Embedded newform 648.2.bb.b.131.13

• Label: 648.2.bb.b.131.13
• Level: $648$
• Weight: $2$
• Character: 648.131
• 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			131.13
Character	\(\chi\)	\(=\)	648.131
Dual form			648.2.bb.b.371.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.31484 - 0.520764i) q^{2} +(-0.404893 - 1.68406i) q^{3} +(1.45761 + 1.36944i) q^{4} +(-0.0450394 + 0.773297i) q^{5} +(-0.344629 + 2.42512i) q^{6} +(0.494858 + 2.08797i) q^{7} +(-1.20337 - 2.55967i) q^{8} +(-2.67212 + 1.36373i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{47}{54}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.31484	−	0.520764i	−0.929733	−	0.368236i
							\(3\)	−0.404893	−	1.68406i	−0.233765	−	0.972293i
\(5\)	−0.0450394	+	0.773297i	−0.0201422	+	0.345829i								0.973148	+	0.230180i	\(0.0739316\pi\)
							−0.993290	+	0.115648i	\(0.963105\pi\)
\(7\)	0.494858	+	2.08797i	0.187039	+	0.789179i	0.982966	+	0.183787i	\(0.0588356\pi\)
							−0.795927	+	0.605392i	\(0.793016\pi\)
\(11\)	−0.691370	−	1.05118i	−0.208456	−	0.316942i	0.716093	−	0.698005i	\(-0.245929\pi\)
							−0.924549	+	0.381063i	\(0.875558\pi\)
\(13\)	−0.0948340	−	0.811357i	−0.0263022	−	0.225030i	0.973694	−	0.227861i	\(-0.0731730\pi\)
							−0.999996	+	0.00283065i	\(0.999099\pi\)
\(17\)	4.33310	−	5.16399i	1.05093	−	1.25245i	0.0842556	−	0.996444i	\(-0.473149\pi\)
							0.966675	−	0.256006i	\(-0.0824068\pi\)
\(19\)	−4.35115	+	3.65104i	−0.998221	+	0.837607i	−0.986737	−	0.162327i	\(-0.948100\pi\)
							−0.0114841	+	0.999934i	\(0.503656\pi\)
\(23\)	6.80141	+	1.61196i	1.41819	+	0.336118i	0.867020	−	0.498274i	\(-0.166033\pi\)
							0.551172	+	0.834392i	\(0.314181\pi\)
\(29\)	2.83771	−	3.81171i	0.526949	−	0.707816i	−0.456352	−	0.889799i	\(-0.650844\pi\)
							0.983302	+	0.181983i	\(0.0582517\pi\)
\(31\)	9.10109	−	2.72469i	1.63460	−	0.489368i	0.667683	−	0.744446i	\(-0.267286\pi\)
							0.966921	+	0.255077i	\(0.0821010\pi\)
\(37\)	3.16598	+	0.558247i	0.520484	+	0.0917753i	0.427717	−	0.903913i	\(-0.359318\pi\)
							0.0927664	+	0.995688i	\(0.470429\pi\)
\(41\)	−7.22382	−	3.11605i	−1.12817	−	0.486645i	−0.251558	−	0.967842i	\(-0.580943\pi\)
							−0.876613	+	0.481197i	\(0.840202\pi\)
\(43\)	−0.739515	−	0.371399i	−0.112775	−	0.0566377i	0.391517	−	0.920171i	\(-0.371950\pi\)
							−0.504292	+	0.863533i	\(0.668247\pi\)
\(47\)	−2.39621	+	8.00389i	−0.349523	+	1.16749i	0.584835	+	0.811152i	\(0.301159\pi\)
							−0.934358	+	0.356335i	\(0.884026\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.131.13	✓	1872
8.3	odd	2	inner	648.2.bb.b.131.39	yes	1872
81.47	odd	54	inner	648.2.bb.b.371.39	yes	1872
648.371	even	54	inner	648.2.bb.b.371.13	yes	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.131.13	✓	1872	1.1	even	1	trivial
648.2.bb.b.131.39	yes	1872	8.3	odd	2	inner
648.2.bb.b.371.13	yes	1872	648.371	even	54	inner
648.2.bb.b.371.39	yes	1872	81.47	odd	54	inner