Embedded newform 105.3.w.a.17.20

• Label: 105.3.w.a.17.20
• 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			17.20
Character	\(\chi\)	\(=\)	105.17
Dual form			105.3.w.a.68.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.55206 + 0.415874i) q^{2} +(-2.99899 - 0.0778404i) q^{3} +(-1.22815 - 0.709073i) q^{4} +(-3.76090 + 3.29479i) q^{5} +(-4.62225 - 1.36802i) q^{6} +(-5.75785 + 3.98085i) q^{7} +(-6.15604 - 6.15604i) q^{8} +(8.98788 + 0.466885i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 6 q^{3} - 16 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.55206	+	0.415874i	0.776032	+	0.207937i	0.625035	−	0.780597i	\(-0.285085\pi\)
							0.150997	+	0.988534i	\(0.451751\pi\)
\(3\)	−2.99899	−	0.0778404i	−0.999663	−	0.0259468i
							\(5\)	−3.76090	+	3.29479i	−0.752181	+	0.658957i
\(7\)	−5.75785	+	3.98085i	−0.822550	+	0.568693i
							\(11\)	−11.9074	−	6.87474i	−1.08249	−	0.624976i	−0.150923	−	0.988545i	\(-0.548225\pi\)
−0.931567	+	0.363569i	\(0.881558\pi\)
\(13\)	13.4181	+	13.4181i	1.03216	+	1.03216i	0.999465	+	0.0326986i	\(0.0104101\pi\)
							0.0326986	+	0.999465i	\(0.489590\pi\)
\(17\)	−11.8463	+	3.17420i	−0.696840	+	0.186718i	−0.589815	−	0.807539i	\(-0.700799\pi\)
							−0.107025	+	0.994256i	\(0.534133\pi\)
\(19\)	0.412457	+	0.714396i	0.0217082	+	0.0375998i	0.876675	−	0.481082i	\(-0.159756\pi\)
							−0.854967	+	0.518682i	\(0.826423\pi\)
\(23\)	1.45610	−	5.43425i	0.0633088	−	0.236272i	−0.927020	−	0.375012i	\(-0.877639\pi\)
							0.990329	+	0.138740i	\(0.0443053\pi\)
\(29\)	4.58981			0.158269			0.0791347	−	0.996864i	\(-0.474784\pi\)
							0.0791347	+	0.996864i	\(0.474784\pi\)
\(31\)	4.73029	+	2.73103i	0.152590	+	0.0880978i	0.574351	−	0.818609i	\(-0.305255\pi\)
							−0.421761	+	0.906707i	\(0.638588\pi\)
\(37\)	−11.0527	+	41.2492i	−0.298721	+	1.11484i	0.639495	+	0.768795i	\(0.279143\pi\)
							−0.938217	+	0.346048i	\(0.887523\pi\)
\(41\)	−68.0906			−1.66075			−0.830373	−	0.557208i	\(-0.811873\pi\)
							−0.830373	+	0.557208i	\(0.811873\pi\)
\(43\)	−24.3339	+	24.3339i	−0.565905	+	0.565905i	−0.930979	−	0.365074i	\(-0.881044\pi\)
							0.365074	+	0.930979i	\(0.381044\pi\)
\(47\)	−6.45346	+	24.0846i	−0.137308	+	0.512439i	0.862670	+	0.505767i	\(0.168791\pi\)
							−0.999978	+	0.00667218i	\(0.997876\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.w.a.17.20	yes	112
3.2	odd	2	inner	105.3.w.a.17.9	✓	112
5.3	odd	4	inner	105.3.w.a.38.20	yes	112
7.5	odd	6	inner	105.3.w.a.47.9	yes	112
15.8	even	4	inner	105.3.w.a.38.9	yes	112
21.5	even	6	inner	105.3.w.a.47.20	yes	112
35.33	even	12	inner	105.3.w.a.68.9	yes	112
105.68	odd	12	inner	105.3.w.a.68.20	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.w.a.17.9	✓	112	3.2	odd	2	inner
105.3.w.a.17.20	yes	112	1.1	even	1	trivial
105.3.w.a.38.9	yes	112	15.8	even	4	inner
105.3.w.a.38.20	yes	112	5.3	odd	4	inner
105.3.w.a.47.9	yes	112	7.5	odd	6	inner
105.3.w.a.47.20	yes	112	21.5	even	6	inner
105.3.w.a.68.9	yes	112	35.33	even	12	inner
105.3.w.a.68.20	yes	112	105.68	odd	12	inner