Embedded newform 210.3.w.b.173.2

• Label: 210.3.w.b.173.2
• 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			173.2
Character	\(\chi\)	\(=\)	210.173
Dual form			210.3.w.b.17.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36603 - 0.366025i) q^{2} +(-2.98701 + 0.278846i) q^{3} +(1.73205 - 1.00000i) q^{4} +(-4.38913 + 2.39490i) q^{5} +(-3.97827 + 1.47423i) q^{6} +(5.69479 - 4.07055i) q^{7} +(2.00000 - 2.00000i) q^{8} +(8.84449 - 1.66583i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 32 q^{2} + 6 q^{3} + 12 q^{5} + 4 q^{7} + 128 q^{8} + 16 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.98701	+	0.278846i	−0.995671	+	0.0929485i
\(5\)	−4.38913	+	2.39490i	−0.877825	+	0.478981i
							\(7\)	5.69479	−	4.07055i	0.813541	−	0.581507i
\(11\)	12.1554	−	7.01794i	1.10504	−	0.637995i								0.167499	−	0.985872i	\(-0.446431\pi\)
							0.937540	+	0.347877i	\(0.113097\pi\)
\(13\)	7.32665	−	7.32665i	0.563588	−	0.563588i	−0.366736	−	0.930325i	\(-0.619525\pi\)
							0.930325	+	0.366736i	\(0.119525\pi\)
\(17\)	−8.77125	−	2.35025i	−0.515956	−	0.138250i	−0.00856047	−	0.999963i	\(-0.502725\pi\)
							−0.507395	+	0.861713i	\(0.669392\pi\)
\(19\)	13.9857	−	24.2240i	0.736090	−	1.27495i	−0.218153	−	0.975914i	\(-0.570003\pi\)
							0.954243	−	0.299031i	\(-0.0966633\pi\)
\(23\)	8.52605	+	31.8197i	0.370698	+	1.38346i	0.859530	+	0.511086i	\(0.170757\pi\)
							−0.488832	+	0.872378i	\(0.662577\pi\)
\(29\)	4.58689			0.158169			0.0790843	−	0.996868i	\(-0.474800\pi\)
							0.0790843	+	0.996868i	\(0.474800\pi\)
\(31\)	−31.4105	+	18.1349i	−1.01324	+	0.584995i	−0.912139	−	0.409881i	\(-0.865570\pi\)
							−0.101103	+	0.994876i	\(0.532237\pi\)
\(37\)	−9.81415	−	36.6269i	−0.265247	−	0.989917i	−0.962099	−	0.272701i	\(-0.912083\pi\)
							0.696851	−	0.717216i	\(-0.254584\pi\)
\(41\)	53.2920			1.29981			0.649903	−	0.760017i	\(-0.274810\pi\)
							0.649903	+	0.760017i	\(0.274810\pi\)
\(43\)	16.6145	+	16.6145i	0.386383	+	0.386383i	0.873395	−	0.487012i	\(-0.161913\pi\)
							−0.487012	+	0.873395i	\(0.661913\pi\)
\(47\)	4.62343	+	17.2549i	0.0983708	+	0.367125i	0.997509	−	0.0705401i	\(-0.0224723\pi\)
							−0.899138	+	0.437665i	\(0.855806\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.w.b.173.2	yes	64
3.2	odd	2		210.3.w.a.173.4	yes	64
5.2	odd	4		210.3.w.a.47.9	yes	64
7.3	odd	6	inner	210.3.w.b.143.5	yes	64
15.2	even	4	inner	210.3.w.b.47.5	yes	64
21.17	even	6		210.3.w.a.143.9	yes	64
35.17	even	12		210.3.w.a.17.4	✓	64
105.17	odd	12	inner	210.3.w.b.17.2	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.w.a.17.4	✓	64	35.17	even	12
210.3.w.a.47.9	yes	64	5.2	odd	4
210.3.w.a.143.9	yes	64	21.17	even	6
210.3.w.a.173.4	yes	64	3.2	odd	2
210.3.w.b.17.2	yes	64	105.17	odd	12	inner
210.3.w.b.47.5	yes	64	15.2	even	4	inner
210.3.w.b.143.5	yes	64	7.3	odd	6	inner
210.3.w.b.173.2	yes	64	1.1	even	1	trivial