Embedded newform 648.2.bd.a.133.45

• Label: 648.2.bd.a.133.45
• Level: $648$
• Weight: $2$
• Character: 648.133
• 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			133.45
Character	\(\chi\)	\(=\)	648.133
Dual form			648.2.bd.a.229.45

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.413699 - 1.35235i) q^{2} +(1.63926 + 0.559320i) q^{3} +(-1.65771 + 1.11893i) q^{4} +(-0.354730 + 0.334671i) q^{5} +(0.0782378 - 2.44824i) q^{6} +(-2.41777 + 0.282597i) q^{7} +(2.19898 + 1.77890i) q^{8} +(2.37432 + 1.83374i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1908 q - 18 q^{2} - 18 q^{4} - 18 q^{6} - 36 q^{7} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{5}{27}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.413699	−	1.35235i	−0.292530	−	0.956256i
							\(3\)	1.63926	+	0.559320i	0.946425	+	0.322924i
\(5\)	−0.354730	+	0.334671i	−0.158640	+	0.149669i								−0.761469	−	0.648201i	\(-0.775522\pi\)
							0.602829	+	0.797870i	\(0.294040\pi\)
\(7\)	−2.41777	+	0.282597i	−0.913831	+	0.106812i	−0.559995	−	0.828496i	\(-0.689197\pi\)
							−0.353836	+	0.935307i	\(0.615123\pi\)
\(11\)	−3.97396	+	1.18973i	−1.19819	+	0.358716i	−0.822898	−	0.568189i	\(-0.807644\pi\)
							−0.375296	+	0.926905i	\(0.622459\pi\)
\(13\)	3.90573	−	0.227483i	1.08325	−	0.0630924i	0.492784	−	0.870152i	\(-0.335979\pi\)
							0.590470	+	0.807059i	\(0.298942\pi\)
\(17\)	5.03057	+	1.83098i	1.22009	+	0.444077i	0.870194	−	0.492710i	\(-0.163994\pi\)
							0.349899	+	0.936787i	\(0.386216\pi\)
\(19\)	1.88416	+	5.17668i	0.432255	+	1.18761i	0.944425	+	0.328727i	\(0.106620\pi\)
							−0.512170	+	0.858884i	\(0.671158\pi\)
\(23\)	0.479777	+	0.0560779i	0.100040	+	0.0116931i	0.165966	−	0.986132i	\(-0.446926\pi\)
							−0.0659254	+	0.997825i	\(0.521000\pi\)
\(29\)	−0.777227	−	1.54759i	−0.144327	−	0.287379i	0.809841	−	0.586649i	\(-0.199553\pi\)
							−0.954169	+	0.299270i	\(0.903257\pi\)
\(31\)	0.978762	−	1.31471i	0.175791	−	0.236128i	−0.705491	−	0.708719i	\(-0.749274\pi\)
							0.881282	+	0.472590i	\(0.156681\pi\)
\(37\)	7.08629	+	8.44511i	1.16498	+	1.38837i	0.906423	+	0.422370i	\(0.138802\pi\)
							0.258555	+	0.965997i	\(0.416754\pi\)
\(41\)	0.941198	−	0.619036i	0.146990	−	0.0966771i	−0.473884	−	0.880588i	\(-0.657148\pi\)
							0.620874	+	0.783910i	\(0.286778\pi\)
\(43\)	−1.65999	+	7.00404i	−0.253146	+	1.06811i	0.686818	+	0.726830i	\(0.259007\pi\)
							−0.939963	+	0.341276i	\(0.889141\pi\)
\(47\)	−5.65115	−	7.59082i	−0.824305	−	1.10723i	−0.992592	−	0.121496i	\(-0.961231\pi\)
							0.168286	−	0.985738i	\(-0.446177\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bd.a.133.45	yes	1908
8.5	even	2	inner	648.2.bd.a.133.6	✓	1908
81.67	even	27	inner	648.2.bd.a.229.6	yes	1908
648.229	even	54	inner	648.2.bd.a.229.45	yes	1908

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bd.a.133.6	✓	1908	8.5	even	2	inner
648.2.bd.a.133.45	yes	1908	1.1	even	1	trivial
648.2.bd.a.229.6	yes	1908	81.67	even	27	inner
648.2.bd.a.229.45	yes	1908	648.229	even	54	inner