Embedded newform 648.2.bd.a.133.49

• Label: 648.2.bd.a.133.49
• Level: $648$
• Weight: $2$
• Character: 648.133
• Analytic conductor: $5.174$
• Analytic rank: $0$
• Dimension: $1908$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	648.bd (of order \(54\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.17430605098\)
Analytic rank:	\(0\)
Dimension:	\(1908\)
Relative dimension:	\(106\) over \(\Q(\zeta_{54})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{54}]$

Embedding invariants

Embedding label			133.49
Character	\(\chi\)	\(=\)	648.133
Dual form			648.2.bd.a.229.49

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.229057 - 1.39554i) q^{2} +(-0.385660 + 1.68857i) q^{3} +(-1.89507 + 0.639318i) q^{4} +(-0.128997 + 0.121702i) q^{5} +(2.44480 + 0.151425i) q^{6} +(3.74809 - 0.438089i) q^{7} +(1.32627 + 2.49820i) q^{8} +(-2.70253 - 1.30243i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1908 q - 18 q^{2} - 18 q^{4} - 18 q^{6} - 36 q^{7} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{5}{27}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.229057	−	1.39554i	−0.161968	−	0.986796i
							\(3\)	−0.385660	+	1.68857i	−0.222661	+	0.974896i
\(5\)	−0.128997	+	0.121702i	−0.0576892	+	0.0544270i								−0.714546	−	0.699588i	\(-0.753367\pi\)
							0.656857	+	0.754015i	\(0.271885\pi\)
\(7\)	3.74809	−	0.438089i	1.41665	−	0.165582i	0.626940	−	0.779068i	\(-0.284307\pi\)
							0.789706	+	0.613485i	\(0.210233\pi\)
\(11\)	1.03121	−	0.308723i	0.310921	−	0.0930836i	−0.127541	−	0.991833i	\(-0.540709\pi\)
							0.438462	+	0.898750i	\(0.355523\pi\)
\(13\)	−5.21590	+	0.303792i	−1.44663	+	0.0842566i	−0.763390	−	0.645937i	\(-0.776467\pi\)
							−0.683240	+	0.730194i	\(0.739430\pi\)
\(17\)	6.40015	+	2.32946i	1.55226	+	0.564978i	0.968947	−	0.247268i	\(-0.0795327\pi\)
							0.583316	+	0.812245i	\(0.301755\pi\)
\(19\)	0.405423	+	1.11389i	0.0930103	+	0.255544i	0.977471	−	0.211072i	\(-0.0676954\pi\)
							−0.884460	+	0.466616i	\(0.845473\pi\)
\(23\)	0.157777	+	0.0184415i	0.0328989	+	0.00384533i	0.132526	−	0.991179i	\(-0.457691\pi\)
							−0.0996276	+	0.995025i	\(0.531765\pi\)
\(29\)	2.32776	+	4.63496i	0.432255	+	0.860690i	0.999315	+	0.0370136i	\(0.0117845\pi\)
							−0.567060	+	0.823677i	\(0.691919\pi\)
\(31\)	−3.63201	+	4.87864i	−0.652328	+	0.876229i	−0.998070	−	0.0621009i	\(-0.980220\pi\)
							0.345742	+	0.938330i	\(0.387627\pi\)
\(37\)	4.35984	+	5.19586i	0.716753	+	0.854193i	0.994311	−	0.106516i	\(-0.0339694\pi\)
							−0.277558	+	0.960709i	\(0.589525\pi\)
\(41\)	8.28477	−	5.44898i	1.29386	−	0.850988i	0.299204	−	0.954189i	\(-0.403279\pi\)
							0.994660	+	0.103201i	\(0.0329086\pi\)
\(43\)	−1.76502	+	7.44719i	−0.269162	+	1.13569i	0.655590	+	0.755117i	\(0.272420\pi\)
							−0.924753	+	0.380569i	\(0.875728\pi\)
\(47\)	0.972869	+	1.30679i	0.141907	+	0.190615i	0.867395	−	0.497621i	\(-0.165793\pi\)
							−0.725487	+	0.688236i	\(0.758385\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bd.a.133.49	yes	1908
8.5	even	2	inner	648.2.bd.a.133.8	✓	1908
81.67	even	27	inner	648.2.bd.a.229.8	yes	1908
648.229	even	54	inner	648.2.bd.a.229.49	yes	1908

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bd.a.133.8	✓	1908	8.5	even	2	inner
648.2.bd.a.133.49	yes	1908	1.1	even	1	trivial
648.2.bd.a.229.8	yes	1908	81.67	even	27	inner
648.2.bd.a.229.49	yes	1908	648.229	even	54	inner