Embedded newform 210.3.v.b.67.2

• Label: 210.3.v.b.67.2
• 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			67.2
Character	\(\chi\)	\(=\)	210.67
Dual form			210.3.v.b.163.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.366025 + 1.36603i) q^{2} +(-0.448288 - 1.67303i) q^{3} +(-1.73205 - 1.00000i) q^{4} +(-3.58272 - 3.48771i) q^{5} +2.44949 q^{6} +(3.00056 + 6.32429i) q^{7} +(2.00000 - 2.00000i) q^{8} +(-2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 16 q^{2} - 8 q^{5} + 24 q^{7} + 64 q^{8}+O(q^{10}) \)

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
\(5\)	−3.58272	−	3.48771i	−0.716543	−	0.697543i
							\(7\)	3.00056	+	6.32429i	0.428652	+	0.903470i
\(11\)	−1.95694	+	3.38951i	−0.177903	+	0.308138i								−0.941162	−	0.337955i	\(-0.890265\pi\)
							0.763259	+	0.646093i	\(0.223598\pi\)
\(13\)	−8.93314	+	8.93314i	−0.687164	+	0.687164i	−0.961604	−	0.274440i	\(-0.911508\pi\)
							0.274440	+	0.961604i	\(0.411508\pi\)
\(17\)	−14.3027	+	3.83238i	−0.841332	+	0.225434i	−0.653651	−	0.756796i	\(-0.726764\pi\)
							−0.187681	+	0.982230i	\(0.560097\pi\)
\(19\)	−28.0079	+	16.1704i	−1.47410	+	0.851072i	−0.999574	−	0.0291696i	\(-0.990714\pi\)
							−0.474526	+	0.880242i	\(0.657380\pi\)
\(23\)	12.3227	+	3.30186i	0.535770	+	0.143559i	0.516552	−	0.856256i	\(-0.327215\pi\)
							0.0192188	+	0.999815i	\(0.493882\pi\)
\(29\)		−	26.6068i		−	0.917478i	−0.888571	−	0.458739i	\(-0.848301\pi\)
							0.888571	−	0.458739i	\(-0.151699\pi\)
\(31\)	−17.0130	+	29.4674i	−0.548807	+	0.950561i	0.449550	+	0.893255i	\(0.351584\pi\)
							−0.998357	+	0.0573060i	\(0.981749\pi\)
\(37\)	1.67475	−	6.25026i	0.0452636	−	0.168926i	−0.939594	−	0.342291i	\(-0.888797\pi\)
							0.984858	+	0.173365i	\(0.0554639\pi\)
\(41\)	−26.0073			−0.634324			−0.317162	−	0.948371i	\(-0.602730\pi\)
							−0.317162	+	0.948371i	\(0.602730\pi\)
\(43\)	21.0534	−	21.0534i	0.489614	−	0.489614i	−0.418570	−	0.908185i	\(-0.637469\pi\)
							0.908185	+	0.418570i	\(0.137469\pi\)
\(47\)	10.7473	−	40.1094i	0.228666	−	0.853391i	−0.752237	−	0.658892i	\(-0.771025\pi\)
							0.980903	−	0.194499i	\(-0.0623081\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.v.b.67.2	yes	32
5.3	odd	4	inner	210.3.v.b.193.6	yes	32
7.2	even	3	inner	210.3.v.b.37.6	✓	32
35.23	odd	12	inner	210.3.v.b.163.2	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.v.b.37.6	✓	32	7.2	even	3	inner
210.3.v.b.67.2	yes	32	1.1	even	1	trivial
210.3.v.b.163.2	yes	32	35.23	odd	12	inner
210.3.v.b.193.6	yes	32	5.3	odd	4	inner