Embedded newform 648.2.bb.b.155.64

• Label: 648.2.bb.b.155.64
• 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			155.64
Character	\(\chi\)	\(=\)	648.155
Dual form			648.2.bb.b.347.64

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.392266 + 1.35872i) q^{2} +(1.07702 + 1.35648i) q^{3} +(-1.69225 + 1.06596i) q^{4} +(0.219193 - 0.144165i) q^{5} +(-1.42060 + 1.99547i) q^{6} +(1.89423 + 1.78711i) q^{7} +(-2.11216 - 1.88116i) q^{8} +(-0.680059 + 2.92190i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.392266	+	1.35872i	0.277374	+	0.960762i
							\(3\)	1.07702	+	1.35648i	0.621817	+	0.783162i
\(5\)	0.219193	−	0.144165i	0.0980259	−	0.0644726i								−0.499550	−	0.866285i	\(-0.666501\pi\)
							0.597576	+	0.801813i	\(0.296131\pi\)
\(7\)	1.89423	+	1.78711i	0.715952	+	0.675466i	0.955357	−	0.295454i	\(-0.0954709\pi\)
							−0.239405	+	0.970920i	\(0.576952\pi\)
\(11\)	1.21248	+	2.41424i	0.365576	+	0.727921i	0.998945	−	0.0459290i	\(-0.0146248\pi\)
							−0.633369	+	0.773850i	\(0.718328\pi\)
\(13\)	2.54356	−	1.09718i	0.705456	−	0.304304i	−0.0129618	−	0.999916i	\(-0.504126\pi\)
							0.718418	+	0.695612i	\(0.244867\pi\)
\(17\)	2.50193	+	2.98168i	0.606806	+	0.723164i	0.978742	−	0.205095i	\(-0.0657503\pi\)
							−0.371936	+	0.928258i	\(0.621306\pi\)
\(19\)	−3.00835	−	2.52431i	−0.690163	−	0.579116i	0.228793	−	0.973475i	\(-0.426522\pi\)
							−0.918956	+	0.394359i	\(0.870967\pi\)
\(23\)	−6.25199	−	6.62672i	−1.30363	−	1.38177i	−0.879307	−	0.476255i	\(-0.841994\pi\)
							−0.424322	−	0.905511i	\(-0.639488\pi\)
\(29\)	−0.0984331	+	0.0115052i	−0.0182786	+	0.00213646i	−0.125227	−	0.992128i	\(-0.539966\pi\)
							0.106949	+	0.994265i	\(0.465892\pi\)
\(31\)	−0.666983	−	2.81422i	−0.119794	−	0.505449i	−0.999549	−	0.0300230i	\(-0.990442\pi\)
							0.879756	−	0.475426i	\(-0.157706\pi\)
\(37\)	7.49128	−	1.32092i	1.23156	−	0.217157i	0.480264	−	0.877124i	\(-0.340541\pi\)
							0.751295	+	0.659967i	\(0.229430\pi\)
\(41\)	−5.60350	−	4.17165i	−0.875120	−	0.651503i	0.0629241	−	0.998018i	\(-0.479957\pi\)
							−0.938044	+	0.346515i	\(0.887365\pi\)
\(43\)	−0.165527	+	2.84199i	−0.0252426	+	0.433399i	0.961884	+	0.273458i	\(0.0881675\pi\)
							−0.987127	+	0.159941i	\(0.948870\pi\)
\(47\)	5.56214	+	1.31825i	0.811322	+	0.192287i	0.615280	−	0.788309i	\(-0.289043\pi\)
							0.196042	+	0.980595i	\(0.437191\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.155.64	yes	1872
8.3	odd	2	inner	648.2.bb.b.155.20	✓	1872
81.23	odd	54	inner	648.2.bb.b.347.20	yes	1872
648.347	even	54	inner	648.2.bb.b.347.64	yes	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.155.20	✓	1872	8.3	odd	2	inner
648.2.bb.b.155.64	yes	1872	1.1	even	1	trivial
648.2.bb.b.347.20	yes	1872	81.23	odd	54	inner
648.2.bb.b.347.64	yes	1872	648.347	even	54	inner