Embedded newform 210.3.v.b.37.4

• Label: 210.3.v.b.37.4
• 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			37.4
Character	\(\chi\)	\(=\)	210.37
Dual form			210.3.v.b.193.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36603 - 0.366025i) q^{2} +(-1.67303 - 0.448288i) q^{3} +(1.73205 - 1.00000i) q^{4} +(3.95091 - 3.06436i) q^{5} -2.44949 q^{6} +(3.71395 + 5.93351i) q^{7} +(2.00000 - 2.00000i) q^{8} +(2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 16 q^{2} - 8 q^{5} + 24 q^{7} + 64 q^{8}+O(q^{10}) \)

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
\(5\)	3.95091	−	3.06436i	0.790182	−	0.612873i
							\(7\)	3.71395	+	5.93351i	0.530564	+	0.847645i
\(11\)	−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	−0.369097	−	0.929391i	\(-0.620333\pi\)
							0.929391	+	0.369097i	\(0.120333\pi\)
\(17\)	4.87353	−	18.1882i	0.286678	−	1.06990i	−0.660926	−	0.750451i	\(-0.729836\pi\)
							0.947604	−	0.319446i	\(-0.103497\pi\)
\(19\)	7.65544	+	4.41987i	0.402918	+	0.232625i	0.687742	−	0.725955i	\(-0.258602\pi\)
							−0.284824	+	0.958580i	\(0.591935\pi\)
\(23\)	−0.107714	−	0.401994i	−0.00468322	−	0.0174780i	0.963545	−	0.267547i	\(-0.0862132\pi\)
							−0.968228	+	0.250069i	\(0.919547\pi\)
\(29\)			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	0.994214	−	0.107415i	\(-0.0342574\pi\)
							−0.590131	+	0.807307i	\(0.700924\pi\)
\(37\)	−51.5702	+	13.8182i	−1.39379	+	0.373464i	−0.876110	−	0.482111i	\(-0.839870\pi\)
							−0.517678	+	0.855575i	\(0.673204\pi\)
\(41\)	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	\(0.460367\pi\)
\(47\)	−31.0816	+	8.32828i	−0.661310	+	0.177198i	−0.573837	−	0.818969i	\(-0.694546\pi\)
							−0.0874729	+	0.996167i	\(0.527879\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.v.b.37.4	✓	32	1.1	even	1	trivial
210.3.v.b.67.7	yes	32	7.4	even	3	inner
210.3.v.b.163.7	yes	32	5.3	odd	4	inner
210.3.v.b.193.4	yes	32	35.18	odd	12	inner