Embedded newform 105.3.w.a.17.1

• Label: 105.3.w.a.17.1
• Level: $105$
• Weight: $3$
• Character: 105.17
• Analytic conductor: $2.861$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.86104277578\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(28\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			17.1
Character	\(\chi\)	\(=\)	105.17
Dual form			105.3.w.a.68.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.72728 - 0.998723i) q^{2} +(2.12159 + 2.12105i) q^{3} +(9.43110 + 5.44505i) q^{4} +(4.76718 + 1.50798i) q^{5} +(-5.78944 - 10.0246i) q^{6} +(-6.13877 + 3.36385i) q^{7} +(-18.8000 - 18.8000i) q^{8} +(0.00230342 + 9.00000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 6 q^{3} - 16 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.72728	−	0.998723i	−1.86364	−	0.499361i	−0.863653	−	0.504086i	\(-0.831829\pi\)
							−0.999989	+	0.00472484i	\(0.998496\pi\)
\(3\)	2.12159	+	2.12105i	0.707197	+	0.707016i
							\(5\)	4.76718	+	1.50798i	0.953436	+	0.301596i
\(7\)	−6.13877	+	3.36385i	−0.876968	+	0.480550i
							\(11\)	2.12166	+	1.22494i	0.192878	+	0.111358i	0.593329	−	0.804960i	\(-0.297813\pi\)
−0.400451	+	0.916318i	\(0.631147\pi\)
\(13\)	2.54324	+	2.54324i	0.195634	+	0.195634i	0.798125	−	0.602492i	\(-0.205825\pi\)
							−0.602492	+	0.798125i	\(0.705825\pi\)
\(17\)	−2.68738	+	0.720080i	−0.158081	+	0.0423577i	−0.336992	−	0.941508i	\(-0.609409\pi\)
							0.178911	+	0.983865i	\(0.442743\pi\)
\(19\)	0.652966	+	1.13097i	0.0343667	+	0.0595248i	0.882697	−	0.469942i	\(-0.155725\pi\)
							−0.848331	+	0.529467i	\(0.822392\pi\)
\(23\)	−5.75265	+	21.4692i	−0.250115	+	0.933443i	0.720628	+	0.693322i	\(0.243854\pi\)
							−0.970743	+	0.240121i	\(0.922813\pi\)
\(29\)	14.4234			0.497358			0.248679	−	0.968586i	\(-0.420004\pi\)
							0.248679	+	0.968586i	\(0.420004\pi\)
\(31\)	5.76876	+	3.33060i	0.186089	+	0.107439i	0.590150	−	0.807293i	\(-0.299068\pi\)
							−0.404061	+	0.914732i	\(0.632402\pi\)
\(37\)	8.72107	−	32.5475i	0.235704	−	0.879661i	−0.742126	−	0.670261i	\(-0.766182\pi\)
							0.977830	−	0.209400i	\(-0.0671511\pi\)
\(41\)	9.57556			0.233550			0.116775	−	0.993158i	\(-0.462744\pi\)
							0.116775	+	0.993158i	\(0.462744\pi\)
\(43\)	27.5904	−	27.5904i	0.641638	−	0.641638i	−0.309320	−	0.950958i	\(-0.600101\pi\)
							0.950958	+	0.309320i	\(0.100101\pi\)
\(47\)	−6.23208	+	23.2584i	−0.132597	+	0.494860i	−0.999996	−	0.00275330i	\(-0.999124\pi\)
							0.867399	+	0.497614i	\(0.165790\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.w.a.17.1	✓	112
3.2	odd	2	inner	105.3.w.a.17.28	yes	112
5.3	odd	4	inner	105.3.w.a.38.1	yes	112
7.5	odd	6	inner	105.3.w.a.47.28	yes	112
15.8	even	4	inner	105.3.w.a.38.28	yes	112
21.5	even	6	inner	105.3.w.a.47.1	yes	112
35.33	even	12	inner	105.3.w.a.68.28	yes	112
105.68	odd	12	inner	105.3.w.a.68.1	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.w.a.17.1	✓	112	1.1	even	1	trivial
105.3.w.a.17.28	yes	112	3.2	odd	2	inner
105.3.w.a.38.1	yes	112	5.3	odd	4	inner
105.3.w.a.38.28	yes	112	15.8	even	4	inner
105.3.w.a.47.1	yes	112	21.5	even	6	inner
105.3.w.a.47.28	yes	112	7.5	odd	6	inner
105.3.w.a.68.1	yes	112	105.68	odd	12	inner
105.3.w.a.68.28	yes	112	35.33	even	12	inner