Embedded newform 210.3.v.b.67.6

• Label: 210.3.v.b.67.6
• 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			67.6
Character	\(\chi\)	\(=\)	210.67
Dual form			210.3.v.b.163.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.366025 + 1.36603i) q^{2} +(0.448288 + 1.67303i) q^{3} +(-1.73205 - 1.00000i) q^{4} +(-0.258026 - 4.99334i) q^{5} -2.44949 q^{6} +(6.75207 - 1.84650i) q^{7} +(2.00000 - 2.00000i) q^{8} +(-2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 16 q^{2} - 8 q^{5} + 24 q^{7} + 64 q^{8}+O(q^{10}) \)

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
\(5\)	−0.258026	−	4.99334i	−0.0516051	−	0.998668i
							\(7\)	6.75207	−	1.84650i	0.964581	−	0.263786i
\(11\)	7.12207	−	12.3358i	0.647461	−	1.12144i								−0.336266	−	0.941767i	\(-0.609164\pi\)
							0.983727	−	0.179669i	\(-0.0575025\pi\)
\(13\)	−2.75603	+	2.75603i	−0.212002	+	0.212002i	−0.805118	−	0.593115i	\(-0.797898\pi\)
							0.593115	+	0.805118i	\(0.297898\pi\)
\(17\)	23.9506	−	6.41754i	1.40886	−	0.377502i	0.527341	−	0.849654i	\(-0.323189\pi\)
							0.881517	+	0.472152i	\(0.156523\pi\)
\(19\)	−0.277055	+	0.159958i	−0.0145818	+	0.00841882i	−0.507273	−	0.861785i	\(-0.669346\pi\)
							0.492691	+	0.870204i	\(0.336013\pi\)
\(23\)	13.6188	+	3.64915i	0.592122	+	0.158659i	0.542423	−	0.840106i	\(-0.317507\pi\)
							0.0496991	+	0.998764i	\(0.484174\pi\)
\(29\)			24.2642i			0.836698i	0.908286	+	0.418349i	\(0.137391\pi\)
							−0.908286	+	0.418349i	\(0.862609\pi\)
\(31\)	−7.62013	+	13.1985i	−0.245811	+	0.425757i	−0.962359	−	0.271781i	\(-0.912387\pi\)
							0.716549	+	0.697537i	\(0.245721\pi\)
\(37\)	0.611341	−	2.28156i	0.0165227	−	0.0616637i	−0.957172	−	0.289519i	\(-0.906505\pi\)
							0.973695	+	0.227855i	\(0.0731713\pi\)
\(41\)	29.0794			0.709254			0.354627	−	0.935008i	\(-0.384608\pi\)
							0.354627	+	0.935008i	\(0.384608\pi\)
\(43\)	−11.7409	+	11.7409i	−0.273045	+	0.273045i	−0.830325	−	0.557280i	\(-0.811845\pi\)
							0.557280	+	0.830325i	\(0.311845\pi\)
\(47\)	21.9076	−	81.7602i	0.466119	−	1.73958i	−0.187035	−	0.982353i	\(-0.559888\pi\)
							0.653154	−	0.757225i	\(-0.273445\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.v.b.67.6	yes	32
5.3	odd	4	inner	210.3.v.b.193.2	yes	32
7.2	even	3	inner	210.3.v.b.37.2	✓	32
35.23	odd	12	inner	210.3.v.b.163.6	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.v.b.37.2	✓	32	7.2	even	3	inner
210.3.v.b.67.6	yes	32	1.1	even	1	trivial
210.3.v.b.163.6	yes	32	35.23	odd	12	inner
210.3.v.b.193.2	yes	32	5.3	odd	4	inner