Embedded newform 72.7.j.a.5.6

• Label: 72.7.j.a.5.6
• Level: $72$
• Weight: $7$
• Character: 72.5
• Analytic conductor: $16.564$
• Analytic rank: $0$
• Dimension: $140$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	72.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(16.5638940206\)
Analytic rank:	\(0\)
Dimension:	\(140\)
Relative dimension:	\(70\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			5.6
Character	\(\chi\)	\(=\)	72.5
Dual form			72.7.j.a.29.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.83675 - 1.60790i) q^{2} +(6.22898 + 26.2717i) q^{3} +(58.8293 + 25.2014i) q^{4} +(-96.7705 + 167.611i) q^{5} +(-6.57280 - 215.900i) q^{6} +(-278.012 - 481.531i) q^{7} +(-420.509 - 292.089i) q^{8} +(-651.400 + 327.291i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 3 q^{2} - q^{4} + 317 q^{6} - 2 q^{7} - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\)	\(37\)	\(55\)	\(65\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.83675	−	1.60790i	−0.979594	−	0.200987i
							\(3\)	6.22898	+	26.2717i	0.230703	+	0.973024i
\(5\)	−96.7705	+	167.611i	−0.774164	+	1.34089i								0.161099	+	0.986938i	\(0.448496\pi\)
							−0.935263	+	0.353954i	\(0.884837\pi\)
\(7\)	−278.012	−	481.531i	−0.810531	−	1.40388i	−0.912493	−	0.409093i	\(-0.865845\pi\)
							0.101961	−	0.994788i	\(-0.467488\pi\)
\(11\)	−717.160	−	1242.16i	−0.538813	−	0.933251i	−0.998968	−	0.0454130i	\(-0.985540\pi\)
							0.460155	−	0.887838i	\(-0.347794\pi\)
\(13\)	2060.65	+	1189.72i	0.937939	+	0.541520i	0.889314	−	0.457297i	\(-0.151183\pi\)
							0.0486257	+	0.998817i	\(0.484516\pi\)
\(17\)			1210.32i			0.246351i	0.992385	+	0.123175i	\(0.0393078\pi\)
							−0.992385	+	0.123175i	\(0.960692\pi\)
\(19\)		−	3328.64i		−	0.485295i	−0.970115	−	0.242647i	\(-0.921984\pi\)
							0.970115	−	0.242647i	\(-0.0780158\pi\)
\(23\)	8308.29	+	4796.79i	0.682854	+	0.394246i	0.800930	−	0.598758i	\(-0.204339\pi\)
							−0.118075	+	0.993005i	\(0.537672\pi\)
\(29\)	18858.7	+	32664.2i	0.773246	+	1.33930i	0.935775	+	0.352598i	\(0.114702\pi\)
							−0.162529	+	0.986704i	\(0.551965\pi\)
\(31\)	15778.2	−	27328.6i	0.529629	−	0.917344i	−0.469774	−	0.882787i	\(-0.655665\pi\)
							0.999403	−	0.0345569i	\(-0.0110020\pi\)
\(37\)		−	82510.5i		−	1.62894i	−0.580208	−	0.814468i	\(-0.697029\pi\)
							0.580208	−	0.814468i	\(-0.302971\pi\)
\(41\)	−53628.1	−	30962.2i	−0.778110	−	0.449242i	0.0576501	−	0.998337i	\(-0.481639\pi\)
							−0.835760	+	0.549095i	\(0.814973\pi\)
\(43\)	−48745.8	+	28143.4i	−0.613101	+	0.353974i	−0.774178	−	0.632968i	\(-0.781836\pi\)
							0.161077	+	0.986942i	\(0.448503\pi\)
\(47\)	−3035.58	+	1752.59i	−0.0292380	+	0.0168806i	−0.514548	−	0.857462i	\(-0.672040\pi\)
							0.485310	+	0.874342i	\(0.338707\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.7.j.a.5.6	✓	140
8.5	even	2	inner	72.7.j.a.5.28	yes	140
9.2	odd	6	inner	72.7.j.a.29.28	yes	140
72.29	odd	6	inner	72.7.j.a.29.6	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.7.j.a.5.6	✓	140	1.1	even	1	trivial
72.7.j.a.5.28	yes	140	8.5	even	2	inner
72.7.j.a.29.6	yes	140	72.29	odd	6	inner
72.7.j.a.29.28	yes	140	9.2	odd	6	inner