Embedded newform 210.3.w.b.173.13

• Label: 210.3.w.b.173.13
• 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			173.13
Character	\(\chi\)	\(=\)	210.173
Dual form			210.3.w.b.17.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36603 - 0.366025i) q^{2} +(2.32144 - 1.90024i) q^{3} +(1.73205 - 1.00000i) q^{4} +(0.695196 + 4.95143i) q^{5} +(2.47561 - 3.44548i) q^{6} +(-2.48720 - 6.54323i) q^{7} +(2.00000 - 2.00000i) q^{8} +(1.77817 - 8.82259i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 32 q^{2} + 6 q^{3} + 12 q^{5} + 4 q^{7} + 128 q^{8} + 16 q^{9}+O(q^{10}) \)

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
							\(3\)	2.32144	−	1.90024i	0.773813	−	0.633414i
\(5\)	0.695196	+	4.95143i	0.139039	+	0.990287i
							\(7\)	−2.48720	−	6.54323i	−0.355315	−	0.934747i
\(11\)	16.0981	−	9.29422i	1.46346	−	0.844929i								0.464291	−	0.885683i	\(-0.346309\pi\)
							0.999169	+	0.0407536i	\(0.0129759\pi\)
\(13\)	−11.8606	+	11.8606i	−0.912351	+	0.912351i	−0.996457	−	0.0841060i	\(-0.973197\pi\)
							0.0841060	+	0.996457i	\(0.473197\pi\)
\(17\)	16.3649	+	4.38496i	0.962642	+	0.257939i	0.705718	−	0.708493i	\(-0.250624\pi\)
							0.256923	+	0.966432i	\(0.417291\pi\)
\(19\)	−2.06928	+	3.58409i	−0.108909	+	0.188636i	−0.915329	−	0.402708i	\(-0.868069\pi\)
							0.806419	+	0.591344i	\(0.201402\pi\)
\(23\)	5.44679	+	20.3277i	0.236817	+	0.883813i	0.977321	+	0.211762i	\(0.0679201\pi\)
							−0.740504	+	0.672052i	\(0.765413\pi\)
\(29\)	−49.5120			−1.70731			−0.853656	−	0.520838i	\(-0.825620\pi\)
							−0.853656	+	0.520838i	\(0.825620\pi\)
\(31\)	−2.73364	+	1.57827i	−0.0881819	+	0.0509119i	−0.543443	−	0.839446i	\(-0.682879\pi\)
							0.455261	+	0.890358i	\(0.349546\pi\)
\(37\)	8.12079	+	30.3072i	0.219481	+	0.819113i	0.984541	+	0.175155i	\(0.0560425\pi\)
							−0.765060	+	0.643959i	\(0.777291\pi\)
\(41\)	−26.6051			−0.648905			−0.324452	−	0.945902i	\(-0.605180\pi\)
							−0.324452	+	0.945902i	\(0.605180\pi\)
\(43\)	−25.3219	−	25.3219i	−0.588880	−	0.588880i	0.348448	−	0.937328i	\(-0.386709\pi\)
							−0.937328	+	0.348448i	\(0.886709\pi\)
\(47\)	13.1425	+	49.0486i	0.279628	+	1.04359i	0.952676	+	0.303987i	\(0.0983181\pi\)
							−0.673048	+	0.739599i	\(0.735015\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.w.b.173.13	yes	64
3.2	odd	2		210.3.w.a.173.12	yes	64
5.2	odd	4		210.3.w.a.47.6	yes	64
7.3	odd	6	inner	210.3.w.b.143.14	yes	64
15.2	even	4	inner	210.3.w.b.47.14	yes	64
21.17	even	6		210.3.w.a.143.6	yes	64
35.17	even	12		210.3.w.a.17.12	✓	64
105.17	odd	12	inner	210.3.w.b.17.13	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.w.a.17.12	✓	64	35.17	even	12
210.3.w.a.47.6	yes	64	5.2	odd	4
210.3.w.a.143.6	yes	64	21.17	even	6
210.3.w.a.173.12	yes	64	3.2	odd	2
210.3.w.b.17.13	yes	64	105.17	odd	12	inner
210.3.w.b.47.14	yes	64	15.2	even	4	inner
210.3.w.b.143.14	yes	64	7.3	odd	6	inner
210.3.w.b.173.13	yes	64	1.1	even	1	trivial