Embedded newform 210.3.w.a.47.1

• Label: 210.3.w.a.47.1
• 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			47.1
Character	\(\chi\)	\(=\)	210.47
Dual form			210.3.w.a.143.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.366025 + 1.36603i) q^{2} +(-2.99647 - 0.145428i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(3.20777 + 3.83539i) q^{5} +(-0.898127 - 4.14649i) q^{6} +(6.79418 + 1.68496i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(8.95770 + 0.871542i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 32 q^{2} - 6 q^{3} - 12 q^{5} + 4 q^{7} - 128 q^{8} - 16 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.99647	−	0.145428i	−0.998824	−	0.0484760i
\(5\)	3.20777	+	3.83539i	0.641554	+	0.767078i
							\(7\)	6.79418	+	1.68496i	0.970597	+	0.240709i
\(11\)	−13.7602	+	7.94448i	−1.25093	+	0.722225i								−0.971294	−	0.237881i	\(-0.923547\pi\)
							−0.279636	+	0.960106i	\(0.590214\pi\)
\(13\)	−5.32844	−	5.32844i	−0.409880	−	0.409880i	0.471817	−	0.881697i	\(-0.343598\pi\)
							−0.881697	+	0.471817i	\(0.843598\pi\)
\(17\)	−4.55096	+	16.9844i	−0.267703	+	0.999083i	0.692872	+	0.721061i	\(0.256345\pi\)
							−0.960575	+	0.278021i	\(0.910321\pi\)
\(19\)	−5.62474	+	9.74233i	−0.296039	+	0.512754i	−0.975226	−	0.221211i	\(-0.928999\pi\)
							0.679187	+	0.733965i	\(0.262332\pi\)
\(23\)	−39.1036	+	10.4778i	−1.70016	+	0.455556i	−0.972979	−	0.230893i	\(-0.925835\pi\)
							−0.727178	+	0.686449i	\(0.759168\pi\)
\(29\)	22.8488			0.787888			0.393944	−	0.919134i	\(-0.371110\pi\)
							0.393944	+	0.919134i	\(0.371110\pi\)
\(31\)	36.0825	−	20.8322i	1.16395	−	0.672007i	0.211703	−	0.977334i	\(-0.432099\pi\)
							0.952248	+	0.305327i	\(0.0987658\pi\)
\(37\)	−28.8409	+	7.72790i	−0.779484	+	0.208862i	−0.626557	−	0.779375i	\(-0.715537\pi\)
							−0.152927	+	0.988238i	\(0.548870\pi\)
\(41\)	55.0717			1.34321			0.671606	−	0.740909i	\(-0.265605\pi\)
							0.671606	+	0.740909i	\(0.265605\pi\)
\(43\)	−14.2521	+	14.2521i	−0.331445	+	0.331445i	−0.853135	−	0.521690i	\(-0.825302\pi\)
							0.521690	+	0.853135i	\(0.325302\pi\)
\(47\)	25.2813	−	6.77411i	0.537900	−	0.144130i	0.0203662	−	0.999793i	\(-0.493517\pi\)
							0.517534	+	0.855663i	\(0.326850\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.w.a.47.1	yes	64
3.2	odd	2		210.3.w.b.47.13	yes	64
5.3	odd	4		210.3.w.b.173.9	yes	64
7.3	odd	6	inner	210.3.w.a.17.6	✓	64
15.8	even	4	inner	210.3.w.a.173.6	yes	64
21.17	even	6		210.3.w.b.17.9	yes	64
35.3	even	12		210.3.w.b.143.13	yes	64
105.38	odd	12	inner	210.3.w.a.143.1	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.w.a.17.6	✓	64	7.3	odd	6	inner
210.3.w.a.47.1	yes	64	1.1	even	1	trivial
210.3.w.a.143.1	yes	64	105.38	odd	12	inner
210.3.w.a.173.6	yes	64	15.8	even	4	inner
210.3.w.b.17.9	yes	64	21.17	even	6
210.3.w.b.47.13	yes	64	3.2	odd	2
210.3.w.b.143.13	yes	64	35.3	even	12
210.3.w.b.173.9	yes	64	5.3	odd	4