Embedded newform 210.3.v.a.163.5

• Label: 210.3.v.a.163.5
• Level: $210$
• Weight: $3$
• Character: 210.163
• Analytic conductor: $5.722$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(5.72208555157\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			163.5
Character	\(\chi\)	\(=\)	210.163
Dual form			210.3.v.a.67.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.366025 + 1.36603i) q^{2} +(0.448288 - 1.67303i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(-4.09442 + 2.86979i) q^{5} +2.44949 q^{6} +(-3.46105 - 6.08450i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(-2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 16 q^{2} - 8 q^{7} - 64 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.366025	+	1.36603i	0.183013	+	0.683013i
							\(3\)	0.448288	−	1.67303i	0.149429	−	0.557678i
\(5\)	−4.09442	+	2.86979i	−0.818884	+	0.573959i
							\(7\)	−3.46105	−	6.08450i	−0.494436	−	0.869214i
\(11\)	−5.62423	−	9.74146i	−0.511294	−	0.885587i								−0.999914	−	0.0130905i	\(-0.995833\pi\)
							0.488620	−	0.872496i	\(-0.337500\pi\)
\(13\)	−5.84142	−	5.84142i	−0.449340	−	0.449340i	0.445795	−	0.895135i	\(-0.352921\pi\)
							−0.895135	+	0.445795i	\(0.852921\pi\)
\(17\)	24.8241	+	6.65160i	1.46024	+	0.391270i	0.899573	−	0.436770i	\(-0.143878\pi\)
							0.560668	+	0.828041i	\(0.310544\pi\)
\(19\)	−27.5095	−	15.8826i	−1.44787	−	0.835926i	−0.449513	−	0.893274i	\(-0.648402\pi\)
							−0.998354	+	0.0573477i	\(0.981736\pi\)
\(23\)	−39.8193	+	10.6696i	−1.73127	+	0.463894i	−0.980476	−	0.196638i	\(-0.936998\pi\)
							−0.750798	+	0.660532i	\(0.770331\pi\)
\(29\)		−	8.29704i		−	0.286105i	−0.989715	−	0.143052i	\(-0.954308\pi\)
							0.989715	−	0.143052i	\(-0.0456917\pi\)
\(31\)	6.00177	+	10.3954i	0.193605	+	0.335334i	0.946442	−	0.322873i	\(-0.104649\pi\)
							−0.752837	+	0.658207i	\(0.771315\pi\)
\(37\)	6.32737	+	23.6141i	0.171010	+	0.638218i	0.997197	+	0.0748217i	\(0.0238388\pi\)
							−0.826187	+	0.563396i	\(0.809495\pi\)
\(41\)	42.9567			1.04772			0.523862	−	0.851803i	\(-0.324491\pi\)
							0.523862	+	0.851803i	\(0.324491\pi\)
\(43\)	−37.4962	−	37.4962i	−0.872005	−	0.872005i	0.120685	−	0.992691i	\(-0.461491\pi\)
							−0.992691	+	0.120685i	\(0.961491\pi\)
\(47\)	9.60246	+	35.8369i	0.204308	+	0.762487i	0.989659	+	0.143437i	\(0.0458154\pi\)
							−0.785352	+	0.619050i	\(0.787518\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.v.a.163.5	yes	32
5.2	odd	4	inner	210.3.v.a.37.3	✓	32
7.4	even	3	inner	210.3.v.a.193.3	yes	32
35.32	odd	12	inner	210.3.v.a.67.5	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.v.a.37.3	✓	32	5.2	odd	4	inner
210.3.v.a.67.5	yes	32	35.32	odd	12	inner
210.3.v.a.163.5	yes	32	1.1	even	1	trivial
210.3.v.a.193.3	yes	32	7.4	even	3	inner