Embedded newform 72.4.l.b.11.13

• Label: 72.4.l.b.11.13
• Level: $72$
• Weight: $4$
• Character: 72.11
• Analytic conductor: $4.248$
• Analytic rank: $0$
• Dimension: $64$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	72.l (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.24813752041\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.13
Character	\(\chi\)	\(=\)	72.11
Dual form			72.4.l.b.59.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.867121 + 2.69223i) q^{2} +(0.0478653 + 5.19593i) q^{3} +(-6.49620 - 4.66898i) q^{4} +(-8.51655 - 14.7511i) q^{5} +(-14.0301 - 4.37664i) q^{6} +(-9.57807 - 5.52990i) q^{7} +(18.2030 - 13.4407i) q^{8} +(-26.9954 + 0.497410i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 3 q^{2} + 6 q^{3} - 17 q^{4} - 3 q^{6} + 42 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.867121	+	2.69223i	−0.306574	+	0.951847i
							\(3\)	0.0478653	+	5.19593i	0.00921169	+	0.999958i
\(5\)	−8.51655	−	14.7511i	−0.761744	−	1.31938i								−0.941951	−	0.335750i	\(-0.891010\pi\)
							0.180207	−	0.983629i	\(-0.442323\pi\)
\(7\)	−9.57807	−	5.52990i	−0.517167	−	0.298587i	0.218608	−	0.975813i	\(-0.429849\pi\)
							−0.735775	+	0.677226i	\(0.763182\pi\)
\(11\)	25.7667	+	14.8764i	0.706269	+	0.407765i	0.809678	−	0.586874i	\(-0.199642\pi\)
							−0.103409	+	0.994639i	\(0.532975\pi\)
\(13\)	−52.0658	+	30.0602i	−1.11081	+	0.641324i	−0.939038	−	0.343814i	\(-0.888281\pi\)
							−0.171767	+	0.985138i	\(0.554948\pi\)
\(17\)		−	91.7888i		−	1.30953i	−0.755831	−	0.654766i	\(-0.772767\pi\)
							0.755831	−	0.654766i	\(-0.227233\pi\)
\(19\)	−136.507			−1.64826			−0.824129	−	0.566402i	\(-0.808335\pi\)
							−0.824129	+	0.566402i	\(0.808335\pi\)
\(23\)	−30.0089	−	51.9770i	−0.272056	−	0.471215i	0.697332	−	0.716748i	\(-0.254370\pi\)
							−0.969388	+	0.245533i	\(0.921037\pi\)
\(29\)	−65.0911	+	112.741i	−0.416797	+	0.721914i	−0.995615	−	0.0935428i	\(-0.970181\pi\)
							0.578818	+	0.815457i	\(0.303514\pi\)
\(31\)	148.423	−	85.6920i	0.859921	−	0.496475i	−0.00406514	−	0.999992i	\(-0.501294\pi\)
							0.863986	+	0.503516i	\(0.167961\pi\)
\(37\)			259.167i			1.15154i	0.817613	+	0.575768i	\(0.195297\pi\)
							−0.817613	+	0.575768i	\(0.804703\pi\)
\(41\)	14.3285	−	8.27259i	0.0545791	−	0.0315113i	−0.472462	−	0.881351i	\(-0.656635\pi\)
							0.527041	+	0.849840i	\(0.323301\pi\)
\(43\)	−85.3289	+	147.794i	−0.302617	+	0.524148i	−0.976728	−	0.214482i	\(-0.931194\pi\)
							0.674111	+	0.738630i	\(0.264527\pi\)
\(47\)	−89.8841	+	155.684i	−0.278956	+	0.483166i	−0.971126	−	0.238569i	\(-0.923322\pi\)
							0.692169	+	0.721735i	\(0.256655\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.4.l.b.11.13	✓	64
3.2	odd	2		216.4.l.b.35.20		64
4.3	odd	2		288.4.p.b.47.17		64
8.3	odd	2	inner	72.4.l.b.11.25	yes	64
8.5	even	2		288.4.p.b.47.18		64
9.4	even	3		216.4.l.b.179.8		64
9.5	odd	6	inner	72.4.l.b.59.25	yes	64
12.11	even	2		864.4.p.b.143.30		64
24.5	odd	2		864.4.p.b.143.3		64
24.11	even	2		216.4.l.b.35.8		64
36.23	even	6		288.4.p.b.239.18		64
36.31	odd	6		864.4.p.b.719.3		64
72.5	odd	6		288.4.p.b.239.17		64
72.13	even	6		864.4.p.b.719.30		64
72.59	even	6	inner	72.4.l.b.59.13	yes	64
72.67	odd	6		216.4.l.b.179.20		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.4.l.b.11.13	✓	64	1.1	even	1	trivial
72.4.l.b.11.25	yes	64	8.3	odd	2	inner
72.4.l.b.59.13	yes	64	72.59	even	6	inner
72.4.l.b.59.25	yes	64	9.5	odd	6	inner
216.4.l.b.35.8		64	24.11	even	2
216.4.l.b.35.20		64	3.2	odd	2
216.4.l.b.179.8		64	9.4	even	3
216.4.l.b.179.20		64	72.67	odd	6
288.4.p.b.47.17		64	4.3	odd	2
288.4.p.b.47.18		64	8.5	even	2
288.4.p.b.239.17		64	72.5	odd	6
288.4.p.b.239.18		64	36.23	even	6
864.4.p.b.143.3		64	24.5	odd	2
864.4.p.b.143.30		64	12.11	even	2
864.4.p.b.719.3		64	36.31	odd	6
864.4.p.b.719.30		64	72.13	even	6