Embedded newform 210.3.v.b.37.1

• Label: 210.3.v.b.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.b.193.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36603 - 0.366025i) q^{2} +(-1.67303 - 0.448288i) q^{3} +(1.73205 - 1.00000i) q^{4} +(-4.95862 + 0.641960i) q^{5} -2.44949 q^{6} +(2.95853 - 6.34406i) q^{7} +(2.00000 - 2.00000i) q^{8} +(2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 16 q^{2} - 8 q^{5} + 24 q^{7} + 64 q^{8}+O(q^{10}) \)

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
\(5\)	−4.95862	+	0.641960i	−0.991724	+	0.128392i
							\(7\)	2.95853	−	6.34406i	0.422648	−	0.906294i
\(11\)	−9.98405	−	17.2929i	−0.907641	−	1.57208i								−0.817333	−	0.576166i	\(-0.804548\pi\)
							−0.0903080	−	0.995914i	\(-0.528785\pi\)
\(13\)	−13.3742	+	13.3742i	−1.02879	+	1.02879i	−0.0292138	+	0.999573i	\(0.509300\pi\)
							−0.999573	+	0.0292138i	\(0.990700\pi\)
\(17\)	5.85604	−	21.8551i	0.344473	−	1.28559i	−0.548753	−	0.835984i	\(-0.684897\pi\)
							0.893227	−	0.449607i	\(-0.148436\pi\)
\(19\)	−14.8761	−	8.58869i	−0.782950	−	0.452037i	0.0545246	−	0.998512i	\(-0.482636\pi\)
							−0.837475	+	0.546476i	\(0.815969\pi\)
\(23\)	−5.43036	−	20.2664i	−0.236103	−	0.881147i	−0.977649	−	0.210245i	\(-0.932574\pi\)
							0.741546	−	0.670902i	\(-0.234093\pi\)
\(29\)			23.1996i			0.799987i	0.916518	+	0.399994i	\(0.130988\pi\)
							−0.916518	+	0.399994i	\(0.869012\pi\)
\(31\)	27.7400	+	48.0470i	0.894837	+	1.54990i	0.834006	+	0.551756i	\(0.186042\pi\)
							0.0608318	+	0.998148i	\(0.480625\pi\)
\(37\)	22.2866	−	5.97166i	0.602339	−	0.161396i	0.0552528	−	0.998472i	\(-0.482404\pi\)
							0.547087	+	0.837076i	\(0.315737\pi\)
\(41\)	−21.6302			−0.527565			−0.263783	−	0.964582i	\(-0.584970\pi\)
							−0.263783	+	0.964582i	\(0.584970\pi\)
\(43\)	14.6809	−	14.6809i	0.341417	−	0.341417i	−0.515483	−	0.856900i	\(-0.672387\pi\)
							0.856900	+	0.515483i	\(0.172387\pi\)
\(47\)	73.1609	−	19.6034i	1.55662	−	0.417094i	0.625025	−	0.780604i	\(-0.285089\pi\)
							0.931590	+	0.363511i	\(0.118422\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.v.b.37.1	✓	32
5.3	odd	4	inner	210.3.v.b.163.8	yes	32
7.4	even	3	inner	210.3.v.b.67.8	yes	32
35.18	odd	12	inner	210.3.v.b.193.1	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.v.b.37.1	✓	32	1.1	even	1	trivial
210.3.v.b.67.8	yes	32	7.4	even	3	inner
210.3.v.b.163.8	yes	32	5.3	odd	4	inner
210.3.v.b.193.1	yes	32	35.18	odd	12	inner