Embedded newform 567.2.bm.a.104.39

• Label: 567.2.bm.a.104.39
• 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			104.39
Character	\(\chi\)	\(=\)	567.104
Dual form			567.2.bm.a.398.39

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.169089 + 0.125882i) q^{2} +(-1.62096 + 0.610312i) q^{3} +(-0.560862 - 1.87341i) q^{4} +(1.67346 + 1.10065i) q^{5} +(-0.350915 - 0.100853i) q^{6} +(-1.70736 - 2.02112i) q^{7} +(0.285190 - 0.783554i) q^{8} +(2.25504 - 1.97859i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1260 q - 36 q^{2} - 36 q^{4} - 18 q^{7} - 36 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.169089	+	0.125882i	0.119564	+	0.0890122i	0.655280	−	0.755386i	\(-0.272551\pi\)
							−0.535716	+	0.844398i	\(0.679958\pi\)
\(3\)	−1.62096	+	0.610312i	−0.935863	+	0.352364i
							\(5\)	1.67346	+	1.10065i	0.748396	+	0.492228i	0.865511	−	0.500891i	\(-0.166994\pi\)
−0.117115	+	0.993118i	\(0.537365\pi\)
\(7\)	−1.70736	−	2.02112i	−0.645322	−	0.763911i
							\(11\)	−1.31915	+	2.62664i	−0.397738	+	0.791962i	−0.999980	−	0.00635185i	\(-0.997978\pi\)
0.602242	+	0.798314i	\(0.294274\pi\)
\(13\)	−2.24409	−	0.968007i	−0.622399	−	0.268477i	0.0614385	−	0.998111i	\(-0.480431\pi\)
							−0.683838	+	0.729634i	\(0.739690\pi\)
\(17\)	3.09578	+	2.59767i	0.750838	+	0.630028i	0.935724	−	0.352732i	\(-0.114747\pi\)
							−0.184886	+	0.982760i	\(0.559192\pi\)
\(19\)	−3.76131	−	4.48255i	−0.862904	−	1.02837i	−0.999289	−	0.0377126i	\(-0.987993\pi\)
							0.136385	−	0.990656i	\(-0.456452\pi\)
\(23\)	−6.72507	−	6.34478i	−1.40227	−	1.32298i	−0.883785	−	0.467894i	\(-0.845013\pi\)
							−0.518490	−	0.855084i	\(-0.673506\pi\)
\(29\)	−0.188084	+	1.60916i	−0.0349263	+	0.298814i	0.964474	+	0.264179i	\(0.0851009\pi\)
							−0.999400	+	0.0346349i	\(0.988973\pi\)
\(31\)	−1.34148	+	5.66016i	−0.240937	+	1.01659i	0.709260	+	0.704947i	\(0.249029\pi\)
							−0.950198	+	0.311648i	\(0.899119\pi\)
\(37\)	1.74771	−	9.91174i	0.287321	−	1.62948i	−0.409553	−	0.912286i	\(-0.634315\pi\)
							0.696874	−	0.717193i	\(-0.254574\pi\)
\(41\)	−7.13618	−	9.58555i	−1.11448	−	1.49701i	−0.845584	−	0.533843i	\(-0.820747\pi\)
							−0.268900	−	0.963168i	\(-0.586660\pi\)
\(43\)	−0.238942	−	4.10248i	−0.0364383	−	0.625622i	−0.966253	−	0.257594i	\(-0.917070\pi\)
							0.929815	−	0.368028i	\(-0.119967\pi\)
\(47\)	0.537385	−	0.127363i	0.0783857	−	0.0185778i	−0.191236	−	0.981544i	\(-0.561250\pi\)
							0.269622	+	0.962966i	\(0.413101\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	567.2.bm.a.104.39	✓	1260
7.6	odd	2	inner	567.2.bm.a.104.40	yes	1260
81.74	odd	54	inner	567.2.bm.a.398.40	yes	1260
567.398	even	54	inner	567.2.bm.a.398.39	yes	1260

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
567.2.bm.a.104.39	✓	1260	1.1	even	1	trivial
567.2.bm.a.104.40	yes	1260	7.6	odd	2	inner
567.2.bm.a.398.39	yes	1260	567.398	even	54	inner
567.2.bm.a.398.40	yes	1260	81.74	odd	54	inner