Embedded newform 210.3.v.a.163.2

• Label: 210.3.v.a.163.2
• 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			163.2
Character	\(\chi\)	\(=\)	210.163
Dual form			210.3.v.a.67.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.366025 + 1.36603i) q^{2} +(-0.448288 + 1.67303i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(-1.45729 - 4.78292i) q^{5} -2.44949 q^{6} +(-1.05182 - 6.92053i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(-2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 16 q^{2} - 8 q^{7} - 64 q^{8}+O(q^{10}) \)

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
\(5\)	−1.45729	−	4.78292i	−0.291457	−	0.956584i
							\(7\)	−1.05182	−	6.92053i	−0.150260	−	0.988647i
\(11\)	−6.50271	−	11.2630i	−0.591156	−	1.02391i								−0.994077	−	0.108677i	\(-0.965339\pi\)
							0.402921	−	0.915235i	\(-0.367995\pi\)
\(13\)	−0.863323	−	0.863323i	−0.0664095	−	0.0664095i	0.673122	−	0.739531i	\(-0.264953\pi\)
							−0.739531	+	0.673122i	\(0.764953\pi\)
\(17\)	−0.902729	−	0.241886i	−0.0531017	−	0.0142286i	0.232170	−	0.972675i	\(-0.425417\pi\)
							−0.285272	+	0.958447i	\(0.592084\pi\)
\(19\)	−5.84532	−	3.37480i	−0.307648	−	0.177621i	0.338225	−	0.941065i	\(-0.390173\pi\)
							−0.645874	+	0.763444i	\(0.723507\pi\)
\(23\)	14.5886	−	3.90901i	0.634287	−	0.169957i	0.0726729	−	0.997356i	\(-0.476847\pi\)
							0.561615	+	0.827399i	\(0.310180\pi\)
\(29\)		−	31.3119i		−	1.07972i	−0.841754	−	0.539861i	\(-0.818477\pi\)
							0.841754	−	0.539861i	\(-0.181523\pi\)
\(31\)	23.4627	+	40.6387i	0.756863	+	1.31092i	0.944443	+	0.328675i	\(0.106602\pi\)
							−0.187580	+	0.982249i	\(0.560064\pi\)
\(37\)	−9.80457	−	36.5911i	−0.264988	−	0.988950i	−0.962258	−	0.272141i	\(-0.912268\pi\)
							0.697269	−	0.716809i	\(-0.254398\pi\)
\(41\)	−64.3783			−1.57020			−0.785101	−	0.619368i	\(-0.787389\pi\)
							−0.785101	+	0.619368i	\(0.787389\pi\)
\(43\)	17.7560	+	17.7560i	0.412931	+	0.412931i	0.882758	−	0.469827i	\(-0.155684\pi\)
							−0.469827	+	0.882758i	\(0.655684\pi\)
\(47\)	−12.9084	−	48.1749i	−0.274647	−	1.02500i	−0.956077	−	0.293114i	\(-0.905308\pi\)
							0.681430	−	0.731883i	\(-0.261358\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.v.a.163.2	yes	32
5.2	odd	4	inner	210.3.v.a.37.8	✓	32
7.4	even	3	inner	210.3.v.a.193.8	yes	32
35.32	odd	12	inner	210.3.v.a.67.2	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.v.a.37.8	✓	32	5.2	odd	4	inner
210.3.v.a.67.2	yes	32	35.32	odd	12	inner
210.3.v.a.163.2	yes	32	1.1	even	1	trivial
210.3.v.a.193.8	yes	32	7.4	even	3	inner