Embedded newform 648.2.bd.a.133.39

• Label: 648.2.bd.a.133.39
• 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			133.39
Character	\(\chi\)	\(=\)	648.133
Dual form			648.2.bd.a.229.39

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.574258 + 1.29237i) q^{2} +(0.775724 - 1.54863i) q^{3} +(-1.34045 - 1.48431i) q^{4} +(-2.01822 + 1.90409i) q^{5} +(1.55594 + 1.89184i) q^{6} +(-4.66303 + 0.545030i) q^{7} +(2.68805 - 0.879989i) q^{8} +(-1.79651 - 2.40262i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1908 q - 18 q^{2} - 18 q^{4} - 18 q^{6} - 36 q^{7} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.574258	+	1.29237i	−0.406062	+	0.913846i
							\(3\)	0.775724	−	1.54863i	0.447864	−	0.894102i
\(5\)	−2.01822	+	1.90409i	−0.902575	+	0.851535i								−0.989478	−	0.144682i	\(-0.953784\pi\)
							0.0869037	+	0.996217i	\(0.472303\pi\)
\(7\)	−4.66303	+	0.545030i	−1.76246	+	0.206002i	−0.934787	−	0.355210i	\(-0.884409\pi\)
							−0.827672	+	0.561212i	\(0.810335\pi\)
\(11\)	3.49260	−	1.04562i	1.05306	−	0.315265i	0.286929	−	0.957952i	\(-0.407366\pi\)
							0.766129	+	0.642687i	\(0.222180\pi\)
\(13\)	5.88656	−	0.342853i	1.63264	−	0.0950903i	0.782848	−	0.622213i	\(-0.213766\pi\)
							0.849789	+	0.527123i	\(0.176729\pi\)
\(17\)	3.93594	+	1.43256i	0.954605	+	0.347448i	0.771917	−	0.635723i	\(-0.219298\pi\)
							0.182688	+	0.983171i	\(0.441520\pi\)
\(19\)	0.176632	+	0.485293i	0.0405222	+	0.111334i	0.958303	−	0.285753i	\(-0.0922435\pi\)
							−0.917781	+	0.397086i	\(0.870021\pi\)
\(23\)	2.66978	+	0.312053i	0.556688	+	0.0650675i	0.389785	−	0.920906i	\(-0.372549\pi\)
							0.166903	+	0.985973i	\(0.446623\pi\)
\(29\)	2.32277	+	4.62502i	0.431328	+	0.858845i	0.999358	+	0.0358369i	\(0.0114097\pi\)
							−0.568029	+	0.823008i	\(0.692294\pi\)
\(31\)	−1.37700	+	1.84963i	−0.247316	+	0.332203i	−0.908395	−	0.418114i	\(-0.862691\pi\)
							0.661079	+	0.750316i	\(0.270099\pi\)
\(37\)	2.72797	+	3.25107i	0.448476	+	0.534473i	0.942158	−	0.335170i	\(-0.108794\pi\)
							−0.493682	+	0.869643i	\(0.664349\pi\)
\(41\)	6.90742	−	4.54308i	1.07876	−	0.709510i	0.119725	−	0.992807i	\(-0.461799\pi\)
							0.959032	+	0.283297i	\(0.0914283\pi\)
\(43\)	−2.53885	+	10.7122i	−0.387171	+	1.63360i	0.335236	+	0.942134i	\(0.391184\pi\)
							−0.722407	+	0.691468i	\(0.756964\pi\)
\(47\)	−1.46529	−	1.96823i	−0.213735	−	0.287096i	0.682307	−	0.731066i	\(-0.260977\pi\)
							−0.896041	+	0.443970i	\(0.853570\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bd.a.133.39	✓	1908
8.5	even	2	inner	648.2.bd.a.133.79	yes	1908
81.67	even	27	inner	648.2.bd.a.229.79	yes	1908
648.229	even	54	inner	648.2.bd.a.229.39	yes	1908

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bd.a.133.39	✓	1908	1.1	even	1	trivial
648.2.bd.a.133.79	yes	1908	8.5	even	2	inner
648.2.bd.a.229.39	yes	1908	648.229	even	54	inner
648.2.bd.a.229.79	yes	1908	81.67	even	27	inner