Embedded newform 648.2.bb.b.11.11

• Label: 648.2.bb.b.11.11
• Level: $648$
• Weight: $2$
• Character: 648.11
• 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			11.11
Character	\(\chi\)	\(=\)	648.11
Dual form			648.2.bb.b.59.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.33569 - 0.464682i) q^{2} +(1.72890 + 0.104486i) q^{3} +(1.56814 + 1.24134i) q^{4} +(1.18213 + 0.138172i) q^{5} +(-2.26072 - 0.942948i) q^{6} +(0.848159 + 1.68882i) q^{7} +(-1.51772 - 2.38674i) q^{8} +(2.97817 + 0.361290i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{13}{54}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.33569	−	0.464682i	−0.944476	−	0.328580i
							\(3\)	1.72890	+	0.104486i	0.998179	+	0.0603249i
\(5\)	1.18213	+	0.138172i	0.528666	+	0.0617922i								0.376239	−	0.926523i	\(-0.377217\pi\)
							0.152427	+	0.988315i	\(0.451291\pi\)
\(7\)	0.848159	+	1.68882i	0.320574	+	0.638315i	0.994983	−	0.100043i	\(-0.0318979\pi\)
							−0.674409	+	0.738358i	\(0.735602\pi\)
\(11\)	−2.04056	−	0.880210i	−0.615251	−	0.265393i	0.0655644	−	0.997848i	\(-0.479115\pi\)
							−0.680816	+	0.732455i	\(0.738374\pi\)
\(13\)	0.967103	+	4.08053i	0.268226	+	1.13174i	0.925700	+	0.378258i	\(0.123477\pi\)
							−0.657474	+	0.753477i	\(0.728375\pi\)
\(17\)	4.35178	−	0.767336i	1.05546	−	0.186106i	0.381120	−	0.924526i	\(-0.375538\pi\)
							0.674342	+	0.738419i	\(0.264427\pi\)
\(19\)	0.578755	−	3.28228i	0.132776	−	0.753007i	−0.843607	−	0.536960i	\(-0.819572\pi\)
							0.976383	−	0.216047i	\(-0.0693164\pi\)
\(23\)	−0.574904	−	0.288727i	−0.119876	−	0.0602038i	0.387851	−	0.921722i	\(-0.373218\pi\)
							−0.507726	+	0.861518i	\(0.669514\pi\)
\(29\)	0.503352	+	1.68131i	0.0934701	+	0.312212i	0.991920	−	0.126864i	\(-0.0404912\pi\)
							−0.898450	+	0.439076i	\(0.855306\pi\)
\(31\)	2.36681	+	3.59857i	0.425092	+	0.646321i	0.983005	−	0.183578i	\(-0.0587680\pi\)
							−0.557913	+	0.829900i	\(0.688398\pi\)
\(37\)	−2.38394	+	6.54981i	−0.391917	+	1.07678i	0.574208	+	0.818709i	\(0.305310\pi\)
							−0.966125	+	0.258074i	\(0.916912\pi\)
\(41\)	−2.18475	+	2.06121i	−0.341201	+	0.321907i	−0.837700	−	0.546131i	\(-0.816100\pi\)
							0.496499	+	0.868037i	\(0.334619\pi\)
\(43\)	−2.19940	−	2.95431i	−0.335406	−	0.450528i	0.602140	−	0.798391i	\(-0.294315\pi\)
							−0.937545	+	0.347863i	\(0.886908\pi\)
\(47\)	−3.28417	−	2.16003i	−0.479045	−	0.315073i	0.286920	−	0.957954i	\(-0.407368\pi\)
							−0.765966	+	0.642881i	\(0.777739\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.11.11	✓	1872
8.3	odd	2	inner	648.2.bb.b.11.40	yes	1872
81.59	odd	54	inner	648.2.bb.b.59.40	yes	1872
648.59	even	54	inner	648.2.bb.b.59.11	yes	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.11.11	✓	1872	1.1	even	1	trivial
648.2.bb.b.11.40	yes	1872	8.3	odd	2	inner
648.2.bb.b.59.11	yes	1872	648.59	even	54	inner
648.2.bb.b.59.40	yes	1872	81.59	odd	54	inner