Embedded newform 210.3.s.a.11.8

• Label: 210.3.s.a.11.8
• 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			11.8
Character	\(\chi\)	\(=\)	210.11
Dual form			210.3.s.a.191.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22474 + 0.707107i) q^{2} +(1.92630 - 2.29987i) q^{3} +(1.00000 - 1.73205i) q^{4} +(-1.93649 + 1.11803i) q^{5} +(-0.732971 + 4.17885i) q^{6} +(-5.85650 - 3.83424i) q^{7} +2.82843i q^{8} +(-1.57876 - 8.86045i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 40 q^{4} - 8 q^{6} + 20 q^{7} - 4 q^{9}+O(q^{10}) \)

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
							\(3\)	1.92630	−	2.29987i	0.642099	−	0.766622i
\(5\)	−1.93649	+	1.11803i	−0.387298	+	0.223607i
							\(7\)	−5.85650	−	3.83424i	−0.836643	−	0.547749i
\(11\)	−4.84850	−	2.79928i	−0.440773	−	0.254480i								0.263152	−	0.964754i	\(-0.415238\pi\)
							−0.703925	+	0.710274i	\(0.748571\pi\)
\(13\)	−1.42084			−0.109295			−0.0546475	−	0.998506i	\(-0.517404\pi\)
							−0.0546475	+	0.998506i	\(0.517404\pi\)
\(17\)	−14.6871	−	8.47962i	−0.863949	−	0.498801i	0.00138394	−	0.999999i	\(-0.499559\pi\)
							−0.865333	+	0.501198i	\(0.832893\pi\)
\(19\)	0.597560	+	1.03500i	0.0314505	+	0.0544739i	0.881322	−	0.472516i	\(-0.156654\pi\)
							−0.849872	+	0.526990i	\(0.823321\pi\)
\(23\)	−17.4993	+	10.1032i	−0.760837	+	0.439269i	−0.829596	−	0.558364i	\(-0.811429\pi\)
							0.0687592	+	0.997633i	\(0.478096\pi\)
\(29\)		−	7.80822i		−	0.269249i	−0.990897	−	0.134624i	\(-0.957017\pi\)
							0.990897	−	0.134624i	\(-0.0429828\pi\)
\(31\)	7.25871	−	12.5725i	0.234152	−	0.405563i	−0.724874	−	0.688882i	\(-0.758102\pi\)
							0.959026	+	0.283318i	\(0.0914353\pi\)
\(37\)	−25.0493	−	43.3866i	−0.677007	−	1.17261i	−0.975878	−	0.218317i	\(-0.929943\pi\)
							0.298871	−	0.954293i	\(-0.403390\pi\)
\(41\)		−	63.5765i		−	1.55065i	−0.631564	−	0.775324i	\(-0.717587\pi\)
							0.631564	−	0.775324i	\(-0.282413\pi\)
\(43\)	70.4952			1.63942			0.819711	−	0.572777i	\(-0.194134\pi\)
							0.819711	+	0.572777i	\(0.194134\pi\)
\(47\)	−24.0322	+	13.8750i	−0.511323	+	0.295213i	−0.733377	−	0.679822i	\(-0.762057\pi\)
							0.222054	+	0.975034i	\(0.428724\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.s.a.11.8	✓	40
3.2	odd	2	inner	210.3.s.a.11.16	yes	40
7.2	even	3	inner	210.3.s.a.191.16	yes	40
21.2	odd	6	inner	210.3.s.a.191.8	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.s.a.11.8	✓	40	1.1	even	1	trivial
210.3.s.a.11.16	yes	40	3.2	odd	2	inner
210.3.s.a.191.8	yes	40	21.2	odd	6	inner
210.3.s.a.191.16	yes	40	7.2	even	3	inner