Embedded newform 210.3.s.a.11.9

• Label: 210.3.s.a.11.9
• 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			11.9
Character	\(\chi\)	\(=\)	210.11
Dual form			210.3.s.a.191.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22474 + 0.707107i) q^{2} +(2.66551 + 1.37661i) q^{3} +(1.00000 - 1.73205i) q^{4} +(-1.93649 + 1.11803i) q^{5} +(-4.23798 + 0.198807i) q^{6} +(3.85860 - 5.84048i) q^{7} +2.82843i q^{8} +(5.20990 + 7.33873i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 40 q^{4} - 8 q^{6} + 20 q^{7} - 4 q^{9}+O(q^{10}) \)

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
							\(3\)	2.66551	+	1.37661i	0.888504	+	0.458869i
\(5\)	−1.93649	+	1.11803i	−0.387298	+	0.223607i
							\(7\)	3.85860	−	5.84048i	0.551229	−	0.834354i
\(11\)	10.1776	+	5.87607i	0.925241	+	0.534188i								0.885303	−	0.465014i	\(-0.153951\pi\)
							0.0399374	+	0.999202i	\(0.487284\pi\)
\(13\)	0.292467			0.0224975			0.0112487	−	0.999937i	\(-0.496419\pi\)
							0.0112487	+	0.999937i	\(0.496419\pi\)
\(17\)	25.0520	+	14.4638i	1.47365	+	0.850812i	0.999560	−	0.0296651i	\(-0.00944409\pi\)
							0.474089	+	0.880477i	\(0.342777\pi\)
\(19\)	−3.00432	−	5.20363i	−0.158122	−	0.273875i	0.776069	−	0.630647i	\(-0.217211\pi\)
							−0.934191	+	0.356772i	\(0.883877\pi\)
\(23\)	−27.2340	+	15.7236i	−1.18409	+	0.683634i	−0.956957	−	0.290230i	\(-0.906268\pi\)
							−0.227132	+	0.973864i	\(0.572935\pi\)
\(29\)		−	27.0852i		−	0.933973i	−0.884265	−	0.466986i	\(-0.845340\pi\)
							0.884265	−	0.466986i	\(-0.154660\pi\)
\(31\)	1.74548	−	3.02326i	0.0563058	−	0.0975244i	−0.836499	−	0.547969i	\(-0.815401\pi\)
							0.892804	+	0.450445i	\(0.148735\pi\)
\(37\)	27.5133	+	47.6544i	0.743602	+	1.28796i	0.950845	+	0.309667i	\(0.100218\pi\)
							−0.207243	+	0.978289i	\(0.566449\pi\)
\(41\)			23.5607i			0.574652i	0.957833	+	0.287326i	\(0.0927662\pi\)
							−0.957833	+	0.287326i	\(0.907234\pi\)
\(43\)	44.4160			1.03293			0.516466	−	0.856308i	\(-0.327247\pi\)
							0.516466	+	0.856308i	\(0.327247\pi\)
\(47\)	2.48256	−	1.43330i	0.0528203	−	0.0304958i	−0.473357	−	0.880871i	\(-0.656958\pi\)
							0.526178	+	0.850375i	\(0.323625\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.s.a.11.9	✓	40
3.2	odd	2	inner	210.3.s.a.11.12	yes	40
7.2	even	3	inner	210.3.s.a.191.12	yes	40
21.2	odd	6	inner	210.3.s.a.191.9	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.s.a.11.9	✓	40	1.1	even	1	trivial
210.3.s.a.11.12	yes	40	3.2	odd	2	inner
210.3.s.a.191.9	yes	40	21.2	odd	6	inner
210.3.s.a.191.12	yes	40	7.2	even	3	inner