Embedded newform 648.2.bb.b.347.17

• Label: 648.2.bb.b.347.17
• Level: $648$
• Weight: $2$
• Character: 648.347
• 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			347.17
Character	\(\chi\)	\(=\)	648.347
Dual form			648.2.bb.b.155.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.24257 + 0.675295i) q^{2} +(1.66780 + 0.467376i) q^{3} +(1.08795 - 1.67820i) q^{4} +(-0.690567 - 0.454193i) q^{5} +(-2.38797 + 0.545511i) q^{6} +(1.79170 - 1.69038i) q^{7} +(-0.218578 + 2.81997i) q^{8} +(2.56312 + 1.55898i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{11}{54}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.24257	+	0.675295i	−0.878629	+	0.477505i
							\(3\)	1.66780	+	0.467376i	0.962905	+	0.269839i
\(5\)	−0.690567	−	0.454193i	−0.308831	−	0.203121i								0.385627	−	0.922655i	\(-0.373985\pi\)
							−0.694458	+	0.719534i	\(0.744356\pi\)
\(7\)	1.79170	−	1.69038i	0.677198	−	0.638903i	−0.268755	−	0.963209i	\(-0.586612\pi\)
							0.945952	+	0.324306i	\(0.105131\pi\)
\(11\)	2.65771	−	5.29194i	0.801331	−	1.59558i	−0.00196389	−	0.999998i	\(-0.500625\pi\)
							0.803294	−	0.595582i	\(-0.203079\pi\)
\(13\)	−4.45793	−	1.92296i	−1.23641	−	0.533334i	−0.325343	−	0.945596i	\(-0.605480\pi\)
							−0.911065	+	0.412262i	\(0.864739\pi\)
\(17\)	−1.63454	+	1.94797i	−0.396435	+	0.472452i	−0.926929	−	0.375236i	\(-0.877562\pi\)
							0.530495	+	0.847688i	\(0.322006\pi\)
\(19\)	4.54105	−	3.81039i	1.04179	−	0.874163i	0.0495811	−	0.998770i	\(-0.484211\pi\)
							0.992206	+	0.124607i	\(0.0397669\pi\)
\(23\)	−4.49074	+	4.75991i	−0.936384	+	0.992509i	−0.999985	−	0.00546656i	\(-0.998260\pi\)
							0.0636013	+	0.997975i	\(0.479741\pi\)
\(29\)	0.241880	+	0.0282717i	0.0449160	+	0.00524992i	0.138521	−	0.990359i	\(-0.455765\pi\)
							−0.0936052	+	0.995609i	\(0.529839\pi\)
\(31\)	1.82303	−	7.69198i	0.327426	−	1.38152i	−0.521609	−	0.853185i	\(-0.674668\pi\)
							0.849036	−	0.528336i	\(-0.177184\pi\)
\(37\)	4.74486	+	0.836646i	0.780050	+	0.137544i	0.549474	−	0.835511i	\(-0.314828\pi\)
							0.230575	+	0.973054i	\(0.425939\pi\)
\(41\)	−3.29326	+	2.45174i	−0.514321	+	0.382898i	−0.822798	−	0.568334i	\(-0.807588\pi\)
							0.308477	+	0.951232i	\(0.400181\pi\)
\(43\)	0.248525	+	4.26702i	0.0378998	+	0.650714i	0.962752	+	0.270387i	\(0.0871516\pi\)
							−0.924852	+	0.380327i	\(0.875811\pi\)
\(47\)	−0.190946	+	0.0452551i	−0.0278524	+	0.00660113i	−0.244518	−	0.969645i	\(-0.578630\pi\)
							0.216666	+	0.976246i	\(0.430482\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.347.17	yes	1872
8.3	odd	2	inner	648.2.bb.b.347.60	yes	1872
81.74	odd	54	inner	648.2.bb.b.155.60	yes	1872
648.155	even	54	inner	648.2.bb.b.155.17	✓	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.155.17	✓	1872	648.155	even	54	inner
648.2.bb.b.155.60	yes	1872	81.74	odd	54	inner
648.2.bb.b.347.17	yes	1872	1.1	even	1	trivial
648.2.bb.b.347.60	yes	1872	8.3	odd	2	inner