Embedded newform 210.3.w.b.47.3

• Label: 210.3.w.b.47.3
• 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			47.3
Character	\(\chi\)	\(=\)	210.47
Dual form			210.3.w.b.143.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.366025 - 1.36603i) q^{2} +(-2.80804 - 1.05588i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(4.87492 - 1.11135i) q^{5} +(-0.414547 + 4.22234i) q^{6} +(0.132847 + 6.99874i) q^{7} +(2.00000 + 2.00000i) q^{8} +(6.77022 + 5.92993i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 32 q^{2} + 6 q^{3} + 12 q^{5} + 4 q^{7} + 128 q^{8} + 16 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.80804	−	1.05588i	−0.936015	−	0.351961i
\(5\)	4.87492	−	1.11135i	0.974985	−	0.222271i
							\(7\)	0.132847	+	6.99874i	0.0189781	+	0.999820i
\(11\)	9.30088	−	5.36987i	0.845535	−	0.488170i								−0.0136071	−	0.999907i	\(-0.504331\pi\)
							0.859142	+	0.511738i	\(0.170998\pi\)
\(13\)	3.28522	+	3.28522i	0.252709	+	0.252709i	0.822081	−	0.569371i	\(-0.192813\pi\)
							−0.569371	+	0.822081i	\(0.692813\pi\)
\(17\)	4.29378	−	16.0246i	0.252575	−	0.942624i	−0.716848	−	0.697229i	\(-0.754416\pi\)
							0.969423	−	0.245394i	\(-0.0789174\pi\)
\(19\)	1.04608	−	1.81186i	0.0550568	−	0.0953611i	−0.837183	−	0.546922i	\(-0.815799\pi\)
							0.892240	+	0.451561i	\(0.149133\pi\)
\(23\)	−12.7041	+	3.40405i	−0.552351	+	0.148002i	−0.524190	−	0.851601i	\(-0.675632\pi\)
							−0.0281612	+	0.999603i	\(0.508965\pi\)
\(29\)	21.8689			0.754099			0.377049	−	0.926193i	\(-0.376939\pi\)
							0.377049	+	0.926193i	\(0.376939\pi\)
\(31\)	40.3894	−	23.3188i	1.30288	−	0.752221i	0.321987	−	0.946744i	\(-0.395649\pi\)
							0.980898	+	0.194523i	\(0.0623160\pi\)
\(37\)	40.7704	−	10.9244i	1.10190	−	0.295254i	0.338363	−	0.941016i	\(-0.390127\pi\)
							0.763539	+	0.645762i	\(0.223460\pi\)
\(41\)	41.7957			1.01941			0.509703	−	0.860350i	\(-0.329755\pi\)
							0.509703	+	0.860350i	\(0.329755\pi\)
\(43\)	−32.4714	+	32.4714i	−0.755148	+	0.755148i	−0.975435	−	0.220287i	\(-0.929301\pi\)
							0.220287	+	0.975435i	\(0.429301\pi\)
\(47\)	−67.3444	+	18.0449i	−1.43286	+	0.383934i	−0.890028	−	0.455907i	\(-0.849315\pi\)
							−0.542833	+	0.839841i	\(0.682648\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.w.b.47.3	yes	64
3.2	odd	2		210.3.w.a.47.16	yes	64
5.3	odd	4		210.3.w.a.173.11	yes	64
7.3	odd	6	inner	210.3.w.b.17.8	yes	64
15.8	even	4	inner	210.3.w.b.173.8	yes	64
21.17	even	6		210.3.w.a.17.11	✓	64
35.3	even	12		210.3.w.a.143.16	yes	64
105.38	odd	12	inner	210.3.w.b.143.3	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.w.a.17.11	✓	64	21.17	even	6
210.3.w.a.47.16	yes	64	3.2	odd	2
210.3.w.a.143.16	yes	64	35.3	even	12
210.3.w.a.173.11	yes	64	5.3	odd	4
210.3.w.b.17.8	yes	64	7.3	odd	6	inner
210.3.w.b.47.3	yes	64	1.1	even	1	trivial
210.3.w.b.143.3	yes	64	105.38	odd	12	inner
210.3.w.b.173.8	yes	64	15.8	even	4	inner