Embedded newform 210.3.v.a.37.5

• Label: 210.3.v.a.37.5
• 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			37.5
Character	\(\chi\)	\(=\)	210.37
Dual form			210.3.v.a.193.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36603 + 0.366025i) q^{2} +(1.67303 + 0.448288i) q^{3} +(1.73205 - 1.00000i) q^{4} +(-4.60928 + 1.93765i) q^{5} -2.44949 q^{6} +(-1.45688 - 6.84672i) q^{7} +(-2.00000 + 2.00000i) q^{8} +(2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 16 q^{2} - 8 q^{7} - 64 q^{8}+O(q^{10}) \)

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
\(5\)	−4.60928	+	1.93765i	−0.921857	+	0.387531i
							\(7\)	−1.45688	−	6.84672i	−0.208125	−	0.978102i
\(11\)	−6.59836	−	11.4287i	−0.599851	−	1.03897i								−0.992843	−	0.119430i	\(-0.961893\pi\)
							0.392992	−	0.919542i	\(-0.371440\pi\)
\(13\)	10.7755	−	10.7755i	0.828886	−	0.828886i	−0.158476	−	0.987363i	\(-0.550658\pi\)
							0.987363	+	0.158476i	\(0.0506582\pi\)
\(17\)	2.10467	−	7.85475i	0.123804	−	0.462044i	−0.875990	−	0.482329i	\(-0.839791\pi\)
							0.999794	+	0.0202855i	\(0.00645751\pi\)
\(19\)	−8.35848	−	4.82577i	−0.439920	−	0.253988i	0.263644	−	0.964620i	\(-0.415076\pi\)
							−0.703564	+	0.710632i	\(0.748409\pi\)
\(23\)	6.06684	+	22.6417i	0.263776	+	0.984424i	0.962996	+	0.269517i	\(0.0868641\pi\)
							−0.699220	+	0.714907i	\(0.746469\pi\)
\(29\)		−	41.4745i		−	1.43015i	−0.699046	−	0.715077i	\(-0.746392\pi\)
							0.699046	−	0.715077i	\(-0.253608\pi\)
\(31\)	−18.0882	−	31.3297i	−0.583490	−	1.01063i	−0.995062	−	0.0992569i	\(-0.968353\pi\)
							0.411572	−	0.911377i	\(-0.364980\pi\)
\(37\)	−4.68858	+	1.25630i	−0.126718	+	0.0339541i	−0.321621	−	0.946869i	\(-0.604228\pi\)
							0.194902	+	0.980823i	\(0.437561\pi\)
\(41\)	−7.31247			−0.178353			−0.0891764	−	0.996016i	\(-0.528424\pi\)
							−0.0891764	+	0.996016i	\(0.528424\pi\)
\(43\)	48.7061	−	48.7061i	1.13270	−	1.13270i	0.142972	−	0.989727i	\(-0.454334\pi\)
							0.989727	−	0.142972i	\(-0.0456660\pi\)
\(47\)	−51.9641	+	13.9237i	−1.10562	+	0.296250i	−0.765051	−	0.643970i	\(-0.777286\pi\)
							−0.340568	+	0.940220i	\(0.610619\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.v.a.37.5	✓	32
5.3	odd	4	inner	210.3.v.a.163.3	yes	32
7.4	even	3	inner	210.3.v.a.67.3	yes	32
35.18	odd	12	inner	210.3.v.a.193.5	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.v.a.37.5	✓	32	1.1	even	1	trivial
210.3.v.a.67.3	yes	32	7.4	even	3	inner
210.3.v.a.163.3	yes	32	5.3	odd	4	inner
210.3.v.a.193.5	yes	32	35.18	odd	12	inner