Embedded newform 210.3.s.a.11.2

• Label: 210.3.s.a.11.2
• 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			11.2
Character	\(\chi\)	\(=\)	210.11
Dual form			210.3.s.a.191.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22474 + 0.707107i) q^{2} +(-2.87585 - 0.854111i) q^{3} +(1.00000 - 1.73205i) q^{4} +(-1.93649 + 1.11803i) q^{5} +(4.12613 - 0.987462i) q^{6} +(6.85537 - 1.41561i) q^{7} +2.82843i q^{8} +(7.54099 + 4.91259i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 40 q^{4} - 8 q^{6} + 20 q^{7} - 4 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.87585	−	0.854111i	−0.958616	−	0.284704i
\(5\)	−1.93649	+	1.11803i	−0.387298	+	0.223607i
							\(7\)	6.85537	−	1.41561i	0.979338	−	0.202230i
\(11\)	1.73712	+	1.00293i	0.157920	+	0.0911753i								0.576878	−	0.816831i	\(-0.304271\pi\)
							−0.418957	+	0.908006i	\(0.637604\pi\)
\(13\)	−17.1531			−1.31947			−0.659736	−	0.751497i	\(-0.729332\pi\)
							−0.659736	+	0.751497i	\(0.729332\pi\)
\(17\)	−16.4195	−	9.47979i	−0.965851	−	0.557634i	−0.0678822	−	0.997693i	\(-0.521624\pi\)
							−0.897969	+	0.440059i	\(0.854958\pi\)
\(19\)	−7.18314	−	12.4416i	−0.378060	−	0.654819i	0.612720	−	0.790300i	\(-0.290075\pi\)
							−0.990780	+	0.135481i	\(0.956742\pi\)
\(23\)	−22.4130	+	12.9402i	−0.974479	+	0.562616i	−0.900599	−	0.434651i	\(-0.856872\pi\)
							−0.0738805	+	0.997267i	\(0.523538\pi\)
\(29\)		−	10.3125i		−	0.355603i	−0.984066	−	0.177801i	\(-0.943102\pi\)
							0.984066	−	0.177801i	\(-0.0568985\pi\)
\(31\)	−24.0092	+	41.5852i	−0.774490	+	1.34146i	0.160590	+	0.987021i	\(0.448660\pi\)
							−0.935081	+	0.354435i	\(0.884673\pi\)
\(37\)	−21.9490	−	38.0168i	−0.593216	−	1.02748i	−0.993796	−	0.111218i	\(-0.964525\pi\)
							0.400580	−	0.916262i	\(-0.368809\pi\)
\(41\)		−	22.8694i		−	0.557790i	−0.960322	−	0.278895i	\(-0.910032\pi\)
							0.960322	−	0.278895i	\(-0.0899682\pi\)
\(43\)	−73.6363			−1.71247			−0.856236	−	0.516584i	\(-0.827203\pi\)
							−0.856236	+	0.516584i	\(0.827203\pi\)
\(47\)	−20.0906	+	11.5993i	−0.427459	+	0.246793i	−0.698263	−	0.715841i	\(-0.746044\pi\)
							0.270805	+	0.962634i	\(0.412710\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.s.a.11.2	✓	40
3.2	odd	2	inner	210.3.s.a.11.19	yes	40
7.2	even	3	inner	210.3.s.a.191.19	yes	40
21.2	odd	6	inner	210.3.s.a.191.2	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.s.a.11.2	✓	40	1.1	even	1	trivial
210.3.s.a.11.19	yes	40	3.2	odd	2	inner
210.3.s.a.191.2	yes	40	21.2	odd	6	inner
210.3.s.a.191.19	yes	40	7.2	even	3	inner