Embedded newform 546.4.s.a.127.5

• Label: 546.4.s.a.127.5
• Level: $546$
• Weight: $4$
• Character: 546.127
• Analytic conductor: $32.215$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	546.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(32.2150428631\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(10\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial:	x^20 + 1388*x^18 + 806954*x^16 + 255183238*x^14 + 47714604791*x^12 + 5370647791638*x^10 + 354442825910071*x^8 + 12732087350994070*x^6 + 214746587484940741*x^4 + 1171569470005573456*x^2 + 422518488802265284
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			127.5
Root			\(16.5747i\) of defining polynomial
Character	\(\chi\)	\(=\)	546.127
Dual form			546.4.s.a.43.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.73205 + 1.00000i) q^{2} +(-1.50000 - 2.59808i) q^{3} +(2.00000 - 3.46410i) q^{4} +17.5747i q^{5} +(5.19615 + 3.00000i) q^{6} +(6.06218 + 3.50000i) q^{7} +8.00000i q^{8} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 30 q^{3} + 40 q^{4} - 90 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).

\(n\)	\(157\)	\(365\)	\(379\)
\(\chi(n)\)	\(1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.73205	+	1.00000i	−0.612372	+	0.353553i
							\(3\)	−1.50000	−	2.59808i	−0.288675	−	0.500000i
\(5\)			17.5747i			1.57193i								0.618272	+	0.785964i	\(0.287833\pi\)
							−0.618272	+	0.785964i	\(0.712167\pi\)
\(7\)	6.06218	+	3.50000i	0.327327	+	0.188982i
							\(11\)	−6.24601	+	3.60613i	−0.171204	+	0.0988445i	−0.583153	−	0.812362i	\(-0.698181\pi\)
0.411950	+	0.911207i	\(0.364848\pi\)
\(13\)	34.3917	+	31.8466i	0.733735	+	0.679436i
							\(17\)	−51.0916	+	88.4932i	−0.728914	+	1.26252i	0.228429	+	0.973561i	\(0.426641\pi\)
−0.957343	+	0.288955i	\(0.906692\pi\)
\(19\)	50.3531	+	29.0713i	0.607989	+	0.351022i	0.772178	−	0.635407i	\(-0.219167\pi\)
							−0.164189	+	0.986429i	\(0.552501\pi\)
\(23\)	−3.10875	−	5.38452i	−0.0281835	−	0.0488152i	0.851590	−	0.524209i	\(-0.175639\pi\)
							−0.879773	+	0.475394i	\(0.842306\pi\)
\(29\)	40.4407	+	70.0453i	0.258953	+	0.448520i	0.965962	−	0.258685i	\(-0.0832890\pi\)
							−0.707008	+	0.707205i	\(0.749956\pi\)
\(31\)		−	64.8623i		−	0.375794i	−0.982189	−	0.187897i	\(-0.939833\pi\)
							0.982189	−	0.187897i	\(-0.0601671\pi\)
\(37\)	326.257	−	188.364i	1.44963	−	0.836944i	0.451170	−	0.892438i	\(-0.351007\pi\)
							0.998459	+	0.0554938i	\(0.0176733\pi\)
\(41\)	−70.9247	+	40.9484i	−0.270160	+	0.155977i	−0.628960	−	0.777437i	\(-0.716519\pi\)
							0.358800	+	0.933414i	\(0.383186\pi\)
\(43\)	186.016	−	322.190i	0.659703	−	1.14264i	−0.320990	−	0.947083i	\(-0.604015\pi\)
							0.980692	−	0.195556i	\(-0.0626512\pi\)
\(47\)			26.6251i			0.0826313i	0.999146	+	0.0413157i	\(0.0131549\pi\)
							−0.999146	+	0.0413157i	\(0.986845\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	546.4.s.a.127.5	yes	20
13.4	even	6	inner	546.4.s.a.43.1	✓	20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.4.s.a.43.1	✓	20	13.4	even	6	inner
546.4.s.a.127.5	yes	20	1.1	even	1	trivial