Embedded newform 210.3.v.a.163.6

• Label: 210.3.v.a.163.6
• 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			163.6
Character	\(\chi\)	\(=\)	210.163
Dual form			210.3.v.a.67.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.366025 + 1.36603i) q^{2} +(0.448288 - 1.67303i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(-2.17682 - 4.50127i) q^{5} +2.44949 q^{6} +(-6.19254 + 3.26382i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(-2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 16 q^{2} - 8 q^{7} - 64 q^{8}+O(q^{10}) \)

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
\(5\)	−2.17682	−	4.50127i	−0.435365	−	0.900254i
							\(7\)	−6.19254	+	3.26382i	−0.884648	+	0.466260i
\(11\)	−0.187724	−	0.325147i	−0.0170658	−	0.0295588i								0.857366	−	0.514707i	\(-0.172099\pi\)
							−0.874432	+	0.485148i	\(0.838766\pi\)
\(13\)	−9.32144	−	9.32144i	−0.717034	−	0.717034i	0.250963	−	0.967997i	\(-0.419253\pi\)
							−0.967997	+	0.250963i	\(0.919253\pi\)
\(17\)	−29.7515	−	7.97190i	−1.75009	−	0.468935i	−0.765445	−	0.643501i	\(-0.777481\pi\)
							−0.984646	+	0.174566i	\(0.944148\pi\)
\(19\)	−6.46747	−	3.73399i	−0.340393	−	0.196526i	0.320053	−	0.947400i	\(-0.396299\pi\)
							−0.660446	+	0.750874i	\(0.729633\pi\)
\(23\)	38.4418	−	10.3004i	1.67138	−	0.447845i	0.705899	−	0.708312i	\(-0.250543\pi\)
							0.965483	+	0.260467i	\(0.0838764\pi\)
\(29\)			22.0507i			0.760370i	0.924911	+	0.380185i	\(0.124140\pi\)
							−0.924911	+	0.380185i	\(0.875860\pi\)
\(31\)	−23.7274	−	41.0971i	−0.765400	−	1.32571i	−0.940035	−	0.341078i	\(-0.889208\pi\)
							0.174635	−	0.984633i	\(-0.444125\pi\)
\(37\)	1.58305	+	5.90803i	0.0427852	+	0.159676i	0.984013	−	0.178095i	\(-0.0569935\pi\)
							−0.941228	+	0.337772i	\(0.890327\pi\)
\(41\)	78.1193			1.90535			0.952675	−	0.303992i	\(-0.0983197\pi\)
							0.952675	+	0.303992i	\(0.0983197\pi\)
\(43\)	−12.2703	−	12.2703i	−0.285357	−	0.285357i	0.549884	−	0.835241i	\(-0.314672\pi\)
							−0.835241	+	0.549884i	\(0.814672\pi\)
\(47\)	3.06391	+	11.4346i	0.0651895	+	0.243290i	0.990830	−	0.135113i	\(-0.0431398\pi\)
							−0.925641	+	0.378404i	\(0.876473\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.v.a.163.6	yes	32
5.2	odd	4	inner	210.3.v.a.37.4	✓	32
7.4	even	3	inner	210.3.v.a.193.4	yes	32
35.32	odd	12	inner	210.3.v.a.67.6	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.v.a.37.4	✓	32	5.2	odd	4	inner
210.3.v.a.67.6	yes	32	35.32	odd	12	inner
210.3.v.a.163.6	yes	32	1.1	even	1	trivial
210.3.v.a.193.4	yes	32	7.4	even	3	inner