Embedded newform 105.4.q.b.4.13

• Label: 105.4.q.b.4.13
• Level: $105$
• Weight: $4$
• Character: 105.4
• Analytic conductor: $6.195$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.19520055060\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(22\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			4.13
Character	\(\chi\)	\(=\)	105.4
Dual form			105.4.q.b.79.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.761268 - 0.439518i) q^{2} +(2.59808 + 1.50000i) q^{3} +(-3.61365 + 6.25902i) q^{4} +(-9.95559 - 5.08784i) q^{5} +2.63711 q^{6} +(-18.5184 + 0.260966i) q^{7} +13.3854i q^{8} +(4.50000 + 7.79423i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + 62 q^{4} - 4 q^{5} + 108 q^{6} + 198 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.761268	−	0.439518i	0.269149	−	0.155393i	−0.359352	−	0.933202i	\(-0.617002\pi\)
							0.628501	+	0.777809i	\(0.283669\pi\)
\(3\)	2.59808	+	1.50000i	0.500000	+	0.288675i
							\(5\)	−9.95559	−	5.08784i	−0.890455	−	0.455070i
\(7\)	−18.5184	+	0.260966i	−0.999901	+	0.0140908i
							\(11\)	−32.2371	+	55.8362i	−0.883622	+	1.53048i	−0.0363361	+	0.999340i	\(0.511569\pi\)
−0.847285	+	0.531138i	\(0.821765\pi\)
\(13\)		−	73.6827i		−	1.57199i	−0.618232	−	0.785996i	\(-0.712151\pi\)
							0.618232	−	0.785996i	\(-0.287849\pi\)
\(17\)	−13.4987	−	7.79346i	−0.192583	−	0.111188i	0.400608	−	0.916249i	\(-0.368799\pi\)
							−0.593191	+	0.805062i	\(0.702132\pi\)
\(19\)	68.6047	+	118.827i	0.828368	+	1.43478i	0.899318	+	0.437296i	\(0.144064\pi\)
							−0.0709491	+	0.997480i	\(0.522603\pi\)
\(23\)	−90.0678	+	52.0007i	−0.816541	+	0.471430i	−0.849222	−	0.528036i	\(-0.822929\pi\)
							0.0326815	+	0.999466i	\(0.489595\pi\)
\(29\)	76.4332			0.489424			0.244712	−	0.969596i	\(-0.421307\pi\)
							0.244712	+	0.969596i	\(0.421307\pi\)
\(31\)	−48.0427	+	83.2125i	−0.278346	+	0.482110i	−0.970974	−	0.239186i	\(-0.923120\pi\)
							0.692628	+	0.721295i	\(0.256453\pi\)
\(37\)	211.796	−	122.280i	0.941054	−	0.543318i	0.0507632	−	0.998711i	\(-0.483835\pi\)
							0.890290	+	0.455393i	\(0.150501\pi\)
\(41\)	−28.9028			−0.110094			−0.0550471	−	0.998484i	\(-0.517531\pi\)
							−0.0550471	+	0.998484i	\(0.517531\pi\)
\(43\)			47.5744i			0.168722i	0.996435	+	0.0843608i	\(0.0268848\pi\)
							−0.996435	+	0.0843608i	\(0.973115\pi\)
\(47\)	−100.768	+	58.1782i	−0.312733	+	0.180557i	−0.648149	−	0.761514i	\(-0.724457\pi\)
							0.335416	+	0.942070i	\(0.391123\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.q.b.4.13	yes	44
5.4	even	2	inner	105.4.q.b.4.10	✓	44
7.2	even	3	inner	105.4.q.b.79.10	yes	44
35.9	even	6	inner	105.4.q.b.79.13	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.q.b.4.10	✓	44	5.4	even	2	inner
105.4.q.b.4.13	yes	44	1.1	even	1	trivial
105.4.q.b.79.10	yes	44	7.2	even	3	inner
105.4.q.b.79.13	yes	44	35.9	even	6	inner