Embedded newform 210.3.w.a.173.3

• Label: 210.3.w.a.173.3
• Level: $210$
• Weight: $3$
• Character: 210.173
• 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			173.3
Character	\(\chi\)	\(=\)	210.173
Dual form			210.3.w.a.17.3

$q$-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 \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(71\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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								−0.296204	−	0.955125i	\(-0.595721\pi\)
							0.679060	+	0.734083i	\(0.262388\pi\)
\(13\)	13.3745	−	13.3745i	1.02881	−	1.02881i	0.0292348	−	0.999573i	\(-0.490693\pi\)
							0.999573	−	0.0292348i	\(-0.00930704\pi\)
\(17\)	−25.3015	−	6.77952i	−1.48832	−	0.398795i	−0.579153	−	0.815219i	\(-0.696617\pi\)
							−0.909171	+	0.416424i	\(0.863283\pi\)
\(19\)	6.44901	−	11.1700i	0.339422	−	0.587896i	−0.644902	−	0.764265i	\(-0.723102\pi\)
							0.984324	+	0.176369i	\(0.0564353\pi\)
\(23\)	−5.04836	−	18.8407i	−0.219494	−	0.819162i	−0.984536	−	0.175182i	\(-0.943949\pi\)
							0.765042	−	0.643980i	\(-0.222718\pi\)
\(29\)	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	0.0339672	−	0.999423i	\(-0.489186\pi\)
							0.882509	+	0.470295i	\(0.155852\pi\)
\(37\)	16.1378	+	60.2271i	0.436157	+	1.62776i	0.738283	+	0.674491i	\(0.235637\pi\)
							−0.302127	+	0.953268i	\(0.597697\pi\)
\(41\)	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	0.0777918	−	0.996970i	\(-0.475213\pi\)
							−0.996970	+	0.0777918i	\(0.975213\pi\)
\(47\)	−0.928232	−	3.46421i	−0.0197496	−	0.0737066i	0.955348	−	0.295484i	\(-0.0954809\pi\)
							−0.975097	+	0.221777i	\(0.928814\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.w.a.173.3	yes	64
3.2	odd	2		210.3.w.b.173.1	yes	64
5.2	odd	4		210.3.w.b.47.6	yes	64
7.3	odd	6	inner	210.3.w.a.143.8	yes	64
15.2	even	4	inner	210.3.w.a.47.8	yes	64
21.17	even	6		210.3.w.b.143.6	yes	64
35.17	even	12		210.3.w.b.17.1	yes	64
105.17	odd	12	inner	210.3.w.a.17.3	✓	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.w.a.17.3	✓	64	105.17	odd	12	inner
210.3.w.a.47.8	yes	64	15.2	even	4	inner
210.3.w.a.143.8	yes	64	7.3	odd	6	inner
210.3.w.a.173.3	yes	64	1.1	even	1	trivial
210.3.w.b.17.1	yes	64	35.17	even	12
210.3.w.b.47.6	yes	64	5.2	odd	4
210.3.w.b.143.6	yes	64	21.17	even	6
210.3.w.b.173.1	yes	64	3.2	odd	2