Embedded newform 210.3.w.b.173.4

• Label: 210.3.w.b.173.4
• Level: $210$
• Weight: $3$
• Character: 210.173
• 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			173.4
Character	\(\chi\)	\(=\)	210.173
Dual form			210.3.w.b.17.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36603 - 0.366025i) q^{2} +(-2.20828 + 2.03064i) q^{3} +(1.73205 - 1.00000i) q^{4} +(-4.28576 - 2.57532i) q^{5} +(-2.27330 + 3.58219i) q^{6} +(-1.16075 + 6.90309i) q^{7} +(2.00000 - 2.00000i) q^{8} +(0.752986 - 8.96845i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 32 q^{2} + 6 q^{3} + 12 q^{5} + 4 q^{7} + 128 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(31\)	\(71\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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
							\(3\)	−2.20828	+	2.03064i	−0.736093	+	0.676881i
\(5\)	−4.28576	−	2.57532i	−0.857151	−	0.515064i
							\(7\)	−1.16075	+	6.90309i	−0.165821	+	0.986156i
\(11\)	−7.83418	+	4.52306i	−0.712198	+	0.411188i								−0.811874	−	0.583832i	\(-0.801553\pi\)
							0.0996765	+	0.995020i	\(0.468219\pi\)
\(13\)	−15.3607	+	15.3607i	−1.18159	+	1.18159i	−0.202260	+	0.979332i	\(0.564829\pi\)
							−0.979332	+	0.202260i	\(0.935171\pi\)
\(17\)	−25.1750	−	6.74562i	−1.48088	−	0.396801i	−0.574235	−	0.818690i	\(-0.694701\pi\)
							−0.906648	+	0.421889i	\(0.861367\pi\)
\(19\)	−0.500582	+	0.867034i	−0.0263464	+	0.0456333i	−0.878898	−	0.477010i	\(-0.841721\pi\)
							0.852552	+	0.522643i	\(0.175054\pi\)
\(23\)	−3.42602	−	12.7861i	−0.148957	−	0.555916i	−0.999547	−	0.0300867i	\(-0.990422\pi\)
							0.850590	−	0.525829i	\(-0.176245\pi\)
\(29\)	−8.69269			−0.299748			−0.149874	−	0.988705i	\(-0.547887\pi\)
							−0.149874	+	0.988705i	\(0.547887\pi\)
\(31\)	29.5586	−	17.0656i	0.953502	−	0.550504i	0.0593347	−	0.998238i	\(-0.481102\pi\)
							0.894167	+	0.447734i	\(0.147769\pi\)
\(37\)	4.54478	+	16.9613i	0.122832	+	0.458414i	0.999753	−	0.0222195i	\(-0.00707326\pi\)
							−0.876921	+	0.480634i	\(0.840407\pi\)
\(41\)	18.4160			0.449172			0.224586	−	0.974454i	\(-0.427897\pi\)
							0.224586	+	0.974454i	\(0.427897\pi\)
\(43\)	24.0320	+	24.0320i	0.558885	+	0.558885i	0.928990	−	0.370105i	\(-0.120678\pi\)
							−0.370105	+	0.928990i	\(0.620678\pi\)
\(47\)	8.15043	+	30.4178i	0.173413	+	0.647188i	0.996816	+	0.0797306i	\(0.0254060\pi\)
							−0.823403	+	0.567457i	\(0.807927\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.w.b.173.4	yes	64
3.2	odd	2		210.3.w.a.173.7	yes	64
5.2	odd	4		210.3.w.a.47.13	yes	64
7.3	odd	6	inner	210.3.w.b.143.2	yes	64
15.2	even	4	inner	210.3.w.b.47.2	yes	64
21.17	even	6		210.3.w.a.143.13	yes	64
35.17	even	12		210.3.w.a.17.7	✓	64
105.17	odd	12	inner	210.3.w.b.17.4	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.w.a.17.7	✓	64	35.17	even	12
210.3.w.a.47.13	yes	64	5.2	odd	4
210.3.w.a.143.13	yes	64	21.17	even	6
210.3.w.a.173.7	yes	64	3.2	odd	2
210.3.w.b.17.4	yes	64	105.17	odd	12	inner
210.3.w.b.47.2	yes	64	15.2	even	4	inner
210.3.w.b.143.2	yes	64	7.3	odd	6	inner
210.3.w.b.173.4	yes	64	1.1	even	1	trivial