Embedded newform 210.3.s.a.11.16

• Label: 210.3.s.a.11.16
• 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			11.16
Character	\(\chi\)	\(=\)	210.11
Dual form			210.3.s.a.191.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.22474 - 0.707107i) q^{2} +(1.02859 - 2.81815i) q^{3} +(1.00000 - 1.73205i) q^{4} +(1.93649 - 1.11803i) q^{5} +(-0.732971 - 4.17885i) q^{6} +(-5.85650 - 3.83424i) q^{7} -2.82843i q^{8} +(-6.88399 - 5.79747i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 40 q^{4} - 8 q^{6} + 20 q^{7} - 4 q^{9}+O(q^{10}) \)

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
							\(3\)	1.02859	−	2.81815i	0.342865	−	0.939385i
\(5\)	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								0.703925	−	0.710274i	\(-0.251429\pi\)
							−0.263152	+	0.964754i	\(0.584762\pi\)
\(13\)	−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	0.865333	−	0.501198i	\(-0.167107\pi\)
							−0.00138394	+	0.999999i	\(0.500441\pi\)
\(19\)	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	−0.0687592	−	0.997633i	\(-0.521904\pi\)
							0.829596	+	0.558364i	\(0.188571\pi\)
\(29\)			7.80822i			0.269249i	0.990897	+	0.134624i	\(0.0429828\pi\)
							−0.990897	+	0.134624i	\(0.957017\pi\)
\(31\)	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	\(0.282413\pi\)
							−0.631564	+	0.775324i	\(0.717587\pi\)
\(43\)	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	−0.222054	−	0.975034i	\(-0.571276\pi\)
							0.733377	+	0.679822i	\(0.237943\pi\)

(See \(a_n\) instead)

Twists

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

    

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