Embedded newform 72.7.j.a.5.8

• Label: 72.7.j.a.5.8
• 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			5.8
Character	\(\chi\)	\(=\)	72.5
Dual form			72.7.j.a.29.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.59716 - 2.50663i) q^{2} +(12.6861 - 23.8341i) q^{3} +(51.4336 + 38.0866i) q^{4} +(-106.929 + 185.206i) q^{5} +(-156.122 + 149.272i) q^{6} +(267.810 + 463.860i) q^{7} +(-295.280 - 418.275i) q^{8} +(-407.126 - 604.723i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 3 q^{2} - q^{4} + 317 q^{6} - 2 q^{7} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.59716	−	2.50663i	−0.949645	−	0.313329i
							\(3\)	12.6861	−	23.8341i	0.469855	−	0.882743i
\(5\)	−106.929	+	185.206i	−0.855429	+	1.48165i								0.0208171	+	0.999783i	\(0.493373\pi\)
							−0.876246	+	0.481864i	\(0.839960\pi\)
\(7\)	267.810	+	463.860i	0.780786	+	1.35236i	0.931484	+	0.363781i	\(0.118514\pi\)
							−0.150698	+	0.988580i	\(0.548152\pi\)
\(11\)	−739.855	−	1281.47i	−0.555864	−	0.962785i	−0.997836	−	0.0657557i	\(-0.979054\pi\)
							0.441972	−	0.897029i	\(-0.354279\pi\)
\(13\)	−1300.36	−	750.761i	−0.591878	−	0.341721i	0.173961	−	0.984752i	\(-0.444343\pi\)
							−0.765840	+	0.643031i	\(0.777677\pi\)
\(17\)		−	4310.39i		−	0.877345i	−0.898647	−	0.438672i	\(-0.855449\pi\)
							0.898647	−	0.438672i	\(-0.144551\pi\)
\(19\)		−	3126.69i		−	0.455852i	−0.973678	−	0.227926i	\(-0.926806\pi\)
							0.973678	−	0.227926i	\(-0.0731944\pi\)
\(23\)	−3087.81	−	1782.75i	−0.253786	−	0.146523i	0.367711	−	0.929940i	\(-0.380142\pi\)
							−0.621497	+	0.783417i	\(0.713475\pi\)
\(29\)	−13276.4	−	22995.4i	−0.544360	−	0.942860i	−0.998647	−	0.0520042i	\(-0.983439\pi\)
							0.454286	−	0.890856i	\(-0.349894\pi\)
\(31\)	−7715.20	+	13363.1i	−0.258977	+	0.448562i	−0.965968	−	0.258661i	\(-0.916719\pi\)
							0.706991	+	0.707223i	\(0.250052\pi\)
\(37\)		−	51209.9i		−	1.01099i	−0.862828	−	0.505497i	\(-0.831309\pi\)
							0.862828	−	0.505497i	\(-0.168691\pi\)
\(41\)	85047.5	+	49102.2i	1.23399	+	0.712442i	0.967858	−	0.251496i	\(-0.0809226\pi\)
							0.266127	+	0.963938i	\(0.414256\pi\)
\(43\)	84337.5	−	48692.3i	1.06076	−	0.612428i	0.135115	−	0.990830i	\(-0.456860\pi\)
							0.925641	+	0.378402i	\(0.123526\pi\)
\(47\)	−125990.	+	72740.4i	−1.21351	+	0.700619i	−0.963522	−	0.267630i	\(-0.913760\pi\)
							−0.249987	+	0.968249i	\(0.580426\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.7.j.a.5.8	✓	140
8.5	even	2	inner	72.7.j.a.5.32	yes	140
9.2	odd	6	inner	72.7.j.a.29.32	yes	140
72.29	odd	6	inner	72.7.j.a.29.8	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.7.j.a.5.8	✓	140	1.1	even	1	trivial
72.7.j.a.5.32	yes	140	8.5	even	2	inner
72.7.j.a.29.8	yes	140	72.29	odd	6	inner
72.7.j.a.29.32	yes	140	9.2	odd	6	inner