Embedded newform 105.3.w.a.17.10

• Label: 105.3.w.a.17.10
• 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			17.10
Character	\(\chi\)	\(=\)	105.17
Dual form			105.3.w.a.68.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40147 - 0.375522i) q^{2} +(-1.48402 - 2.60724i) q^{3} +(-1.64101 - 0.947435i) q^{4} +(-3.73472 - 3.32444i) q^{5} +(1.10073 + 4.21124i) q^{6} +(2.97299 + 6.33730i) q^{7} +(6.04782 + 6.04782i) q^{8} +(-4.59537 + 7.73838i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 6 q^{3} - 16 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40147	−	0.375522i	−0.700734	−	0.187761i	−0.109175	−	0.994023i	\(-0.534821\pi\)
							−0.591559	+	0.806261i	\(0.701487\pi\)
\(3\)	−1.48402	−	2.60724i	−0.494673	−	0.869079i
							\(5\)	−3.73472	−	3.32444i	−0.746943	−	0.664888i
\(7\)	2.97299	+	6.33730i	0.424712	+	0.905328i
							\(11\)	−3.76549	−	2.17401i	−0.342318	−	0.197637i	0.318979	−	0.947762i	\(-0.396660\pi\)
−0.661296	+	0.750125i	\(0.729993\pi\)
\(13\)	4.76059	+	4.76059i	0.366199	+	0.366199i	0.866089	−	0.499890i	\(-0.166626\pi\)
							−0.499890	+	0.866089i	\(0.666626\pi\)
\(17\)	−19.0658	+	5.10867i	−1.12152	+	0.300510i	−0.771498	−	0.636231i	\(-0.780492\pi\)
							−0.350021	+	0.936742i	\(0.613826\pi\)
\(19\)	−12.5949	−	21.8150i	−0.662890	−	1.14816i	−0.979853	−	0.199721i	\(-0.935996\pi\)
							0.316963	−	0.948438i	\(-0.397337\pi\)
\(23\)	−8.21961	+	30.6760i	−0.357374	+	1.33374i	0.520095	+	0.854108i	\(0.325896\pi\)
							−0.877470	+	0.479632i	\(0.840770\pi\)
\(29\)	−28.8306			−0.994158			−0.497079	−	0.867705i	\(-0.665594\pi\)
							−0.497079	+	0.867705i	\(0.665594\pi\)
\(31\)	−12.3444	−	7.12704i	−0.398206	−	0.229904i	0.287504	−	0.957780i	\(-0.407175\pi\)
							−0.685710	+	0.727875i	\(0.740508\pi\)
\(37\)	13.1470	−	49.0654i	0.355325	−	1.32609i	−0.524750	−	0.851257i	\(-0.675841\pi\)
							0.880075	−	0.474835i	\(-0.157492\pi\)
\(41\)	1.56196			0.0380967			0.0190483	−	0.999819i	\(-0.493936\pi\)
							0.0190483	+	0.999819i	\(0.493936\pi\)
\(43\)	−46.2033	+	46.2033i	−1.07449	+	1.07449i	−0.0775021	+	0.996992i	\(0.524694\pi\)
							−0.996992	+	0.0775021i	\(0.975306\pi\)
\(47\)	−3.56182	+	13.2929i	−0.0757834	+	0.282828i	−0.993410	−	0.114616i	\(-0.963436\pi\)
							0.917626	+	0.397444i	\(0.130103\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.w.a.17.10	✓	112
3.2	odd	2	inner	105.3.w.a.17.19	yes	112
5.3	odd	4	inner	105.3.w.a.38.10	yes	112
7.5	odd	6	inner	105.3.w.a.47.19	yes	112
15.8	even	4	inner	105.3.w.a.38.19	yes	112
21.5	even	6	inner	105.3.w.a.47.10	yes	112
35.33	even	12	inner	105.3.w.a.68.19	yes	112
105.68	odd	12	inner	105.3.w.a.68.10	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.w.a.17.10	✓	112	1.1	even	1	trivial
105.3.w.a.17.19	yes	112	3.2	odd	2	inner
105.3.w.a.38.10	yes	112	5.3	odd	4	inner
105.3.w.a.38.19	yes	112	15.8	even	4	inner
105.3.w.a.47.10	yes	112	21.5	even	6	inner
105.3.w.a.47.19	yes	112	7.5	odd	6	inner
105.3.w.a.68.10	yes	112	105.68	odd	12	inner
105.3.w.a.68.19	yes	112	35.33	even	12	inner