Embedded newform 756.2.cf.a.659.15

• Label: 756.2.cf.a.659.15
• 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			659.15
Character	\(\chi\)	\(=\)	756.659
Dual form			756.2.cf.a.491.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.27469 + 0.612508i) q^{2} +(-1.51360 - 0.842029i) q^{3} +(1.24967 - 1.56151i) q^{4} +(-1.15117 + 3.16281i) q^{5} +(2.44512 + 0.146234i) q^{6} +(0.984808 + 0.173648i) q^{7} +(-0.636500 + 2.75588i) q^{8} +(1.58197 + 2.54899i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.27469	+	0.612508i	−0.901342	+	0.433108i
							\(3\)	−1.51360	−	0.842029i	−0.873878	−	0.486146i
\(5\)	−1.15117	+	3.16281i	−0.514819	+	1.41445i								0.361341	+	0.932434i	\(0.382319\pi\)
							−0.876160	+	0.482020i	\(0.839903\pi\)
\(7\)	0.984808	+	0.173648i	0.372222	+	0.0656328i
							\(11\)	2.52494	−	0.919002i	0.761297	−	0.277089i	0.0679456	−	0.997689i	\(-0.478356\pi\)
0.693351	+	0.720600i	\(0.256133\pi\)
\(13\)	1.75210	−	1.47018i	0.485944	−	0.407756i	−0.366626	−	0.930368i	\(-0.619487\pi\)
							0.852570	+	0.522613i	\(0.175043\pi\)
\(17\)	−4.09050	+	2.36165i	−0.992092	+	0.572784i	−0.905899	−	0.423494i	\(-0.860803\pi\)
							−0.0861929	+	0.996278i	\(0.527470\pi\)
\(19\)	3.50980	+	2.02638i	0.805204	+	0.464885i	0.845287	−	0.534312i	\(-0.179429\pi\)
							−0.0400839	+	0.999196i	\(0.512763\pi\)
\(23\)	−0.542808	−	3.07842i	−0.113183	−	0.641894i	−0.987634	−	0.156779i	\(-0.949889\pi\)
							0.874450	−	0.485115i	\(-0.161222\pi\)
\(29\)	−0.207583	+	0.247388i	−0.0385472	+	0.0459388i	−0.784974	−	0.619529i	\(-0.787324\pi\)
							0.746426	+	0.665468i	\(0.231768\pi\)
\(31\)	−2.34834	+	0.414075i	−0.421774	+	0.0743701i	−0.380507	−	0.924778i	\(-0.624251\pi\)
							−0.0412669	+	0.999148i	\(0.513139\pi\)
\(37\)	−1.29177	−	2.23741i	−0.212366	−	0.367829i	0.740089	−	0.672509i	\(-0.234784\pi\)
							−0.952455	+	0.304681i	\(0.901450\pi\)
\(41\)	5.06829	+	6.04015i	0.791534	+	0.943313i	0.999393	−	0.0348456i	\(-0.0110939\pi\)
							−0.207859	+	0.978159i	\(0.566649\pi\)
\(43\)	3.74287	+	10.2835i	0.570782	+	1.56821i	0.803272	+	0.595613i	\(0.203091\pi\)
							−0.232490	+	0.972599i	\(0.574687\pi\)
\(47\)	−1.41707	+	8.03658i	−0.206700	+	1.17226i	0.688041	+	0.725671i	\(0.258471\pi\)
							−0.894742	+	0.446584i	\(0.852640\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.cf.a.659.15	yes	648
4.3	odd	2	inner	756.2.cf.a.659.46	yes	648
27.5	odd	18	inner	756.2.cf.a.491.46	yes	648
108.59	even	18	inner	756.2.cf.a.491.15	✓	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.cf.a.491.15	✓	648	108.59	even	18	inner
756.2.cf.a.491.46	yes	648	27.5	odd	18	inner
756.2.cf.a.659.15	yes	648	1.1	even	1	trivial
756.2.cf.a.659.46	yes	648	4.3	odd	2	inner