Embedded newform 756.2.cf.a.659.80

• Label: 756.2.cf.a.659.80
• 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			659.80
Character	\(\chi\)	\(=\)	756.659
Dual form			756.2.cf.a.491.80

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.961296 + 1.03726i) q^{2} +(-1.23300 - 1.21644i) q^{3} +(-0.151819 + 1.99423i) q^{4} +(0.0717474 - 0.197124i) q^{5} +(0.0764880 - 2.44830i) q^{6} +(-0.984808 - 0.173648i) q^{7} +(-2.21448 + 1.75957i) q^{8} +(0.0405601 + 2.99973i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.961296	+	1.03726i	0.679739	+	0.733454i
							\(3\)	−1.23300	−	1.21644i	−0.711871	−	0.702310i
\(5\)	0.0717474	−	0.197124i	0.0320864	−	0.0881567i								−0.922615	−	0.385722i	\(-0.873952\pi\)
							0.954701	+	0.297565i	\(0.0961746\pi\)
\(7\)	−0.984808	−	0.173648i	−0.372222	−	0.0656328i
							\(11\)	5.85555	−	2.13125i	1.76551	−	0.642595i	0.765515	−	0.643418i	\(-0.222484\pi\)
1.00000		0.000823427i	\(0.000262105\pi\)
\(13\)	−3.71117	+	3.11405i	−1.02929	+	0.863681i	−0.990767	−	0.135577i	\(-0.956711\pi\)
							−0.0385277	+	0.999258i	\(0.512267\pi\)
\(17\)	−4.41652	+	2.54988i	−1.07116	+	0.618436i	−0.928499	−	0.371335i	\(-0.878900\pi\)
							−0.142664	+	0.989771i	\(0.545567\pi\)
\(19\)	5.39829	+	3.11670i	1.23845	+	0.715021i	0.968778	−	0.247931i	\(-0.0797505\pi\)
							0.269675	+	0.962952i	\(0.413084\pi\)
\(23\)	1.34858	+	7.64816i	0.281198	+	1.59475i	0.718559	+	0.695466i	\(0.244802\pi\)
							−0.437362	+	0.899286i	\(0.644087\pi\)
\(29\)	0.785750	−	0.936421i	0.145910	−	0.173889i	−0.688139	−	0.725579i	\(-0.741572\pi\)
							0.834050	+	0.551690i	\(0.186017\pi\)
\(31\)	−6.62883	+	1.16884i	−1.19057	+	0.209930i	−0.733619	−	0.679561i	\(-0.762170\pi\)
							−0.456953	+	0.889491i	\(0.651059\pi\)
\(37\)	0.533310	+	0.923721i	0.0876757	+	0.151859i	0.906528	−	0.422145i	\(-0.138723\pi\)
							−0.818853	+	0.574004i	\(0.805389\pi\)
\(41\)	0.152505	+	0.181749i	0.0238174	+	0.0283844i	0.777822	−	0.628485i	\(-0.216325\pi\)
							−0.754004	+	0.656869i	\(0.771880\pi\)
\(43\)	−1.44134	−	3.96006i	−0.219803	−	0.603903i	0.779957	−	0.625833i	\(-0.215241\pi\)
							−0.999760	+	0.0219304i	\(0.993019\pi\)
\(47\)	−0.716624	+	4.06418i	−0.104530	+	0.592821i	0.886876	+	0.462007i	\(0.152870\pi\)
							−0.991407	+	0.130815i	\(0.958241\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.cf.a.659.80	yes	648
4.3	odd	2	inner	756.2.cf.a.659.19	yes	648
27.5	odd	18	inner	756.2.cf.a.491.19	✓	648
108.59	even	18	inner	756.2.cf.a.491.80	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.cf.a.491.19	✓	648	27.5	odd	18	inner
756.2.cf.a.491.80	yes	648	108.59	even	18	inner
756.2.cf.a.659.19	yes	648	4.3	odd	2	inner
756.2.cf.a.659.80	yes	648	1.1	even	1	trivial