.974282	+	0.225330i	\(0.0723460\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.p.a.59.17	✓	88
3.2	odd	2	inner	105.4.p.a.59.27	yes	88
5.4	even	2	inner	105.4.p.a.59.28	yes	88
7.5	odd	6	inner	105.4.p.a.89.18	yes	88
15.14	odd	2	inner	105.4.p.a.59.18	yes	88
21.5	even	6	inner	105.4.p.a.89.28	yes	88
35.19	odd	6	inner	105.4.p.a.89.27	yes	88
105.89	even	6	inner	105.4.p.a.89.17	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.p.a.59.17	✓	88	1.1	even	1	trivial
105.4.p.a.59.18	yes	88	15.14	odd	2	inner
105.4.p.a.59.27	yes	88	3.2	odd	2	inner
105.4.p.a.59.28	yes	88	5.4	even	2	inner
105.4.p.a.89.17	yes	88	105.89	even	6	inner
105.4.p.a.89.18	yes	88	7.5	odd	6	inner
105.4.p.a.89.27	yes	88	35.19	odd	6	inner
105.4.p.a.89.28	yes	88	21.5	even	6	inner