Embedded newform 210.3.w.b.47.4

• Label: 210.3.w.b.47.4
• Level: $210$
• Weight: $3$
• Character: 210.47
• 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			47.4
Character	\(\chi\)	\(=\)	210.47
Dual form			210.3.w.b.143.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.366025 - 1.36603i) q^{2} +(-1.76063 - 2.42903i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(2.77440 + 4.15965i) q^{5} +(-2.67368 + 3.29415i) q^{6} +(-5.04081 - 4.85698i) q^{7} +(2.00000 + 2.00000i) q^{8} +(-2.80035 + 8.55325i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 32 q^{2} + 6 q^{3} + 12 q^{5} + 4 q^{7} + 128 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.366025	−	1.36603i	−0.183013	−	0.683013i
							\(3\)	−1.76063	−	2.42903i	−0.586877	−	0.809676i
\(5\)	2.77440	+	4.15965i	0.554880	+	0.831931i
							\(7\)	−5.04081	−	4.85698i	−0.720115	−	0.693855i
\(11\)	−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	\(-0.111816\pi\)
							−0.344100	+	0.938933i	\(0.611816\pi\)
\(17\)	−3.28852	+	12.2729i	−0.193442	+	0.721936i	0.799223	+	0.601035i	\(0.205245\pi\)
							−0.992665	+	0.120901i	\(0.961422\pi\)
\(19\)	−0.306444	+	0.530777i	−0.0161286	+	0.0279356i	−0.873977	−	0.485967i	\(-0.838467\pi\)
							0.857848	+	0.513903i	\(0.171801\pi\)
\(23\)	12.1940	−	3.26736i	0.530172	−	0.142059i	0.0162043	−	0.999869i	\(-0.494842\pi\)
							0.513968	+	0.857810i	\(0.328175\pi\)
\(29\)	29.8506			1.02933			0.514666	−	0.857391i	\(-0.327916\pi\)
							0.514666	+	0.857391i	\(0.327916\pi\)
\(31\)	−17.4831	+	10.0939i	−0.563972	+	0.325610i	−0.754738	−	0.656026i	\(-0.772236\pi\)
							0.190766	+	0.981636i	\(0.438903\pi\)
\(37\)	13.2325	−	3.54563i	0.357634	−	0.0958278i	−0.0755282	−	0.997144i	\(-0.524064\pi\)
							0.433162	+	0.901316i	\(0.357398\pi\)
\(41\)	−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	−0.0919759	+	0.995761i	\(0.529318\pi\)
							−0.995761	+	0.0919759i	\(0.970682\pi\)
\(47\)	−50.2630	+	13.4679i	−1.06942	+	0.286552i	−0.750257	−	0.661147i	\(-0.770070\pi\)
							−0.319168	+	0.947698i	\(0.603403\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.w.a.17.13	✓	64	21.17	even	6
210.3.w.a.47.15	yes	64	3.2	odd	2
210.3.w.a.143.15	yes	64	35.3	even	12
210.3.w.a.173.13	yes	64	5.3	odd	4
210.3.w.b.17.10	yes	64	7.3	odd	6	inner
210.3.w.b.47.4	yes	64	1.1	even	1	trivial
210.3.w.b.143.4	yes	64	105.38	odd	12	inner
210.3.w.b.173.10	yes	64	15.8	even	4	inner