Embedded newform 567.2.bm.a.104.52

• Label: 567.2.bm.a.104.52
• Level: $567$
• Weight: $2$
• Character: 567.104
• Analytic conductor: $4.528$
• Analytic rank: $0$
• Dimension: $1260$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 567 = 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	567.bm (of order \(54\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.52751779461\)
Analytic rank:	\(0\)
Dimension:	\(1260\)
Relative dimension:	\(70\) over \(\Q(\zeta_{54})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{54}]$

Embedding invariants

Embedding label			104.52
Character	\(\chi\)	\(=\)	567.104
Dual form			567.2.bm.a.398.52

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.16421 + 0.866725i) q^{2} +(1.68424 - 0.404131i) q^{3} +(0.0305746 + 0.102126i) q^{4} +(-2.88972 - 1.90060i) q^{5} +(2.31109 + 0.989281i) q^{6} +(0.453018 + 2.60668i) q^{7} +(0.939906 - 2.58237i) q^{8} +(2.67336 - 1.36131i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1260 q - 36 q^{2} - 36 q^{4} - 18 q^{7} - 36 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(407\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{11}{54}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.16421	+	0.866725i	0.823223	+	0.612867i	0.924151	−	0.382027i	\(-0.124774\pi\)
							−0.100928	+	0.994894i	\(0.532181\pi\)
\(3\)	1.68424	−	0.404131i	0.972399	−	0.233325i
							\(5\)	−2.88972	−	1.90060i	−1.29232	−	0.849975i	−0.297815	−	0.954624i	\(-0.596258\pi\)
−0.994509	+	0.104649i	\(0.966628\pi\)
\(7\)	0.453018	+	2.60668i	0.171225	+	0.985232i
							\(11\)	1.50991	−	3.00647i	0.455254	−	0.906486i	−0.542535	−	0.840033i	\(-0.682535\pi\)
0.997790	−	0.0664531i	\(-0.0211683\pi\)
\(13\)	6.12402	+	2.64164i	1.69850	+	0.732660i	0.999852	+	0.0171919i	\(0.00547262\pi\)
							0.698645	+	0.715468i	\(0.253787\pi\)
\(17\)	−1.24870	−	1.04778i	−0.302854	−	0.254125i	0.478677	−	0.877991i	\(-0.341117\pi\)
							−0.781531	+	0.623866i	\(0.785561\pi\)
\(19\)	−4.77804	−	5.69425i	−1.09616	−	1.30635i	−0.948312	−	0.317339i	\(-0.897211\pi\)
							−0.147845	−	0.989011i	\(-0.547234\pi\)
\(23\)	−0.137404	−	0.129634i	−0.0286508	−	0.0270306i	0.671783	−	0.740748i	\(-0.265529\pi\)
							−0.700434	+	0.713717i	\(0.747010\pi\)
\(29\)	−0.561187	+	4.80126i	−0.104210	+	0.891572i	0.835475	+	0.549529i	\(0.185193\pi\)
							−0.939684	+	0.342043i	\(0.888881\pi\)
\(31\)	−1.01521	+	4.28351i	−0.182337	+	0.769341i	0.802613	+	0.596501i	\(0.203443\pi\)
							−0.984950	+	0.172841i	\(0.944705\pi\)
\(37\)	−1.13116	+	6.41510i	−0.185961	+	1.05464i	0.738754	+	0.673976i	\(0.235415\pi\)
							−0.924715	+	0.380661i	\(0.875696\pi\)
\(41\)	3.28702	+	4.41524i	0.513347	+	0.689545i	0.980920	−	0.194410i	\(-0.0622791\pi\)
							−0.467574	+	0.883954i	\(0.654872\pi\)
\(43\)	0.172505	+	2.96179i	0.0263067	+	0.451669i	0.985582	+	0.169197i	\(0.0541174\pi\)
							−0.959276	+	0.282472i	\(0.908846\pi\)
\(47\)	−0.214259	+	0.0507803i	−0.0312529	+	0.00740706i	−0.246213	−	0.969216i	\(-0.579186\pi\)
							0.214960	+	0.976623i	\(0.431038\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	567.2.bm.a.104.52	yes	1260
7.6	odd	2	inner	567.2.bm.a.104.51	✓	1260
81.74	odd	54	inner	567.2.bm.a.398.51	yes	1260
567.398	even	54	inner	567.2.bm.a.398.52	yes	1260

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
567.2.bm.a.104.51	✓	1260	7.6	odd	2	inner
567.2.bm.a.104.52	yes	1260	1.1	even	1	trivial
567.2.bm.a.398.51	yes	1260	81.74	odd	54	inner
567.2.bm.a.398.52	yes	1260	567.398	even	54	inner