Embedded newform 756.2.cf.a.659.74

• Label: 756.2.cf.a.659.74
• 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			659.74
Character	\(\chi\)	\(=\)	756.659
Dual form			756.2.cf.a.491.74

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.827195 - 1.14706i) q^{2} +(1.29047 - 1.15529i) q^{3} +(-0.631498 - 1.89769i) q^{4} +(0.639184 - 1.75614i) q^{5} +(-0.257718 - 2.43589i) q^{6} +(0.984808 + 0.173648i) q^{7} +(-2.69913 - 0.845390i) q^{8} +(0.330617 - 2.98173i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.827195	−	1.14706i	0.584915	−	0.811095i
							\(3\)	1.29047	−	1.15529i	0.745052	−	0.667006i
\(5\)	0.639184	−	1.75614i	0.285852	−	0.785372i								−0.710784	−	0.703411i	\(-0.751660\pi\)
							0.996636	−	0.0819609i	\(-0.0261183\pi\)
\(7\)	0.984808	+	0.173648i	0.372222	+	0.0656328i
							\(11\)	0.0740830	−	0.0269640i	0.0223369	−	0.00812996i	−0.330828	−	0.943691i	\(-0.607328\pi\)
0.353164	+	0.935561i	\(0.385106\pi\)
\(13\)	−0.167701	+	0.140718i	−0.0465120	+	0.0390282i	−0.665747	−	0.746177i	\(-0.731887\pi\)
							0.619235	+	0.785205i	\(0.287443\pi\)
\(17\)	−3.21755	+	1.85765i	−0.780370	+	0.450547i	−0.836561	−	0.547873i	\(-0.815438\pi\)
							0.0561916	+	0.998420i	\(0.482104\pi\)
\(19\)	3.43365	+	1.98242i	0.787733	+	0.454798i	0.839164	−	0.543879i	\(-0.183045\pi\)
							−0.0514311	+	0.998677i	\(0.516378\pi\)
\(23\)	0.995430	+	5.64537i	0.207562	+	1.17714i	0.893358	+	0.449346i	\(0.148343\pi\)
							−0.685796	+	0.727794i	\(0.740546\pi\)
\(29\)	1.80334	−	2.14914i	0.334872	−	0.399085i	−0.572163	−	0.820140i	\(-0.693896\pi\)
							0.907035	+	0.421055i	\(0.138340\pi\)
\(31\)	6.34231	−	1.11832i	1.13911	−	0.200856i	0.427895	−	0.903828i	\(-0.359255\pi\)
							0.711217	+	0.702972i	\(0.248144\pi\)
\(37\)	−4.32034	−	7.48305i	−0.710259	−	1.23021i	−0.964760	−	0.263133i	\(-0.915244\pi\)
							0.254500	−	0.967073i	\(-0.418089\pi\)
\(41\)	−1.30970	−	1.56084i	−0.204541	−	0.243762i	0.654016	−	0.756481i	\(-0.273083\pi\)
							−0.858557	+	0.512718i	\(0.828638\pi\)
\(43\)	0.501406	+	1.37760i	0.0764637	+	0.210082i	0.972035	−	0.234835i	\(-0.0754551\pi\)
							−0.895571	+	0.444918i	\(0.853233\pi\)
\(47\)	−0.871864	+	4.94459i	−0.127174	+	0.721242i	0.852818	+	0.522208i	\(0.174892\pi\)
							−0.979993	+	0.199034i	\(0.936220\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.cf.a.659.74	yes	648
4.3	odd	2	inner	756.2.cf.a.659.83	yes	648
27.5	odd	18	inner	756.2.cf.a.491.83	yes	648
108.59	even	18	inner	756.2.cf.a.491.74	✓	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.cf.a.491.74	✓	648	108.59	even	18	inner
756.2.cf.a.491.83	yes	648	27.5	odd	18	inner
756.2.cf.a.659.74	yes	648	1.1	even	1	trivial
756.2.cf.a.659.83	yes	648	4.3	odd	2	inner