Embedded newform 756.2.cf.a.659.59

• Label: 756.2.cf.a.659.59
• 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			659.59
Character	\(\chi\)	\(=\)	756.659
Dual form			756.2.cf.a.491.59

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.211879 - 1.39825i) q^{2} +(-1.50719 + 0.853450i) q^{3} +(-1.91021 - 0.592521i) q^{4} +(0.248693 - 0.683279i) q^{5} +(0.873995 + 2.28826i) q^{6} +(0.984808 + 0.173648i) q^{7} +(-1.23323 + 2.54542i) q^{8} +(1.54325 - 2.57262i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.211879	−	1.39825i	0.149821	−	0.988713i
							\(3\)	−1.50719	+	0.853450i	−0.870177	+	0.492739i
\(5\)	0.248693	−	0.683279i	0.111219	−	0.305572i								−0.871579	−	0.490255i	\(-0.836904\pi\)
							0.982798	+	0.184683i	\(0.0591259\pi\)
\(7\)	0.984808	+	0.173648i	0.372222	+	0.0656328i
							\(11\)	−0.984832	+	0.358450i	−0.296938	+	0.108077i	−0.486193	−	0.873851i	\(-0.661615\pi\)
0.189255	+	0.981928i	\(0.439393\pi\)
\(13\)	−2.69961	+	2.26524i	−0.748738	+	0.628265i	−0.935169	−	0.354203i	\(-0.884752\pi\)
							0.186431	+	0.982468i	\(0.440308\pi\)
\(17\)	−3.34338	+	1.93030i	−0.810889	+	0.468167i	−0.847265	−	0.531171i	\(-0.821752\pi\)
							0.0363753	+	0.999338i	\(0.488419\pi\)
\(19\)	−2.50054	−	1.44369i	−0.573663	−	0.331204i	0.184948	−	0.982748i	\(-0.440788\pi\)
							−0.758611	+	0.651544i	\(0.774122\pi\)
\(23\)	0.0714328	+	0.405115i	0.0148948	+	0.0844724i	0.991349	−	0.131252i	\(-0.0418998\pi\)
							−0.976454	+	0.215725i	\(0.930789\pi\)
\(29\)	−2.28576	+	2.72407i	−0.424455	+	0.505846i	−0.935314	−	0.353818i	\(-0.884883\pi\)
							0.510859	+	0.859665i	\(0.329328\pi\)
\(31\)	3.49380	−	0.616051i	0.627504	−	0.110646i	0.149153	−	0.988814i	\(-0.452345\pi\)
							0.478352	+	0.878168i	\(0.341234\pi\)
\(37\)	3.18251	+	5.51226i	0.523201	+	0.906210i	0.999635	+	0.0270003i	\(0.00859552\pi\)
							−0.476435	+	0.879210i	\(0.658071\pi\)
\(41\)	5.89415	+	7.02437i	0.920511	+	1.09702i	0.995007	+	0.0998011i	\(0.0318207\pi\)
							−0.0744963	+	0.997221i	\(0.523735\pi\)
\(43\)	2.35770	+	6.47772i	0.359545	+	0.987843i	0.979187	+	0.202958i	\(0.0650555\pi\)
							−0.619642	+	0.784885i	\(0.712722\pi\)
\(47\)	−1.21462	+	6.88845i	−0.177171	+	1.00478i	0.758438	+	0.651745i	\(0.225963\pi\)
							−0.935609	+	0.353039i	\(0.885148\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.cf.a.659.59	yes	648
4.3	odd	2	inner	756.2.cf.a.659.99	yes	648
27.5	odd	18	inner	756.2.cf.a.491.99	yes	648
108.59	even	18	inner	756.2.cf.a.491.59	✓	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.cf.a.491.59	✓	648	108.59	even	18	inner
756.2.cf.a.491.99	yes	648	27.5	odd	18	inner
756.2.cf.a.659.59	yes	648	1.1	even	1	trivial
756.2.cf.a.659.99	yes	648	4.3	odd	2	inner