Embedded newform 567.2.bm.a.104.35

• Label: 567.2.bm.a.104.35
• 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			104.35
Character	\(\chi\)	\(=\)	567.104
Dual form			567.2.bm.a.398.35

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.140879 - 0.104880i) q^{2} +(-1.72167 - 0.189342i) q^{3} +(-0.564760 - 1.88643i) q^{4} +(0.0783509 + 0.0515322i) q^{5} +(0.222688 + 0.207244i) q^{6} +(0.998637 + 2.45005i) q^{7} +(-0.238426 + 0.655070i) q^{8} +(2.92830 + 0.651970i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1260 q - 36 q^{2} - 36 q^{4} - 18 q^{7} - 36 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.140879	−	0.104880i	−0.0996163	−	0.0741616i	0.546198	−	0.837656i	\(-0.316075\pi\)
							−0.645814	+	0.763494i	\(0.723482\pi\)
\(3\)	−1.72167	−	0.189342i	−0.994007	−	0.109317i
							\(5\)	0.0783509	+	0.0515322i	0.0350396	+	0.0230459i	0.566908	−	0.823781i	\(-0.308140\pi\)
−0.531868	+	0.846827i	\(0.678510\pi\)
\(7\)	0.998637	+	2.45005i	0.377449	+	0.926030i
							\(11\)	1.86713	−	3.71776i	0.562960	−	1.12095i	−0.414544	−	0.910029i	\(-0.636059\pi\)
0.977504	−	0.210917i	\(-0.0676449\pi\)
\(13\)	1.51348	+	0.652850i	0.419763	+	0.181068i	0.595475	−	0.803374i	\(-0.296964\pi\)
							−0.175713	+	0.984441i	\(0.556223\pi\)
\(17\)	3.37697	+	2.83362i	0.819036	+	0.687253i	0.952746	−	0.303767i	\(-0.0982446\pi\)
							−0.133710	+	0.991021i	\(0.542689\pi\)
\(19\)	−4.53414	−	5.40358i	−1.04020	−	1.23967i	−0.970247	−	0.242117i	\(-0.922158\pi\)
							−0.0699570	−	0.997550i	\(-0.522286\pi\)
\(23\)	2.52387	+	2.38115i	0.526264	+	0.496504i	0.902896	−	0.429859i	\(-0.141437\pi\)
							−0.376633	+	0.926363i	\(0.622918\pi\)
\(29\)	0.989056	−	8.46192i	0.183663	−	1.57134i	−0.515825	−	0.856694i	\(-0.672514\pi\)
							0.699488	−	0.714645i	\(-0.253412\pi\)
\(31\)	1.83204	−	7.72997i	0.329043	−	1.38834i	−0.517345	−	0.855777i	\(-0.673080\pi\)
							0.846388	−	0.532566i	\(-0.178772\pi\)
\(37\)	−0.169295	+	0.960122i	−0.0278320	+	0.157843i	−0.995556	−	0.0941681i	\(-0.969981\pi\)
							0.967724	+	0.252011i	\(0.0810920\pi\)
\(41\)	2.23213	+	2.99827i	0.348600	+	0.468251i	0.941494	−	0.337029i	\(-0.109422\pi\)
							−0.592894	+	0.805280i	\(0.702015\pi\)
\(43\)	−0.367151	−	6.30374i	−0.0559900	−	0.961312i	−0.902975	−	0.429694i	\(-0.858621\pi\)
							0.846985	−	0.531618i	\(-0.178416\pi\)
\(47\)	−3.45097	+	0.817894i	−0.503375	+	0.119302i	−0.474455	−	0.880280i	\(-0.657355\pi\)
							−0.0289200	+	0.999582i	\(0.509207\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	567.2.bm.a.104.35	✓	1260
7.6	odd	2	inner	567.2.bm.a.104.36	yes	1260
81.74	odd	54	inner	567.2.bm.a.398.36	yes	1260
567.398	even	54	inner	567.2.bm.a.398.35	yes	1260

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
567.2.bm.a.104.35	✓	1260	1.1	even	1	trivial
567.2.bm.a.104.36	yes	1260	7.6	odd	2	inner
567.2.bm.a.398.35	yes	1260	567.398	even	54	inner
567.2.bm.a.398.36	yes	1260	81.74	odd	54	inner