Embedded newform 756.2.cf.a.659.73

• Label: 756.2.cf.a.659.73
• 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			659.73
Character	\(\chi\)	\(=\)	756.659
Dual form			756.2.cf.a.491.73

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.781907 - 1.17840i) q^{2} +(1.69439 + 0.359245i) q^{3} +(-0.777241 - 1.84280i) q^{4} +(1.22710 - 3.37142i) q^{5} +(1.74819 - 1.71576i) q^{6} +(-0.984808 - 0.173648i) q^{7} +(-2.77928 - 0.524996i) q^{8} +(2.74189 + 1.21740i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.781907	−	1.17840i	0.552892	−	0.833253i
							\(3\)	1.69439	+	0.359245i	0.978254	+	0.207410i
\(5\)	1.22710	−	3.37142i	0.548774	−	1.50775i								−0.286593	−	0.958053i	\(-0.592523\pi\)
							0.835367	−	0.549693i	\(-0.185255\pi\)
\(7\)	−0.984808	−	0.173648i	−0.372222	−	0.0656328i
							\(11\)	−4.02487	+	1.46493i	−1.21354	+	0.441694i	−0.867931	−	0.496684i	\(-0.834551\pi\)
−0.345612	+	0.938378i	\(0.612329\pi\)
\(13\)	1.16243	−	0.975392i	0.322399	−	0.270525i	−0.467195	−	0.884154i	\(-0.654735\pi\)
							0.789594	+	0.613629i	\(0.210291\pi\)
\(17\)	2.93280	−	1.69325i	0.711308	−	0.410674i	−0.100237	−	0.994964i	\(-0.531960\pi\)
							0.811545	+	0.584290i	\(0.198627\pi\)
\(19\)	1.42431	+	0.822327i	0.326760	+	0.188655i	0.654402	−	0.756147i	\(-0.272921\pi\)
							−0.327642	+	0.944802i	\(0.606254\pi\)
\(23\)	−0.881887	−	5.00143i	−0.183886	−	1.04287i	−0.927378	−	0.374125i	\(-0.877943\pi\)
							0.743492	−	0.668745i	\(-0.233168\pi\)
\(29\)	−3.95188	+	4.70966i	−0.733845	+	0.874563i	−0.995897	−	0.0904910i	\(-0.971156\pi\)
							0.262052	+	0.965054i	\(0.415601\pi\)
\(31\)	6.28066	−	1.10745i	1.12804	−	0.198904i	0.421672	−	0.906749i	\(-0.361444\pi\)
							0.706368	+	0.707845i	\(0.250332\pi\)
\(37\)	5.36903	+	9.29944i	0.882663	+	1.52882i	0.848369	+	0.529406i	\(0.177585\pi\)
							0.0342947	+	0.999412i	\(0.489082\pi\)
\(41\)	1.64389	+	1.95911i	0.256732	+	0.305961i	0.878980	−	0.476859i	\(-0.158225\pi\)
							−0.622248	+	0.782820i	\(0.713780\pi\)
\(43\)	2.34980	+	6.45601i	0.358340	+	0.984532i	0.979605	+	0.200932i	\(0.0643969\pi\)
							−0.621265	+	0.783601i	\(0.713381\pi\)
\(47\)	0.809719	−	4.59215i	0.118110	−	0.669833i	−0.867054	−	0.498215i	\(-0.833989\pi\)
							0.985163	−	0.171619i	\(-0.0548997\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.cf.a.659.73	yes	648
4.3	odd	2	inner	756.2.cf.a.659.84	yes	648
27.5	odd	18	inner	756.2.cf.a.491.84	yes	648
108.59	even	18	inner	756.2.cf.a.491.73	✓	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.cf.a.491.73	✓	648	108.59	even	18	inner
756.2.cf.a.491.84	yes	648	27.5	odd	18	inner
756.2.cf.a.659.73	yes	648	1.1	even	1	trivial
756.2.cf.a.659.84	yes	648	4.3	odd	2	inner