Embedded newform 72.7.j.a.5.9

• Label: 72.7.j.a.5.9
• Level: $72$
• Weight: $7$
• Character: 72.5
• 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			5.9
Character	\(\chi\)	\(=\)	72.5
Dual form			72.7.j.a.29.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.46395 - 2.87915i) q^{2} +(-26.9713 + 1.24421i) q^{3} +(47.4210 + 42.9797i) q^{4} +(-7.54460 + 13.0676i) q^{5} +(204.895 + 68.3678i) q^{6} +(-156.472 - 271.018i) q^{7} +(-230.202 - 457.330i) q^{8} +(725.904 - 67.1160i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 3 q^{2} - q^{4} + 317 q^{6} - 2 q^{7} - 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(37\)	\(55\)	\(65\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.46395	−	2.87915i	−0.932993	−	0.359894i
							\(3\)	−26.9713	+	1.24421i	−0.998938	+	0.0460819i
\(5\)	−7.54460	+	13.0676i	−0.0603568	+	0.104541i								−0.894625	−	0.446818i	\(-0.852557\pi\)
							0.834268	+	0.551359i	\(0.185891\pi\)
\(7\)	−156.472	−	271.018i	−0.456187	−	0.790140i	0.542568	−	0.840012i	\(-0.317452\pi\)
							−0.998756	+	0.0498721i	\(0.984119\pi\)
\(11\)	−66.1554	−	114.585i	−0.0497035	−	0.0860891i	0.840103	−	0.542427i	\(-0.182494\pi\)
							−0.889807	+	0.456337i	\(0.849161\pi\)
\(13\)	−3147.73	−	1817.35i	−1.43274	−	0.827194i	−0.435413	−	0.900231i	\(-0.643398\pi\)
							−0.997329	+	0.0730367i	\(0.976731\pi\)
\(17\)		−	9667.61i		−	1.96776i	−0.178828	−	0.983880i	\(-0.557231\pi\)
							0.178828	−	0.983880i	\(-0.442769\pi\)
\(19\)			8076.76i			1.17754i	0.808300	+	0.588771i	\(0.200388\pi\)
							−0.808300	+	0.588771i	\(0.799612\pi\)
\(23\)	12722.1	+	7345.13i	1.04563	+	0.603693i	0.921422	−	0.388564i	\(-0.127029\pi\)
							0.124205	+	0.992257i	\(0.460362\pi\)
\(29\)	2821.53	+	4887.03i	0.115689	+	0.200378i	0.918055	−	0.396454i	\(-0.129759\pi\)
							−0.802366	+	0.596832i	\(0.796426\pi\)
\(31\)	−10111.6	+	17513.8i	−0.339418	+	0.587890i	−0.984323	−	0.176373i	\(-0.943564\pi\)
							0.644905	+	0.764263i	\(0.276897\pi\)
\(37\)			24983.5i			0.493228i	0.969114	+	0.246614i	\(0.0793180\pi\)
							−0.969114	+	0.246614i	\(0.920682\pi\)
\(41\)	25019.0	+	14444.8i	0.363010	+	0.209584i	0.670401	−	0.741999i	\(-0.266122\pi\)
							−0.307390	+	0.951584i	\(0.599456\pi\)
\(43\)	−90890.9	+	52475.9i	−1.14318	+	0.660016i	−0.947216	−	0.320595i	\(-0.896117\pi\)
							−0.195965	+	0.980611i	\(0.562784\pi\)
\(47\)	−132187.	+	76318.3i	−1.27320	+	0.735081i	−0.975589	−	0.219606i	\(-0.929523\pi\)
							−0.297610	+	0.954688i	\(0.596189\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.7.j.a.5.9	✓	140
8.5	even	2	inner	72.7.j.a.5.34	yes	140
9.2	odd	6	inner	72.7.j.a.29.34	yes	140
72.29	odd	6	inner	72.7.j.a.29.9	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.7.j.a.5.9	✓	140	1.1	even	1	trivial
72.7.j.a.5.34	yes	140	8.5	even	2	inner
72.7.j.a.29.9	yes	140	72.29	odd	6	inner
72.7.j.a.29.34	yes	140	9.2	odd	6	inner