Embedded newform 648.2.bb.b.347.25

• Label: 648.2.bb.b.347.25
• 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			347.25
Character	\(\chi\)	\(=\)	648.347
Dual form			648.2.bb.b.155.25

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.04169 - 0.956491i) q^{2} +(1.63974 - 0.557908i) q^{3} +(0.170251 + 1.99274i) q^{4} +(-1.60979 - 1.05877i) q^{5} +(-2.24174 - 0.987224i) q^{6} +(-1.00602 + 0.949130i) q^{7} +(1.72869 - 2.23867i) q^{8} +(2.37748 - 1.82965i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.04169	−	0.956491i	−0.736589	−	0.676341i
							\(3\)	1.63974	−	0.557908i	0.946703	−	0.322108i
\(5\)	−1.60979	−	1.05877i	−0.719918	−	0.473498i								0.135934	−	0.990718i	\(-0.456596\pi\)
							−0.855852	+	0.517220i	\(0.826967\pi\)
\(7\)	−1.00602	+	0.949130i	−0.380240	+	0.358738i	−0.852505	−	0.522719i	\(-0.824918\pi\)
							0.472266	+	0.881456i	\(0.343436\pi\)
\(11\)	0.968063	−	1.92757i	0.291882	−	0.581185i	−0.699124	−	0.715000i	\(-0.746427\pi\)
							0.991006	+	0.133815i	\(0.0427229\pi\)
\(13\)	5.48198	+	2.36470i	1.52043	+	0.655849i	0.982351	−	0.187046i	\(-0.0598914\pi\)
							0.538078	+	0.842895i	\(0.319151\pi\)
\(17\)	2.40897	−	2.87090i	0.584260	−	0.696295i	−0.390232	−	0.920717i	\(-0.627605\pi\)
							0.974492	+	0.224422i	\(0.0720495\pi\)
\(19\)	−0.268535	+	0.225327i	−0.0616061	+	0.0516937i	−0.673071	−	0.739578i	\(-0.735025\pi\)
							0.611465	+	0.791272i	\(0.290581\pi\)
\(23\)	−0.552084	+	0.585174i	−0.115117	+	0.122017i	−0.782369	−	0.622815i	\(-0.785989\pi\)
							0.667252	+	0.744832i	\(0.267470\pi\)
\(29\)	−6.60001	−	0.771431i	−1.22559	−	0.143251i	−0.521432	−	0.853293i	\(-0.674602\pi\)
							−0.704159	+	0.710042i	\(0.748676\pi\)
\(31\)	2.41758	−	10.2006i	0.434211	−	1.83208i	−0.111052	−	0.993815i	\(-0.535422\pi\)
							0.545263	−	0.838265i	\(-0.316430\pi\)
\(37\)	−7.36298	−	1.29829i	−1.21047	−	0.213438i	−0.468248	−	0.883597i	\(-0.655115\pi\)
							−0.742217	+	0.670159i	\(0.766226\pi\)
\(41\)	5.62821	−	4.19004i	0.878978	−	0.654375i	−0.0600575	−	0.998195i	\(-0.519128\pi\)
							0.939036	+	0.343820i	\(0.111721\pi\)
\(43\)	0.0630875	+	1.08317i	0.00962075	+	0.165182i	0.999696	+	0.0246455i	\(0.00784569\pi\)
							−0.990076	+	0.140536i	\(0.955117\pi\)
\(47\)	3.10073	−	0.734887i	0.452288	−	0.107194i	0.00184304	−	0.999998i	\(-0.499413\pi\)
							0.450445	+	0.892804i	\(0.351265\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.347.25	yes	1872
8.3	odd	2	inner	648.2.bb.b.347.18	yes	1872
81.74	odd	54	inner	648.2.bb.b.155.18	✓	1872
648.155	even	54	inner	648.2.bb.b.155.25	yes	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.155.18	✓	1872	81.74	odd	54	inner
648.2.bb.b.155.25	yes	1872	648.155	even	54	inner
648.2.bb.b.347.18	yes	1872	8.3	odd	2	inner
648.2.bb.b.347.25	yes	1872	1.1	even	1	trivial