Embedded newform 567.2.bm.a.104.63

• Label: 567.2.bm.a.104.63
• 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			104.63
Character	\(\chi\)	\(=\)	567.104
Dual form			567.2.bm.a.398.63

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.82857 + 1.36132i) q^{2} +(-1.59311 - 0.679700i) q^{3} +(0.916874 + 3.06257i) q^{4} +(-1.51959 - 0.999447i) q^{5} +(-1.98783 - 3.41162i) q^{6} +(0.330764 - 2.62499i) q^{7} +(-0.933193 + 2.56393i) q^{8} +(2.07602 + 2.16568i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1260 q - 36 q^{2} - 36 q^{4} - 18 q^{7} - 36 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.82857	+	1.36132i	1.29300	+	0.962600i	0.999979	+	0.00649273i	\(0.00206672\pi\)
							0.293017	+	0.956107i	\(0.405341\pi\)
\(3\)	−1.59311	−	0.679700i	−0.919784	−	0.392425i
							\(5\)	−1.51959	−	0.999447i	−0.679579	−	0.446966i	0.162222	−	0.986754i	\(-0.448134\pi\)
−0.841801	+	0.539788i	\(0.818504\pi\)
\(7\)	0.330764	−	2.62499i	0.125017	−	0.992155i
							\(11\)	2.17460	−	4.32999i	0.655667	−	1.30554i	−0.282470	−	0.959276i	\(-0.591154\pi\)
0.938137	−	0.346264i	\(-0.112550\pi\)
\(13\)	2.75921	+	1.19021i	0.765267	+	0.330104i	0.742685	−	0.669641i	\(-0.233552\pi\)
							0.0225821	+	0.999745i	\(0.492811\pi\)
\(17\)	4.41278	+	3.70276i	1.07026	+	0.898051i	0.995076	−	0.0991185i	\(-0.0316023\pi\)
							0.0751804	+	0.997170i	\(0.476047\pi\)
\(19\)	−0.529206	−	0.630683i	−0.121408	−	0.144689i	0.701917	−	0.712259i	\(-0.252328\pi\)
							−0.823325	+	0.567571i	\(0.807883\pi\)
\(23\)	−4.23219	−	3.99286i	−0.882472	−	0.832570i	0.104323	−	0.994543i	\(-0.466733\pi\)
							−0.986795	+	0.161974i	\(0.948214\pi\)
\(29\)	−0.770835	+	6.59492i	−0.143141	+	1.22465i	0.710468	+	0.703730i	\(0.248483\pi\)
							−0.853608	+	0.520916i	\(0.825591\pi\)
\(31\)	1.42451	−	6.01050i	0.255850	−	1.07952i	−0.681690	−	0.731641i	\(-0.738755\pi\)
							0.937541	−	0.347876i	\(-0.113097\pi\)
\(37\)	0.124077	−	0.703676i	0.0203981	−	0.115684i	−0.972909	−	0.231190i	\(-0.925738\pi\)
							0.993307	+	0.115507i	\(0.0368492\pi\)
\(41\)	−3.15274	−	4.23486i	−0.492375	−	0.661374i	0.484598	−	0.874737i	\(-0.338966\pi\)
							−0.976973	+	0.213362i	\(0.931558\pi\)
\(43\)	0.270195	+	4.63908i	0.0412044	+	0.707452i	0.954142	+	0.299354i	\(0.0967710\pi\)
							−0.912938	+	0.408099i	\(0.866192\pi\)
\(47\)	−3.76267	+	0.891770i	−0.548842	+	0.130078i	−0.495676	−	0.868508i	\(-0.665080\pi\)
							−0.0531665	+	0.998586i	\(0.516931\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	567.2.bm.a.104.63	✓	1260
7.6	odd	2	inner	567.2.bm.a.104.64	yes	1260
81.74	odd	54	inner	567.2.bm.a.398.64	yes	1260
567.398	even	54	inner	567.2.bm.a.398.63	yes	1260

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
567.2.bm.a.104.63	✓	1260	1.1	even	1	trivial
567.2.bm.a.104.64	yes	1260	7.6	odd	2	inner
567.2.bm.a.398.63	yes	1260	567.398	even	54	inner
567.2.bm.a.398.64	yes	1260	81.74	odd	54	inner