Embedded newform 105.4.x.a.2.10

• Label: 105.4.x.a.2.10
• Level: $105$
• Weight: $4$
• Character: 105.2
• Analytic conductor: $6.195$
• Analytic rank: $0$
• Dimension: $176$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	105.x (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.19520055060\)
Analytic rank:	\(0\)
Dimension:	\(176\)
Relative dimension:	\(44\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			2.10
Character	\(\chi\)	\(=\)	105.2
Dual form			105.4.x.a.53.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.904480 - 3.37557i) q^{2} +(2.72714 - 4.42297i) q^{3} +(-3.64816 + 2.10627i) q^{4} +(10.0776 + 4.84170i) q^{5} +(-17.3967 - 5.20516i) q^{6} +(-16.2054 - 8.96573i) q^{7} +(-9.35916 - 9.35916i) q^{8} +(-12.1254 - 24.1242i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 2 q^{3} + 4 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.904480	−	3.37557i	−0.319782	−	1.19344i	−0.919454	−	0.393197i	\(-0.871369\pi\)
							0.599672	−	0.800246i	\(-0.295298\pi\)
\(3\)	2.72714	−	4.42297i	0.524839	−	0.851202i
							\(5\)	10.0776	+	4.84170i	0.901368	+	0.433055i
\(7\)	−16.2054	−	8.96573i	−0.875011	−	0.484104i
							\(11\)	40.5537	−	23.4137i	1.11158	−	0.641772i	0.172344	−	0.985037i	\(-0.444866\pi\)
0.939239	+	0.343264i	\(0.111533\pi\)
\(13\)	−10.8175	+	10.8175i	−0.230788	+	0.230788i	−0.813021	−	0.582234i	\(-0.802179\pi\)
							0.582234	+	0.813021i	\(0.302179\pi\)
\(17\)	−97.1979	−	26.0441i	−1.38670	−	0.371566i	−0.513151	−	0.858298i	\(-0.671522\pi\)
							−0.873551	+	0.486732i	\(0.838189\pi\)
\(19\)	70.9851	+	40.9833i	0.857111	+	0.494853i	0.863044	−	0.505129i	\(-0.168555\pi\)
							−0.00593305	+	0.999982i	\(0.501889\pi\)
\(23\)	57.5031	−	15.4079i	0.521315	−	0.139686i	0.0114400	−	0.999935i	\(-0.496358\pi\)
							0.509875	+	0.860249i	\(0.329692\pi\)
\(29\)	170.773			1.09351			0.546754	−	0.837293i	\(-0.315863\pi\)
							0.546754	+	0.837293i	\(0.315863\pi\)
\(31\)	44.8313	+	77.6501i	0.259740	+	0.449883i	0.966172	−	0.257898i	\(-0.0830298\pi\)
							−0.706432	+	0.707781i	\(0.749696\pi\)
\(37\)	50.1777	−	13.4451i	0.222950	−	0.0597393i	−0.145615	−	0.989341i	\(-0.546516\pi\)
							0.368565	+	0.929602i	\(0.379849\pi\)
\(41\)		−	417.721i		−	1.59115i	−0.605856	−	0.795574i	\(-0.707169\pi\)
							0.605856	−	0.795574i	\(-0.292831\pi\)
\(43\)	70.4671	−	70.4671i	0.249910	−	0.249910i	−0.571024	−	0.820934i	\(-0.693453\pi\)
							0.820934	+	0.571024i	\(0.193453\pi\)
\(47\)	136.993	+	511.266i	0.425160	+	1.58672i	0.763573	+	0.645721i	\(0.223443\pi\)
							−0.338413	+	0.940998i	\(0.609890\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.x.a.2.10	✓	176
3.2	odd	2	inner	105.4.x.a.2.35	yes	176
5.3	odd	4	inner	105.4.x.a.23.10	yes	176
7.4	even	3	inner	105.4.x.a.32.35	yes	176
15.8	even	4	inner	105.4.x.a.23.35	yes	176
21.11	odd	6	inner	105.4.x.a.32.10	yes	176
35.18	odd	12	inner	105.4.x.a.53.35	yes	176
105.53	even	12	inner	105.4.x.a.53.10	yes	176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.x.a.2.10	✓	176	1.1	even	1	trivial
105.4.x.a.2.35	yes	176	3.2	odd	2	inner
105.4.x.a.23.10	yes	176	5.3	odd	4	inner
105.4.x.a.23.35	yes	176	15.8	even	4	inner
105.4.x.a.32.10	yes	176	21.11	odd	6	inner
105.4.x.a.32.35	yes	176	7.4	even	3	inner
105.4.x.a.53.10	yes	176	105.53	even	12	inner
105.4.x.a.53.35	yes	176	35.18	odd	12	inner