Embedded newform 210.3.v.b.163.1

• Label: 210.3.v.b.163.1
• 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			163.1
Character	\(\chi\)	\(=\)	210.163
Dual form			210.3.v.b.67.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.366025 - 1.36603i) q^{2} +(-0.448288 + 1.67303i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(-3.76724 - 3.28753i) q^{5} +2.44949 q^{6} +(5.65016 + 4.13228i) q^{7} +(2.00000 + 2.00000i) q^{8} +(-2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 16 q^{2} - 8 q^{5} + 24 q^{7} + 64 q^{8}+O(q^{10}) \)

Character values

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

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

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
\(5\)	−3.76724	−	3.28753i	−0.753449	−	0.657507i
							\(7\)	5.65016	+	4.13228i	0.807166	+	0.590325i
\(11\)	2.47138	+	4.28056i	0.224671	+	0.389141i								0.956221	−	0.292647i	\(-0.0945360\pi\)
							−0.731550	+	0.681788i	\(0.761203\pi\)
\(13\)	7.82868	+	7.82868i	0.602206	+	0.602206i	0.940897	−	0.338692i	\(-0.109984\pi\)
							−0.338692	+	0.940897i	\(0.609984\pi\)
\(17\)	3.41115	+	0.914015i	0.200656	+	0.0537656i	0.357747	−	0.933819i	\(-0.383545\pi\)
							−0.157091	+	0.987584i	\(0.550212\pi\)
\(19\)	26.8487	+	15.5011i	1.41309	+	0.815848i	0.995678	−	0.0928696i	\(-0.0296040\pi\)
							0.417412	+	0.908718i	\(0.362937\pi\)
\(23\)	17.2084	−	4.61097i	0.748191	−	0.200477i	0.135475	−	0.990781i	\(-0.456744\pi\)
							0.612715	+	0.790304i	\(0.290077\pi\)
\(29\)			24.0299i			0.828619i	0.910136	+	0.414309i	\(0.135977\pi\)
							−0.910136	+	0.414309i	\(0.864023\pi\)
\(31\)	7.79698	+	13.5048i	0.251515	+	0.435638i	0.963943	−	0.266108i	\(-0.0857377\pi\)
							−0.712428	+	0.701745i	\(0.752404\pi\)
\(37\)	−8.92003	−	33.2900i	−0.241082	−	0.899730i	−0.975312	−	0.220830i	\(-0.929124\pi\)
							0.734231	−	0.678900i	\(-0.237543\pi\)
\(41\)	−19.3822			−0.472736			−0.236368	−	0.971664i	\(-0.575957\pi\)
							−0.236368	+	0.971664i	\(0.575957\pi\)
\(43\)	−11.5955	−	11.5955i	−0.269662	−	0.269662i	0.559302	−	0.828964i	\(-0.311069\pi\)
							−0.828964	+	0.559302i	\(0.811069\pi\)
\(47\)	8.13382	+	30.3558i	0.173060	+	0.645869i	0.996874	+	0.0790084i	\(0.0251754\pi\)
							−0.823814	+	0.566860i	\(0.808158\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.v.b.163.1	yes	32
5.2	odd	4	inner	210.3.v.b.37.8	✓	32
7.4	even	3	inner	210.3.v.b.193.8	yes	32
35.32	odd	12	inner	210.3.v.b.67.1	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.v.b.37.8	✓	32	5.2	odd	4	inner
210.3.v.b.67.1	yes	32	35.32	odd	12	inner
210.3.v.b.163.1	yes	32	1.1	even	1	trivial
210.3.v.b.193.8	yes	32	7.4	even	3	inner