Embedded newform 648.2.bb.b.155.41

• Label: 648.2.bb.b.155.41
• 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			155.41
Character	\(\chi\)	\(=\)	648.155
Dual form			648.2.bb.b.347.41

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.440051 - 1.34401i) q^{2} +(0.652639 - 1.60439i) q^{3} +(-1.61271 + 1.18286i) q^{4} +(-0.484875 + 0.318908i) q^{5} +(-2.44350 - 0.171139i) q^{6} +(-3.43317 - 3.23903i) q^{7} +(2.29945 + 1.64698i) q^{8} +(-2.14813 - 2.09417i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.440051	−	1.34401i	−0.311163	−	0.950357i
							\(3\)	0.652639	−	1.60439i	0.376801	−	0.926294i
\(5\)	−0.484875	+	0.318908i	−0.216843	+	0.142620i								−0.653284	−	0.757113i	\(-0.726609\pi\)
							0.436441	+	0.899733i	\(0.356239\pi\)
\(7\)	−3.43317	−	3.23903i	−1.29762	−	1.22424i	−0.959030	−	0.283303i	\(-0.908570\pi\)
							−0.338587	−	0.940935i	\(-0.609949\pi\)
\(11\)	0.127707	+	0.254285i	0.0385050	+	0.0766699i	0.912063	−	0.410051i	\(-0.134489\pi\)
							−0.873557	+	0.486721i	\(0.838193\pi\)
\(13\)	−0.136627	+	0.0589351i	−0.0378935	+	0.0163457i	−0.414942	−	0.909848i	\(-0.636198\pi\)
							0.377049	+	0.926193i	\(0.376939\pi\)
\(17\)	−2.86822	−	3.41821i	−0.695646	−	0.829038i	0.296381	−	0.955070i	\(-0.404220\pi\)
							−0.992026	+	0.126032i	\(0.959776\pi\)
\(19\)	3.61268	+	3.03140i	0.828805	+	0.695450i	0.955016	−	0.296553i	\(-0.0958373\pi\)
							−0.126211	+	0.992003i	\(0.540282\pi\)
\(23\)	2.17962	+	2.31026i	0.454481	+	0.481722i	0.913578	−	0.406664i	\(-0.133308\pi\)
							−0.459096	+	0.888387i	\(0.651827\pi\)
\(29\)	−3.01492	+	0.352393i	−0.559856	+	0.0654378i	−0.391315	−	0.920257i	\(-0.627980\pi\)
							−0.168541	+	0.985695i	\(0.553906\pi\)
\(31\)	2.22088	+	9.37062i	0.398881	+	1.68301i	0.687993	+	0.725718i	\(0.258492\pi\)
							−0.289111	+	0.957295i	\(0.593360\pi\)
\(37\)	−7.26526	+	1.28106i	−1.19440	+	0.210605i	−0.735277	−	0.677767i	\(-0.762948\pi\)
							−0.459124	+	0.888372i	\(0.651837\pi\)
\(41\)	−5.85526	−	4.35908i	−0.914438	−	0.680774i	0.0334287	−	0.999441i	\(-0.489357\pi\)
							−0.947867	+	0.318667i	\(0.896765\pi\)
\(43\)	0.465059	−	7.98476i	0.0709209	−	1.21766i	−0.755705	−	0.654912i	\(-0.772705\pi\)
							0.826626	−	0.562752i	\(-0.190257\pi\)
\(47\)	−5.66400	−	1.34239i	−0.826179	−	0.195808i	−0.204287	−	0.978911i	\(-0.565488\pi\)
							−0.621892	+	0.783103i	\(0.713636\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.155.41	✓	1872
8.3	odd	2	inner	648.2.bb.b.155.84	yes	1872
81.23	odd	54	inner	648.2.bb.b.347.84	yes	1872
648.347	even	54	inner	648.2.bb.b.347.41	yes	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.155.41	✓	1872	1.1	even	1	trivial
648.2.bb.b.155.84	yes	1872	8.3	odd	2	inner
648.2.bb.b.347.41	yes	1872	648.347	even	54	inner
648.2.bb.b.347.84	yes	1872	81.23	odd	54	inner