Embedded newform 210.3.s.a.11.4

• Label: 210.3.s.a.11.4
• 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			11.4
Character	\(\chi\)	\(=\)	210.11
Dual form			210.3.s.a.191.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22474 + 0.707107i) q^{2} +(-0.690423 - 2.91947i) q^{3} +(1.00000 - 1.73205i) q^{4} +(-1.93649 + 1.11803i) q^{5} +(2.90997 + 3.08741i) q^{6} +(0.399356 + 6.98860i) q^{7} +2.82843i q^{8} +(-8.04663 + 4.03134i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 40 q^{4} - 8 q^{6} + 20 q^{7} - 4 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.690423	−	2.91947i	−0.230141	−	0.973157i
\(5\)	−1.93649	+	1.11803i	−0.387298	+	0.223607i
							\(7\)	0.399356	+	6.98860i	0.0570508	+	0.998371i
\(11\)	0.415399	+	0.239831i	0.0377635	+	0.0218028i								0.518763	−	0.854918i	\(-0.326393\pi\)
							−0.480999	+	0.876721i	\(0.659726\pi\)
\(13\)	8.95456			0.688812			0.344406	−	0.938821i	\(-0.388080\pi\)
							0.344406	+	0.938821i	\(0.388080\pi\)
\(17\)	22.6674	+	13.0870i	1.33338	+	0.769826i	0.985816	−	0.167831i	\(-0.0536764\pi\)
							0.347562	+	0.937657i	\(0.387010\pi\)
\(19\)	1.48682	+	2.57526i	0.0782539	+	0.135540i	0.902497	−	0.430697i	\(-0.141732\pi\)
							−0.824243	+	0.566237i	\(0.808399\pi\)
\(23\)	27.1234	−	15.6597i	1.17928	−	0.680856i	0.223430	−	0.974720i	\(-0.428275\pi\)
							0.955847	+	0.293864i	\(0.0949414\pi\)
\(29\)			22.9607i			0.791749i	0.918305	+	0.395874i	\(0.129558\pi\)
							−0.918305	+	0.395874i	\(0.870442\pi\)
\(31\)	−19.2316	+	33.3101i	−0.620375	+	1.07452i	0.369041	+	0.929413i	\(0.379686\pi\)
							−0.989416	+	0.145107i	\(0.953647\pi\)
\(37\)	19.3588	+	33.5305i	0.523212	+	0.906230i	0.999635	+	0.0270135i	\(0.00859971\pi\)
							−0.476423	+	0.879216i	\(0.658067\pi\)
\(41\)		−	15.5940i		−	0.380341i	−0.981751	−	0.190170i	\(-0.939096\pi\)
							0.981751	−	0.190170i	\(-0.0609041\pi\)
\(43\)	11.0661			0.257352			0.128676	−	0.991687i	\(-0.458927\pi\)
							0.128676	+	0.991687i	\(0.458927\pi\)
\(47\)	−15.0347	+	8.68028i	−0.319887	+	0.184687i	−0.651342	−	0.758784i	\(-0.725794\pi\)
							0.331455	+	0.943471i	\(0.392460\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.s.a.11.4	✓	40
3.2	odd	2	inner	210.3.s.a.11.20	yes	40
7.2	even	3	inner	210.3.s.a.191.20	yes	40
21.2	odd	6	inner	210.3.s.a.191.4	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.s.a.11.4	✓	40	1.1	even	1	trivial
210.3.s.a.11.20	yes	40	3.2	odd	2	inner
210.3.s.a.191.4	yes	40	21.2	odd	6	inner
210.3.s.a.191.20	yes	40	7.2	even	3	inner