Embedded newform 72.7.p.b.43.15

• Label: 72.7.p.b.43.15
• Level: $72$
• Weight: $7$
• Character: 72.43
• Analytic conductor: $16.564$
• Analytic rank: $0$
• Dimension: $136$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	72.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(16.5638940206\)
Analytic rank:	\(0\)
Dimension:	\(136\)
Relative dimension:	\(68\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			43.15
Character	\(\chi\)	\(=\)	72.43
Dual form			72.7.p.b.67.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.06984 + 5.21124i) q^{2} +(-7.87021 + 25.8275i) q^{3} +(9.68594 - 63.2628i) q^{4} +(110.415 - 63.7480i) q^{5} +(-86.8224 - 197.782i) q^{6} +(81.2276 + 46.8968i) q^{7} +(270.886 + 434.471i) q^{8} +(-605.120 - 406.536i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 136 q - 17 q^{2} + 42 q^{3} + 127 q^{4} - 561 q^{6} + 1774 q^{8} + 654 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.06984	+	5.21124i	−0.758730	+	0.651405i
							\(3\)	−7.87021	+	25.8275i	−0.291489	+	0.956574i
\(5\)	110.415	−	63.7480i	0.883319	−	0.509984i								0.0115674	−	0.999933i	\(-0.496318\pi\)
							0.871751	+	0.489949i	\(0.162985\pi\)
\(7\)	81.2276	+	46.8968i	0.236815	+	0.136725i	0.613712	−	0.789530i	\(-0.289676\pi\)
							−0.376897	+	0.926255i	\(0.623009\pi\)
\(11\)	−729.124	+	1262.88i	−0.547801	+	0.948820i	0.450623	+	0.892714i	\(0.351202\pi\)
							−0.998425	+	0.0561057i	\(0.982132\pi\)
\(13\)	−1805.75	+	1042.55i	−0.821918	+	0.474535i	−0.851077	−	0.525040i	\(-0.824050\pi\)
							0.0291595	+	0.999575i	\(0.490717\pi\)
\(17\)	−1504.34			−0.306195			−0.153098	−	0.988211i	\(-0.548925\pi\)
							−0.153098	+	0.988211i	\(0.548925\pi\)
\(19\)	−1248.33			−0.181999			−0.0909993	−	0.995851i	\(-0.529006\pi\)
							−0.0909993	+	0.995851i	\(0.529006\pi\)
\(23\)	10278.5	−	5934.27i	0.844781	−	0.487735i	−0.0141052	−	0.999901i	\(-0.504490\pi\)
							0.858887	+	0.512166i	\(0.171157\pi\)
\(29\)	−36667.1	−	21169.8i	−1.50343	−	0.868005i	−0.999992	−	0.00397379i	\(-0.998735\pi\)
							−0.503437	−	0.864032i	\(-0.667932\pi\)
\(31\)	−29992.7	+	17316.3i	−1.00677	+	0.581260i	−0.910245	−	0.414071i	\(-0.864107\pi\)
							−0.0965264	+	0.995330i	\(0.530773\pi\)
\(37\)			62410.1i			1.23211i	0.787703	+	0.616056i	\(0.211270\pi\)
							−0.787703	+	0.616056i	\(0.788730\pi\)
\(41\)	−52947.4	−	91707.6i	−0.768233	−	1.33062i	−0.938520	−	0.345225i	\(-0.887803\pi\)
							0.170287	−	0.985395i	\(-0.445531\pi\)
\(43\)	−14581.2	+	25255.4i	−0.183395	+	0.317650i	−0.943035	−	0.332695i	\(-0.892042\pi\)
							0.759639	+	0.650345i	\(0.225376\pi\)
\(47\)	−54658.8	−	31557.3i	−0.526462	−	0.303953i	0.213113	−	0.977028i	\(-0.431640\pi\)
							−0.739574	+	0.673075i	\(0.764973\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.7.p.b.43.15	✓	136
8.3	odd	2	inner	72.7.p.b.43.33	yes	136
9.4	even	3	inner	72.7.p.b.67.33	yes	136
72.67	odd	6	inner	72.7.p.b.67.15	yes	136

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.7.p.b.43.15	✓	136	1.1	even	1	trivial
72.7.p.b.43.33	yes	136	8.3	odd	2	inner
72.7.p.b.67.15	yes	136	72.67	odd	6	inner
72.7.p.b.67.33	yes	136	9.4	even	3	inner