Embedded newform 210.3.w.b.47.1

• Label: 210.3.w.b.47.1
• Level: $210$
• Weight: $3$
• Character: 210.47
• 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			47.1
Character	\(\chi\)	\(=\)	210.47
Dual form			210.3.w.b.143.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.366025 - 1.36603i) q^{2} +(-2.99350 + 0.197383i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(-3.67935 + 3.38562i) q^{5} +(1.36533 + 4.01695i) q^{6} +(6.78461 - 1.72310i) q^{7} +(2.00000 + 2.00000i) q^{8} +(8.92208 - 1.18173i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 32 q^{2} + 6 q^{3} + 12 q^{5} + 4 q^{7} + 128 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(31\)	\(71\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.366025	−	1.36603i	−0.183013	−	0.683013i
							\(3\)	−2.99350	+	0.197383i	−0.997833	+	0.0657945i
\(5\)	−3.67935	+	3.38562i	−0.735869	+	0.677124i
							\(7\)	6.78461	−	1.72310i	0.969230	−	0.246158i
\(11\)	−5.25762	+	3.03549i	−0.477966	+	0.275954i								−0.719568	−	0.694422i	\(-0.755660\pi\)
							0.241603	+	0.970375i	\(0.422327\pi\)
\(13\)	−4.95646	−	4.95646i	−0.381266	−	0.381266i	0.490292	−	0.871558i	\(-0.336890\pi\)
							−0.871558	+	0.490292i	\(0.836890\pi\)
\(17\)	6.62741	−	24.7338i	0.389847	−	1.45493i	−0.440534	−	0.897736i	\(-0.645211\pi\)
							0.830382	−	0.557195i	\(-0.188122\pi\)
\(19\)	8.47334	−	14.6763i	0.445965	−	0.772435i	−0.552154	−	0.833742i	\(-0.686194\pi\)
							0.998119	+	0.0613078i	\(0.0195271\pi\)
\(23\)	27.7996	−	7.44889i	1.20868	−	0.323865i	0.402436	−	0.915448i	\(-0.368163\pi\)
							0.806244	+	0.591583i	\(0.201497\pi\)
\(29\)	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	−0.955986	−	0.293411i	\(-0.905210\pi\)
							−0.223892	+	0.974614i	\(0.571876\pi\)
\(37\)	46.2903	−	12.4034i	1.25109	−	0.335228i	0.428332	−	0.903621i	\(-0.359101\pi\)
							0.822757	+	0.568393i	\(0.192435\pi\)
\(41\)	2.99309			0.0730021			0.0365011	−	0.999334i	\(-0.488379\pi\)
							0.0365011	+	0.999334i	\(0.488379\pi\)
\(43\)	22.2278	−	22.2278i	0.516926	−	0.516926i	−0.399714	−	0.916640i	\(-0.630891\pi\)
							0.916640	+	0.399714i	\(0.130891\pi\)
\(47\)	61.1668	−	16.3896i	1.30142	−	0.348715i	0.459434	−	0.888212i	\(-0.348052\pi\)
							0.841987	+	0.539498i	\(0.181386\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.w.a.17.8	✓	64	21.17	even	6
210.3.w.a.47.14	yes	64	3.2	odd	2
210.3.w.a.143.14	yes	64	35.3	even	12
210.3.w.a.173.8	yes	64	5.3	odd	4
210.3.w.b.17.6	yes	64	7.3	odd	6	inner
210.3.w.b.47.1	yes	64	1.1	even	1	trivial
210.3.w.b.143.1	yes	64	105.38	odd	12	inner
210.3.w.b.173.6	yes	64	15.8	even	4	inner