Embedded newform 210.3.v.a.37.8

• Label: 210.3.v.a.37.8
• 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			37.8
Character	\(\chi\)	\(=\)	210.37
Dual form			210.3.v.a.193.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36603 + 0.366025i) q^{2} +(1.67303 + 0.448288i) q^{3} +(1.73205 - 1.00000i) q^{4} +(4.87077 - 1.12941i) q^{5} -2.44949 q^{6} +(6.92053 - 1.05182i) q^{7} +(-2.00000 + 2.00000i) q^{8} +(2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 16 q^{2} - 8 q^{7} - 64 q^{8}+O(q^{10}) \)

Character values

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

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

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
\(5\)	4.87077	−	1.12941i	0.974155	−	0.225882i
							\(7\)	6.92053	−	1.05182i	0.988647	−	0.150260i
\(11\)	−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	−0.739531	−	0.673122i	\(-0.764953\pi\)
							0.673122	+	0.739531i	\(0.264953\pi\)
\(17\)	0.241886	−	0.902729i	0.0142286	−	0.0531017i	−0.958447	−	0.285272i	\(-0.907916\pi\)
							0.972675	+	0.232170i	\(0.0745827\pi\)
\(19\)	5.84532	+	3.37480i	0.307648	+	0.177621i	0.645874	−	0.763444i	\(-0.276493\pi\)
							−0.338225	+	0.941065i	\(0.609827\pi\)
\(23\)	−3.90901	−	14.5886i	−0.169957	−	0.634287i	−0.997356	−	0.0726729i	\(-0.976847\pi\)
							0.827399	−	0.561615i	\(-0.189820\pi\)
\(29\)			31.3119i			1.07972i	0.841754	+	0.539861i	\(0.181523\pi\)
							−0.841754	+	0.539861i	\(0.818477\pi\)
\(31\)	23.4627	+	40.6387i	0.756863	+	1.31092i	0.944443	+	0.328675i	\(0.106602\pi\)
							−0.187580	+	0.982249i	\(0.560064\pi\)
\(37\)	36.5911	−	9.80457i	0.988950	−	0.264988i	0.272141	−	0.962258i	\(-0.412268\pi\)
							0.716809	+	0.697269i	\(0.245602\pi\)
\(41\)	−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	−0.469827	−	0.882758i	\(-0.655684\pi\)
							0.882758	+	0.469827i	\(0.155684\pi\)
\(47\)	48.1749	−	12.9084i	1.02500	−	0.274647i	0.293114	−	0.956077i	\(-0.405308\pi\)
							0.731883	+	0.681430i	\(0.238642\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.v.a.37.8	✓	32	1.1	even	1	trivial
210.3.v.a.67.2	yes	32	7.4	even	3	inner
210.3.v.a.163.2	yes	32	5.3	odd	4	inner
210.3.v.a.193.8	yes	32	35.18	odd	12	inner