Embedded newform 756.2.cf.a.659.97

• Label: 756.2.cf.a.659.97
• 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			659.97
Character	\(\chi\)	\(=\)	756.659
Dual form			756.2.cf.a.491.97

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.32034 + 0.506659i) q^{2} +(-1.02352 - 1.39728i) q^{3} +(1.48659 + 1.33792i) q^{4} +(-0.779214 + 2.14087i) q^{5} +(-0.643455 - 2.36346i) q^{6} +(-0.984808 - 0.173648i) q^{7} +(1.28494 + 2.51971i) q^{8} +(-0.904796 + 2.86030i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.32034	+	0.506659i	0.933621	+	0.358262i
							\(3\)	−1.02352	−	1.39728i	−0.590932	−	0.806721i
\(5\)	−0.779214	+	2.14087i	−0.348475	+	0.957428i								0.634375	+	0.773025i	\(0.281257\pi\)
							−0.982851	+	0.184403i	\(0.940965\pi\)
\(7\)	−0.984808	−	0.173648i	−0.372222	−	0.0656328i
							\(11\)	−5.69009	+	2.07102i	−1.71563	+	0.624437i	−0.997446	−	0.0714196i	\(-0.977247\pi\)
−0.718181	+	0.695857i	\(0.755025\pi\)
\(13\)	2.72452	−	2.28615i	0.755647	−	0.634063i	−0.181343	−	0.983420i	\(-0.558044\pi\)
							0.936990	+	0.349357i	\(0.113600\pi\)
\(17\)	−5.84222	+	3.37301i	−1.41695	+	0.818075i	−0.996029	−	0.0890246i	\(-0.971625\pi\)
							−0.420917	+	0.907099i	\(0.638292\pi\)
\(19\)	−1.21441	−	0.701137i	−0.278604	−	0.160852i	0.354187	−	0.935174i	\(-0.384757\pi\)
							−0.632791	+	0.774323i	\(0.718091\pi\)
\(23\)	−0.430394	−	2.44088i	−0.0897433	−	0.508959i	−0.996232	−	0.0867292i	\(-0.972359\pi\)
							0.906489	−	0.422230i	\(-0.138753\pi\)
\(29\)	−1.90412	+	2.26924i	−0.353586	+	0.421387i	−0.913293	−	0.407303i	\(-0.866469\pi\)
							0.559707	+	0.828690i	\(0.310914\pi\)
\(31\)	−1.27834	+	0.225405i	−0.229596	+	0.0404840i	−0.287262	−	0.957852i	\(-0.592745\pi\)
							0.0576664	+	0.998336i	\(0.481634\pi\)
\(37\)	4.61467	+	7.99284i	0.758646	+	1.31401i	0.943541	+	0.331256i	\(0.107472\pi\)
							−0.184895	+	0.982758i	\(0.559194\pi\)
\(41\)	−1.27581	−	1.52045i	−0.199248	−	0.237455i	0.657164	−	0.753748i	\(-0.271756\pi\)
							−0.856412	+	0.516293i	\(0.827312\pi\)
\(43\)	2.91000	+	7.99516i	0.443771	+	1.21925i	0.936993	+	0.349347i	\(0.113597\pi\)
							−0.493223	+	0.869903i	\(0.664181\pi\)
\(47\)	1.14774	−	6.50915i	0.167415	−	0.949457i	−0.779124	−	0.626869i	\(-0.784336\pi\)
							0.946539	−	0.322588i	\(-0.104553\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.cf.a.659.97	yes	648
4.3	odd	2	inner	756.2.cf.a.659.36	yes	648
27.5	odd	18	inner	756.2.cf.a.491.36	✓	648
108.59	even	18	inner	756.2.cf.a.491.97	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.cf.a.491.36	✓	648	27.5	odd	18	inner
756.2.cf.a.491.97	yes	648	108.59	even	18	inner
756.2.cf.a.659.36	yes	648	4.3	odd	2	inner
756.2.cf.a.659.97	yes	648	1.1	even	1	trivial