Embedded newform 648.2.bb.b.155.103

• Label: 648.2.bb.b.155.103
• 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			155.103
Character	\(\chi\)	\(=\)	648.155
Dual form			648.2.bb.b.347.103

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41359 + 0.0421237i) q^{2} +(1.42632 + 0.982653i) q^{3} +(1.99645 + 0.119091i) q^{4} +(-3.16551 + 2.08199i) q^{5} +(1.97484 + 1.44915i) q^{6} +(-1.35474 - 1.27813i) q^{7} +(2.81714 + 0.252443i) q^{8} +(1.06879 + 2.80316i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41359	+	0.0421237i	0.999556	+	0.0297860i
							\(3\)	1.42632	+	0.982653i	0.823487	+	0.567335i
\(5\)	−3.16551	+	2.08199i	−1.41566	+	0.931094i								−0.415894	+	0.909413i	\(0.636531\pi\)
							−0.999765	+	0.0216808i	\(0.993098\pi\)
\(7\)	−1.35474	−	1.27813i	−0.512043	−	0.483087i	0.386306	−	0.922371i	\(-0.373751\pi\)
							−0.898349	+	0.439283i	\(0.855232\pi\)
\(11\)	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	−0.387341	−	0.921936i	\(-0.626606\pi\)
							0.404783	+	0.914413i	\(0.367347\pi\)
\(17\)	−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	0.719540	−	0.694451i	\(-0.244353\pi\)
							−0.735114	+	0.677944i	\(0.762871\pi\)
\(29\)	8.41268	−	0.983301i	1.56219	−	0.182594i	0.709387	−	0.704819i	\(-0.248972\pi\)
							0.852808	+	0.522225i	\(0.174898\pi\)
\(31\)	0.371204	+	1.56623i	0.0666702	+	0.281304i	0.996161	−	0.0875420i	\(-0.0279012\pi\)
							−0.929491	+	0.368846i	\(0.879753\pi\)
\(37\)	5.21394	−	0.919359i	0.857167	−	0.151142i	0.272242	−	0.962229i	\(-0.412235\pi\)
							0.584925	+	0.811087i	\(0.301124\pi\)
\(41\)	−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	0.739695	−	0.672943i	\(-0.234970\pi\)
							0.358999	+	0.933338i	\(0.383118\pi\)

(See \(a_n\) instead)

Twists

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

    

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