Embedded newform 210.3.w.a.17.16

• Label: 210.3.w.a.17.16
• Level: $210$
• Weight: $3$
• Character: 210.17
• Analytic conductor: $5.722$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	210.w (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.72208555157\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			17.16
Character	\(\chi\)	\(=\)	210.17
Dual form			210.3.w.a.173.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36603 - 0.366025i) q^{2} +(2.97842 + 0.359186i) q^{3} +(1.73205 + 1.00000i) q^{4} +(2.39011 - 4.39174i) q^{5} +(-3.93713 - 1.58084i) q^{6} +(6.65191 + 2.17994i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(8.74197 + 2.13962i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 32 q^{2} - 6 q^{3} - 12 q^{5} + 4 q^{7} - 128 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(71\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.36603	−	0.366025i	−0.683013	−	0.183013i
							\(3\)	2.97842	+	0.359186i	0.992807	+	0.119729i
\(5\)	2.39011	−	4.39174i	0.478021	−	0.878348i
							\(7\)	6.65191	+	2.17994i	0.950273	+	0.311419i
\(11\)	−5.82368	−	3.36230i	−0.529425	−	0.305664i								0.211357	−	0.977409i	\(-0.432212\pi\)
							−0.740782	+	0.671745i	\(0.765545\pi\)
\(13\)	−3.78650	−	3.78650i	−0.291269	−	0.291269i	0.546312	−	0.837582i	\(-0.316031\pi\)
							−0.837582	+	0.546312i	\(0.816031\pi\)
\(17\)	3.69500	−	0.990073i	0.217353	−	0.0582396i	−0.148499	−	0.988913i	\(-0.547444\pi\)
							0.365852	+	0.930673i	\(0.380778\pi\)
\(19\)	2.17576	+	3.76853i	0.114514	+	0.198344i	0.917585	−	0.397539i	\(-0.130136\pi\)
							−0.803072	+	0.595883i	\(0.796802\pi\)
\(23\)	−3.48168	+	12.9938i	−0.151377	+	0.564948i	0.848011	+	0.529979i	\(0.177800\pi\)
							−0.999388	+	0.0349695i	\(0.988867\pi\)
\(29\)	19.0062			0.655386			0.327693	−	0.944784i	\(-0.393729\pi\)
							0.327693	+	0.944784i	\(0.393729\pi\)
\(31\)	0.140971	+	0.0813895i	0.00454744	+	0.00262547i	0.502272	−	0.864710i	\(-0.332498\pi\)
							−0.497725	+	0.867335i	\(0.665831\pi\)
\(37\)	13.9748	−	52.1548i	0.377698	−	1.40959i	−0.471664	−	0.881779i	\(-0.656346\pi\)
							0.849362	−	0.527811i	\(-0.176987\pi\)
\(41\)	64.3886			1.57045			0.785226	−	0.619209i	\(-0.212547\pi\)
							0.785226	+	0.619209i	\(0.212547\pi\)
\(43\)	−49.3467	+	49.3467i	−1.14760	+	1.14760i	−0.160573	+	0.987024i	\(0.551334\pi\)
							−0.987024	+	0.160573i	\(0.948666\pi\)
\(47\)	8.05536	−	30.0630i	0.171391	−	0.639638i	−0.825748	−	0.564040i	\(-0.809247\pi\)
							0.997138	−	0.0755987i	\(-0.0240868\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.w.a.17.16	✓	64
3.2	odd	2		210.3.w.b.17.14	yes	64
5.3	odd	4		210.3.w.b.143.7	yes	64
7.5	odd	6	inner	210.3.w.a.47.12	yes	64
15.8	even	4	inner	210.3.w.a.143.12	yes	64
21.5	even	6		210.3.w.b.47.7	yes	64
35.33	even	12		210.3.w.b.173.14	yes	64
105.68	odd	12	inner	210.3.w.a.173.16	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.w.a.17.16	✓	64	1.1	even	1	trivial
210.3.w.a.47.12	yes	64	7.5	odd	6	inner
210.3.w.a.143.12	yes	64	15.8	even	4	inner
210.3.w.a.173.16	yes	64	105.68	odd	12	inner
210.3.w.b.17.14	yes	64	3.2	odd	2
210.3.w.b.47.7	yes	64	21.5	even	6
210.3.w.b.143.7	yes	64	5.3	odd	4
210.3.w.b.173.14	yes	64	35.33	even	12