Embedded newform 72.7.p.b.43.2

• Label: 72.7.p.b.43.2
• 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			43.2
Character	\(\chi\)	\(=\)	72.43
Dual form			72.7.p.b.67.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.99923 + 0.111052i) q^{2} +(14.5758 + 22.7277i) q^{3} +(63.9753 - 1.77667i) q^{4} +(100.062 - 57.7710i) q^{5} +(-119.119 - 180.185i) q^{6} +(-427.342 - 246.726i) q^{7} +(-511.556 + 21.3166i) q^{8} +(-304.094 + 662.547i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 136 q - 17 q^{2} + 42 q^{3} + 127 q^{4} - 561 q^{6} + 1774 q^{8} + 654 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.99923	+	0.111052i	−0.999904	+	0.0138815i
							\(3\)	14.5758	+	22.7277i	0.539843	+	0.841766i
\(5\)	100.062	−	57.7710i	0.800498	−	0.462168i								−0.0431472	−	0.999069i	\(-0.513738\pi\)
							0.843645	+	0.536901i	\(0.180405\pi\)
\(7\)	−427.342	−	246.726i	−1.24590	−	0.719318i	−0.275607	−	0.961270i	\(-0.588879\pi\)
							−0.970288	+	0.241953i	\(0.922212\pi\)
\(11\)	231.186	−	400.425i	0.173693	−	0.300845i	−0.766015	−	0.642823i	\(-0.777763\pi\)
							0.939708	+	0.341977i	\(0.111097\pi\)
\(13\)	−762.075	+	439.984i	−0.346871	+	0.200266i	−0.663306	−	0.748348i	\(-0.730847\pi\)
							0.316435	+	0.948614i	\(0.397514\pi\)
\(17\)	−9092.00			−1.85060			−0.925300	−	0.379236i	\(-0.876187\pi\)
							−0.925300	+	0.379236i	\(0.876187\pi\)
\(19\)	5407.98			0.788450			0.394225	−	0.919014i	\(-0.371013\pi\)
							0.394225	+	0.919014i	\(0.371013\pi\)
\(23\)	−19876.5	+	11475.7i	−1.63364	+	0.943182i	−0.650682	+	0.759350i	\(0.725517\pi\)
							−0.982958	+	0.183832i	\(0.941150\pi\)
\(29\)	−6288.08	−	3630.42i	−0.257824	−	0.148855i	0.365517	−	0.930805i	\(-0.380892\pi\)
							−0.623342	+	0.781950i	\(0.714225\pi\)
\(31\)	10787.8	−	6228.34i	0.362116	−	0.209068i	−0.307892	−	0.951421i	\(-0.599624\pi\)
							0.670009	+	0.742353i	\(0.266290\pi\)
\(37\)		−	75500.3i		−	1.49054i	−0.666763	−	0.745270i	\(-0.732321\pi\)
							0.666763	−	0.745270i	\(-0.267679\pi\)
\(41\)	−52250.8	−	90501.0i	−0.758125	−	1.31311i	−0.943805	−	0.330502i	\(-0.892782\pi\)
							0.185680	−	0.982610i	\(-0.440551\pi\)
\(43\)	−14376.2	+	24900.3i	−0.180817	+	0.313184i	−0.942159	−	0.335166i	\(-0.891208\pi\)
							0.761342	+	0.648351i	\(0.224541\pi\)
\(47\)	−38561.4	−	22263.4i	−0.371415	−	0.214437i	0.302661	−	0.953098i	\(-0.402125\pi\)
							−0.674076	+	0.738662i	\(0.735458\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.7.p.b.43.2	✓	136
8.3	odd	2	inner	72.7.p.b.43.45	yes	136
9.4	even	3	inner	72.7.p.b.67.45	yes	136
72.67	odd	6	inner	72.7.p.b.67.2	yes	136

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.7.p.b.43.2	✓	136	1.1	even	1	trivial
72.7.p.b.43.45	yes	136	8.3	odd	2	inner
72.7.p.b.67.2	yes	136	72.67	odd	6	inner
72.7.p.b.67.45	yes	136	9.4	even	3	inner