Embedded newform 210.3.v.a.37.1

• Label: 210.3.v.a.37.1
• 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			37.1
Character	\(\chi\)	\(=\)	210.37
Dual form			210.3.v.a.193.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36603 + 0.366025i) q^{2} +(-1.67303 - 0.448288i) q^{3} +(1.73205 - 1.00000i) q^{4} +(-4.60436 - 1.94932i) q^{5} +2.44949 q^{6} +(-4.28537 - 5.53495i) q^{7} +(-2.00000 + 2.00000i) q^{8} +(2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 16 q^{2} - 8 q^{7} - 64 q^{8}+O(q^{10}) \)

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
\(5\)	−4.60436	−	1.94932i	−0.920873	−	0.389864i
							\(7\)	−4.28537	−	5.53495i	−0.612195	−	0.790707i
\(11\)	8.47882	+	14.6857i	0.770802	+	1.33507i								0.937124	+	0.348996i	\(0.113477\pi\)
							−0.166322	+	0.986071i	\(0.553189\pi\)
\(13\)	3.05181	−	3.05181i	0.234754	−	0.234754i	−0.579920	−	0.814674i	\(-0.696916\pi\)
							0.814674	+	0.579920i	\(0.196916\pi\)
\(17\)	−2.87910	+	10.7449i	−0.169359	+	0.632055i	0.828085	+	0.560602i	\(0.189430\pi\)
							−0.997444	+	0.0714531i	\(0.977236\pi\)
\(19\)	−5.74405	−	3.31633i	−0.302318	−	0.174544i	0.341166	−	0.940003i	\(-0.389178\pi\)
							−0.643484	+	0.765460i	\(0.722512\pi\)
\(23\)	4.49666	+	16.7817i	0.195507	+	0.729641i	0.992135	+	0.125172i	\(0.0399482\pi\)
							−0.796628	+	0.604469i	\(0.793385\pi\)
\(29\)			35.8861i			1.23745i	0.785607	+	0.618725i	\(0.212351\pi\)
							−0.785607	+	0.618725i	\(0.787649\pi\)
\(31\)	22.7901	+	39.4736i	0.735165	+	1.27334i	0.954651	+	0.297727i	\(0.0962286\pi\)
							−0.219486	+	0.975616i	\(0.570438\pi\)
\(37\)	17.7095	−	4.74524i	0.478635	−	0.128250i	−0.0114320	−	0.999935i	\(-0.503639\pi\)
							0.490067	+	0.871685i	\(0.336972\pi\)
\(41\)	−42.9609			−1.04783			−0.523913	−	0.851772i	\(-0.675528\pi\)
							−0.523913	+	0.851772i	\(0.675528\pi\)
\(43\)	−1.98628	+	1.98628i	−0.0461926	+	0.0461926i	−0.729826	−	0.683633i	\(-0.760399\pi\)
							0.683633	+	0.729826i	\(0.260399\pi\)
\(47\)	42.2437	−	11.3192i	0.898801	−	0.240833i	0.220300	−	0.975432i	\(-0.429296\pi\)
							0.678501	+	0.734599i	\(0.262630\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.v.a.37.1	✓	32
5.3	odd	4	inner	210.3.v.a.163.7	yes	32
7.4	even	3	inner	210.3.v.a.67.7	yes	32
35.18	odd	12	inner	210.3.v.a.193.1	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.v.a.37.1	✓	32	1.1	even	1	trivial
210.3.v.a.67.7	yes	32	7.4	even	3	inner
210.3.v.a.163.7	yes	32	5.3	odd	4	inner
210.3.v.a.193.1	yes	32	35.18	odd	12	inner