Embedded newform 756.2.cf.a.659.47

• Label: 756.2.cf.a.659.47
• 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			659.47
Character	\(\chi\)	\(=\)	756.659
Dual form			756.2.cf.a.491.47

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.270123 + 1.38818i) q^{2} +(-1.73156 - 0.0414000i) q^{3} +(-1.85407 - 0.749955i) q^{4} +(-0.0583884 + 0.160421i) q^{5} +(0.525203 - 2.39252i) q^{6} +(-0.984808 - 0.173648i) q^{7} +(1.54190 - 2.37119i) q^{8} +(2.99657 + 0.143373i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.270123	+	1.38818i	−0.191005	+	0.981589i
							\(3\)	−1.73156	−	0.0414000i	−0.999714	−	0.0239023i
\(5\)	−0.0583884	+	0.160421i	−0.0261121	+	0.0717424i								−0.952064	−	0.305900i	\(-0.901043\pi\)
							0.925952	+	0.377642i	\(0.123265\pi\)
\(7\)	−0.984808	−	0.173648i	−0.372222	−	0.0656328i
							\(11\)	−1.14378	+	0.416302i	−0.344863	+	0.125520i	−0.508644	−	0.860977i	\(-0.669853\pi\)
0.163781	+	0.986497i	\(0.447631\pi\)
\(13\)	2.79988	−	2.34938i	0.776546	−	0.651600i	−0.165830	−	0.986154i	\(-0.553030\pi\)
							0.942376	+	0.334555i	\(0.108586\pi\)
\(17\)	−0.976411	+	0.563731i	−0.236814	+	0.136725i	−0.613712	−	0.789530i	\(-0.710324\pi\)
							0.376897	+	0.926255i	\(0.376991\pi\)
\(19\)	1.62255	+	0.936781i	0.372239	+	0.214912i	0.674436	−	0.738333i	\(-0.264387\pi\)
							−0.302197	+	0.953245i	\(0.597720\pi\)
\(23\)	0.245953	+	1.39487i	0.0512847	+	0.290850i	0.999654	−	0.0263195i	\(-0.00837872\pi\)
							−0.948369	+	0.317169i	\(0.897268\pi\)
\(29\)	−2.77440	+	3.30640i	−0.515193	+	0.613983i	−0.959437	−	0.281923i	\(-0.909028\pi\)
							0.444244	+	0.895906i	\(0.353472\pi\)
\(31\)	−2.93017	+	0.516668i	−0.526274	+	0.0927963i	−0.430472	−	0.902604i	\(-0.641653\pi\)
							−0.0958022	+	0.995400i	\(0.530542\pi\)
\(37\)	−1.20683	−	2.09030i	−0.198402	−	0.343643i	0.749608	−	0.661882i	\(-0.230242\pi\)
							−0.948011	+	0.318239i	\(0.896909\pi\)
\(41\)	1.85938	+	2.21592i	0.290386	+	0.346069i	0.891439	−	0.453141i	\(-0.149696\pi\)
							−0.601053	+	0.799209i	\(0.705252\pi\)
\(43\)	2.37451	+	6.52390i	0.362109	+	0.994886i	0.978283	+	0.207275i	\(0.0664596\pi\)
							−0.616174	+	0.787610i	\(0.711318\pi\)
\(47\)	−0.989730	+	5.61304i	−0.144367	+	0.818746i	0.823506	+	0.567307i	\(0.192015\pi\)
							−0.967873	+	0.251439i	\(0.919096\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.cf.a.659.47	yes	648
4.3	odd	2	inner	756.2.cf.a.659.14	yes	648
27.5	odd	18	inner	756.2.cf.a.491.14	✓	648
108.59	even	18	inner	756.2.cf.a.491.47	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.cf.a.491.14	✓	648	27.5	odd	18	inner
756.2.cf.a.491.47	yes	648	108.59	even	18	inner
756.2.cf.a.659.14	yes	648	4.3	odd	2	inner
756.2.cf.a.659.47	yes	648	1.1	even	1	trivial