Embedded newform 648.2.bd.a.13.4

• Label: 648.2.bd.a.13.4
• Level: $648$
• Weight: $2$
• Character: 648.13
• Analytic conductor: $5.174$
• Analytic rank: $0$
• Dimension: $1908$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	648.bd (of order \(54\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.17430605098\)
Analytic rank:	\(0\)
Dimension:	\(1908\)
Relative dimension:	\(106\) over \(\Q(\zeta_{54})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{54}]$

Embedding invariants

Embedding label			13.4
Character	\(\chi\)	\(=\)	648.13
Dual form			648.2.bd.a.349.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40484 - 0.162553i) q^{2} +(-0.0416517 - 1.73155i) q^{3} +(1.94715 + 0.456721i) q^{4} +(1.45600 - 0.435896i) q^{5} +(-0.222954 + 2.43932i) q^{6} +(1.13129 + 2.62263i) q^{7} +(-2.66120 - 0.958136i) q^{8} +(-2.99653 + 0.144244i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1908 q - 18 q^{2} - 18 q^{4} - 18 q^{6} - 36 q^{7} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{4}{27}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40484	−	0.162553i	−0.993372	−	0.114942i
							\(3\)	−0.0416517	−	1.73155i	−0.0240476	−	0.999711i
\(5\)	1.45600	−	0.435896i	0.651141	−	0.194939i								0.0558376	−	0.998440i	\(-0.482217\pi\)
							0.595303	+	0.803501i	\(0.297032\pi\)
\(7\)	1.13129	+	2.62263i	0.427589	+	0.991262i	0.986757	+	0.162208i	\(0.0518615\pi\)
							−0.559168	+	0.829054i	\(0.688879\pi\)
\(11\)	−0.570613	−	2.40761i	−0.172046	−	0.725920i	−0.988839	−	0.148989i	\(-0.952398\pi\)
							0.816793	−	0.576931i	\(-0.195750\pi\)
\(13\)	0.904969	−	1.37594i	0.250993	−	0.381617i	−0.687911	−	0.725795i	\(-0.741472\pi\)
							0.938904	+	0.344178i	\(0.111843\pi\)
\(17\)	1.06626	−	0.388088i	0.258607	−	0.0941253i	−0.209463	−	0.977817i	\(-0.567172\pi\)
							0.468070	+	0.883691i	\(0.344949\pi\)
\(19\)	0.828111	−	2.27522i	0.189982	−	0.521971i	−0.807732	−	0.589550i	\(-0.799305\pi\)
							0.997714	+	0.0675791i	\(0.0215275\pi\)
\(23\)	3.31171	−	7.67740i	0.690538	−	1.60085i	−0.104667	−	0.994507i	\(-0.533378\pi\)
							0.795205	−	0.606341i	\(-0.207363\pi\)
\(29\)	6.06457	−	0.353221i	1.12616	−	0.0655915i	0.515068	−	0.857149i	\(-0.327767\pi\)
							0.611094	+	0.791558i	\(0.290730\pi\)
\(31\)	1.90018	−	0.222100i	0.341283	−	0.0398903i	0.0562749	−	0.998415i	\(-0.482078\pi\)
							0.285008	+	0.958525i	\(0.408004\pi\)
\(37\)	0.431096	−	0.513760i	0.0708717	−	0.0844616i	−0.729444	−	0.684041i	\(-0.760221\pi\)
							0.800315	+	0.599579i	\(0.204665\pi\)
\(41\)	0.910814	−	0.457428i	0.142245	−	0.0714383i	−0.376253	−	0.926517i	\(-0.622788\pi\)
							0.518498	+	0.855079i	\(0.326491\pi\)
\(43\)	7.16052	−	6.75560i	1.09197	−	1.03022i	0.0925641	−	0.995707i	\(-0.470494\pi\)
							0.999404	−	0.0345123i	\(-0.0109878\pi\)
\(47\)	−12.0748	−	1.41134i	−1.76129	−	0.205866i	−0.826958	−	0.562264i	\(-0.809931\pi\)
							−0.934334	+	0.356398i	\(0.884005\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bd.a.13.4	✓	1908
8.5	even	2	inner	648.2.bd.a.13.27	yes	1908
81.25	even	27	inner	648.2.bd.a.349.27	yes	1908
648.349	even	54	inner	648.2.bd.a.349.4	yes	1908

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bd.a.13.4	✓	1908	1.1	even	1	trivial
648.2.bd.a.13.27	yes	1908	8.5	even	2	inner
648.2.bd.a.349.4	yes	1908	648.349	even	54	inner
648.2.bd.a.349.27	yes	1908	81.25	even	27	inner