Embedded newform 864.2.bt.a.11.2

• Label: 864.2.bt.a.11.2
• Level: $864$
• Weight: $2$
• Character: 864.11
• Analytic conductor: $6.899$
• Analytic rank: $0$
• Dimension: $3408$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	864.bt (of order \(72\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.89907473464\)
Analytic rank:	\(0\)
Dimension:	\(3408\)
Relative dimension:	\(142\) over \(\Q(\zeta_{72})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{72}]$

Embedding invariants

Embedding label			11.2
Character	\(\chi\)	\(=\)	864.11
Dual form			864.2.bt.a.707.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41363 + 0.0407906i) q^{2} +(0.204734 - 1.71991i) q^{3} +(1.99667 - 0.115325i) q^{4} +(-1.35883 - 1.24514i) q^{5} +(-0.219261 + 2.43966i) q^{6} +(1.85804 + 2.65356i) q^{7} +(-2.81784 + 0.244472i) q^{8} +(-2.91617 - 0.704247i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3408 q - 24 q^{2} - 24 q^{3} - 24 q^{4} - 24 q^{5} - 24 q^{6} - 24 q^{7} - 36 q^{8} - 24 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(353\)	\(703\)
\(\chi(n)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41363	+	0.0407906i	−0.999584	+	0.0288433i
							\(3\)	0.204734	−	1.71991i	0.118203	−	0.992989i
\(5\)	−1.35883	−	1.24514i	−0.607686	−	0.556842i								0.311709	−	0.950178i	\(-0.399099\pi\)
							−0.919395	+	0.393336i	\(0.871321\pi\)
\(7\)	1.85804	+	2.65356i	0.702275	+	1.00295i	0.998850	+	0.0479430i	\(0.0152666\pi\)
							−0.296575	+	0.955010i	\(0.595845\pi\)
\(11\)	−4.50585	−	0.196730i	−1.35857	−	0.0593163i	−0.647298	−	0.762237i	\(-0.724101\pi\)
							−0.711268	+	0.702921i	\(0.751879\pi\)
\(13\)	2.92850	+	1.52448i	0.812221	+	0.422815i	0.816621	−	0.577175i	\(-0.195845\pi\)
							−0.00439969	+	0.999990i	\(0.501400\pi\)
\(17\)	3.54485	+	6.13987i	0.859753	+	1.48914i	0.872164	+	0.489214i	\(0.162716\pi\)
							−0.0124107	+	0.999923i	\(0.503951\pi\)
\(19\)	−0.433272	+	3.29103i	−0.0993995	+	0.755014i	0.867062	+	0.498201i	\(0.166006\pi\)
							−0.966461	+	0.256813i	\(0.917327\pi\)
\(23\)	3.78347	+	2.64921i	0.788908	+	0.552399i	0.897150	−	0.441725i	\(-0.145633\pi\)
							−0.108243	+	0.994125i	\(0.534522\pi\)
\(29\)	−1.08162	+	3.43047i	−0.200852	+	0.637022i	0.798533	+	0.601951i	\(0.205610\pi\)
							−0.999385	+	0.0350702i	\(0.988835\pi\)
\(31\)	3.03585	−	0.535302i	0.545255	−	0.0961431i	0.105766	−	0.994391i	\(-0.466271\pi\)
							0.439489	+	0.898248i	\(0.355160\pi\)
\(37\)	−11.8487	+	1.55991i	−1.94791	+	0.256447i	−0.998364	−	0.0571806i	\(-0.981789\pi\)
							−0.949546	+	0.313628i	\(0.898456\pi\)
\(41\)	4.13398	+	0.361676i	0.645619	+	0.0564844i	0.405263	−	0.914200i	\(-0.367180\pi\)
							0.240357	+	0.970685i	\(0.422736\pi\)
\(43\)	−0.0286828	+	0.656944i	−0.00437408	+	0.100183i	0.995605	+	0.0936560i	\(0.0298554\pi\)
							−0.999979	+	0.00652695i	\(0.997922\pi\)
\(47\)	9.20807	+	1.62363i	1.34314	+	0.236831i	0.798577	−	0.601892i	\(-0.205586\pi\)
							0.544558	+	0.838723i	\(0.316697\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.bt.a.11.2	✓	3408
27.5	odd	18	inner	864.2.bt.a.491.80	yes	3408
32.3	odd	8	inner	864.2.bt.a.227.80	yes	3408
864.707	even	72	inner	864.2.bt.a.707.2	yes	3408

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.2.bt.a.11.2	✓	3408	1.1	even	1	trivial
864.2.bt.a.227.80	yes	3408	32.3	odd	8	inner
864.2.bt.a.491.80	yes	3408	27.5	odd	18	inner
864.2.bt.a.707.2	yes	3408	864.707	even	72	inner