Embedded newform 648.2.bb.b.11.5

• Label: 648.2.bb.b.11.5
• 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			11.5
Character	\(\chi\)	\(=\)	648.11
Dual form			648.2.bb.b.59.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40370 + 0.172115i) q^{2} +(1.49726 - 0.870756i) q^{3} +(1.94075 - 0.483196i) q^{4} +(-1.56914 - 0.183407i) q^{5} +(-1.95183 + 1.47998i) q^{6} +(-1.80991 - 3.60382i) q^{7} +(-2.64107 + 1.01229i) q^{8} +(1.48357 - 2.60749i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40370	+	0.172115i	−0.992566	+	0.121704i
							\(3\)	1.49726	−	0.870756i	0.864443	−	0.502731i
\(5\)	−1.56914	−	0.183407i	−0.701742	−	0.0820219i								−0.242263	−	0.970211i	\(-0.577890\pi\)
							−0.459479	+	0.888189i	\(0.651964\pi\)
\(7\)	−1.80991	−	3.60382i	−0.684080	−	1.36212i	−0.920607	−	0.390491i	\(-0.872305\pi\)
							0.236527	−	0.971625i	\(-0.423991\pi\)
\(11\)	2.44489	+	1.05462i	0.737163	+	0.317981i	0.731350	−	0.682002i	\(-0.238890\pi\)
							0.00581288	+	0.999983i	\(0.498150\pi\)
\(13\)	0.349308	+	1.47385i	0.0968806	+	0.408772i	0.999778	−	0.0210853i	\(-0.00671217\pi\)
							−0.902897	+	0.429857i	\(0.858564\pi\)
\(17\)	−2.23994	+	0.394961i	−0.543264	+	0.0957921i	−0.438544	−	0.898710i	\(-0.644506\pi\)
							−0.104720	+	0.994502i	\(0.533395\pi\)
\(19\)	0.583246	−	3.30775i	0.133806	−	0.758851i	−0.841878	−	0.539668i	\(-0.818550\pi\)
							0.975684	−	0.219183i	\(-0.0703391\pi\)
\(23\)	−6.16229	−	3.09482i	−1.28493	−	0.645314i	−0.330512	−	0.943802i	\(-0.607222\pi\)
							−0.954414	+	0.298488i	\(0.903518\pi\)
\(29\)	0.282331	+	0.943050i	0.0524275	+	0.175120i	0.980193	−	0.198043i	\(-0.0634585\pi\)
							−0.927766	+	0.373163i	\(0.878273\pi\)
\(31\)	0.272929	+	0.414969i	0.0490196	+	0.0745306i	0.859166	−	0.511697i	\(-0.170983\pi\)
							−0.810147	+	0.586227i	\(0.800613\pi\)
\(37\)	1.07049	−	2.94114i	0.175987	−	0.483520i	−0.820067	−	0.572267i	\(-0.806064\pi\)
							0.996054	+	0.0887473i	\(0.0282864\pi\)
\(41\)	−4.08373	+	3.85280i	−0.637771	+	0.601706i	−0.935663	−	0.352894i	\(-0.885198\pi\)
							0.297893	+	0.954599i	\(0.403716\pi\)
\(43\)	−6.55742	−	8.80815i	−0.999998	−	1.34323i	−0.938559	−	0.345119i	\(-0.887838\pi\)
							−0.0614391	−	0.998111i	\(-0.519569\pi\)
\(47\)	7.83591	+	5.15376i	1.14299	+	0.751753i	0.972338	−	0.233579i	\(-0.0750436\pi\)
							0.170648	+	0.985332i	\(0.445414\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.11.5	✓	1872
8.3	odd	2	inner	648.2.bb.b.11.55	yes	1872
81.59	odd	54	inner	648.2.bb.b.59.55	yes	1872
648.59	even	54	inner	648.2.bb.b.59.5	yes	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.11.5	✓	1872	1.1	even	1	trivial
648.2.bb.b.11.55	yes	1872	8.3	odd	2	inner
648.2.bb.b.59.5	yes	1872	648.59	even	54	inner
648.2.bb.b.59.55	yes	1872	81.59	odd	54	inner