Embedded newform 72.8.l.b.11.11

• Label: 72.8.l.b.11.11
• Level: $72$
• Weight: $8$
• Character: 72.11
• Analytic conductor: $22.492$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	72.l (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(22.4917218349\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(80\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.11
Character	\(\chi\)	\(=\)	72.11
Dual form			72.8.l.b.59.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-10.0761 + 5.14514i) q^{2} +(-12.6662 + 45.0174i) q^{3} +(75.0550 - 103.686i) q^{4} +(-217.264 - 376.312i) q^{5} +(-103.995 - 518.769i) q^{6} +(-1159.78 - 669.601i) q^{7} +(-222.782 + 1430.92i) q^{8} +(-1866.14 - 1140.40i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 3 q^{2} - 90 q^{3} - 257 q^{4} - 459 q^{6} + 3018 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−10.0761	+	5.14514i	−0.890609	+	0.454771i
							\(3\)	−12.6662	+	45.0174i	−0.270846	+	0.962623i
\(5\)	−217.264	−	376.312i	−0.777306	−	1.34633i								−0.933489	−	0.358605i	\(-0.883253\pi\)
							0.156184	−	0.987728i	\(-0.450081\pi\)
\(7\)	−1159.78	−	669.601i	−1.27801	−	0.737858i	−0.301526	−	0.953458i	\(-0.597496\pi\)
							−0.976482	+	0.215599i	\(0.930829\pi\)
\(11\)	−2960.53	−	1709.26i	−0.670649	−	0.387199i	0.125674	−	0.992072i	\(-0.459891\pi\)
							−0.796322	+	0.604872i	\(0.793224\pi\)
\(13\)	2937.09	−	1695.73i	0.370780	−	0.214070i	−0.303019	−	0.952984i	\(-0.597995\pi\)
							0.673799	+	0.738915i	\(0.264661\pi\)
\(17\)			15117.8i			0.746308i	0.927769	+	0.373154i	\(0.121724\pi\)
							−0.927769	+	0.373154i	\(0.878276\pi\)
\(19\)	−1263.20			−0.0422507			−0.0211253	−	0.999777i	\(-0.506725\pi\)
							−0.0211253	+	0.999777i	\(0.506725\pi\)
\(23\)	−12242.0	−	21203.8i	−0.209800	−	0.363384i	0.741851	−	0.670564i	\(-0.233948\pi\)
							−0.951651	+	0.307180i	\(0.900615\pi\)
\(29\)	−98508.9	+	170622.i	−0.750037	+	1.29910i	0.197768	+	0.980249i	\(0.436631\pi\)
							−0.947804	+	0.318853i	\(0.896702\pi\)
\(31\)	191829.	−	110753.i	1.15651	−	0.667710i	0.206043	−	0.978543i	\(-0.433941\pi\)
							0.950465	+	0.310833i	\(0.100608\pi\)
\(37\)			77806.7i			0.252529i	0.991997	+	0.126264i	\(0.0402988\pi\)
							−0.991997	+	0.126264i	\(0.959701\pi\)
\(41\)	−659189.	+	380583.i	−1.49371	+	0.862394i	−0.999974	−	0.00721719i	\(-0.997703\pi\)
							−0.493737	+	0.869611i	\(0.664369\pi\)
\(43\)	72126.7	−	124927.i	0.138343	−	0.239617i	−0.788527	−	0.615001i	\(-0.789156\pi\)
							0.926869	+	0.375384i	\(0.122489\pi\)
\(47\)	574932.	−	995812.i	0.807745	−	1.39906i	−0.106677	−	0.994294i	\(-0.534021\pi\)
							0.914422	−	0.404761i	\(-0.132645\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.8.l.b.11.11	✓	160
8.3	odd	2	inner	72.8.l.b.11.38	yes	160
9.5	odd	6	inner	72.8.l.b.59.38	yes	160
72.59	even	6	inner	72.8.l.b.59.11	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.8.l.b.11.11	✓	160	1.1	even	1	trivial
72.8.l.b.11.38	yes	160	8.3	odd	2	inner
72.8.l.b.59.11	yes	160	72.59	even	6	inner
72.8.l.b.59.38	yes	160	9.5	odd	6	inner