Embedded newform 210.3.w.a.47.3

• Label: 210.3.w.a.47.3
• 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			47.3
Character	\(\chi\)	\(=\)	210.47
Dual form			210.3.w.a.143.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.366025 + 1.36603i) q^{2} +(-2.45437 + 1.72513i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(2.65653 - 4.23590i) q^{5} +(-3.25493 - 2.72129i) q^{6} +(3.47039 - 6.07917i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(3.04785 - 8.46821i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 32 q^{2} - 6 q^{3} - 12 q^{5} + 4 q^{7} - 128 q^{8} - 16 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.45437	+	1.72513i	−0.818123	+	0.575044i
\(5\)	2.65653	−	4.23590i	0.531307	−	0.847179i
							\(7\)	3.47039	−	6.07917i	0.495771	−	0.868454i
\(11\)	5.23131	−	3.02030i	0.475574	−	0.274573i								−0.242996	−	0.970027i	\(-0.578130\pi\)
							0.718570	+	0.695455i	\(0.244797\pi\)
\(13\)	7.33446	+	7.33446i	0.564189	+	0.564189i	0.930495	−	0.366305i	\(-0.119377\pi\)
							−0.366305	+	0.930495i	\(0.619377\pi\)
\(17\)	6.03451	−	22.5211i	0.354971	−	1.32477i	−0.525549	−	0.850763i	\(-0.676140\pi\)
							0.880520	−	0.474008i	\(-0.157193\pi\)
\(19\)	−17.0920	+	29.6042i	−0.899578	+	1.55811i	−0.0715438	+	0.997437i	\(0.522793\pi\)
							−0.828034	+	0.560678i	\(0.810541\pi\)
\(23\)	26.5519	−	7.11456i	1.15443	−	0.309329i	0.369691	−	0.929155i	\(-0.379464\pi\)
							0.784739	+	0.619826i	\(0.212797\pi\)
\(29\)	−6.86667			−0.236782			−0.118391	−	0.992967i	\(-0.537774\pi\)
							−0.118391	+	0.992967i	\(0.537774\pi\)
\(31\)	38.9114	−	22.4655i	1.25520	−	0.724693i	0.283066	−	0.959100i	\(-0.408648\pi\)
							0.972138	+	0.234407i	\(0.0753150\pi\)
\(37\)	34.1241	−	9.14352i	0.922272	−	0.247122i	0.233716	−	0.972305i	\(-0.424911\pi\)
							0.688556	+	0.725183i	\(0.258245\pi\)
\(41\)	−18.2221			−0.444442			−0.222221	−	0.974996i	\(-0.571331\pi\)
							−0.222221	+	0.974996i	\(0.571331\pi\)
\(43\)	17.2571	−	17.2571i	0.401329	−	0.401329i	−0.477372	−	0.878701i	\(-0.658411\pi\)
							0.878701	+	0.477372i	\(0.158411\pi\)
\(47\)	−23.3212	+	6.24890i	−0.496196	+	0.132955i	−0.498234	−	0.867043i	\(-0.666018\pi\)
							0.00203792	+	0.999998i	\(0.499351\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.w.a.47.3	yes	64
3.2	odd	2		210.3.w.b.47.11	yes	64
5.3	odd	4		210.3.w.b.173.5	yes	64
7.3	odd	6	inner	210.3.w.a.17.2	✓	64
15.8	even	4	inner	210.3.w.a.173.2	yes	64
21.17	even	6		210.3.w.b.17.5	yes	64
35.3	even	12		210.3.w.b.143.11	yes	64
105.38	odd	12	inner	210.3.w.a.143.3	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.w.a.17.2	✓	64	7.3	odd	6	inner
210.3.w.a.47.3	yes	64	1.1	even	1	trivial
210.3.w.a.143.3	yes	64	105.38	odd	12	inner
210.3.w.a.173.2	yes	64	15.8	even	4	inner
210.3.w.b.17.5	yes	64	21.17	even	6
210.3.w.b.47.11	yes	64	3.2	odd	2
210.3.w.b.143.11	yes	64	35.3	even	12
210.3.w.b.173.5	yes	64	5.3	odd	4