Embedded newform 648.2.bb.b.155.62

• Label: 648.2.bb.b.155.62
• 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			155.62
Character	\(\chi\)	\(=\)	648.155
Dual form			648.2.bb.b.347.62

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.365067 - 1.36628i) q^{2} +(1.42632 + 0.982653i) q^{3} +(-1.73345 - 0.997569i) q^{4} +(3.16551 - 2.08199i) q^{5} +(1.86328 - 1.59002i) q^{6} +(1.35474 + 1.27813i) q^{7} +(-1.99579 + 2.00420i) q^{8} +(1.06879 + 2.80316i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.365067	−	1.36628i	0.258141	−	0.966107i
							\(3\)	1.42632	+	0.982653i	0.823487	+	0.567335i
\(5\)	3.16551	−	2.08199i	1.41566	−	0.931094i								0.415894	−	0.909413i	\(-0.363469\pi\)
							0.999765	−	0.0216808i	\(-0.00690174\pi\)
\(7\)	1.35474	+	1.27813i	0.512043	+	0.483087i	0.898349	−	0.439283i	\(-0.144768\pi\)
							−0.386306	+	0.922371i	\(0.626249\pi\)
\(11\)	2.67890	+	5.33414i	0.807720	+	1.60830i	0.793629	+	0.608402i	\(0.208189\pi\)
							0.0140911	+	0.999901i	\(0.495515\pi\)
\(13\)	−0.0628852	+	0.0271260i	−0.0174412	+	0.00752341i	−0.404783	−	0.914413i	\(-0.632653\pi\)
							0.387341	+	0.921936i	\(0.373394\pi\)
\(17\)	−2.02678	−	2.41542i	−0.491566	−	0.585826i	0.462049	−	0.886855i	\(-0.347115\pi\)
							−0.953615	+	0.301029i	\(0.902670\pi\)
\(19\)	−2.74060	−	2.29964i	−0.628737	−	0.527573i	0.271799	−	0.962354i	\(-0.412381\pi\)
							−0.900536	+	0.434781i	\(0.856826\pi\)
\(23\)	0.0746894	+	0.0791662i	0.0155738	+	0.0165073i	0.735114	−	0.677944i	\(-0.237129\pi\)
							−0.719540	+	0.694451i	\(0.755647\pi\)
\(29\)	−8.41268	+	0.983301i	−1.56219	+	0.182594i	−0.852808	−	0.522225i	\(-0.825102\pi\)
							−0.709387	+	0.704819i	\(0.751028\pi\)
\(31\)	−0.371204	−	1.56623i	−0.0666702	−	0.281304i	0.929491	−	0.368846i	\(-0.120247\pi\)
							−0.996161	+	0.0875420i	\(0.972099\pi\)
\(37\)	−5.21394	+	0.919359i	−0.857167	+	0.151142i	−0.584925	−	0.811087i	\(-0.698876\pi\)
							−0.272242	+	0.962229i	\(0.587765\pi\)
\(41\)	−4.52851	−	3.37135i	−0.707234	−	0.526517i	0.182380	−	0.983228i	\(-0.441620\pi\)
							−0.889615	+	0.456711i	\(0.849027\pi\)
\(43\)	0.296936	−	5.09820i	0.0452824	−	0.777469i	−0.896872	−	0.442290i	\(-0.854166\pi\)
							0.942155	−	0.335179i	\(-0.108797\pi\)
\(47\)	−7.53227	−	1.78518i	−1.09869	−	0.260395i	−0.358999	−	0.933338i	\(-0.616882\pi\)
							−0.739695	+	0.672943i	\(0.765030\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.155.62	✓	1872
8.3	odd	2	inner	648.2.bb.b.155.103	yes	1872
81.23	odd	54	inner	648.2.bb.b.347.103	yes	1872
648.347	even	54	inner	648.2.bb.b.347.62	yes	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.155.62	✓	1872	1.1	even	1	trivial
648.2.bb.b.155.103	yes	1872	8.3	odd	2	inner
648.2.bb.b.347.62	yes	1872	648.347	even	54	inner
648.2.bb.b.347.103	yes	1872	81.23	odd	54	inner