Embedded newform 210.3.v.b.163.5

• Label: 210.3.v.b.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.b.67.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.366025 - 1.36603i) q^{2} +(0.448288 - 1.67303i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(-4.98617 + 0.371566i) q^{5} -2.44949 q^{6} +(-6.80563 - 1.63812i) q^{7} +(2.00000 + 2.00000i) q^{8} +(-2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 16 q^{2} - 8 q^{5} + 24 q^{7} + 64 q^{8}+O(q^{10}) \)

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
\(5\)	−4.98617	+	0.371566i	−0.997235	+	0.0743132i
							\(7\)	−6.80563	−	1.63812i	−0.972233	−	0.234017i
\(11\)	9.66984	+	16.7487i	0.879077	+	1.52261i								0.852356	+	0.522962i	\(0.175173\pi\)
							0.0267208	+	0.999643i	\(0.491493\pi\)
\(13\)	10.3969	+	10.3969i	0.799765	+	0.799765i	0.983058	−	0.183293i	\(-0.0586759\pi\)
							−0.183293	+	0.983058i	\(0.558676\pi\)
\(17\)	−23.1080	−	6.19178i	−1.35930	−	0.364222i	−0.495738	−	0.868472i	\(-0.665102\pi\)
							−0.863558	+	0.504250i	\(0.831769\pi\)
\(19\)	−2.70091	−	1.55937i	−0.142153	−	0.0820723i	0.427236	−	0.904140i	\(-0.359487\pi\)
							−0.569390	+	0.822068i	\(0.692821\pi\)
\(23\)	−34.2872	+	9.18722i	−1.49075	+	0.399444i	−0.909989	−	0.414632i	\(-0.863910\pi\)
							−0.580758	+	0.814076i	\(0.697244\pi\)
\(29\)			36.5420i			1.26007i	0.776567	+	0.630034i	\(0.216959\pi\)
							−0.776567	+	0.630034i	\(0.783041\pi\)
\(31\)	−1.49997	−	2.59803i	−0.0483862	−	0.0838074i	0.840818	−	0.541318i	\(-0.182075\pi\)
							−0.889204	+	0.457511i	\(0.848741\pi\)
\(37\)	−0.113231	−	0.422582i	−0.00306029	−	0.0114211i	0.964379	−	0.264525i	\(-0.0852152\pi\)
							−0.967439	+	0.253104i	\(0.918549\pi\)
\(41\)	−64.3827			−1.57031			−0.785155	−	0.619299i	\(-0.787417\pi\)
							−0.785155	+	0.619299i	\(0.787417\pi\)
\(43\)	−38.9059	−	38.9059i	−0.904787	−	0.904787i	0.0910583	−	0.995846i	\(-0.470975\pi\)
							−0.995846	+	0.0910583i	\(0.970975\pi\)
\(47\)	−2.88858	−	10.7803i	−0.0614592	−	0.229369i	0.928364	−	0.371673i	\(-0.121216\pi\)
							−0.989823	+	0.142304i	\(0.954549\pi\)

(See \(a_n\) instead)

Twists

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

    

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