Embedded newform 72.5.p.b.67.40

• Label: 72.5.p.b.67.40
• Level: $72$
• Weight: $5$
• Character: 72.67
• 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			67.40
Character	\(\chi\)	\(=\)	72.67
Dual form			72.5.p.b.43.40

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.83772 + 1.12779i) q^{2} +(8.90483 - 1.30534i) q^{3} +(13.4562 + 8.65626i) q^{4} +(-23.2381 - 13.4165i) q^{5} +(35.6464 + 5.03320i) q^{6} +(52.2165 - 30.1472i) q^{7} +(41.8787 + 48.3960i) q^{8} +(77.5922 - 23.2478i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 7 q^{2} - 18 q^{3} + 31 q^{4} - 9 q^{6} - 386 q^{8} + 30 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.83772	+	1.12779i	0.959430	+	0.281947i
							\(3\)	8.90483	−	1.30534i	0.989426	−	0.145038i
\(5\)	−23.2381	−	13.4165i	−0.929523	−	0.536660i								−0.0428621	−	0.999081i	\(-0.513648\pi\)
							−0.886661	+	0.462421i	\(0.846981\pi\)
\(7\)	52.2165	−	30.1472i	1.06564	−	0.615249i	0.138655	−	0.990341i	\(-0.455722\pi\)
							0.926988	+	0.375092i	\(0.122389\pi\)
\(11\)	92.8475	+	160.817i	0.767335	+	1.32906i	0.939003	+	0.343908i	\(0.111751\pi\)
							−0.171669	+	0.985155i	\(0.554916\pi\)
\(13\)	−166.026	−	95.8552i	−0.982402	−	0.567190i	−0.0794078	−	0.996842i	\(-0.525303\pi\)
							−0.902995	+	0.429652i	\(0.858636\pi\)
\(17\)	−425.178			−1.47120			−0.735602	−	0.677414i	\(-0.763101\pi\)
							−0.735602	+	0.677414i	\(0.763101\pi\)
\(19\)	−114.203			−0.316351			−0.158175	−	0.987411i	\(-0.550561\pi\)
							−0.158175	+	0.987411i	\(0.550561\pi\)
\(23\)	239.080	+	138.033i	0.451946	+	0.260931i	0.708652	−	0.705558i	\(-0.249304\pi\)
							−0.256705	+	0.966490i	\(0.582637\pi\)
\(29\)	−757.999	+	437.631i	−0.901306	+	0.520369i	−0.877624	−	0.479350i	\(-0.840872\pi\)
							−0.0236825	+	0.999720i	\(0.507539\pi\)
\(31\)	−910.769	−	525.833i	−0.947730	−	0.547172i	−0.0553551	−	0.998467i	\(-0.517629\pi\)
							−0.892375	+	0.451294i	\(0.850962\pi\)
\(37\)			194.549i			0.142110i	0.997472	+	0.0710551i	\(0.0226366\pi\)
							−0.997472	+	0.0710551i	\(0.977363\pi\)
\(41\)	1420.33	−	2460.08i	0.844930	−	1.46346i	−0.0407524	−	0.999169i	\(-0.512975\pi\)
							0.885682	−	0.464292i	\(-0.153691\pi\)
\(43\)	−449.077	−	777.825i	−0.242876	−	0.420673i	0.718656	−	0.695365i	\(-0.244757\pi\)
							−0.961532	+	0.274692i	\(0.911424\pi\)
\(47\)	−348.382	+	201.138i	−0.157710	+	0.0910540i	−0.576778	−	0.816901i	\(-0.695690\pi\)
							0.419068	+	0.907955i	\(0.362357\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.5.p.b.67.40	yes	88
3.2	odd	2		216.5.p.b.91.5		88
4.3	odd	2		288.5.t.b.175.3		88
8.3	odd	2	inner	72.5.p.b.67.18	yes	88
8.5	even	2		288.5.t.b.175.4		88
9.2	odd	6		216.5.p.b.19.27		88
9.7	even	3	inner	72.5.p.b.43.18	✓	88
12.11	even	2		864.5.t.b.847.36		88
24.5	odd	2		864.5.t.b.847.9		88
24.11	even	2		216.5.p.b.91.27		88
36.7	odd	6		288.5.t.b.79.4		88
36.11	even	6		864.5.t.b.559.9		88
72.11	even	6		216.5.p.b.19.5		88
72.29	odd	6		864.5.t.b.559.36		88
72.43	odd	6	inner	72.5.p.b.43.40	yes	88
72.61	even	6		288.5.t.b.79.3		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.5.p.b.43.18	✓	88	9.7	even	3	inner
72.5.p.b.43.40	yes	88	72.43	odd	6	inner
72.5.p.b.67.18	yes	88	8.3	odd	2	inner
72.5.p.b.67.40	yes	88	1.1	even	1	trivial
216.5.p.b.19.5		88	72.11	even	6
216.5.p.b.19.27		88	9.2	odd	6
216.5.p.b.91.5		88	3.2	odd	2
216.5.p.b.91.27		88	24.11	even	2
288.5.t.b.79.3		88	72.61	even	6
288.5.t.b.79.4		88	36.7	odd	6
288.5.t.b.175.3		88	4.3	odd	2
288.5.t.b.175.4		88	8.5	even	2
864.5.t.b.559.9		88	36.11	even	6
864.5.t.b.559.36		88	72.29	odd	6
864.5.t.b.847.9		88	24.5	odd	2
864.5.t.b.847.36		88	12.11	even	2