Embedded newform 210.3.w.a.17.5

• Label: 210.3.w.a.17.5
• 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			17.5
Character	\(\chi\)	\(=\)	210.17
Dual form			210.3.w.a.173.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36603 - 0.366025i) q^{2} +(-2.25410 - 1.97966i) q^{3} +(1.73205 + 1.00000i) q^{4} +(-3.83488 + 3.20838i) q^{5} +(2.35455 + 3.52932i) q^{6} +(6.92770 - 1.00349i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(1.16192 + 8.92468i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 32 q^{2} - 6 q^{3} - 12 q^{5} + 4 q^{7} - 128 q^{8} - 16 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.25410	−	1.97966i	−0.751366	−	0.659886i
\(5\)	−3.83488	+	3.20838i	−0.766975	+	0.641677i
							\(7\)	6.92770	−	1.00349i	0.989671	−	0.143356i
\(11\)	−8.28471	−	4.78318i	−0.753155	−	0.434834i								0.0736777	−	0.997282i	\(-0.476526\pi\)
							−0.826833	+	0.562448i	\(0.809860\pi\)
\(13\)	9.08652	+	9.08652i	0.698963	+	0.698963i	0.964187	−	0.265224i	\(-0.0854459\pi\)
							−0.265224	+	0.964187i	\(0.585446\pi\)
\(17\)	24.4806	−	6.55954i	1.44003	−	0.385856i	0.547488	−	0.836814i	\(-0.315584\pi\)
							0.892545	+	0.450958i	\(0.148918\pi\)
\(19\)	−10.4928	−	18.1740i	−0.552250	−	0.956526i	−0.998112	−	0.0614237i	\(-0.980436\pi\)
							0.445861	−	0.895102i	\(-0.352897\pi\)
\(23\)	8.88865	−	33.1729i	0.386463	−	1.44230i	−0.449384	−	0.893338i	\(-0.648357\pi\)
							0.835848	−	0.548962i	\(-0.184977\pi\)
\(29\)	−6.08119			−0.209696			−0.104848	−	0.994488i	\(-0.533436\pi\)
							−0.104848	+	0.994488i	\(0.533436\pi\)
\(31\)	7.22620	+	4.17205i	0.233103	+	0.134582i	0.612003	−	0.790856i	\(-0.290364\pi\)
							−0.378900	+	0.925438i	\(0.623697\pi\)
\(37\)	3.80688	−	14.2075i	0.102889	−	0.383986i	−0.895208	−	0.445648i	\(-0.852973\pi\)
							0.998097	+	0.0616620i	\(0.0196401\pi\)
\(41\)	55.3266			1.34943			0.674714	−	0.738079i	\(-0.264267\pi\)
							0.674714	+	0.738079i	\(0.264267\pi\)
\(43\)	57.2947	−	57.2947i	1.33243	−	1.33243i	0.429248	−	0.903187i	\(-0.358779\pi\)
							0.903187	−	0.429248i	\(-0.141221\pi\)
\(47\)	−19.9579	+	74.4838i	−0.424636	+	1.58476i	0.340081	+	0.940396i	\(0.389545\pi\)
							−0.764717	+	0.644366i	\(0.777121\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.w.a.17.5	✓	64
3.2	odd	2		210.3.w.b.17.7	yes	64
5.3	odd	4		210.3.w.b.143.12	yes	64
7.5	odd	6	inner	210.3.w.a.47.2	yes	64
15.8	even	4	inner	210.3.w.a.143.2	yes	64
21.5	even	6		210.3.w.b.47.12	yes	64
35.33	even	12		210.3.w.b.173.7	yes	64
105.68	odd	12	inner	210.3.w.a.173.5	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.w.a.17.5	✓	64	1.1	even	1	trivial
210.3.w.a.47.2	yes	64	7.5	odd	6	inner
210.3.w.a.143.2	yes	64	15.8	even	4	inner
210.3.w.a.173.5	yes	64	105.68	odd	12	inner
210.3.w.b.17.7	yes	64	3.2	odd	2
210.3.w.b.47.12	yes	64	21.5	even	6
210.3.w.b.143.12	yes	64	5.3	odd	4
210.3.w.b.173.7	yes	64	35.33	even	12