Embedded newform 210.3.w.b.173.9

• Label: 210.3.w.b.173.9
• 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			173.9
Character	\(\chi\)	\(=\)	210.173
Dual form			210.3.w.b.17.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36603 - 0.366025i) q^{2} +(0.145428 - 2.99647i) q^{3} +(1.73205 - 1.00000i) q^{4} +(-4.92543 - 0.860317i) q^{5} +(-0.898127 - 4.14649i) q^{6} +(1.68496 - 6.79418i) q^{7} +(2.00000 - 2.00000i) q^{8} +(-8.95770 - 0.871542i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 32 q^{2} + 6 q^{3} + 12 q^{5} + 4 q^{7} + 128 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

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

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
							\(3\)	0.145428	−	2.99647i	0.0484760	−	0.998824i
\(5\)	−4.92543	−	0.860317i	−0.985086	−	0.172063i
							\(7\)	1.68496	−	6.79418i	0.240709	−	0.970597i
\(11\)	−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	\(-0.656402\pi\)
							0.881697	+	0.471817i	\(0.156402\pi\)
\(17\)	16.9844	+	4.55096i	0.999083	+	0.267703i	0.721061	−	0.692872i	\(-0.243655\pi\)
							0.278021	+	0.960575i	\(0.410321\pi\)
\(19\)	5.62474	−	9.74233i	0.296039	−	0.512754i	−0.679187	−	0.733965i	\(-0.737668\pi\)
							0.975226	+	0.221211i	\(0.0710009\pi\)
\(23\)	−10.4778	−	39.1036i	−0.455556	−	1.70016i	−0.686449	−	0.727178i	\(-0.740832\pi\)
							0.230893	−	0.972979i	\(-0.425835\pi\)
\(29\)	−22.8488			−0.787888			−0.393944	−	0.919134i	\(-0.628890\pi\)
							−0.393944	+	0.919134i	\(0.628890\pi\)
\(31\)	36.0825	−	20.8322i	1.16395	−	0.672007i	0.211703	−	0.977334i	\(-0.432099\pi\)
							0.952248	+	0.305327i	\(0.0987658\pi\)
\(37\)	7.72790	+	28.8409i	0.208862	+	0.779484i	0.988238	+	0.152927i	\(0.0488699\pi\)
							−0.779375	+	0.626557i	\(0.784463\pi\)
\(41\)	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	0.521690	−	0.853135i	\(-0.325302\pi\)
							−0.853135	+	0.521690i	\(0.825302\pi\)
\(47\)	−6.77411	−	25.2813i	−0.144130	−	0.537900i	−0.999793	−	0.0203662i	\(-0.993517\pi\)
							0.855663	−	0.517534i	\(-0.173150\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.w.a.17.6	✓	64	35.17	even	12
210.3.w.a.47.1	yes	64	5.2	odd	4
210.3.w.a.143.1	yes	64	21.17	even	6
210.3.w.a.173.6	yes	64	3.2	odd	2
210.3.w.b.17.9	yes	64	105.17	odd	12	inner
210.3.w.b.47.13	yes	64	15.2	even	4	inner
210.3.w.b.143.13	yes	64	7.3	odd	6	inner
210.3.w.b.173.9	yes	64	1.1	even	1	trivial