Embedded newform 210.3.v.b.67.7

• Label: 210.3.v.b.67.7
• 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			67.7
Character	\(\chi\)	\(=\)	210.67
Dual form			210.3.v.b.163.7

$q$-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 \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(71\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{1}{4}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.v.b.37.4	✓	32	7.2	even	3	inner
210.3.v.b.67.7	yes	32	1.1	even	1	trivial
210.3.v.b.163.7	yes	32	35.23	odd	12	inner
210.3.v.b.193.4	yes	32	5.3	odd	4	inner