Embedded newform 210.3.v.b.67.3

• Label: 210.3.v.b.67.3
• Level: $210$
• Weight: $3$
• Character: 210.67
• 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			67.3
Character	\(\chi\)	\(=\)	210.67
Dual form			210.3.v.b.163.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.366025 + 1.36603i) q^{2} +(-0.448288 - 1.67303i) q^{3} +(-1.73205 - 1.00000i) q^{4} +(1.65824 + 4.71702i) q^{5} +2.44949 q^{6} +(-6.62205 - 2.26902i) q^{7} +(2.00000 - 2.00000i) q^{8} +(-2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 16 q^{2} - 8 q^{5} + 24 q^{7} + 64 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(31\)	\(71\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.366025	+	1.36603i	−0.183013	+	0.683013i
							\(3\)	−0.448288	−	1.67303i	−0.149429	−	0.557678i
\(5\)	1.65824	+	4.71702i	0.331648	+	0.943403i
							\(7\)	−6.62205	−	2.26902i	−0.946007	−	0.324145i
\(11\)	−1.33229	+	2.30759i	−0.121117	+	0.209781i								−0.920208	−	0.391429i	\(-0.871981\pi\)
							0.799092	+	0.601209i	\(0.205314\pi\)
\(13\)	−9.16064	+	9.16064i	−0.704664	+	0.704664i	−0.965408	−	0.260744i	\(-0.916032\pi\)
							0.260744	+	0.965408i	\(0.416032\pi\)
\(17\)	−2.70344	+	0.724383i	−0.159026	+	0.0426108i	−0.337453	−	0.941342i	\(-0.609566\pi\)
							0.178428	+	0.983953i	\(0.442899\pi\)
\(19\)	−16.5863	+	9.57608i	−0.872961	+	0.504004i	−0.868331	−	0.495985i	\(-0.834807\pi\)
							−0.00462986	+	0.999989i	\(0.501474\pi\)
\(23\)	−34.9499	−	9.36480i	−1.51956	−	0.407165i	−0.599964	−	0.800027i	\(-0.704818\pi\)
							−0.919597	+	0.392862i	\(0.871485\pi\)
\(29\)			12.4700i			0.429999i	0.976614	+	0.215000i	\(0.0689750\pi\)
							−0.976614	+	0.215000i	\(0.931025\pi\)
\(31\)	5.33782	−	9.24538i	0.172188	−	0.298238i	−0.766997	−	0.641651i	\(-0.778250\pi\)
							0.939184	+	0.343413i	\(0.111583\pi\)
\(37\)	−5.88355	+	21.9577i	−0.159015	+	0.593452i	0.839713	+	0.543030i	\(0.182723\pi\)
							−0.998728	+	0.0504215i	\(0.983944\pi\)
\(41\)	−1.17994			−0.0287791			−0.0143895	−	0.999896i	\(-0.504580\pi\)
							−0.0143895	+	0.999896i	\(0.504580\pi\)
\(43\)	2.70577	−	2.70577i	0.0629250	−	0.0629250i	−0.674944	−	0.737869i	\(-0.735832\pi\)
							0.737869	+	0.674944i	\(0.235832\pi\)
\(47\)	19.2185	−	71.7245i	0.408905	−	1.52605i	−0.387834	−	0.921729i	\(-0.626777\pi\)
							0.796739	−	0.604324i	\(-0.206557\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.v.b.67.3	yes	32
5.3	odd	4	inner	210.3.v.b.193.7	yes	32
7.2	even	3	inner	210.3.v.b.37.7	✓	32
35.23	odd	12	inner	210.3.v.b.163.3	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.v.b.37.7	✓	32	7.2	even	3	inner
210.3.v.b.67.3	yes	32	1.1	even	1	trivial
210.3.v.b.163.3	yes	32	35.23	odd	12	inner
210.3.v.b.193.7	yes	32	5.3	odd	4	inner