Embedded newform 648.2.bb.b.347.59

• Label: 648.2.bb.b.347.59
• 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			347.59
Character	\(\chi\)	\(=\)	648.347
Dual form			648.2.bb.b.155.59

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.243700 - 1.39306i) q^{2} +(-0.418174 - 1.68081i) q^{3} +(-1.88122 - 0.678976i) q^{4} +(-0.258586 - 0.170075i) q^{5} +(-2.44338 + 0.172926i) q^{6} +(0.0104041 - 0.00981575i) q^{7} +(-1.40431 + 2.45518i) q^{8} +(-2.65026 + 1.40574i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.243700	−	1.39306i	0.172322	−	0.985041i
							\(3\)	−0.418174	−	1.68081i	−0.241433	−	0.970418i
\(5\)	−0.258586	−	0.170075i	−0.115643	−	0.0760597i								0.490370	−	0.871514i	\(-0.336862\pi\)
							−0.606013	+	0.795455i	\(0.707232\pi\)
\(7\)	0.0104041	−	0.00981575i	0.00393237	−	0.00371000i	−0.684273	−	0.729226i	\(-0.739880\pi\)
							0.688205	+	0.725516i	\(0.258399\pi\)
\(11\)	−1.41514	+	2.81777i	−0.426680	+	0.849589i	0.572872	+	0.819645i	\(0.305829\pi\)
							−0.999552	+	0.0299443i	\(0.990467\pi\)
\(13\)	−2.26854	−	0.978551i	−0.629179	−	0.271401i	0.0575188	−	0.998344i	\(-0.481681\pi\)
							−0.686698	+	0.726943i	\(0.740940\pi\)
\(17\)	−1.51006	+	1.79963i	−0.366245	+	0.436473i	−0.917423	−	0.397914i	\(-0.869734\pi\)
							0.551178	+	0.834388i	\(0.314179\pi\)
\(19\)	−4.81770	+	4.04253i	−1.10526	+	0.927420i	−0.997767	−	0.0667865i	\(-0.978725\pi\)
							−0.107489	+	0.994206i	\(0.534281\pi\)
\(23\)	4.28568	−	4.54255i	0.893626	−	0.947188i	−0.105257	−	0.994445i	\(-0.533567\pi\)
							0.998883	+	0.0472572i	\(0.0150480\pi\)
\(29\)	−4.55258	−	0.532120i	−0.845393	−	0.0988122i	−0.317641	−	0.948211i	\(-0.602891\pi\)
							−0.527752	+	0.849399i	\(0.676965\pi\)
\(31\)	−0.306448	+	1.29301i	−0.0550397	+	0.232231i	−0.993612	−	0.112853i	\(-0.964001\pi\)
							0.938572	+	0.345083i	\(0.112149\pi\)
\(37\)	−8.30995	−	1.46527i	−1.36615	−	0.240889i	−0.557986	−	0.829850i	\(-0.688426\pi\)
							−0.808161	+	0.588961i	\(0.799537\pi\)
\(41\)	1.30776	−	0.973589i	0.204237	−	0.152049i	−0.490181	−	0.871621i	\(-0.663069\pi\)
							0.694418	+	0.719572i	\(0.255662\pi\)
\(43\)	0.0545200	+	0.936072i	0.00831422	+	0.142750i	0.999911	+	0.0133788i	\(0.00425872\pi\)
							−0.991596	+	0.129371i	\(0.958704\pi\)
\(47\)	3.27955	−	0.777267i	0.478371	−	0.113376i	0.0156456	−	0.999878i	\(-0.495020\pi\)
							0.462726	+	0.886502i	\(0.346871\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.347.59	yes	1872
8.3	odd	2	inner	648.2.bb.b.347.16	yes	1872
81.74	odd	54	inner	648.2.bb.b.155.16	✓	1872
648.155	even	54	inner	648.2.bb.b.155.59	yes	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.155.16	✓	1872	81.74	odd	54	inner
648.2.bb.b.155.59	yes	1872	648.155	even	54	inner
648.2.bb.b.347.16	yes	1872	8.3	odd	2	inner
648.2.bb.b.347.59	yes	1872	1.1	even	1	trivial