Embedded newform 210.3.v.b.163.7

• Label: 210.3.v.b.163.7
• 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			163.7
Character	\(\chi\)	\(=\)	210.163
Dual form			210.3.v.b.67.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.366025 - 1.36603i) q^{2} +(0.448288 - 1.67303i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(0.678362 - 4.95377i) q^{5} -2.44949 q^{6} +(5.93351 - 3.71395i) q^{7} +(2.00000 + 2.00000i) q^{8} +(-2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 16 q^{2} - 8 q^{5} + 24 q^{7} + 64 q^{8}+O(q^{10}) \)

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
\(5\)	0.678362	−	4.95377i	0.135672	−	0.990754i
							\(7\)	5.93351	−	3.71395i	0.847645	−	0.530564i
\(11\)	−3.58312	−	6.20615i	−0.325738	−	0.564195i								0.655923	−	0.754828i	\(-0.272280\pi\)
							−0.981662	+	0.190632i	\(0.938946\pi\)
\(13\)	7.28383	+	7.28383i	0.560295	+	0.560295i	0.929391	−	0.369097i	\(-0.120333\pi\)
							−0.369097	+	0.929391i	\(0.620333\pi\)
\(17\)	−18.1882	−	4.87353i	−1.06990	−	0.286678i	−0.319446	−	0.947604i	\(-0.603497\pi\)
							−0.750451	+	0.660926i	\(0.770164\pi\)
\(19\)	−7.65544	−	4.41987i	−0.402918	−	0.232625i	0.284824	−	0.958580i	\(-0.408065\pi\)
							−0.687742	+	0.725955i	\(0.741398\pi\)
\(23\)	0.401994	−	0.107714i	0.0174780	−	0.00468322i	−0.250069	−	0.968228i	\(-0.580453\pi\)
							0.267547	+	0.963545i	\(0.413787\pi\)
\(29\)		−	27.7751i		−	0.957761i	−0.877880	−	0.478880i	\(-0.841043\pi\)
							0.877880	−	0.478880i	\(-0.158957\pi\)
\(31\)	12.5266	+	21.6967i	0.404083	+	0.699892i	0.994214	−	0.107415i	\(-0.0342574\pi\)
							−0.590131	+	0.807307i	\(0.700924\pi\)
\(37\)	13.8182	+	51.5702i	0.373464	+	1.39379i	0.855575	+	0.517678i	\(0.173204\pi\)
							−0.482111	+	0.876110i	\(0.660130\pi\)
\(41\)	46.7769			1.14090			0.570450	−	0.821333i	\(-0.306769\pi\)
							0.570450	+	0.821333i	\(0.306769\pi\)
\(43\)	−37.3270	−	37.3270i	−0.868070	−	0.868070i	0.124189	−	0.992259i	\(-0.460367\pi\)
							−0.992259	+	0.124189i	\(0.960367\pi\)
\(47\)	8.32828	+	31.0816i	0.177198	+	0.661310i	0.996167	+	0.0874729i	\(0.0278791\pi\)
							−0.818969	+	0.573837i	\(0.805454\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.v.b.163.7	yes	32
5.2	odd	4	inner	210.3.v.b.37.4	✓	32
7.4	even	3	inner	210.3.v.b.193.4	yes	32
35.32	odd	12	inner	210.3.v.b.67.7	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.v.b.37.4	✓	32	5.2	odd	4	inner
210.3.v.b.67.7	yes	32	35.32	odd	12	inner
210.3.v.b.163.7	yes	32	1.1	even	1	trivial
210.3.v.b.193.4	yes	32	7.4	even	3	inner