Embedded newform 756.2.cf.a.659.81

• Label: 756.2.cf.a.659.81
• Level: $756$
• Weight: $2$
• Character: 756.659
• Analytic conductor: $6.037$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	756.cf (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.03669039281\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(108\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			659.81
Character	\(\chi\)	\(=\)	756.659
Dual form			756.2.cf.a.491.81

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.961880 + 1.03672i) q^{2} +(-1.03042 + 1.39220i) q^{3} +(-0.149574 + 1.99440i) q^{4} +(-0.337395 + 0.926986i) q^{5} +(-2.43447 + 0.270870i) q^{6} +(0.984808 + 0.173648i) q^{7} +(-2.21150 + 1.76331i) q^{8} +(-0.876451 - 2.86912i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(325\)	\(379\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)	\(1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.961880	+	1.03672i	0.680152	+	0.733071i
							\(3\)	−1.03042	+	1.39220i	−0.594916	+	0.803788i
\(5\)	−0.337395	+	0.926986i	−0.150888	+	0.414561i								−0.991990	−	0.126315i	\(-0.959685\pi\)
							0.841102	+	0.540876i	\(0.181907\pi\)
\(7\)	0.984808	+	0.173648i	0.372222	+	0.0656328i
							\(11\)	−4.05547	+	1.47607i	−1.22277	+	0.445052i	−0.871116	−	0.491077i	\(-0.836603\pi\)
−0.351655	+	0.936130i	\(0.614381\pi\)
\(13\)	1.89865	−	1.59316i	0.526591	−	0.441862i	−0.340331	−	0.940306i	\(-0.610539\pi\)
							0.866922	+	0.498443i	\(0.166095\pi\)
\(17\)	−4.64721	+	2.68307i	−1.12711	+	0.650739i	−0.943207	−	0.332205i	\(-0.892207\pi\)
							−0.183906	+	0.982944i	\(0.558874\pi\)
\(19\)	−3.40172	−	1.96398i	−0.780408	−	0.450569i	0.0561670	−	0.998421i	\(-0.482112\pi\)
							−0.836575	+	0.547853i	\(0.815445\pi\)
\(23\)	1.28913	+	7.31104i	0.268803	+	1.52446i	0.757982	+	0.652275i	\(0.226185\pi\)
							−0.489179	+	0.872183i	\(0.662704\pi\)
\(29\)	5.80611	−	6.91945i	1.07817	−	1.28491i	0.121859	−	0.992547i	\(-0.461114\pi\)
							0.956308	−	0.292362i	\(-0.0944413\pi\)
\(31\)	−0.573485	+	0.101121i	−0.103001	+	0.0181619i	−0.224911	−	0.974379i	\(-0.572209\pi\)
							0.121910	+	0.992541i	\(0.461098\pi\)
\(37\)	−0.657893	−	1.13950i	−0.108157	−	0.187333i	0.806867	−	0.590734i	\(-0.201162\pi\)
							−0.915024	+	0.403400i	\(0.867828\pi\)
\(41\)	−6.21257	−	7.40386i	−0.970241	−	1.15629i	−0.987687	−	0.156441i	\(-0.949998\pi\)
							0.0174462	−	0.999848i	\(-0.494446\pi\)
\(43\)	0.107479	+	0.295296i	0.0163904	+	0.0450322i	0.947618	−	0.319406i	\(-0.103483\pi\)
							−0.931228	+	0.364438i	\(0.881261\pi\)
\(47\)	−1.36505	+	7.74157i	−0.199113	+	1.12922i	0.707326	+	0.706887i	\(0.249901\pi\)
							−0.906439	+	0.422337i	\(0.861210\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.cf.a.659.81	yes	648
4.3	odd	2	inner	756.2.cf.a.659.20	yes	648
27.5	odd	18	inner	756.2.cf.a.491.20	✓	648
108.59	even	18	inner	756.2.cf.a.491.81	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.cf.a.491.20	✓	648	27.5	odd	18	inner
756.2.cf.a.491.81	yes	648	108.59	even	18	inner
756.2.cf.a.659.20	yes	648	4.3	odd	2	inner
756.2.cf.a.659.81	yes	648	1.1	even	1	trivial