Embedded newform 210.3.v.b.193.7

• Label: 210.3.v.b.193.7
• Level: $210$
• Weight: $3$
• Character: 210.193
• 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			193.7
Character	\(\chi\)	\(=\)	210.193
Dual form			210.3.v.b.37.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36603 + 0.366025i) q^{2} +(1.67303 - 0.448288i) q^{3} +(1.73205 + 1.00000i) q^{4} +(3.25594 + 3.79459i) q^{5} +2.44949 q^{6} +(-2.26902 + 6.62205i) q^{7} +(2.00000 + 2.00000i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 16 q^{2} - 8 q^{5} + 24 q^{7} + 64 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.36603	+	0.366025i	0.683013	+	0.183013i
							\(3\)	1.67303	−	0.448288i	0.557678	−	0.149429i
\(5\)	3.25594	+	3.79459i	0.651187	+	0.758917i
							\(7\)	−2.26902	+	6.62205i	−0.324145	+	0.946007i
\(11\)	−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	0.260744	−	0.965408i	\(-0.416032\pi\)
							−0.965408	+	0.260744i	\(0.916032\pi\)
\(17\)	0.724383	+	2.70344i	0.0426108	+	0.159026i	0.983953	−	0.178428i	\(-0.0571011\pi\)
							−0.941342	+	0.337453i	\(0.890434\pi\)
\(19\)	16.5863	−	9.57608i	0.872961	−	0.504004i	0.00462986	−	0.999989i	\(-0.498526\pi\)
							0.868331	+	0.495985i	\(0.165193\pi\)
\(23\)	9.36480	−	34.9499i	0.407165	−	1.51956i	−0.392862	−	0.919597i	\(-0.628515\pi\)
							0.800027	−	0.599964i	\(-0.204818\pi\)
\(29\)		−	12.4700i		−	0.429999i	−0.976614	−	0.215000i	\(-0.931025\pi\)
							0.976614	−	0.215000i	\(-0.0689750\pi\)
\(31\)	5.33782	−	9.24538i	0.172188	−	0.298238i	−0.766997	−	0.641651i	\(-0.778250\pi\)
							0.939184	+	0.343413i	\(0.111583\pi\)
\(37\)	21.9577	+	5.88355i	0.593452	+	0.159015i	0.543030	−	0.839713i	\(-0.317277\pi\)
							0.0504215	+	0.998728i	\(0.483944\pi\)
\(41\)	−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	0.737869	−	0.674944i	\(-0.235832\pi\)
							−0.674944	+	0.737869i	\(0.735832\pi\)
\(47\)	−71.7245	−	19.2185i	−1.52605	−	0.408905i	−0.604324	−	0.796739i	\(-0.706557\pi\)
							−0.921729	+	0.387834i	\(0.873223\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.v.b.37.7	✓	32	35.2	odd	12	inner
210.3.v.b.67.3	yes	32	5.2	odd	4	inner
210.3.v.b.163.3	yes	32	7.2	even	3	inner
210.3.v.b.193.7	yes	32	1.1	even	1	trivial