Embedded newform 210.3.w.a.17.9

• Label: 210.3.w.a.17.9
• Level: $210$
• Weight: $3$
• Character: 210.17
• Analytic conductor: $5.722$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	210.w (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.72208555157\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			17.9
Character	\(\chi\)	\(=\)	210.17
Dual form			210.3.w.a.173.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36603 - 0.366025i) q^{2} +(0.587638 - 2.94188i) q^{3} +(1.73205 + 1.00000i) q^{4} +(-3.75994 - 3.29588i) q^{5} +(-1.87953 + 3.80360i) q^{6} +(5.65896 - 4.12022i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(-8.30936 - 3.45752i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 32 q^{2} - 6 q^{3} - 12 q^{5} + 4 q^{7} - 128 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(71\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.36603	−	0.366025i	−0.683013	−	0.183013i
							\(3\)	0.587638	−	2.94188i	0.195879	−	0.980628i
\(5\)	−3.75994	−	3.29588i	−0.751988	−	0.659177i
							\(7\)	5.65896	−	4.12022i	0.808423	−	0.588602i
\(11\)	10.7431	+	6.20251i	0.976641	+	0.563864i								0.901254	−	0.433290i	\(-0.142647\pi\)
							0.0753870	+	0.997154i	\(0.475981\pi\)
\(13\)	−17.5103	−	17.5103i	−1.34695	−	1.34695i	−0.888958	−	0.457989i	\(-0.848570\pi\)
							−0.457989	−	0.888958i	\(-0.651430\pi\)
\(17\)	−9.60968	+	2.57491i	−0.565275	+	0.151465i	−0.530129	−	0.847917i	\(-0.677856\pi\)
							−0.0351465	+	0.999382i	\(0.511190\pi\)
\(19\)	6.00440	+	10.3999i	0.316021	+	0.547365i	0.979654	−	0.200694i	\(-0.0643196\pi\)
							−0.663633	+	0.748058i	\(0.730986\pi\)
\(23\)	0.588343	−	2.19573i	0.0255801	−	0.0954664i	−0.951956	−	0.306236i	\(-0.900930\pi\)
							0.977536	+	0.210770i	\(0.0675970\pi\)
\(29\)	−19.7542			−0.681181			−0.340590	−	0.940212i	\(-0.610627\pi\)
							−0.340590	+	0.940212i	\(0.610627\pi\)
\(31\)	−45.3975	−	26.2103i	−1.46444	−	0.845493i	−0.465225	−	0.885193i	\(-0.654026\pi\)
							−0.999212	+	0.0396998i	\(0.987360\pi\)
\(37\)	−1.97085	+	7.35531i	−0.0532662	+	0.198792i	−0.987431	−	0.158049i	\(-0.949480\pi\)
							0.934165	+	0.356841i	\(0.116146\pi\)
\(41\)	−18.4715			−0.450524			−0.225262	−	0.974298i	\(-0.572324\pi\)
							−0.225262	+	0.974298i	\(0.572324\pi\)
\(43\)	33.8663	−	33.8663i	0.787588	−	0.787588i	−0.193511	−	0.981098i	\(-0.561987\pi\)
							0.981098	+	0.193511i	\(0.0619874\pi\)
\(47\)	6.09028	−	22.7292i	0.129580	−	0.483600i	−0.870381	−	0.492379i	\(-0.836128\pi\)
							0.999961	+	0.00877822i	\(0.00279423\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.w.a.17.9	✓	64
3.2	odd	2		210.3.w.b.17.11	yes	64
5.3	odd	4		210.3.w.b.143.16	yes	64
7.5	odd	6	inner	210.3.w.a.47.4	yes	64
15.8	even	4	inner	210.3.w.a.143.4	yes	64
21.5	even	6		210.3.w.b.47.16	yes	64
35.33	even	12		210.3.w.b.173.11	yes	64
105.68	odd	12	inner	210.3.w.a.173.9	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.w.a.17.9	✓	64	1.1	even	1	trivial
210.3.w.a.47.4	yes	64	7.5	odd	6	inner
210.3.w.a.143.4	yes	64	15.8	even	4	inner
210.3.w.a.173.9	yes	64	105.68	odd	12	inner
210.3.w.b.17.11	yes	64	3.2	odd	2
210.3.w.b.47.16	yes	64	21.5	even	6
210.3.w.b.143.16	yes	64	5.3	odd	4
210.3.w.b.173.11	yes	64	35.33	even	12