Embedded newform 210.3.q.a.149.1

• Label: 210.3.q.a.149.1
• 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			149.1
Character	\(\chi\)	\(=\)	210.149
Dual form			210.3.q.a.179.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.707107 + 1.22474i) q^{2} +(-2.99996 + 0.0151730i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(-3.19601 - 3.84519i) q^{5} +(2.10271 - 3.68492i) q^{6} +(-6.79135 + 1.69633i) q^{7} +2.82843 q^{8} +(8.99954 - 0.0910371i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 64 q^{4} + 8 q^{6} - 4 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.99996	+	0.0151730i	−0.999987	+	0.00505768i
\(5\)	−3.19601	−	3.84519i	−0.639202	−	0.769039i
							\(7\)	−6.79135	+	1.69633i	−0.970193	+	0.242333i
\(11\)	9.12651	−	5.26919i	0.829682	−	0.479017i								−0.0240616	−	0.999710i	\(-0.507660\pi\)
							0.853744	+	0.520693i	\(0.174326\pi\)
\(13\)			15.0249i			1.15576i	0.816121	+	0.577882i	\(0.196120\pi\)
							−0.816121	+	0.577882i	\(0.803880\pi\)
\(17\)	9.29631	+	16.1017i	0.546842	+	0.947157i	0.998489	+	0.0549606i	\(0.0175033\pi\)
							−0.451647	+	0.892197i	\(0.649163\pi\)
\(19\)	0.584159	−	1.01179i	0.0307452	−	0.0532523i	−0.850243	−	0.526390i	\(-0.823545\pi\)
							0.880989	+	0.473137i	\(0.156879\pi\)
\(23\)	21.5754	−	37.3698i	0.938063	−	1.62477i	0.168984	−	0.985619i	\(-0.445951\pi\)
							0.769079	−	0.639154i	\(-0.220715\pi\)
\(29\)			12.2080i			0.420967i	0.977597	+	0.210484i	\(0.0675038\pi\)
							−0.977597	+	0.210484i	\(0.932496\pi\)
\(31\)	25.0391	+	43.3691i	0.807714	+	1.39900i	0.914443	+	0.404714i	\(0.132629\pi\)
							−0.106729	+	0.994288i	\(0.534038\pi\)
\(37\)	37.6647	+	21.7457i	1.01796	+	0.587722i	0.913514	−	0.406808i	\(-0.133358\pi\)
							0.104451	+	0.994530i	\(0.466692\pi\)
\(41\)			22.5970i			0.551147i	0.961280	+	0.275573i	\(0.0888677\pi\)
							−0.961280	+	0.275573i	\(0.911132\pi\)
\(43\)		−	69.7934i		−	1.62310i	−0.584281	−	0.811551i	\(-0.698624\pi\)
							0.584281	−	0.811551i	\(-0.301376\pi\)
\(47\)	−12.2369	+	21.1950i	−0.260360	+	0.450957i	−0.966338	−	0.257277i	\(-0.917175\pi\)
							0.705977	+	0.708234i	\(0.250508\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.q.a.149.1	✓	64
3.2	odd	2	inner	210.3.q.a.149.22	yes	64
5.4	even	2	inner	210.3.q.a.149.32	yes	64
7.4	even	3	inner	210.3.q.a.179.11	yes	64
15.14	odd	2	inner	210.3.q.a.149.11	yes	64
21.11	odd	6	inner	210.3.q.a.179.32	yes	64
35.4	even	6	inner	210.3.q.a.179.22	yes	64
105.74	odd	6	inner	210.3.q.a.179.1	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.q.a.149.1	✓	64	1.1	even	1	trivial
210.3.q.a.149.11	yes	64	15.14	odd	2	inner
210.3.q.a.149.22	yes	64	3.2	odd	2	inner
210.3.q.a.149.32	yes	64	5.4	even	2	inner
210.3.q.a.179.1	yes	64	105.74	odd	6	inner
210.3.q.a.179.11	yes	64	7.4	even	3	inner
210.3.q.a.179.22	yes	64	35.4	even	6	inner
210.3.q.a.179.32	yes	64	21.11	odd	6	inner