Embedded newform 210.3.w.b.173.5

• Label: 210.3.w.b.173.5
• Level: $210$
• Weight: $3$
• Character: 210.173
• 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			173.5
Character	\(\chi\)	\(=\)	210.173
Dual form			210.3.w.b.17.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36603 - 0.366025i) q^{2} +(-1.72513 - 2.45437i) q^{3} +(1.73205 - 1.00000i) q^{4} +(2.34013 - 4.41857i) q^{5} +(-3.25493 - 2.72129i) q^{6} +(-6.07917 - 3.47039i) q^{7} +(2.00000 - 2.00000i) q^{8} +(-3.04785 + 8.46821i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 32 q^{2} + 6 q^{3} + 12 q^{5} + 4 q^{7} + 128 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(31\)	\(71\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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
							\(3\)	−1.72513	−	2.45437i	−0.575044	−	0.818123i
\(5\)	2.34013	−	4.41857i	0.468025	−	0.883715i
							\(7\)	−6.07917	−	3.47039i	−0.868454	−	0.495771i
\(11\)	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	\(-0.880623\pi\)
							0.366305	+	0.930495i	\(0.380623\pi\)
\(17\)	−22.5211	−	6.03451i	−1.32477	−	0.354971i	−0.474008	−	0.880520i	\(-0.657193\pi\)
							−0.850763	+	0.525549i	\(0.823860\pi\)
\(19\)	17.0920	−	29.6042i	0.899578	−	1.55811i	0.0715438	−	0.997437i	\(-0.477207\pi\)
							0.828034	−	0.560678i	\(-0.189459\pi\)
\(23\)	7.11456	+	26.5519i	0.309329	+	1.15443i	0.929155	+	0.369691i	\(0.120536\pi\)
							−0.619826	+	0.784739i	\(0.712797\pi\)
\(29\)	6.86667			0.236782			0.118391	−	0.992967i	\(-0.462226\pi\)
							0.118391	+	0.992967i	\(0.462226\pi\)
\(31\)	38.9114	−	22.4655i	1.25520	−	0.724693i	0.283066	−	0.959100i	\(-0.408648\pi\)
							0.972138	+	0.234407i	\(0.0753150\pi\)
\(37\)	−9.14352	−	34.1241i	−0.247122	−	0.922272i	−0.972305	−	0.233716i	\(-0.924911\pi\)
							0.725183	−	0.688556i	\(-0.241755\pi\)
\(41\)	−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	0.878701	−	0.477372i	\(-0.158411\pi\)
							−0.477372	+	0.878701i	\(0.658411\pi\)
\(47\)	6.24890	+	23.3212i	0.132955	+	0.496196i	0.999998	−	0.00203792i	\(-0.000648691\pi\)
							−0.867043	+	0.498234i	\(0.833982\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.w.a.17.2	✓	64	35.17	even	12
210.3.w.a.47.3	yes	64	5.2	odd	4
210.3.w.a.143.3	yes	64	21.17	even	6
210.3.w.a.173.2	yes	64	3.2	odd	2
210.3.w.b.17.5	yes	64	105.17	odd	12	inner
210.3.w.b.47.11	yes	64	15.2	even	4	inner
210.3.w.b.143.11	yes	64	7.3	odd	6	inner
210.3.w.b.173.5	yes	64	1.1	even	1	trivial