Embedded newform 72.5.p.b.43.8

• Label: 72.5.p.b.43.8
• Level: $72$
• Weight: $5$
• Character: 72.43
• Analytic conductor: $7.443$
• Analytic rank: $0$
• Dimension: $88$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	72.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.44263734204\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(44\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			43.8
Character	\(\chi\)	\(=\)	72.43
Dual form			72.5.p.b.67.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.42315 + 2.06931i) q^{2} +(-7.80714 - 4.47756i) q^{3} +(7.43587 - 14.1671i) q^{4} +(14.7009 - 8.48755i) q^{5} +(35.9905 - 0.828075i) q^{6} +(-16.5269 - 9.54179i) q^{7} +(3.86217 + 63.8834i) q^{8} +(40.9029 + 69.9139i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 7 q^{2} - 18 q^{3} + 31 q^{4} - 9 q^{6} - 386 q^{8} + 30 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(37\)	\(55\)	\(65\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.42315	+	2.06931i	−0.855787	+	0.517329i
							\(3\)	−7.80714	−	4.47756i	−0.867460	−	0.497507i
\(5\)	14.7009	−	8.48755i	0.588035	−	0.339502i								−0.176285	−	0.984339i	\(-0.556408\pi\)
							0.764320	+	0.644837i	\(0.223075\pi\)
\(7\)	−16.5269	−	9.54179i	−0.337283	−	0.194730i	0.321787	−	0.946812i	\(-0.395717\pi\)
							−0.659070	+	0.752082i	\(0.729050\pi\)
\(11\)	82.8239	−	143.455i	0.684495	−	1.18558i	−0.289101	−	0.957299i	\(-0.593356\pi\)
							0.973595	−	0.228281i	\(-0.0733104\pi\)
\(13\)	−260.072	+	150.152i	−1.53888	+	0.888476i	−0.539981	+	0.841677i	\(0.681568\pi\)
							−0.998904	+	0.0467981i	\(0.985098\pi\)
\(17\)	−351.052			−1.21471			−0.607357	−	0.794429i	\(-0.707770\pi\)
							−0.607357	+	0.794429i	\(0.707770\pi\)
\(19\)	−65.0013			−0.180059			−0.0900295	−	0.995939i	\(-0.528696\pi\)
							−0.0900295	+	0.995939i	\(0.528696\pi\)
\(23\)	−268.140	+	154.811i	−0.506882	+	0.292648i	−0.731551	−	0.681787i	\(-0.761203\pi\)
							0.224669	+	0.974435i	\(0.427870\pi\)
\(29\)	693.254	+	400.250i	0.824321	+	0.475922i	0.851904	−	0.523697i	\(-0.175448\pi\)
							−0.0275831	+	0.999620i	\(0.508781\pi\)
\(31\)	−1503.86	+	868.252i	−1.56489	+	0.903488i	−0.568137	+	0.822934i	\(0.692336\pi\)
							−0.996750	+	0.0805539i	\(0.974331\pi\)
\(37\)			1622.77i			1.18537i	0.805436	+	0.592683i	\(0.201931\pi\)
							−0.805436	+	0.592683i	\(0.798069\pi\)
\(41\)	2.04717	+	3.54581i	0.00121783	+	0.00210934i	0.866634	−	0.498945i	\(-0.166279\pi\)
							−0.865416	+	0.501054i	\(0.832946\pi\)
\(43\)	781.189	−	1353.06i	0.422493	−	0.731779i	−0.573690	−	0.819073i	\(-0.694489\pi\)
							0.996183	+	0.0872938i	\(0.0278219\pi\)
\(47\)	−1137.68	−	656.841i	−0.515021	−	0.297348i	0.219874	−	0.975528i	\(-0.429435\pi\)
							−0.734895	+	0.678181i	\(0.762769\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.5.p.b.43.8	✓	88
3.2	odd	2		216.5.p.b.19.37		88
4.3	odd	2		288.5.t.b.79.36		88
8.3	odd	2	inner	72.5.p.b.43.21	yes	88
8.5	even	2		288.5.t.b.79.35		88
9.4	even	3	inner	72.5.p.b.67.21	yes	88
9.5	odd	6		216.5.p.b.91.24		88
12.11	even	2		864.5.t.b.559.16		88
24.5	odd	2		864.5.t.b.559.29		88
24.11	even	2		216.5.p.b.19.24		88
36.23	even	6		864.5.t.b.847.29		88
36.31	odd	6		288.5.t.b.175.35		88
72.5	odd	6		864.5.t.b.847.16		88
72.13	even	6		288.5.t.b.175.36		88
72.59	even	6		216.5.p.b.91.37		88
72.67	odd	6	inner	72.5.p.b.67.8	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.5.p.b.43.8	✓	88	1.1	even	1	trivial
72.5.p.b.43.21	yes	88	8.3	odd	2	inner
72.5.p.b.67.8	yes	88	72.67	odd	6	inner
72.5.p.b.67.21	yes	88	9.4	even	3	inner
216.5.p.b.19.24		88	24.11	even	2
216.5.p.b.19.37		88	3.2	odd	2
216.5.p.b.91.24		88	9.5	odd	6
216.5.p.b.91.37		88	72.59	even	6
288.5.t.b.79.35		88	8.5	even	2
288.5.t.b.79.36		88	4.3	odd	2
288.5.t.b.175.35		88	36.31	odd	6
288.5.t.b.175.36		88	72.13	even	6
864.5.t.b.559.16		88	12.11	even	2
864.5.t.b.559.29		88	24.5	odd	2
864.5.t.b.847.16		88	72.5	odd	6
864.5.t.b.847.29		88	36.23	even	6