Embedded newform 567.2.bm.a.398.21

• Label: 567.2.bm.a.398.21
• Level: $567$
• Weight: $2$
• Character: 567.398
• 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			398.21
Character	\(\chi\)	\(=\)	567.398
Dual form			567.2.bm.a.104.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.19184 + 0.887291i) q^{2} +(-0.856404 - 1.50551i) q^{3} +(0.0595871 - 0.199035i) q^{4} +(3.59255 - 2.36286i) q^{5} +(2.35652 + 1.03445i) q^{6} +(0.605732 + 2.57548i) q^{7} +(-0.910801 - 2.50240i) q^{8} +(-1.53314 + 2.57866i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1260 q - 36 q^{2} - 36 q^{4} - 18 q^{7} - 36 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(325\)	\(407\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{43}{54}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.19184	+	0.887291i	−0.842757	+	0.627409i	−0.929500	−	0.368821i	\(-0.879761\pi\)
							0.0867434	+	0.996231i	\(0.472354\pi\)
\(3\)	−0.856404	−	1.50551i	−0.494445	−	0.869209i
							\(5\)	3.59255	−	2.36286i	1.60664	−	1.05670i	0.652306	−	0.757955i	\(-0.273802\pi\)
0.954332	−	0.298747i	\(-0.0965688\pi\)
\(7\)	0.605732	+	2.57548i	0.228945	+	0.973439i
							\(11\)	2.45431	+	4.88694i	0.740003	+	1.47347i	0.875771	+	0.482726i	\(0.160353\pi\)
−0.135768	+	0.990741i	\(0.543350\pi\)
\(13\)	−0.235928	+	0.101769i	−0.0654346	+	0.0282257i	−0.428546	−	0.903520i	\(-0.640974\pi\)
							0.363111	+	0.931746i	\(0.381715\pi\)
\(17\)	0.337325	−	0.283049i	0.0818132	−	0.0686495i	−0.600964	−	0.799276i	\(-0.705216\pi\)
							0.682777	+	0.730627i	\(0.260772\pi\)
\(19\)	−1.78112	+	2.12266i	−0.408617	+	0.486971i	−0.930627	−	0.365968i	\(-0.880738\pi\)
							0.522010	+	0.852939i	\(0.325182\pi\)
\(23\)	1.40297	−	1.32363i	0.292539	−	0.275996i	−0.525954	−	0.850513i	\(-0.676292\pi\)
							0.818493	+	0.574517i	\(0.194810\pi\)
\(29\)	0.952584	+	8.14987i	0.176890	+	1.51339i	0.731821	+	0.681497i	\(0.238671\pi\)
							−0.554931	+	0.831897i	\(0.687255\pi\)
\(31\)	−0.260243	−	1.09805i	−0.0467411	−	0.197216i	0.944665	−	0.328037i	\(-0.106387\pi\)
							−0.991406	+	0.130821i	\(0.958239\pi\)
\(37\)	−1.43462	−	8.13616i	−0.235851	−	1.33758i	−0.840814	−	0.541323i	\(-0.817923\pi\)
							0.604964	−	0.796253i	\(-0.293188\pi\)
\(41\)	0.437357	−	0.587473i	0.0683037	−	0.0917478i	−0.766658	−	0.642055i	\(-0.778082\pi\)
							0.834962	+	0.550307i	\(0.185489\pi\)
\(43\)	−0.563597	+	9.67659i	−0.0859478	+	1.47567i	0.629500	+	0.777000i	\(0.283260\pi\)
							−0.715448	+	0.698666i	\(0.753777\pi\)
\(47\)	11.0662	+	2.62274i	1.61417	+	0.382566i	0.935906	−	0.352249i	\(-0.114583\pi\)
							0.678267	+	0.734816i	\(0.262731\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	567.2.bm.a.398.21	yes	1260
7.6	odd	2	inner	567.2.bm.a.398.22	yes	1260
81.23	odd	54	inner	567.2.bm.a.104.22	yes	1260
567.104	even	54	inner	567.2.bm.a.104.21	✓	1260

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
567.2.bm.a.104.21	✓	1260	567.104	even	54	inner
567.2.bm.a.104.22	yes	1260	81.23	odd	54	inner
567.2.bm.a.398.21	yes	1260	1.1	even	1	trivial
567.2.bm.a.398.22	yes	1260	7.6	odd	2	inner