Embedded newform 210.3.s.a.11.1

• Label: 210.3.s.a.11.1
• Level: $210$
• Weight: $3$
• Character: 210.11
• Analytic conductor: $5.722$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	210.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.72208555157\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(20\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.1
Character	\(\chi\)	\(=\)	210.11
Dual form			210.3.s.a.191.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22474 + 0.707107i) q^{2} +(-2.97598 - 0.378837i) q^{3} +(1.00000 - 1.73205i) q^{4} +(1.93649 - 1.11803i) q^{5} +(3.91270 - 1.64036i) q^{6} +(-2.08959 + 6.68084i) q^{7} +2.82843i q^{8} +(8.71297 + 2.25482i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 40 q^{4} - 8 q^{6} + 20 q^{7} - 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(31\)	\(71\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.22474	+	0.707107i	−0.612372	+	0.353553i
							\(3\)	−2.97598	−	0.378837i	−0.991995	−	0.126279i
\(5\)	1.93649	−	1.11803i	0.387298	−	0.223607i
							\(7\)	−2.08959	+	6.68084i	−0.298512	+	0.954406i
\(11\)	−14.2957	−	8.25361i	−1.29961	−	0.750328i								−0.319271	−	0.947664i	\(-0.603438\pi\)
							−0.980336	+	0.197335i	\(0.936771\pi\)
\(13\)	13.0314			1.00241			0.501206	−	0.865328i	\(-0.332890\pi\)
							0.501206	+	0.865328i	\(0.332890\pi\)
\(17\)	−4.10654	−	2.37091i	−0.241561	−	0.139466i	0.374333	−	0.927294i	\(-0.377872\pi\)
							−0.615894	+	0.787829i	\(0.711205\pi\)
\(19\)	−16.8648	−	29.2107i	−0.887623	−	1.53741i	−0.842678	−	0.538418i	\(-0.819022\pi\)
							−0.0449448	−	0.998989i	\(-0.514311\pi\)
\(23\)	10.0856	−	5.82293i	0.438505	−	0.253171i	−0.264458	−	0.964397i	\(-0.585193\pi\)
							0.702963	+	0.711226i	\(0.251860\pi\)
\(29\)		−	39.6383i		−	1.36684i	−0.730027	−	0.683419i	\(-0.760492\pi\)
							0.730027	−	0.683419i	\(-0.239508\pi\)
\(31\)	26.8474	−	46.5011i	0.866046	−	1.50004i	4.20833e−5	−	1.00000i	\(-0.499987\pi\)
							0.866004	−	0.500036i	\(-0.166680\pi\)
\(37\)	−19.9373	−	34.5325i	−0.538847	−	0.933310i	−0.998966	−	0.0454534i	\(-0.985527\pi\)
							0.460119	−	0.887857i	\(-0.347807\pi\)
\(41\)			63.1758i			1.54087i	0.637517	+	0.770436i	\(0.279962\pi\)
							−0.637517	+	0.770436i	\(0.720038\pi\)
\(43\)	−13.9597			−0.324644			−0.162322	−	0.986738i	\(-0.551898\pi\)
							−0.162322	+	0.986738i	\(0.551898\pi\)
\(47\)	−9.23264	+	5.33046i	−0.196439	+	0.113414i	−0.594993	−	0.803731i	\(-0.702845\pi\)
							0.398554	+	0.917145i	\(0.369512\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.s.a.11.1	✓	40
3.2	odd	2	inner	210.3.s.a.11.17	yes	40
7.2	even	3	inner	210.3.s.a.191.17	yes	40
21.2	odd	6	inner	210.3.s.a.191.1	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.s.a.11.1	✓	40	1.1	even	1	trivial
210.3.s.a.11.17	yes	40	3.2	odd	2	inner
210.3.s.a.191.1	yes	40	21.2	odd	6	inner
210.3.s.a.191.17	yes	40	7.2	even	3	inner