Embedded newform 72.7.j.a.29.15

• Label: 72.7.j.a.29.15
• Level: $72$
• Weight: $7$
• Character: 72.29
• Analytic conductor: $16.564$
• Analytic rank: $0$
• Dimension: $140$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(16.5638940206\)
Analytic rank:	\(0\)
Dimension:	\(140\)
Relative dimension:	\(70\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			29.15
Character	\(\chi\)	\(=\)	72.29
Dual form			72.7.j.a.5.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.45319 - 4.72825i) q^{2} +(26.7644 + 3.55934i) q^{3} +(19.2873 + 61.0246i) q^{4} +(-100.826 - 174.637i) q^{5} +(-155.886 - 149.518i) q^{6} +(-187.285 + 324.386i) q^{7} +(164.075 - 484.998i) q^{8} +(703.662 + 190.527i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 3 q^{2} - q^{4} + 317 q^{6} - 2 q^{7} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.45319	−	4.72825i	−0.806649	−	0.591031i
							\(3\)	26.7644	+	3.55934i	0.991273	+	0.131827i
\(5\)	−100.826	−	174.637i	−0.806612	−	1.39709i								−0.915197	−	0.403006i	\(-0.867965\pi\)
							0.108586	−	0.994087i	\(-0.465368\pi\)
\(7\)	−187.285	+	324.386i	−0.546019	+	0.945733i	0.452523	+	0.891753i	\(0.350524\pi\)
							−0.998542	+	0.0539800i	\(0.982809\pi\)
\(11\)	132.086	−	228.780i	0.0992384	−	0.171886i	−0.812131	−	0.583475i	\(-0.801693\pi\)
							0.911370	+	0.411589i	\(0.135026\pi\)
\(13\)	−2361.06	+	1363.16i	−1.07467	+	0.620463i	−0.929455	−	0.368936i	\(-0.879722\pi\)
							−0.145219	+	0.989400i	\(0.546389\pi\)
\(17\)			1477.50i			0.300732i	0.988630	+	0.150366i	\(0.0480452\pi\)
							−0.988630	+	0.150366i	\(0.951955\pi\)
\(19\)		−	5769.94i		−	0.841222i	−0.907241	−	0.420611i	\(-0.861816\pi\)
							0.907241	−	0.420611i	\(-0.138184\pi\)
\(23\)	−13416.0	+	7745.74i	−1.10266	+	0.636619i	−0.936917	−	0.349551i	\(-0.886334\pi\)
							−0.165738	+	0.986170i	\(0.553001\pi\)
\(29\)	−19389.4	+	33583.4i	−0.795006	+	1.37699i	0.127829	+	0.991796i	\(0.459199\pi\)
							−0.922835	+	0.385195i	\(0.874134\pi\)
\(31\)	22570.7	+	39093.6i	0.757635	+	1.31226i	0.944054	+	0.329792i	\(0.106979\pi\)
							−0.186419	+	0.982470i	\(0.559688\pi\)
\(37\)		−	5204.50i		−	0.102748i	−0.998679	−	0.0513741i	\(-0.983640\pi\)
							0.998679	−	0.0513741i	\(-0.0163601\pi\)
\(41\)	−84700.7	+	48902.0i	−1.22895	+	0.709537i	−0.966811	−	0.255492i	\(-0.917762\pi\)
							−0.262143	+	0.965029i	\(0.584429\pi\)
\(43\)	−21634.2	−	12490.5i	−0.272104	−	0.157099i	0.357739	−	0.933821i	\(-0.383548\pi\)
							−0.629843	+	0.776722i	\(0.716881\pi\)
\(47\)	−122856.	−	70931.1i	−1.18332	−	0.683193i	−0.226544	−	0.974001i	\(-0.572743\pi\)
							−0.956781	+	0.290808i	\(0.906076\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.7.j.a.29.15	yes	140
8.5	even	2	inner	72.7.j.a.29.10	yes	140
9.5	odd	6	inner	72.7.j.a.5.10	✓	140
72.5	odd	6	inner	72.7.j.a.5.15	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.7.j.a.5.10	✓	140	9.5	odd	6	inner
72.7.j.a.5.15	yes	140	72.5	odd	6	inner
72.7.j.a.29.10	yes	140	8.5	even	2	inner
72.7.j.a.29.15	yes	140	1.1	even	1	trivial