Embedded newform 72.7.p.b.43.10

• Label: 72.7.p.b.43.10
• 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			43.10
Character	\(\chi\)	\(=\)	72.43
Dual form			72.7.p.b.67.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.43575 - 2.95121i) q^{2} +(4.82015 - 26.5663i) q^{3} +(46.5807 + 43.8890i) q^{4} +(29.7357 - 17.1679i) q^{5} +(-114.244 + 183.315i) q^{6} +(464.017 + 267.900i) q^{7} +(-216.836 - 463.817i) q^{8} +(-682.532 - 256.107i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 136 q - 17 q^{2} + 42 q^{3} + 127 q^{4} - 561 q^{6} + 1774 q^{8} + 654 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.43575	−	2.95121i	−0.929468	−	0.368902i
							\(3\)	4.82015	−	26.5663i	0.178524	−	0.983936i
\(5\)	29.7357	−	17.1679i	0.237886	−	0.137343i								−0.376319	−	0.926490i	\(-0.622810\pi\)
							0.614205	+	0.789147i	\(0.289477\pi\)
\(7\)	464.017	+	267.900i	1.35282	+	0.781051i	0.988644	−	0.150279i	\(-0.0480173\pi\)
							0.364176	+	0.931330i	\(0.381351\pi\)
\(11\)	−1306.01	+	2262.08i	−0.981226	+	1.69953i	−0.323591	+	0.946197i	\(0.604890\pi\)
							−0.657635	+	0.753337i	\(0.728443\pi\)
\(13\)	−1343.19	+	775.493i	−0.611376	+	0.352978i	−0.773504	−	0.633791i	\(-0.781498\pi\)
							0.162128	+	0.986770i	\(0.448164\pi\)
\(17\)	−5191.90			−1.05677			−0.528384	−	0.849005i	\(-0.677202\pi\)
							−0.528384	+	0.849005i	\(0.677202\pi\)
\(19\)	3935.64			0.573792			0.286896	−	0.957962i	\(-0.407377\pi\)
							0.286896	+	0.957962i	\(0.407377\pi\)
\(23\)	−9273.74	+	5354.20i	−0.762204	+	0.440059i	−0.830087	−	0.557634i	\(-0.811709\pi\)
							0.0678823	+	0.997693i	\(0.478376\pi\)
\(29\)	36834.6	+	21266.5i	1.51030	+	0.871971i	0.999928	+	0.0120154i	\(0.00382470\pi\)
							0.510370	+	0.859955i	\(0.329509\pi\)
\(31\)	2488.68	−	1436.84i	0.0835379	−	0.0482306i	−0.457649	−	0.889133i	\(-0.651308\pi\)
							0.541187	+	0.840902i	\(0.317975\pi\)
\(37\)			67376.1i			1.33015i	0.746776	+	0.665075i	\(0.231601\pi\)
							−0.746776	+	0.665075i	\(0.768399\pi\)
\(41\)	−19510.4	−	33792.9i	−0.283083	−	0.490314i	0.689060	−	0.724705i	\(-0.258024\pi\)
							−0.972142	+	0.234391i	\(0.924690\pi\)
\(43\)	44583.2	−	77220.3i	0.560745	−	0.971239i	−0.436686	−	0.899614i	\(-0.643848\pi\)
							0.997432	−	0.0716253i	\(-0.0228186\pi\)
\(47\)	−114168.	−	65914.8i	−1.09964	−	0.634877i	−0.163513	−	0.986541i	\(-0.552282\pi\)
							−0.936126	+	0.351664i	\(0.885616\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.7.p.b.43.10	✓	136
8.3	odd	2	inner	72.7.p.b.43.53	yes	136
9.4	even	3	inner	72.7.p.b.67.53	yes	136
72.67	odd	6	inner	72.7.p.b.67.10	yes	136

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.7.p.b.43.10	✓	136	1.1	even	1	trivial
72.7.p.b.43.53	yes	136	8.3	odd	2	inner
72.7.p.b.67.10	yes	136	72.67	odd	6	inner
72.7.p.b.67.53	yes	136	9.4	even	3	inner