Embedded newform 210.3.v.b.37.8

• Label: 210.3.v.b.37.8
• 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			37.8
Character	\(\chi\)	\(=\)	210.37
Dual form			210.3.v.b.193.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36603 - 0.366025i) q^{2} +(1.67303 + 0.448288i) q^{3} +(1.73205 - 1.00000i) q^{4} +(4.73071 + 1.61876i) q^{5} +2.44949 q^{6} +(-4.13228 + 5.65016i) q^{7} +(2.00000 - 2.00000i) q^{8} +(2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 16 q^{2} - 8 q^{5} + 24 q^{7} + 64 q^{8}+O(q^{10}) \)

Character values

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

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

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
\(5\)	4.73071	+	1.61876i	0.946142	+	0.323752i
							\(7\)	−4.13228	+	5.65016i	−0.590325	+	0.807166i
\(11\)	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	−0.338692	−	0.940897i	\(-0.609984\pi\)
							0.940897	+	0.338692i	\(0.109984\pi\)
\(17\)	−0.914015	+	3.41115i	−0.0537656	+	0.200656i	−0.987584	−	0.157091i	\(-0.949788\pi\)
							0.933819	+	0.357747i	\(0.116455\pi\)
\(19\)	−26.8487	−	15.5011i	−1.41309	−	0.815848i	−0.417412	−	0.908718i	\(-0.637063\pi\)
							−0.995678	+	0.0928696i	\(0.970396\pi\)
\(23\)	−4.61097	−	17.2084i	−0.200477	−	0.748191i	−0.990781	−	0.135475i	\(-0.956744\pi\)
							0.790304	−	0.612715i	\(-0.209923\pi\)
\(29\)		−	24.0299i		−	0.828619i	−0.910136	−	0.414309i	\(-0.864023\pi\)
							0.910136	−	0.414309i	\(-0.135977\pi\)
\(31\)	7.79698	+	13.5048i	0.251515	+	0.435638i	0.963943	−	0.266108i	\(-0.0857377\pi\)
							−0.712428	+	0.701745i	\(0.752404\pi\)
\(37\)	33.2900	−	8.92003i	0.899730	−	0.241082i	0.220830	−	0.975312i	\(-0.429124\pi\)
							0.678900	+	0.734231i	\(0.262457\pi\)
\(41\)	−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	−0.828964	−	0.559302i	\(-0.811069\pi\)
							0.559302	+	0.828964i	\(0.311069\pi\)
\(47\)	−30.3558	+	8.13382i	−0.645869	+	0.173060i	−0.566860	−	0.823814i	\(-0.691842\pi\)
							−0.0790084	+	0.996874i	\(0.525175\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.v.b.37.8	✓	32	1.1	even	1	trivial
210.3.v.b.67.1	yes	32	7.4	even	3	inner
210.3.v.b.163.1	yes	32	5.3	odd	4	inner
210.3.v.b.193.8	yes	32	35.18	odd	12	inner