Embedded newform 210.3.w.a.17.12

• Label: 210.3.w.a.17.12
• 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			17.12
Character	\(\chi\)	\(=\)	210.17
Dual form			210.3.w.a.173.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36603 - 0.366025i) q^{2} +(1.06031 - 2.80638i) q^{3} +(1.73205 + 1.00000i) q^{4} +(-0.695196 + 4.95143i) q^{5} +(-2.47561 + 3.44548i) q^{6} +(-2.48720 + 6.54323i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(-6.75151 - 5.95123i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 32 q^{2} - 6 q^{3} - 12 q^{5} + 4 q^{7} - 128 q^{8} - 16 q^{9}+O(q^{10}) \)

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
							\(3\)	1.06031	−	2.80638i	0.353435	−	0.935459i
\(5\)	−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	\(-0.653691\pi\)
							−0.999169	+	0.0407536i	\(0.987024\pi\)
\(13\)	−11.8606	−	11.8606i	−0.912351	−	0.912351i	0.0841060	−	0.996457i	\(-0.473197\pi\)
							−0.996457	+	0.0841060i	\(0.973197\pi\)
\(17\)	−16.3649	+	4.38496i	−0.962642	+	0.257939i	−0.705718	−	0.708493i	\(-0.749376\pi\)
							−0.256923	+	0.966432i	\(0.582709\pi\)
\(19\)	−2.06928	−	3.58409i	−0.108909	−	0.188636i	0.806419	−	0.591344i	\(-0.201402\pi\)
							−0.915329	+	0.402708i	\(0.868069\pi\)
\(23\)	−5.44679	+	20.3277i	−0.236817	+	0.883813i	0.740504	+	0.672052i	\(0.234587\pi\)
							−0.977321	+	0.211762i	\(0.932080\pi\)
\(29\)	49.5120			1.70731			0.853656	−	0.520838i	\(-0.174380\pi\)
							0.853656	+	0.520838i	\(0.174380\pi\)
\(31\)	−2.73364	−	1.57827i	−0.0881819	−	0.0509119i	0.455261	−	0.890358i	\(-0.349546\pi\)
							−0.543443	+	0.839446i	\(0.682879\pi\)
\(37\)	8.12079	−	30.3072i	0.219481	−	0.819113i	−0.765060	−	0.643959i	\(-0.777291\pi\)
							0.984541	−	0.175155i	\(-0.0560425\pi\)
\(41\)	26.6051			0.648905			0.324452	−	0.945902i	\(-0.394820\pi\)
							0.324452	+	0.945902i	\(0.394820\pi\)
\(43\)	−25.3219	+	25.3219i	−0.588880	+	0.588880i	−0.937328	−	0.348448i	\(-0.886709\pi\)
							0.348448	+	0.937328i	\(0.386709\pi\)
\(47\)	−13.1425	+	49.0486i	−0.279628	+	1.04359i	0.673048	+	0.739599i	\(0.264985\pi\)
							−0.952676	+	0.303987i	\(0.901682\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.w.a.17.12	✓	64
3.2	odd	2		210.3.w.b.17.13	yes	64
5.3	odd	4		210.3.w.b.143.14	yes	64
7.5	odd	6	inner	210.3.w.a.47.6	yes	64
15.8	even	4	inner	210.3.w.a.143.6	yes	64
21.5	even	6		210.3.w.b.47.14	yes	64
35.33	even	12		210.3.w.b.173.13	yes	64
105.68	odd	12	inner	210.3.w.a.173.12	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.w.a.17.12	✓	64	1.1	even	1	trivial
210.3.w.a.47.6	yes	64	7.5	odd	6	inner
210.3.w.a.143.6	yes	64	15.8	even	4	inner
210.3.w.a.173.12	yes	64	105.68	odd	12	inner
210.3.w.b.17.13	yes	64	3.2	odd	2
210.3.w.b.47.14	yes	64	21.5	even	6
210.3.w.b.143.14	yes	64	5.3	odd	4
210.3.w.b.173.13	yes	64	35.33	even	12