Embedded newform 210.3.w.a.17.3

• Label: 210.3.w.a.17.3
• 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			17.3
Character	\(\chi\)	\(=\)	210.17
Dual form			210.3.w.a.173.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36603 - 0.366025i) q^{2} +(-2.49614 + 1.66411i) q^{3} +(1.73205 + 1.00000i) q^{4} +(-4.84166 - 1.24831i) q^{5} +(4.01890 - 1.35957i) q^{6} +(-0.400482 - 6.98853i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(3.46146 - 8.30773i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 32 q^{2} - 6 q^{3} - 12 q^{5} + 4 q^{7} - 128 q^{8} - 16 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.49614	+	1.66411i	−0.832048	+	0.554704i
\(5\)	−4.84166	−	1.24831i	−0.968333	−	0.249663i
							\(7\)	−0.400482	−	6.98853i	−0.0572117	−	0.998362i
\(11\)	4.21141	+	2.43146i	0.382856	+	0.221042i								0.679060	−	0.734083i	\(-0.262388\pi\)
							−0.296204	+	0.955125i	\(0.595721\pi\)
\(13\)	13.3745	+	13.3745i	1.02881	+	1.02881i	0.999573	+	0.0292348i	\(0.00930704\pi\)
							0.0292348	+	0.999573i	\(0.490693\pi\)
\(17\)	−25.3015	+	6.77952i	−1.48832	+	0.398795i	−0.909171	−	0.416424i	\(-0.863283\pi\)
							−0.579153	+	0.815219i	\(0.696617\pi\)
\(19\)	6.44901	+	11.1700i	0.339422	+	0.587896i	0.984324	−	0.176369i	\(-0.0564353\pi\)
							−0.644902	+	0.764265i	\(0.723102\pi\)
\(23\)	−5.04836	+	18.8407i	−0.219494	+	0.819162i	0.765042	+	0.643980i	\(0.222718\pi\)
							−0.984536	+	0.175182i	\(0.943949\pi\)
\(29\)	41.0872			1.41680			0.708400	−	0.705811i	\(-0.249417\pi\)
							0.708400	+	0.705811i	\(0.249417\pi\)
\(31\)	28.4108	+	16.4030i	0.916477	+	0.529128i	0.882509	−	0.470295i	\(-0.155852\pi\)
							0.0339672	+	0.999423i	\(0.489186\pi\)
\(37\)	16.1378	−	60.2271i	0.436157	−	1.62776i	−0.302127	−	0.953268i	\(-0.597697\pi\)
							0.738283	−	0.674491i	\(-0.235637\pi\)
\(41\)	50.2823			1.22640			0.613199	−	0.789929i	\(-0.289883\pi\)
							0.613199	+	0.789929i	\(0.289883\pi\)
\(43\)	−39.5246	+	39.5246i	−0.919178	+	0.919178i	−0.996970	−	0.0777918i	\(-0.975213\pi\)
							0.0777918	+	0.996970i	\(0.475213\pi\)
\(47\)	−0.928232	+	3.46421i	−0.0197496	+	0.0737066i	−0.975097	−	0.221777i	\(-0.928814\pi\)
							0.955348	+	0.295484i	\(0.0954809\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.w.a.17.3	✓	64
3.2	odd	2		210.3.w.b.17.1	yes	64
5.3	odd	4		210.3.w.b.143.6	yes	64
7.5	odd	6	inner	210.3.w.a.47.8	yes	64
15.8	even	4	inner	210.3.w.a.143.8	yes	64
21.5	even	6		210.3.w.b.47.6	yes	64
35.33	even	12		210.3.w.b.173.1	yes	64
105.68	odd	12	inner	210.3.w.a.173.3	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.w.a.17.3	✓	64	1.1	even	1	trivial
210.3.w.a.47.8	yes	64	7.5	odd	6	inner
210.3.w.a.143.8	yes	64	15.8	even	4	inner
210.3.w.a.173.3	yes	64	105.68	odd	12	inner
210.3.w.b.17.1	yes	64	3.2	odd	2
210.3.w.b.47.6	yes	64	21.5	even	6
210.3.w.b.143.6	yes	64	5.3	odd	4
210.3.w.b.173.1	yes	64	35.33	even	12