Embedded newform 105.4.q.b.4.16

• Label: 105.4.q.b.4.16
• 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			4.16
Character	\(\chi\)	\(=\)	105.4
Dual form			105.4.q.b.79.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.01833 - 1.16528i) q^{2} +(-2.59808 - 1.50000i) q^{3} +(-1.28423 + 2.22435i) q^{4} +(2.89467 + 10.7991i) q^{5} -6.99170 q^{6} +(-18.1257 - 3.80267i) q^{7} +24.6305i q^{8} +(4.50000 + 7.79423i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + 62 q^{4} - 4 q^{5} + 108 q^{6} + 198 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.01833	−	1.16528i	0.713587	−	0.411990i	−0.0988005	−	0.995107i	\(-0.531501\pi\)
							0.812388	+	0.583117i	\(0.198167\pi\)
\(3\)	−2.59808	−	1.50000i	−0.500000	−	0.288675i
							\(5\)	2.89467	+	10.7991i	0.258907	+	0.965902i
\(7\)	−18.1257	−	3.80267i	−0.978694	−	0.205325i
							\(11\)	−17.6913	+	30.6423i	−0.484921	+	0.839909i	−0.999850	−	0.0173246i	\(-0.994485\pi\)
0.514928	+	0.857233i	\(0.327818\pi\)
\(13\)			8.12235i			0.173287i	0.996239	+	0.0866437i	\(0.0276142\pi\)
							−0.996239	+	0.0866437i	\(0.972386\pi\)
\(17\)	10.0752	+	5.81695i	0.143742	+	0.0829892i	0.570146	−	0.821543i	\(-0.306887\pi\)
							−0.426404	+	0.904533i	\(0.640220\pi\)
\(19\)	36.3540	+	62.9670i	0.438957	+	0.760296i	0.997609	−	0.0691061i	\(-0.0220147\pi\)
							−0.558652	+	0.829402i	\(0.688681\pi\)
\(23\)	144.288	−	83.3050i	1.30810	−	0.755230i	0.326318	−	0.945260i	\(-0.394192\pi\)
							0.981778	+	0.190030i	\(0.0608586\pi\)
\(29\)	−259.901			−1.66422			−0.832111	−	0.554609i	\(-0.812868\pi\)
							−0.832111	+	0.554609i	\(0.812868\pi\)
\(31\)	112.403	−	194.688i	0.651232	−	1.12797i	−0.331593	−	0.943423i	\(-0.607586\pi\)
							0.982824	−	0.184544i	\(-0.0590807\pi\)
\(37\)	−311.299	+	179.729i	−1.38317	+	0.798574i	−0.992534	−	0.121971i	\(-0.961079\pi\)
							−0.390637	+	0.920545i	\(0.627745\pi\)
\(41\)	294.571			1.12205			0.561027	−	0.827797i	\(-0.310406\pi\)
							0.561027	+	0.827797i	\(0.310406\pi\)
\(43\)			167.151i			0.592799i	0.955064	+	0.296399i	\(0.0957859\pi\)
							−0.955064	+	0.296399i	\(0.904214\pi\)
\(47\)	−53.3797	+	30.8188i	−0.165664	+	0.0956464i	−0.580540	−	0.814232i	\(-0.697158\pi\)
							0.414875	+	0.909878i	\(0.363825\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.q.b.4.16	yes	44
5.4	even	2	inner	105.4.q.b.4.7	✓	44
7.2	even	3	inner	105.4.q.b.79.7	yes	44
35.9	even	6	inner	105.4.q.b.79.16	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.q.b.4.7	✓	44	5.4	even	2	inner
105.4.q.b.4.16	yes	44	1.1	even	1	trivial
105.4.q.b.79.7	yes	44	7.2	even	3	inner
105.4.q.b.79.16	yes	44	35.9	even	6	inner