Embedded newform 210.3.v.b.37.2

• Label: 210.3.v.b.37.2
• 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			37.2
Character	\(\chi\)	\(=\)	210.37
Dual form			210.3.v.b.193.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36603 - 0.366025i) q^{2} +(-1.67303 - 0.448288i) q^{3} +(1.73205 - 1.00000i) q^{4} +(-4.19534 + 2.72013i) q^{5} -2.44949 q^{6} +(-1.84650 + 6.75207i) q^{7} +(2.00000 - 2.00000i) q^{8} +(2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 16 q^{2} - 8 q^{5} + 24 q^{7} + 64 q^{8}+O(q^{10}) \)

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
\(5\)	−4.19534	+	2.72013i	−0.839069	+	0.544025i
							\(7\)	−1.84650	+	6.75207i	−0.263786	+	0.964581i
\(11\)	7.12207	+	12.3358i	0.647461	+	1.12144i								0.983727	+	0.179669i	\(0.0575025\pi\)
							−0.336266	+	0.941767i	\(0.609164\pi\)
\(13\)	−2.75603	+	2.75603i	−0.212002	+	0.212002i	−0.805118	−	0.593115i	\(-0.797898\pi\)
							0.593115	+	0.805118i	\(0.297898\pi\)
\(17\)	−6.41754	+	23.9506i	−0.377502	+	1.40886i	0.472152	+	0.881517i	\(0.343477\pi\)
							−0.849654	+	0.527341i	\(0.823189\pi\)
\(19\)	0.277055	+	0.159958i	0.0145818	+	0.00841882i	0.507273	−	0.861785i	\(-0.330654\pi\)
							−0.492691	+	0.870204i	\(0.663987\pi\)
\(23\)	−3.64915	−	13.6188i	−0.158659	−	0.592122i	−0.998764	−	0.0496991i	\(-0.984174\pi\)
							0.840106	−	0.542423i	\(-0.182493\pi\)
\(29\)			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	0.716549	−	0.697537i	\(-0.245721\pi\)
							−0.962359	+	0.271781i	\(0.912387\pi\)
\(37\)	−2.28156	+	0.611341i	−0.0616637	+	0.0165227i	−0.289519	−	0.957172i	\(-0.593495\pi\)
							0.227855	+	0.973695i	\(0.426829\pi\)
\(41\)	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	\(0.311845\pi\)
\(47\)	−81.7602	+	21.9076i	−1.73958	+	0.466119i	−0.982353	−	0.187035i	\(-0.940112\pi\)
							−0.757225	+	0.653154i	\(0.773445\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.v.b.37.2	✓	32	1.1	even	1	trivial
210.3.v.b.67.6	yes	32	7.4	even	3	inner
210.3.v.b.163.6	yes	32	5.3	odd	4	inner
210.3.v.b.193.2	yes	32	35.18	odd	12	inner