Embedded newform 210.3.v.a.37.6

• Label: 210.3.v.a.37.6
• 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			37.6
Character	\(\chi\)	\(=\)	210.37
Dual form			210.3.v.a.193.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36603 + 0.366025i) q^{2} +(1.67303 + 0.448288i) q^{3} +(1.73205 - 1.00000i) q^{4} +(-2.69832 - 4.20940i) q^{5} -2.44949 q^{6} +(1.39786 + 6.85901i) q^{7} +(-2.00000 + 2.00000i) q^{8} +(2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 16 q^{2} - 8 q^{7} - 64 q^{8}+O(q^{10}) \)

Character values

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

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

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
\(5\)	−2.69832	−	4.20940i	−0.539663	−	0.841881i
							\(7\)	1.39786	+	6.85901i	0.199695	+	0.979858i
\(11\)	6.05758	+	10.4920i	0.550689	+	0.953822i								0.998225	+	0.0595558i	\(0.0189684\pi\)
							−0.447536	+	0.894266i	\(0.647698\pi\)
\(13\)	12.6272	−	12.6272i	0.971326	−	0.971326i	−0.0282743	−	0.999600i	\(-0.509001\pi\)
							0.999600	+	0.0282743i	\(0.00900118\pi\)
\(17\)	4.41196	−	16.4657i	0.259527	−	0.968569i	−0.705988	−	0.708223i	\(-0.749497\pi\)
							0.965516	−	0.260345i	\(-0.0838364\pi\)
\(19\)	30.0470	+	17.3477i	1.58142	+	0.913035i	0.994652	+	0.103282i	\(0.0329344\pi\)
							0.586771	+	0.809753i	\(0.300399\pi\)
\(23\)	1.08501	+	4.04932i	0.0471745	+	0.176058i	0.985493	−	0.169713i	\(-0.0542841\pi\)
							−0.938319	+	0.345771i	\(0.887617\pi\)
\(29\)		−	30.5436i		−	1.05323i	−0.850104	−	0.526614i	\(-0.823461\pi\)
							0.850104	−	0.526614i	\(-0.176539\pi\)
\(31\)	11.3001	+	19.5724i	0.364519	+	0.631366i	0.988699	−	0.149915i	\(-0.0478999\pi\)
							−0.624179	+	0.781281i	\(0.714567\pi\)
\(37\)	40.8732	−	10.9520i	1.10468	−	0.295999i	0.340012	−	0.940421i	\(-0.389569\pi\)
							0.764670	+	0.644422i	\(0.222902\pi\)
\(41\)	68.1394			1.66194			0.830969	−	0.556319i	\(-0.187787\pi\)
							0.830969	+	0.556319i	\(0.187787\pi\)
\(43\)	−48.6732	+	48.6732i	−1.13193	+	1.13193i	−0.142079	+	0.989855i	\(0.545379\pi\)
							−0.989855	+	0.142079i	\(0.954621\pi\)
\(47\)	−58.4823	+	15.6703i	−1.24430	+	0.333410i	−0.820134	−	0.572172i	\(-0.806101\pi\)
							−0.424171	+	0.905582i	\(0.639434\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.v.a.37.6	✓	32
5.3	odd	4	inner	210.3.v.a.163.4	yes	32
7.4	even	3	inner	210.3.v.a.67.4	yes	32
35.18	odd	12	inner	210.3.v.a.193.6	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.v.a.37.6	✓	32	1.1	even	1	trivial
210.3.v.a.67.4	yes	32	7.4	even	3	inner
210.3.v.a.163.4	yes	32	5.3	odd	4	inner
210.3.v.a.193.6	yes	32	35.18	odd	12	inner