Embedded newform 210.3.v.b.37.3

• Label: 210.3.v.b.37.3
• Level: $210$
• Weight: $3$
• Character: 210.37
• 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			37.3
Character	\(\chi\)	\(=\)	210.37
Dual form			210.3.v.b.193.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36603 - 0.366025i) q^{2} +(-1.67303 - 0.448288i) q^{3} +(1.73205 - 1.00000i) q^{4} +(2.17130 + 4.50394i) q^{5} -2.44949 q^{6} +(1.63812 - 6.80563i) q^{7} +(2.00000 - 2.00000i) q^{8} +(2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 16 q^{2} - 8 q^{5} + 24 q^{7} + 64 q^{8}+O(q^{10}) \)

Character values

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

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

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
							\(3\)	−1.67303	−	0.448288i	−0.557678	−	0.149429i
\(5\)	2.17130	+	4.50394i	0.434260	+	0.900787i
							\(7\)	1.63812	−	6.80563i	0.234017	−	0.972233i
\(11\)	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	−0.183293	−	0.983058i	\(-0.558676\pi\)
							0.983058	+	0.183293i	\(0.0586759\pi\)
\(17\)	6.19178	−	23.1080i	0.364222	−	1.35930i	−0.504250	−	0.863558i	\(-0.668231\pi\)
							0.868472	−	0.495738i	\(-0.165102\pi\)
\(19\)	2.70091	+	1.55937i	0.142153	+	0.0820723i	0.569390	−	0.822068i	\(-0.307179\pi\)
							−0.427236	+	0.904140i	\(0.640513\pi\)
\(23\)	9.18722	+	34.2872i	0.399444	+	1.49075i	0.814076	+	0.580758i	\(0.197244\pi\)
							−0.414632	+	0.909989i	\(0.636090\pi\)
\(29\)		−	36.5420i		−	1.26007i	−0.776567	−	0.630034i	\(-0.783041\pi\)
							0.776567	−	0.630034i	\(-0.216959\pi\)
\(31\)	−1.49997	−	2.59803i	−0.0483862	−	0.0838074i	0.840818	−	0.541318i	\(-0.182075\pi\)
							−0.889204	+	0.457511i	\(0.848741\pi\)
\(37\)	0.422582	−	0.113231i	0.0114211	−	0.00306029i	−0.253104	−	0.967439i	\(-0.581451\pi\)
							0.264525	+	0.964379i	\(0.414785\pi\)
\(41\)	−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	−0.995846	−	0.0910583i	\(-0.970975\pi\)
							0.0910583	+	0.995846i	\(0.470975\pi\)
\(47\)	10.7803	−	2.88858i	0.229369	−	0.0614592i	−0.142304	−	0.989823i	\(-0.545451\pi\)
							0.371673	+	0.928364i	\(0.378784\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.v.b.37.3	✓	32	1.1	even	1	trivial
210.3.v.b.67.5	yes	32	7.4	even	3	inner
210.3.v.b.163.5	yes	32	5.3	odd	4	inner
210.3.v.b.193.3	yes	32	35.18	odd	12	inner