Embedded newform 105.4.p.a.59.3

• Label: 105.4.p.a.59.3
• 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			59.3
Character	\(\chi\)	\(=\)	105.59
Dual form			105.4.p.a.89.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.57122 - 4.45348i) q^{2} +(-4.78955 + 2.01500i) q^{3} +(-9.22230 + 15.9735i) q^{4} +(-9.15758 + 6.41394i) q^{5} +(21.2887 + 16.1492i) q^{6} +(-17.8760 - 4.84251i) q^{7} +53.7107 q^{8} +(18.8796 - 19.3018i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 164 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.57122	−	4.45348i	−0.909062	−	1.57454i	−0.815370	−	0.578941i	\(-0.803466\pi\)
							−0.0936923	−	0.995601i	\(-0.529867\pi\)
\(3\)	−4.78955	+	2.01500i	−0.921749	+	0.387786i
							\(5\)	−9.15758	+	6.41394i	−0.819079	+	0.573681i
\(7\)	−17.8760	−	4.84251i	−0.965211	−	0.261471i
							\(11\)	−34.2443	−	19.7710i	−0.938641	−	0.541924i	−0.0491067	−	0.998794i	\(-0.515637\pi\)
−0.889534	+	0.456869i	\(0.848971\pi\)
\(13\)	74.9149			1.59828			0.799140	−	0.601145i	\(-0.205288\pi\)
							0.799140	+	0.601145i	\(0.205288\pi\)
\(17\)	−5.91020	−	3.41226i	−0.0843196	−	0.0486820i	0.457247	−	0.889340i	\(-0.348835\pi\)
							−0.541567	+	0.840658i	\(0.682169\pi\)
\(19\)	−55.2999	+	31.9274i	−0.667719	+	0.385508i	−0.795212	−	0.606332i	\(-0.792640\pi\)
							0.127492	+	0.991840i	\(0.459307\pi\)
\(23\)	32.7188	+	56.6707i	0.296624	+	0.513767i	0.975361	−	0.220614i	\(-0.0708059\pi\)
							−0.678738	+	0.734381i	\(0.737473\pi\)
\(29\)		−	58.3303i		−	0.373506i	−0.982407	−	0.186753i	\(-0.940204\pi\)
							0.982407	−	0.186753i	\(-0.0597964\pi\)
\(31\)	193.256	+	111.577i	1.11967	+	0.646443i	0.941317	−	0.337523i	\(-0.109589\pi\)
							0.178356	+	0.983966i	\(0.442922\pi\)
\(37\)	147.718	−	85.2850i	0.656342	−	0.378939i	−0.134540	−	0.990908i	\(-0.542956\pi\)
							0.790882	+	0.611969i	\(0.209622\pi\)
\(41\)	159.741			0.608471			0.304236	−	0.952597i	\(-0.401599\pi\)
							0.304236	+	0.952597i	\(0.401599\pi\)
\(43\)		−	120.964i		−	0.428997i	−0.976724	−	0.214499i	\(-0.931188\pi\)
							0.976724	−	0.214499i	\(-0.0688117\pi\)
\(47\)	81.6238	−	47.1255i	0.253320	−	0.146255i	−0.367963	−	0.929840i	\(-0.619945\pi\)
							0.621284	+	0.783586i	\(0.286611\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.p.a.59.3	✓	88
3.2	odd	2	inner	105.4.p.a.59.41	yes	88
5.4	even	2	inner	105.4.p.a.59.42	yes	88
7.5	odd	6	inner	105.4.p.a.89.4	yes	88
15.14	odd	2	inner	105.4.p.a.59.4	yes	88
21.5	even	6	inner	105.4.p.a.89.42	yes	88
35.19	odd	6	inner	105.4.p.a.89.41	yes	88
105.89	even	6	inner	105.4.p.a.89.3	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.p.a.59.3	✓	88	1.1	even	1	trivial
105.4.p.a.59.4	yes	88	15.14	odd	2	inner
105.4.p.a.59.41	yes	88	3.2	odd	2	inner
105.4.p.a.59.42	yes	88	5.4	even	2	inner
105.4.p.a.89.3	yes	88	105.89	even	6	inner
105.4.p.a.89.4	yes	88	7.5	odd	6	inner
105.4.p.a.89.41	yes	88	35.19	odd	6	inner
105.4.p.a.89.42	yes	88	21.5	even	6	inner