Embedded newform 648.2.bb.b.155.79

• Label: 648.2.bb.b.155.79
• Level: $648$
• Weight: $2$
• Character: 648.155
• 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			155.79
Character	\(\chi\)	\(=\)	648.155
Dual form			648.2.bb.b.347.79

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.940884 + 1.05581i) q^{2} +(1.70993 - 0.275948i) q^{3} +(-0.229475 + 1.98679i) q^{4} +(-2.15207 + 1.41544i) q^{5} +(1.90019 + 1.54573i) q^{6} +(1.15580 + 1.09044i) q^{7} +(-2.31359 + 1.62706i) q^{8} +(2.84771 - 0.943701i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{43}{54}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.940884	+	1.05581i	0.665305	+	0.746571i
							\(3\)	1.70993	−	0.275948i	0.987227	−	0.159318i
\(5\)	−2.15207	+	1.41544i	−0.962437	+	0.633005i								−0.930425	−	0.366483i	\(-0.880562\pi\)
							−0.0320118	+	0.999487i	\(0.510191\pi\)
\(7\)	1.15580	+	1.09044i	0.436852	+	0.412149i	0.872995	−	0.487728i	\(-0.162174\pi\)
							−0.436143	+	0.899877i	\(0.643656\pi\)
\(11\)	0.0977803	+	0.194697i	0.0294819	+	0.0587033i	0.907891	−	0.419205i	\(-0.137691\pi\)
							−0.878410	+	0.477909i	\(0.841395\pi\)
\(13\)	−2.10183	+	0.906643i	−0.582944	+	0.251457i	−0.667067	−	0.744998i	\(-0.732450\pi\)
							0.0841229	+	0.996455i	\(0.473191\pi\)
\(17\)	3.39492	+	4.04591i	0.823389	+	0.981277i	0.999995	−	0.00300473i	\(-0.000956436\pi\)
							−0.176606	+	0.984282i	\(0.556512\pi\)
\(19\)	−2.51809	−	2.11292i	−0.577688	−	0.484738i	0.306499	−	0.951871i	\(-0.400842\pi\)
							−0.884187	+	0.467133i	\(0.845287\pi\)
\(23\)	2.30738	+	2.44568i	0.481123	+	0.509960i	0.921779	−	0.387716i	\(-0.126736\pi\)
							−0.440656	+	0.897676i	\(0.645254\pi\)
\(29\)	5.78954	−	0.676700i	1.07509	−	0.125660i	0.439924	−	0.898035i	\(-0.355005\pi\)
							0.635167	+	0.772375i	\(0.280931\pi\)
\(31\)	−0.779542	−	3.28915i	−0.140010	−	0.590748i	−0.997135	−	0.0756403i	\(-0.975900\pi\)
							0.857125	−	0.515108i	\(-0.172248\pi\)
\(37\)	−6.62351	+	1.16790i	−1.08890	+	0.192002i	−0.689147	−	0.724621i	\(-0.742015\pi\)
							−0.399751	+	0.916624i	\(0.630904\pi\)
\(41\)	3.98601	+	2.96748i	0.622510	+	0.463442i	0.861622	−	0.507550i	\(-0.169449\pi\)
							−0.239112	+	0.970992i	\(0.576856\pi\)
\(43\)	0.594653	−	10.2098i	0.0906837	−	1.55698i	−0.579617	−	0.814889i	\(-0.696798\pi\)
							0.670300	−	0.742090i	\(-0.266165\pi\)
\(47\)	0.899980	+	0.213299i	0.131276	+	0.0311129i	0.295728	−	0.955272i	\(-0.404438\pi\)
							−0.164453	+	0.986385i	\(0.552586\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.155.79	yes	1872
8.3	odd	2	inner	648.2.bb.b.155.36	✓	1872
81.23	odd	54	inner	648.2.bb.b.347.36	yes	1872
648.347	even	54	inner	648.2.bb.b.347.79	yes	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.155.36	✓	1872	8.3	odd	2	inner
648.2.bb.b.155.79	yes	1872	1.1	even	1	trivial
648.2.bb.b.347.36	yes	1872	81.23	odd	54	inner
648.2.bb.b.347.79	yes	1872	648.347	even	54	inner