Embedded newform 648.2.bb.b.347.31

• Label: 648.2.bb.b.347.31
• Level: $648$
• Weight: $2$
• Character: 648.347
• Analytic conductor: $5.174$
• Analytic rank: $0$
• Dimension: $1872$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(5.17430605098\)
Analytic rank:	\(0\)
Dimension:	\(1872\)
Relative dimension:	\(104\) over \(\Q(\zeta_{54})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{54}]$

Embedding invariants

Embedding label			347.31
Character	\(\chi\)	\(=\)	648.347
Dual form			648.2.bb.b.155.31

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.849438 + 1.13069i) q^{2} +(0.956229 - 1.44417i) q^{3} +(-0.556910 - 1.92090i) q^{4} +(1.24556 + 0.819217i) q^{5} +(0.820648 + 2.30793i) q^{6} +(1.85241 - 1.74766i) q^{7} +(2.64500 + 1.00199i) q^{8} +(-1.17125 - 2.76191i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{11}{54}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.849438	+	1.13069i	−0.600643	+	0.799517i
							\(3\)	0.956229	−	1.44417i	0.552079	−	0.833792i
\(5\)	1.24556	+	0.819217i	0.557031	+	0.366365i								0.796593	−	0.604516i	\(-0.206633\pi\)
							−0.239562	+	0.970881i	\(0.577004\pi\)
\(7\)	1.85241	−	1.74766i	0.700147	−	0.660554i	−0.251462	−	0.967867i	\(-0.580911\pi\)
							0.951608	+	0.307313i	\(0.0994299\pi\)
\(11\)	−0.248670	+	0.495144i	−0.0749770	+	0.149291i	−0.927998	−	0.372585i	\(-0.878471\pi\)
							0.853021	+	0.521876i	\(0.174768\pi\)
\(13\)	2.59398	+	1.11893i	0.719441	+	0.310336i	0.724140	−	0.689653i	\(-0.242237\pi\)
							−0.00469972	+	0.999989i	\(0.501496\pi\)
\(17\)	−2.14789	+	2.55976i	−0.520941	+	0.620833i	−0.960803	−	0.277232i	\(-0.910583\pi\)
							0.439862	+	0.898065i	\(0.355027\pi\)
\(19\)	6.16382	−	5.17206i	1.41408	−	1.18655i	0.459653	−	0.888098i	\(-0.347974\pi\)
							0.954424	−	0.298453i	\(-0.0964708\pi\)
\(23\)	−1.36541	+	1.44725i	−0.284708	+	0.301772i	−0.853905	−	0.520428i	\(-0.825772\pi\)
							0.569198	+	0.822201i	\(0.307254\pi\)
\(29\)	−1.03535	−	0.121015i	−0.192259	−	0.0224719i	0.0194184	−	0.999811i	\(-0.493819\pi\)
							−0.211677	+	0.977340i	\(0.567893\pi\)
\(31\)	−1.01091	+	4.26535i	−0.181564	+	0.766079i	0.803699	+	0.595036i	\(0.202862\pi\)
							−0.985263	+	0.171044i	\(0.945286\pi\)
\(37\)	−5.17617	−	0.912699i	−0.850957	−	0.150047i	−0.268874	−	0.963175i	\(-0.586651\pi\)
							−0.582084	+	0.813129i	\(0.697762\pi\)
\(41\)	7.75568	−	5.77389i	1.21123	−	0.901730i	0.214056	−	0.976821i	\(-0.431332\pi\)
							0.997177	+	0.0750918i	\(0.0239250\pi\)
\(43\)	−0.429357	−	7.37178i	−0.0654764	−	1.12419i	−0.857766	−	0.514040i	\(-0.828148\pi\)
							0.792290	−	0.610145i	\(-0.208889\pi\)
\(47\)	12.2789	−	2.91014i	1.79106	−	0.424488i	0.805328	−	0.592830i	\(-0.201989\pi\)
							0.985729	+	0.168341i	\(0.0538411\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.347.31	yes	1872
8.3	odd	2	inner	648.2.bb.b.347.73	yes	1872
81.74	odd	54	inner	648.2.bb.b.155.73	yes	1872
648.155	even	54	inner	648.2.bb.b.155.31	✓	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.155.31	✓	1872	648.155	even	54	inner
648.2.bb.b.155.73	yes	1872	81.74	odd	54	inner
648.2.bb.b.347.31	yes	1872	1.1	even	1	trivial
648.2.bb.b.347.73	yes	1872	8.3	odd	2	inner