Embedded newform 210.3.v.b.193.1

• Label: 210.3.v.b.193.1
• Level: $210$
• Weight: $3$
• Character: 210.193
• 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			193.1
Character	\(\chi\)	\(=\)	210.193
Dual form			210.3.v.b.37.1

$q$-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 \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(71\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{3}{4}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.v.b.37.1	✓	32	35.2	odd	12	inner
210.3.v.b.67.8	yes	32	5.2	odd	4	inner
210.3.v.b.163.8	yes	32	7.2	even	3	inner
210.3.v.b.193.1	yes	32	1.1	even	1	trivial