Embedded newform 648.2.bb.b.11.1

• Label: 648.2.bb.b.11.1
• Level: $648$
• Weight: $2$
• Character: 648.11
• 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			11.1
Character	\(\chi\)	\(=\)	648.11
Dual form			648.2.bb.b.59.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41419 + 0.00803184i) q^{2} +(0.351152 - 1.69608i) q^{3} +(1.99987 - 0.0227171i) q^{4} +(-0.912599 - 0.106667i) q^{5} +(-0.482973 + 2.40140i) q^{6} +(1.19613 + 2.38169i) q^{7} +(-2.82802 + 0.0481890i) q^{8} +(-2.75338 - 1.19116i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{13}{54}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41419	+	0.00803184i	−0.999984	+	0.00567937i
							\(3\)	0.351152	−	1.69608i	0.202738	−	0.979233i
\(5\)	−0.912599	−	0.106667i	−0.408127	−	0.0477032i								−0.0904470	−	0.995901i	\(-0.528830\pi\)
							−0.317680	+	0.948198i	\(0.602904\pi\)
\(7\)	1.19613	+	2.38169i	0.452095	+	0.900195i	0.998052	+	0.0623818i	\(0.0198696\pi\)
							−0.545958	+	0.837813i	\(0.683834\pi\)
\(11\)	0.381495	+	0.164561i	0.115025	+	0.0496170i	0.452815	−	0.891605i	\(-0.350420\pi\)
							−0.337789	+	0.941222i	\(0.609679\pi\)
\(13\)	1.32022	+	5.57046i	0.366164	+	1.54497i	0.774762	+	0.632252i	\(0.217869\pi\)
							−0.408599	+	0.912714i	\(0.633982\pi\)
\(17\)	0.113818	−	0.0200692i	0.0276049	−	0.00486748i	−0.159829	−	0.987145i	\(-0.551094\pi\)
							0.187434	+	0.982277i	\(0.439983\pi\)
\(19\)	−1.18883	+	6.74218i	−0.272736	+	1.54676i	0.473327	+	0.880887i	\(0.343053\pi\)
							−0.746063	+	0.665876i	\(0.768058\pi\)
\(23\)	2.20148	+	1.10562i	0.459039	+	0.230538i	0.663264	−	0.748386i	\(-0.269171\pi\)
							−0.204224	+	0.978924i	\(0.565467\pi\)
\(29\)	1.91933	+	6.41102i	0.356411	+	1.19050i	0.928821	+	0.370529i	\(0.120824\pi\)
							−0.572410	+	0.819968i	\(0.693991\pi\)
\(31\)	−2.10779	−	3.20474i	−0.378571	−	0.575589i	0.595071	−	0.803673i	\(-0.297124\pi\)
							−0.973641	+	0.228085i	\(0.926754\pi\)
\(37\)	1.27461	−	3.50196i	0.209544	−	0.575718i	−0.789744	−	0.613436i	\(-0.789787\pi\)
							0.999288	+	0.0377183i	\(0.0120089\pi\)
\(41\)	5.32735	−	5.02609i	0.831992	−	0.784944i	−0.146937	−	0.989146i	\(-0.546941\pi\)
							0.978929	+	0.204202i	\(0.0654600\pi\)
\(43\)	4.06173	+	5.45584i	0.619407	+	0.832009i	0.995519	−	0.0945660i	\(-0.0301463\pi\)
							−0.376111	+	0.926575i	\(0.622739\pi\)
\(47\)	−3.78631	−	2.49029i	−0.552289	−	0.363247i	0.242481	−	0.970156i	\(-0.422039\pi\)
							−0.794771	+	0.606910i	\(0.792409\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.11.1	✓	1872
8.3	odd	2	inner	648.2.bb.b.11.51	yes	1872
81.59	odd	54	inner	648.2.bb.b.59.51	yes	1872
648.59	even	54	inner	648.2.bb.b.59.1	yes	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.11.1	✓	1872	1.1	even	1	trivial
648.2.bb.b.11.51	yes	1872	8.3	odd	2	inner
648.2.bb.b.59.1	yes	1872	648.59	even	54	inner
648.2.bb.b.59.51	yes	1872	81.59	odd	54	inner