Embedded newform 72.7.j.a.5.17

• Label: 72.7.j.a.5.17
• 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			5.17
Character	\(\chi\)	\(=\)	72.5
Dual form			72.7.j.a.29.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.34800 - 4.86856i) q^{2} +(15.8845 + 21.8331i) q^{3} +(16.5943 + 61.8112i) q^{4} +(122.026 - 211.355i) q^{5} +(5.46045 - 215.931i) q^{6} +(-28.5227 - 49.4028i) q^{7} +(195.591 - 473.168i) q^{8} +(-224.364 + 693.615i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 3 q^{2} - q^{4} + 317 q^{6} - 2 q^{7} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.34800	−	4.86856i	−0.793501	−	0.608569i
							\(3\)	15.8845	+	21.8331i	0.588315	+	0.808632i
\(5\)	122.026	−	211.355i	0.976209	−	1.69084i								0.300323	−	0.953838i	\(-0.402906\pi\)
							0.675886	−	0.737006i	\(-0.263761\pi\)
\(7\)	−28.5227	−	49.4028i	−0.0831566	−	0.144031i	0.821448	−	0.570284i	\(-0.193167\pi\)
							−0.904604	+	0.426253i	\(0.859833\pi\)
\(11\)	−529.794	−	917.630i	−0.398042	−	0.689429i	0.595442	−	0.803398i	\(-0.296977\pi\)
							−0.993484	+	0.113969i	\(0.963644\pi\)
\(13\)	1072.25	+	619.067i	0.488054	+	0.281778i	0.723767	−	0.690045i	\(-0.242409\pi\)
							−0.235713	+	0.971823i	\(0.575742\pi\)
\(17\)		−	6617.79i		−	1.34700i	−0.739189	−	0.673498i	\(-0.764791\pi\)
							0.739189	−	0.673498i	\(-0.235209\pi\)
\(19\)			3342.42i			0.487304i	0.969863	+	0.243652i	\(0.0783455\pi\)
							−0.969863	+	0.243652i	\(0.921654\pi\)
\(23\)	5525.65	+	3190.23i	0.454150	+	0.262204i	0.709581	−	0.704623i	\(-0.248884\pi\)
							−0.255431	+	0.966827i	\(0.582217\pi\)
\(29\)	−10797.1	−	18701.2i	−0.442705	−	0.766788i	0.555184	−	0.831728i	\(-0.312648\pi\)
							−0.997889	+	0.0649394i	\(0.979315\pi\)
\(31\)	10636.5	−	18422.9i	0.357037	−	0.618405i	−0.630428	−	0.776248i	\(-0.717121\pi\)
							0.987464	+	0.157843i	\(0.0504538\pi\)
\(37\)		−	35838.8i		−	0.707536i	−0.935333	−	0.353768i	\(-0.884900\pi\)
							0.935333	−	0.353768i	\(-0.115100\pi\)
\(41\)	13932.0	+	8043.65i	0.202145	+	0.116708i	0.597655	−	0.801753i	\(-0.296099\pi\)
							−0.395511	+	0.918461i	\(0.629432\pi\)
\(43\)	122952.	−	70986.1i	1.54642	−	0.892829i	0.548014	−	0.836469i	\(-0.315384\pi\)
							0.998411	−	0.0563597i	\(-0.0179493\pi\)
\(47\)	−11817.8	+	6823.00i	−0.113826	+	0.0657176i	−0.555832	−	0.831295i	\(-0.687600\pi\)
							0.442006	+	0.897012i	\(0.354267\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.7.j.a.5.17	✓	140
8.5	even	2	inner	72.7.j.a.5.39	yes	140
9.2	odd	6	inner	72.7.j.a.29.39	yes	140
72.29	odd	6	inner	72.7.j.a.29.17	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.7.j.a.5.17	✓	140	1.1	even	1	trivial
72.7.j.a.5.39	yes	140	8.5	even	2	inner
72.7.j.a.29.17	yes	140	72.29	odd	6	inner
72.7.j.a.29.39	yes	140	9.2	odd	6	inner