Embedded newform 648.2.bd.a.133.86

• Label: 648.2.bd.a.133.86
• Level: $648$
• Weight: $2$
• Character: 648.133
• Analytic conductor: $5.174$
• Analytic rank: $0$
• Dimension: $1908$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	648.bd (of order \(54\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.17430605098\)
Analytic rank:	\(0\)
Dimension:	\(1908\)
Relative dimension:	\(106\) over \(\Q(\zeta_{54})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{54}]$

Embedding invariants

Embedding label			133.86
Character	\(\chi\)	\(=\)	648.133
Dual form			648.2.bd.a.229.86

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.16158 + 0.806678i) q^{2} +(0.872911 + 1.49600i) q^{3} +(0.698542 + 1.87404i) q^{4} +(-0.321381 + 0.303208i) q^{5} +(-0.192835 + 2.44189i) q^{6} +(-4.51253 + 0.527439i) q^{7} +(-0.700336 + 2.74035i) q^{8} +(-1.47605 + 2.61176i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1908 q - 18 q^{2} - 18 q^{4} - 18 q^{6} - 36 q^{7} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{5}{27}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.16158	+	0.806678i	0.821362	+	0.570407i
							\(3\)	0.872911	+	1.49600i	0.503976	+	0.863718i
\(5\)	−0.321381	+	0.303208i	−0.143726	+	0.135599i								−0.754747	−	0.656016i	\(-0.772241\pi\)
							0.611021	+	0.791614i	\(0.290759\pi\)
\(7\)	−4.51253	+	0.527439i	−1.70557	+	0.199353i	−0.912303	−	0.409515i	\(-0.865698\pi\)
							−0.793271	+	0.608868i	\(0.791624\pi\)
\(11\)	4.81375	−	1.44114i	1.45140	−	0.434521i	0.538474	−	0.842642i	\(-0.319001\pi\)
							0.912928	+	0.408121i	\(0.133816\pi\)
\(13\)	0.307645	−	0.0179183i	0.0853253	−	0.00496963i	−0.0154290	−	0.999881i	\(-0.504911\pi\)
							0.100754	+	0.994911i	\(0.467874\pi\)
\(17\)	0.824301	+	0.300021i	0.199922	+	0.0727658i	0.440041	−	0.897978i	\(-0.354964\pi\)
							−0.240118	+	0.970744i	\(0.577186\pi\)
\(19\)	0.0936536	+	0.257311i	0.0214856	+	0.0590313i	0.949972	−	0.312336i	\(-0.101111\pi\)
							−0.928486	+	0.371367i	\(0.878889\pi\)
\(23\)	−2.02800	−	0.237040i	−0.422868	−	0.0494262i	−0.0980020	−	0.995186i	\(-0.531245\pi\)
							−0.324866	+	0.945760i	\(0.605319\pi\)
\(29\)	0.115900	+	0.230777i	0.0215222	+	0.0428541i	0.904137	−	0.427243i	\(-0.140515\pi\)
							−0.882615	+	0.470097i	\(0.844219\pi\)
\(31\)	3.02954	−	4.06938i	0.544122	−	0.730883i	−0.441980	−	0.897025i	\(-0.645724\pi\)
							0.986102	+	0.166142i	\(0.0531311\pi\)
\(37\)	3.87260	+	4.61518i	0.636651	+	0.758731i	0.983837	−	0.179065i	\(-0.0573073\pi\)
							−0.347186	+	0.937796i	\(0.612863\pi\)
\(41\)	1.75101	−	1.15165i	0.273461	−	0.179858i	−0.405370	−	0.914153i	\(-0.632857\pi\)
							0.678831	+	0.734295i	\(0.262487\pi\)
\(43\)	2.07713	−	8.76410i	0.316759	−	1.33651i	−0.548904	−	0.835885i	\(-0.684955\pi\)
							0.865663	−	0.500627i	\(-0.166897\pi\)
\(47\)	5.08513	+	6.83052i	0.741743	+	0.996333i	0.999546	+	0.0301357i	\(0.00959395\pi\)
							−0.257803	+	0.966197i	\(0.582999\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bd.a.133.86	✓	1908
8.5	even	2	inner	648.2.bd.a.133.89	yes	1908
81.67	even	27	inner	648.2.bd.a.229.89	yes	1908
648.229	even	54	inner	648.2.bd.a.229.86	yes	1908

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bd.a.133.86	✓	1908	1.1	even	1	trivial
648.2.bd.a.133.89	yes	1908	8.5	even	2	inner
648.2.bd.a.229.86	yes	1908	648.229	even	54	inner
648.2.bd.a.229.89	yes	1908	81.67	even	27	inner