Embedded newform 210.3.w.a.173.13

• Label: 210.3.w.a.173.13
• Level: $210$
• Weight: $3$
• Character: 210.173
• 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			173.13
Character	\(\chi\)	\(=\)	210.173
Dual form			210.3.w.a.17.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36603 + 0.366025i) q^{2} +(2.42903 - 1.76063i) q^{3} +(1.73205 - 1.00000i) q^{4} +(-4.98956 - 0.322873i) q^{5} +(-2.67368 + 3.29415i) q^{6} +(-4.85698 + 5.04081i) q^{7} +(-2.00000 + 2.00000i) q^{8} +(2.80035 - 8.55325i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 32 q^{2} - 6 q^{3} - 12 q^{5} + 4 q^{7} - 128 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(31\)	\(71\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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
							\(3\)	2.42903	−	1.76063i	0.809676	−	0.586877i
\(5\)	−4.98956	−	0.322873i	−0.997913	−	0.0645746i
							\(7\)	−4.85698	+	5.04081i	−0.693855	+	0.720115i
\(11\)	−17.0019	+	9.81605i	−1.54563	+	0.892369i								−0.547160	+	0.837028i	\(0.684291\pi\)
							−0.998468	+	0.0553407i	\(0.982375\pi\)
\(13\)	−7.73283	+	7.73283i	−0.594833	+	0.594833i	−0.938933	−	0.344100i	\(-0.888184\pi\)
							0.344100	+	0.938933i	\(0.388184\pi\)
\(17\)	12.2729	+	3.28852i	0.721936	+	0.193442i	0.601035	−	0.799223i	\(-0.294755\pi\)
							0.120901	+	0.992665i	\(0.461422\pi\)
\(19\)	0.306444	−	0.530777i	0.0161286	−	0.0279356i	−0.857848	−	0.513903i	\(-0.828199\pi\)
							0.873977	+	0.485967i	\(0.161533\pi\)
\(23\)	3.26736	+	12.1940i	0.142059	+	0.530172i	0.999869	+	0.0162043i	\(0.00515821\pi\)
							−0.857810	+	0.513968i	\(0.828175\pi\)
\(29\)	−29.8506			−1.02933			−0.514666	−	0.857391i	\(-0.672084\pi\)
							−0.514666	+	0.857391i	\(0.672084\pi\)
\(31\)	−17.4831	+	10.0939i	−0.563972	+	0.325610i	−0.754738	−	0.656026i	\(-0.772236\pi\)
							0.190766	+	0.981636i	\(0.438903\pi\)
\(37\)	−3.54563	−	13.2325i	−0.0958278	−	0.357634i	0.901316	−	0.433162i	\(-0.142602\pi\)
							−0.997144	+	0.0755282i	\(0.975936\pi\)
\(41\)	−75.0186			−1.82972			−0.914861	−	0.403768i	\(-0.867700\pi\)
							−0.914861	+	0.403768i	\(0.867700\pi\)
\(43\)	−46.7727	−	46.7727i	−1.08774	−	1.08774i	−0.995761	−	0.0919759i	\(-0.970682\pi\)
							−0.0919759	−	0.995761i	\(-0.529318\pi\)
\(47\)	13.4679	+	50.2630i	0.286552	+	1.06942i	0.947698	+	0.319168i	\(0.103403\pi\)
							−0.661147	+	0.750257i	\(0.729930\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.w.a.173.13	yes	64
3.2	odd	2		210.3.w.b.173.10	yes	64
5.2	odd	4		210.3.w.b.47.4	yes	64
7.3	odd	6	inner	210.3.w.a.143.15	yes	64
15.2	even	4	inner	210.3.w.a.47.15	yes	64
21.17	even	6		210.3.w.b.143.4	yes	64
35.17	even	12		210.3.w.b.17.10	yes	64
105.17	odd	12	inner	210.3.w.a.17.13	✓	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.w.a.17.13	✓	64	105.17	odd	12	inner
210.3.w.a.47.15	yes	64	15.2	even	4	inner
210.3.w.a.143.15	yes	64	7.3	odd	6	inner
210.3.w.a.173.13	yes	64	1.1	even	1	trivial
210.3.w.b.17.10	yes	64	35.17	even	12
210.3.w.b.47.4	yes	64	5.2	odd	4
210.3.w.b.143.4	yes	64	21.17	even	6
210.3.w.b.173.10	yes	64	3.2	odd	2