Embedded newform 546.4.l.h.211.5

• Label: 546.4.l.h.211.5
• Level: $546$
• Weight: $4$
• Character: 546.211
• Analytic conductor: $32.215$
• Analytic rank: $0$
• Dimension: $10$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(32.2150428631\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Relative dimension:	\(5\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial:	x^10 - x^9 + 218*x^8 - 1187*x^7 + 37612*x^6 - 176472*x^5 + 2657151*x^4 - 12165606*x^3 + 134848233*x^2 - 452924784*x + 1979894016
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 3^{3} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			211.5
Root			\(4.94804 + 8.57026i\) of defining polynomial
Character	\(\chi\)	\(=\)	546.211
Dual form			546.4.l.h.295.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{2} +(1.50000 + 2.59808i) q^{3} +(-2.00000 + 3.46410i) q^{4} +7.89608 q^{5} +(3.00000 - 5.19615i) q^{6} +(3.50000 - 6.06218i) q^{7} +8.00000 q^{8} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{2} + 15 q^{3} - 20 q^{4} - 18 q^{5} + 30 q^{6} + 35 q^{7} + 80 q^{8} - 45 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(157\)	\(365\)	\(379\)
\(\chi(n)\)	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	−	1.73205i	−0.353553	−	0.612372i
							\(3\)	1.50000	+	2.59808i	0.288675	+	0.500000i
\(5\)	7.89608			0.706247										0.353123	−	0.935577i	\(-0.385120\pi\)
							0.353123	+	0.935577i	\(0.385120\pi\)
\(7\)	3.50000	−	6.06218i	0.188982	−	0.327327i
							\(11\)	4.74363	+	8.21620i	0.130023	+	0.225207i	0.923685	−	0.383152i	\(-0.125161\pi\)
−0.793662	+	0.608359i	\(0.791828\pi\)
\(13\)	−36.6387	−	29.2337i	−0.781672	−	0.623690i
							\(17\)	3.93225	−	6.81085i	0.0561006	−	0.0971691i	−0.836611	−	0.547797i	\(-0.815467\pi\)
0.892712	+	0.450628i	\(0.148800\pi\)
\(19\)	76.1933	−	131.971i	0.919997	−	1.59348i	0.120581	−	0.992704i	\(-0.461524\pi\)
							0.799416	−	0.600778i	\(-0.205142\pi\)
\(23\)	−64.0760	−	110.983i	−0.580903	−	1.00615i	−0.995373	−	0.0960906i	\(-0.969366\pi\)
							0.414469	−	0.910063i	\(-0.363967\pi\)
\(29\)	10.4058	+	18.0233i	0.0666312	+	0.115409i	0.897416	−	0.441185i	\(-0.145442\pi\)
							−0.830785	+	0.556593i	\(0.812108\pi\)
\(31\)	197.738			1.14564			0.572819	−	0.819682i	\(-0.305850\pi\)
							0.572819	+	0.819682i	\(0.305850\pi\)
\(37\)	62.6003	+	108.427i	0.278147	+	0.481764i	0.970924	−	0.239387i	\(-0.0769466\pi\)
							−0.692778	+	0.721151i	\(0.743613\pi\)
\(41\)	−64.8606	−	112.342i	−0.247062	−	0.427923i	0.715648	−	0.698462i	\(-0.246132\pi\)
							−0.962709	+	0.270538i	\(0.912798\pi\)
\(43\)	−150.259	+	260.256i	−0.532890	+	0.922992i	0.466373	+	0.884588i	\(0.345561\pi\)
							−0.999262	+	0.0384036i	\(0.987773\pi\)
\(47\)	283.129			0.878695			0.439347	−	0.898317i	\(-0.355210\pi\)
							0.439347	+	0.898317i	\(0.355210\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	546.4.l.h.211.5	✓	10
13.9	even	3	inner	546.4.l.h.295.5	yes	10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.4.l.h.211.5	✓	10	1.1	even	1	trivial
546.4.l.h.295.5	yes	10	13.9	even	3	inner