Embedded newform 648.2.bd.a.229.39

• Label: 648.2.bd.a.229.39
• Level: $648$
• Weight: $2$
• Character: 648.229
• Analytic conductor: $5.174$
• Analytic rank: $0$
• Dimension: $1908$
• 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			229.39
Character	\(\chi\)	\(=\)	648.229
Dual form			648.2.bd.a.133.39

$q$-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 \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{22}{27}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bd.a.133.39	✓	1908	648.133	even	54	inner
648.2.bd.a.133.79	yes	1908	81.52	even	27	inner
648.2.bd.a.229.39	yes	1908	1.1	even	1	trivial
648.2.bd.a.229.79	yes	1908	8.5	even	2	inner