Embedded newform 756.2.cf.a.155.17

• Label: 756.2.cf.a.155.17
• Level: $756$
• Weight: $2$
• Character: 756.155
• 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			155.17
Character	\(\chi\)	\(=\)	756.155
Dual form			756.2.cf.a.239.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.24559 + 0.669697i) q^{2} +(1.67938 - 0.423888i) q^{3} +(1.10301 - 1.66834i) q^{4} +(-3.97359 + 0.700651i) q^{5} +(-1.80795 + 1.65267i) q^{6} +(0.642788 - 0.766044i) q^{7} +(-0.256623 + 2.81676i) q^{8} +(2.64064 - 1.42374i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(325\)	\(379\)
\(\chi(n)\)	\(e\left(\frac{7}{18}\right)\)	\(1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.24559	+	0.669697i	−0.880768	+	0.473547i
							\(3\)	1.67938	−	0.423888i	0.969591	−	0.244732i
\(5\)	−3.97359	+	0.700651i	−1.77704	+	0.313341i								−0.963409	−	0.268037i	\(-0.913625\pi\)
							−0.813634	+	0.581377i	\(0.802514\pi\)
\(7\)	0.642788	−	0.766044i	0.242951	−	0.289538i
							\(11\)	−0.441140	+	2.50183i	−0.133009	+	0.754329i	0.843217	+	0.537573i	\(0.180659\pi\)
−0.976226	+	0.216756i	\(0.930452\pi\)
\(13\)	3.19415	+	1.16257i	0.885897	+	0.322440i	0.744587	−	0.667525i	\(-0.232646\pi\)
							0.141310	+	0.989965i	\(0.454869\pi\)
\(17\)	5.00010	−	2.88681i	1.21270	−	0.700154i	0.249355	−	0.968412i	\(-0.419781\pi\)
							0.963347	+	0.268258i	\(0.0864480\pi\)
\(19\)	−4.67482	−	2.69901i	−1.07248	−	0.619196i	−0.143621	−	0.989633i	\(-0.545875\pi\)
							−0.928857	+	0.370437i	\(0.879208\pi\)
\(23\)	0.468468	−	0.393092i	0.0976824	−	0.0819653i	−0.592639	−	0.805469i	\(-0.701914\pi\)
							0.690321	+	0.723503i	\(0.257469\pi\)
\(29\)	2.25604	+	6.19841i	0.418935	+	1.15102i	0.952309	+	0.305136i	\(0.0987018\pi\)
							−0.533374	+	0.845880i	\(0.679076\pi\)
\(31\)	1.25104	+	1.49093i	0.224693	+	0.267778i	0.866599	−	0.499004i	\(-0.166301\pi\)
							−0.641907	+	0.766783i	\(0.721856\pi\)
\(37\)	4.30240	+	7.45197i	0.707310	+	1.22510i	0.965852	+	0.259096i	\(0.0834246\pi\)
							−0.258542	+	0.966000i	\(0.583242\pi\)
\(41\)	3.95182	−	10.8575i	0.617170	−	1.69566i	−0.0966376	−	0.995320i	\(-0.530809\pi\)
							0.713808	−	0.700342i	\(-0.246969\pi\)
\(43\)	6.26309	+	1.10435i	0.955113	+	0.168412i	0.629422	−	0.777064i	\(-0.283292\pi\)
							0.325691	+	0.945476i	\(0.394403\pi\)
\(47\)	7.93166	+	6.65545i	1.15695	+	0.970797i	0.999859	−	0.0167869i	\(-0.00534369\pi\)
							0.157092	+	0.987584i	\(0.449788\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.cf.a.155.17	✓	648
4.3	odd	2	inner	756.2.cf.a.155.99	yes	648
27.23	odd	18	inner	756.2.cf.a.239.99	yes	648
108.23	even	18	inner	756.2.cf.a.239.17	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.cf.a.155.17	✓	648	1.1	even	1	trivial
756.2.cf.a.155.99	yes	648	4.3	odd	2	inner
756.2.cf.a.239.17	yes	648	108.23	even	18	inner
756.2.cf.a.239.99	yes	648	27.23	odd	18	inner