Embedded newform 210.3.v.b.37.6

• Label: 210.3.v.b.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.b.193.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36603 - 0.366025i) q^{2} +(1.67303 + 0.448288i) q^{3} +(1.73205 - 1.00000i) q^{4} +(-1.22909 + 4.84658i) q^{5} +2.44949 q^{6} +(6.32429 + 3.00056i) q^{7} +(2.00000 - 2.00000i) q^{8} +(2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 16 q^{2} - 8 q^{5} + 24 q^{7} + 64 q^{8}+O(q^{10}) \)

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
\(5\)	−1.22909	+	4.84658i	−0.245818	+	0.969316i
							\(7\)	6.32429	+	3.00056i	0.903470	+	0.428652i
\(11\)	−1.95694	−	3.38951i	−0.177903	−	0.308138i								0.763259	−	0.646093i	\(-0.223598\pi\)
							−0.941162	+	0.337955i	\(0.890265\pi\)
\(13\)	−8.93314	+	8.93314i	−0.687164	+	0.687164i	−0.961604	−	0.274440i	\(-0.911508\pi\)
							0.274440	+	0.961604i	\(0.411508\pi\)
\(17\)	3.83238	−	14.3027i	0.225434	−	0.841332i	−0.756796	−	0.653651i	\(-0.773236\pi\)
							0.982230	−	0.187681i	\(-0.0600971\pi\)
\(19\)	28.0079	+	16.1704i	1.47410	+	0.851072i	0.999574	−	0.0291696i	\(-0.00928630\pi\)
							0.474526	+	0.880242i	\(0.342620\pi\)
\(23\)	−3.30186	−	12.3227i	−0.143559	−	0.535770i	−0.999815	−	0.0192188i	\(-0.993882\pi\)
							0.856256	−	0.516552i	\(-0.172785\pi\)
\(29\)		−	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	−0.998357	−	0.0573060i	\(-0.981749\pi\)
							0.449550	−	0.893255i	\(-0.351584\pi\)
\(37\)	−6.25026	+	1.67475i	−0.168926	+	0.0452636i	−0.342291	−	0.939594i	\(-0.611203\pi\)
							0.173365	+	0.984858i	\(0.444536\pi\)
\(41\)	−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	\(0.137469\pi\)
\(47\)	−40.1094	+	10.7473i	−0.853391	+	0.228666i	−0.658892	−	0.752237i	\(-0.728975\pi\)
							−0.194499	+	0.980903i	\(0.562308\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.v.b.37.6	✓	32	1.1	even	1	trivial
210.3.v.b.67.2	yes	32	7.4	even	3	inner
210.3.v.b.163.2	yes	32	5.3	odd	4	inner
210.3.v.b.193.6	yes	32	35.18	odd	12	inner