Embedded newform 210.3.w.b.17.6

• Label: 210.3.w.b.17.6
• Level: $210$
• Weight: $3$
• Character: 210.17
• Analytic conductor: $5.722$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	210.w (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.72208555157\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			17.6
Character	\(\chi\)	\(=\)	210.17
Dual form			210.3.w.b.173.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36603 + 0.366025i) q^{2} +(-1.66769 - 2.49375i) q^{3} +(1.73205 + 1.00000i) q^{4} +(1.09236 + 4.87922i) q^{5} +(-1.36533 - 4.01695i) q^{6} +(-1.72310 + 6.78461i) q^{7} +(2.00000 + 2.00000i) q^{8} +(-3.43763 + 8.31761i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 32 q^{2} + 6 q^{3} + 12 q^{5} + 4 q^{7} + 128 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(71\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)

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
							\(3\)	−1.66769	−	2.49375i	−0.555896	−	0.831252i
\(5\)	1.09236	+	4.87922i	0.218472	+	0.975843i
							\(7\)	−1.72310	+	6.78461i	−0.246158	+	0.969230i
\(11\)	5.25762	+	3.03549i	0.477966	+	0.275954i								0.719568	−	0.694422i	\(-0.244340\pi\)
							−0.241603	+	0.970375i	\(0.577673\pi\)
\(13\)	4.95646	+	4.95646i	0.381266	+	0.381266i	0.871558	−	0.490292i	\(-0.163110\pi\)
							−0.490292	+	0.871558i	\(0.663110\pi\)
\(17\)	24.7338	−	6.62741i	1.45493	−	0.389847i	0.557195	−	0.830382i	\(-0.311878\pi\)
							0.897736	+	0.440534i	\(0.145211\pi\)
\(19\)	−8.47334	−	14.6763i	−0.445965	−	0.772435i	0.552154	−	0.833742i	\(-0.313806\pi\)
							−0.998119	+	0.0613078i	\(0.980473\pi\)
\(23\)	−7.44889	+	27.7996i	−0.323865	+	1.20868i	0.591583	+	0.806244i	\(0.298503\pi\)
							−0.915448	+	0.402436i	\(0.868163\pi\)
\(29\)	29.7644			1.02636			0.513179	−	0.858282i	\(-0.328468\pi\)
							0.513179	+	0.858282i	\(0.328468\pi\)
\(31\)	−36.5762	−	21.1173i	−1.17988	−	0.681203i	−0.223892	−	0.974614i	\(-0.571876\pi\)
							−0.955986	+	0.293411i	\(0.905210\pi\)
\(37\)	−12.4034	+	46.2903i	−0.335228	+	1.25109i	0.568393	+	0.822757i	\(0.307565\pi\)
							−0.903621	+	0.428332i	\(0.859101\pi\)
\(41\)	−2.99309			−0.0730021			−0.0365011	−	0.999334i	\(-0.511621\pi\)
							−0.0365011	+	0.999334i	\(0.511621\pi\)
\(43\)	22.2278	−	22.2278i	0.516926	−	0.516926i	−0.399714	−	0.916640i	\(-0.630891\pi\)
							0.916640	+	0.399714i	\(0.130891\pi\)
\(47\)	16.3896	−	61.1668i	0.348715	−	1.30142i	−0.539498	−	0.841987i	\(-0.681386\pi\)
							0.888212	−	0.459434i	\(-0.151948\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.w.b.17.6	yes	64
3.2	odd	2		210.3.w.a.17.8	✓	64
5.3	odd	4		210.3.w.a.143.14	yes	64
7.5	odd	6	inner	210.3.w.b.47.1	yes	64
15.8	even	4	inner	210.3.w.b.143.1	yes	64
21.5	even	6		210.3.w.a.47.14	yes	64
35.33	even	12		210.3.w.a.173.8	yes	64
105.68	odd	12	inner	210.3.w.b.173.6	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.w.a.17.8	✓	64	3.2	odd	2
210.3.w.a.47.14	yes	64	21.5	even	6
210.3.w.a.143.14	yes	64	5.3	odd	4
210.3.w.a.173.8	yes	64	35.33	even	12
210.3.w.b.17.6	yes	64	1.1	even	1	trivial
210.3.w.b.47.1	yes	64	7.5	odd	6	inner
210.3.w.b.143.1	yes	64	15.8	even	4	inner
210.3.w.b.173.6	yes	64	105.68	odd	12	inner