Embedded newform 648.2.bb.b.155.27

• Label: 648.2.bb.b.155.27
• 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			155.27
Character	\(\chi\)	\(=\)	648.155
Dual form			648.2.bb.b.347.27

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.987659 - 1.01219i) q^{2} +(-1.64667 + 0.537094i) q^{3} +(-0.0490596 + 1.99940i) q^{4} +(-1.64663 + 1.08300i) q^{5} +(2.16999 + 1.13628i) q^{6} +(-3.20307 - 3.02194i) q^{7} +(2.07223 - 1.92507i) q^{8} +(2.42306 - 1.76884i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.987659	−	1.01219i	−0.698380	−	0.715727i
							\(3\)	−1.64667	+	0.537094i	−0.950707	+	0.310091i
\(5\)	−1.64663	+	1.08300i	−0.736393	+	0.484333i								−0.861467	−	0.507814i	\(-0.830454\pi\)
							0.125074	+	0.992147i	\(0.460083\pi\)
\(7\)	−3.20307	−	3.02194i	−1.21065	−	1.14219i	−0.985823	−	0.167786i	\(-0.946338\pi\)
							−0.224823	−	0.974400i	\(-0.572180\pi\)
\(11\)	0.555299	+	1.10569i	0.167429	+	0.333378i	0.961663	−	0.274233i	\(-0.0884241\pi\)
							−0.794234	+	0.607612i	\(0.792128\pi\)
\(13\)	−5.40045	+	2.32953i	−1.49781	+	0.646094i	−0.978079	−	0.208233i	\(-0.933229\pi\)
							−0.519735	+	0.854327i	\(0.673969\pi\)
\(17\)	−0.839181	−	1.00010i	−0.203531	−	0.242559i	0.654617	−	0.755960i	\(-0.272830\pi\)
							−0.858149	+	0.513401i	\(0.828385\pi\)
\(19\)	0.869010	+	0.729186i	0.199365	+	0.167287i	0.737005	−	0.675888i	\(-0.236240\pi\)
							−0.537640	+	0.843174i	\(0.680684\pi\)
\(23\)	−2.24693	−	2.38160i	−0.468516	−	0.496598i	0.449423	−	0.893319i	\(-0.351630\pi\)
							−0.917939	+	0.396721i	\(0.870148\pi\)
\(29\)	9.66022	−	1.12912i	1.79386	−	0.209672i	0.847060	−	0.531498i	\(-0.178371\pi\)
							0.946799	+	0.321826i	\(0.104297\pi\)
\(31\)	−0.734321	−	3.09834i	−0.131888	−	0.556479i	−0.998360	−	0.0572528i	\(-0.981766\pi\)
							0.866472	−	0.499226i	\(-0.166382\pi\)
\(37\)	4.78133	−	0.843077i	0.786045	−	0.138601i	0.233802	−	0.972284i	\(-0.424883\pi\)
							0.552243	+	0.833683i	\(0.313772\pi\)
\(41\)	8.90955	+	6.63291i	1.39144	+	1.03589i	0.992927	+	0.118723i	\(0.0378800\pi\)
							0.398510	+	0.917164i	\(0.369527\pi\)
\(43\)	0.456032	−	7.82977i	0.0695443	−	1.19403i	−0.765348	−	0.643616i	\(-0.777433\pi\)
							0.834892	−	0.550413i	\(-0.185530\pi\)
\(47\)	9.28431	+	2.20042i	1.35426	+	0.320965i	0.842745	−	0.538312i	\(-0.180938\pi\)
							0.511511	+	0.859277i	\(0.329086\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.155.27	✓	1872
8.3	odd	2	inner	648.2.bb.b.155.70	yes	1872
81.23	odd	54	inner	648.2.bb.b.347.70	yes	1872
648.347	even	54	inner	648.2.bb.b.347.27	yes	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.155.27	✓	1872	1.1	even	1	trivial
648.2.bb.b.155.70	yes	1872	8.3	odd	2	inner
648.2.bb.b.347.27	yes	1872	648.347	even	54	inner
648.2.bb.b.347.70	yes	1872	81.23	odd	54	inner