Embedded newform 210.3.q.a.149.9

• Label: 210.3.q.a.149.9
• 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			149.9
Character	\(\chi\)	\(=\)	210.149
Dual form			210.3.q.a.179.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.707107 + 1.22474i) q^{2} +(0.00338259 - 3.00000i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(-4.99937 + 0.0792566i) q^{5} +(3.67184 + 2.12546i) q^{6} +(1.98724 + 6.71199i) q^{7} +2.82843 q^{8} +(-8.99998 - 0.0202955i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 64 q^{4} + 8 q^{6} - 4 q^{9}+O(q^{10}) \)

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
							\(3\)	0.00338259	−	3.00000i	0.00112753	−	0.999999i
\(5\)	−4.99937	+	0.0792566i	−0.999874	+	0.0158513i
							\(7\)	1.98724	+	6.71199i	0.283892	+	0.958856i
\(11\)	1.77949	−	1.02739i	0.161772	−	0.0933991i								−0.416928	−	0.908939i	\(-0.636893\pi\)
							0.578700	+	0.815540i	\(0.303560\pi\)
\(13\)			1.58348i			0.121806i	0.998144	+	0.0609030i	\(0.0193980\pi\)
							−0.998144	+	0.0609030i	\(0.980602\pi\)
\(17\)	15.2509	+	26.4153i	0.897112	+	1.55384i	0.831169	+	0.556020i	\(0.187672\pi\)
							0.0659432	+	0.997823i	\(0.478994\pi\)
\(19\)	−7.36738	+	12.7607i	−0.387757	+	0.671614i	−0.992147	−	0.125074i	\(-0.960083\pi\)
							0.604391	+	0.796688i	\(0.293417\pi\)
\(23\)	−22.1767	+	38.4111i	−0.964203	+	1.67005i	−0.252461	+	0.967607i	\(0.581240\pi\)
							−0.711742	+	0.702441i	\(0.752093\pi\)
\(29\)		−	18.5482i		−	0.639594i	−0.947486	−	0.319797i	\(-0.896385\pi\)
							0.947486	−	0.319797i	\(-0.103615\pi\)
\(31\)	−0.0675789	−	0.117050i	−0.00217997	−	0.00377581i	0.864933	−	0.501887i	\(-0.167361\pi\)
							−0.867113	+	0.498111i	\(0.834027\pi\)
\(37\)	−46.2572	−	26.7066i	−1.25020	−	0.721801i	−0.279047	−	0.960277i	\(-0.590019\pi\)
							−0.971148	+	0.238477i	\(0.923352\pi\)
\(41\)			53.4028i			1.30251i	0.758860	+	0.651254i	\(0.225757\pi\)
							−0.758860	+	0.651254i	\(0.774243\pi\)
\(43\)		−	25.3638i		−	0.589856i	−0.955520	−	0.294928i	\(-0.904704\pi\)
							0.955520	−	0.294928i	\(-0.0952956\pi\)
\(47\)	15.3229	−	26.5400i	0.326018	−	0.564681i	−0.655699	−	0.755022i	\(-0.727626\pi\)
							0.981718	+	0.190341i	\(0.0609595\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.q.a.149.9	yes	64
3.2	odd	2	inner	210.3.q.a.149.30	yes	64
5.4	even	2	inner	210.3.q.a.149.24	yes	64
7.4	even	3	inner	210.3.q.a.179.3	yes	64
15.14	odd	2	inner	210.3.q.a.149.3	✓	64
21.11	odd	6	inner	210.3.q.a.179.24	yes	64
35.4	even	6	inner	210.3.q.a.179.30	yes	64
105.74	odd	6	inner	210.3.q.a.179.9	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.q.a.149.3	✓	64	15.14	odd	2	inner
210.3.q.a.149.9	yes	64	1.1	even	1	trivial
210.3.q.a.149.24	yes	64	5.4	even	2	inner
210.3.q.a.149.30	yes	64	3.2	odd	2	inner
210.3.q.a.179.3	yes	64	7.4	even	3	inner
210.3.q.a.179.9	yes	64	105.74	odd	6	inner
210.3.q.a.179.24	yes	64	21.11	odd	6	inner
210.3.q.a.179.30	yes	64	35.4	even	6	inner