Embedded newform 210.3.q.a.149.13

• Label: 210.3.q.a.149.13
• Level: $210$
• Weight: $3$
• Character: 210.149
• Analytic conductor: $5.722$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	210.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.72208555157\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			149.13
Character	\(\chi\)	\(=\)	210.149
Dual form			210.3.q.a.179.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.707107 + 1.22474i) q^{2} +(2.36093 + 1.85094i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(3.08146 - 3.93759i) q^{5} +(-3.93637 + 1.58273i) q^{6} +(0.123332 - 6.99891i) q^{7} +2.82843 q^{8} +(2.14801 + 8.73991i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 64 q^{4} + 8 q^{6} - 4 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.707107	+	1.22474i	−0.353553	+	0.612372i
							\(3\)	2.36093	+	1.85094i	0.786978	+	0.616981i
\(5\)	3.08146	−	3.93759i	0.616291	−	0.787518i
							\(7\)	0.123332	−	6.99891i	0.0176189	−	0.999845i
\(11\)	12.6874	−	7.32507i	1.15340	−	0.665915i								0.203686	−	0.979036i	\(-0.434708\pi\)
							0.949713	+	0.313121i	\(0.101375\pi\)
\(13\)		−	13.9938i		−	1.07644i	−0.842804	−	0.538221i	\(-0.819096\pi\)
							0.842804	−	0.538221i	\(-0.180904\pi\)
\(17\)	0.794498	+	1.37611i	0.0467352	+	0.0809477i	0.888447	−	0.458980i	\(-0.151785\pi\)
							−0.841712	+	0.539927i	\(0.818452\pi\)
\(19\)	−6.75399	+	11.6983i	−0.355473	+	0.615698i	−0.987199	−	0.159494i	\(-0.949014\pi\)
							0.631725	+	0.775192i	\(0.282347\pi\)
\(23\)	−2.35231	+	4.07433i	−0.102274	+	0.177145i	−0.912621	−	0.408806i	\(-0.865945\pi\)
							0.810347	+	0.585950i	\(0.199279\pi\)
\(29\)			41.4490i			1.42928i	0.699494	+	0.714638i	\(0.253409\pi\)
							−0.699494	+	0.714638i	\(0.746591\pi\)
\(31\)	18.3544	+	31.7908i	0.592079	+	1.02551i	0.993952	+	0.109815i	\(0.0350258\pi\)
							−0.401873	+	0.915695i	\(0.631641\pi\)
\(37\)	−8.16507	−	4.71410i	−0.220678	−	0.127408i	0.385586	−	0.922672i	\(-0.373999\pi\)
							−0.606264	+	0.795264i	\(0.707332\pi\)
\(41\)			28.7694i			0.701693i	0.936433	+	0.350846i	\(0.114106\pi\)
							−0.936433	+	0.350846i	\(0.885894\pi\)
\(43\)		−	72.9004i		−	1.69536i	−0.530510	−	0.847679i	\(-0.678000\pi\)
							0.530510	−	0.847679i	\(-0.322000\pi\)
\(47\)	4.73110	−	8.19450i	0.100662	−	0.174351i	−0.811296	−	0.584636i	\(-0.801237\pi\)
							0.911957	+	0.410285i	\(0.134571\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.q.a.149.13	yes	64
3.2	odd	2	inner	210.3.q.a.149.23	yes	64
5.4	even	2	inner	210.3.q.a.149.20	yes	64
7.4	even	3	inner	210.3.q.a.179.10	yes	64
15.14	odd	2	inner	210.3.q.a.149.10	✓	64
21.11	odd	6	inner	210.3.q.a.179.20	yes	64
35.4	even	6	inner	210.3.q.a.179.23	yes	64
105.74	odd	6	inner	210.3.q.a.179.13	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.q.a.149.10	✓	64	15.14	odd	2	inner
210.3.q.a.149.13	yes	64	1.1	even	1	trivial
210.3.q.a.149.20	yes	64	5.4	even	2	inner
210.3.q.a.149.23	yes	64	3.2	odd	2	inner
210.3.q.a.179.10	yes	64	7.4	even	3	inner
210.3.q.a.179.13	yes	64	105.74	odd	6	inner
210.3.q.a.179.20	yes	64	21.11	odd	6	inner
210.3.q.a.179.23	yes	64	35.4	even	6	inner