Embedded newform 210.3.v.a.37.4

• Label: 210.3.v.a.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.a.193.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36603 + 0.366025i) q^{2} +(-1.67303 - 0.448288i) q^{3} +(1.73205 - 1.00000i) q^{4} +(4.98663 - 0.365451i) q^{5} +2.44949 q^{6} +(-3.26382 - 6.19254i) q^{7} +(-2.00000 + 2.00000i) q^{8} +(2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 16 q^{2} - 8 q^{7} - 64 q^{8}+O(q^{10}) \)

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
\(5\)	4.98663	−	0.365451i	0.997325	−	0.0730902i
							\(7\)	−3.26382	−	6.19254i	−0.466260	−	0.884648i
\(11\)	−0.187724	−	0.325147i	−0.0170658	−	0.0295588i								0.857366	−	0.514707i	\(-0.172099\pi\)
							−0.874432	+	0.485148i	\(0.838766\pi\)
\(13\)	−9.32144	+	9.32144i	−0.717034	+	0.717034i	−0.967997	−	0.250963i	\(-0.919253\pi\)
							0.250963	+	0.967997i	\(0.419253\pi\)
\(17\)	7.97190	−	29.7515i	0.468935	−	1.75009i	−0.174566	−	0.984646i	\(-0.555852\pi\)
							0.643501	−	0.765445i	\(-0.277481\pi\)
\(19\)	6.46747	+	3.73399i	0.340393	+	0.196526i	0.660446	−	0.750874i	\(-0.270367\pi\)
							−0.320053	+	0.947400i	\(0.603701\pi\)
\(23\)	−10.3004	−	38.4418i	−0.447845	−	1.67138i	−0.708312	−	0.705899i	\(-0.750543\pi\)
							0.260467	−	0.965483i	\(-0.416124\pi\)
\(29\)		−	22.0507i		−	0.760370i	−0.924911	−	0.380185i	\(-0.875860\pi\)
							0.924911	−	0.380185i	\(-0.124140\pi\)
\(31\)	−23.7274	−	41.0971i	−0.765400	−	1.32571i	−0.940035	−	0.341078i	\(-0.889208\pi\)
							0.174635	−	0.984633i	\(-0.444125\pi\)
\(37\)	−5.90803	+	1.58305i	−0.159676	+	0.0427852i	−0.337772	−	0.941228i	\(-0.609673\pi\)
							0.178095	+	0.984013i	\(0.443006\pi\)
\(41\)	78.1193			1.90535			0.952675	−	0.303992i	\(-0.0983197\pi\)
							0.952675	+	0.303992i	\(0.0983197\pi\)
\(43\)	−12.2703	+	12.2703i	−0.285357	+	0.285357i	−0.835241	−	0.549884i	\(-0.814672\pi\)
							0.549884	+	0.835241i	\(0.314672\pi\)
\(47\)	−11.4346	+	3.06391i	−0.243290	+	0.0651895i	−0.378404	−	0.925641i	\(-0.623527\pi\)
							0.135113	+	0.990830i	\(0.456860\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.v.a.37.4	✓	32
5.3	odd	4	inner	210.3.v.a.163.6	yes	32
7.4	even	3	inner	210.3.v.a.67.6	yes	32
35.18	odd	12	inner	210.3.v.a.193.4	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.v.a.37.4	✓	32	1.1	even	1	trivial
210.3.v.a.67.6	yes	32	7.4	even	3	inner
210.3.v.a.163.6	yes	32	5.3	odd	4	inner
210.3.v.a.193.4	yes	32	35.18	odd	12	inner