Embedded newform 864.2.bt.a.11.15

• Label: 864.2.bt.a.11.15
• 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			11.15
Character	\(\chi\)	\(=\)	864.11
Dual form			864.2.bt.a.707.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.35292 - 0.411830i) q^{2} +(1.69193 + 0.370633i) q^{3} +(1.66079 + 1.11435i) q^{4} +(-2.86206 - 2.62259i) q^{5} +(-2.13641 - 1.19823i) q^{6} +(2.22100 + 3.17192i) q^{7} +(-1.78800 - 2.19159i) q^{8} +(2.72526 + 1.25417i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3408 q - 24 q^{2} - 24 q^{3} - 24 q^{4} - 24 q^{5} - 24 q^{6} - 24 q^{7} - 36 q^{8} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.35292	−	0.411830i	−0.956660	−	0.291208i
							\(3\)	1.69193	+	0.370633i	0.976837	+	0.213985i
\(5\)	−2.86206	−	2.62259i	−1.27995	−	1.17286i								−0.975419	−	0.220356i	\(-0.929278\pi\)
							−0.304531	−	0.952502i	\(-0.598500\pi\)
\(7\)	2.22100	+	3.17192i	0.839459	+	1.19887i	0.978019	+	0.208518i	\(0.0668640\pi\)
							−0.138559	+	0.990354i	\(0.544247\pi\)
\(11\)	3.74729	+	0.163610i	1.12985	+	0.0493304i	0.600902	−	0.799323i	\(-0.294808\pi\)
							0.528950	+	0.848653i	\(0.322586\pi\)
\(13\)	−2.52086	−	1.31228i	−0.699160	−	0.363960i	0.0747962	−	0.997199i	\(-0.476169\pi\)
							−0.773956	+	0.633239i	\(0.781725\pi\)
\(17\)	3.16671	+	5.48490i	0.768040	+	1.33028i	0.938624	+	0.344942i	\(0.112101\pi\)
							−0.170584	+	0.985343i	\(0.554565\pi\)
\(19\)	0.0213424	−	0.162112i	0.00489629	−	0.0371910i	−0.988822	−	0.149099i	\(-0.952363\pi\)
							0.993719	+	0.111908i	\(0.0356961\pi\)
\(23\)	−5.14760	−	3.60439i	−1.07335	−	0.751566i	−0.103341	−	0.994646i	\(-0.532953\pi\)
							−0.970006	+	0.243080i	\(0.921842\pi\)
\(29\)	−0.850249	+	2.69664i	−0.157887	+	0.500754i	−0.999208	−	0.0397812i	\(-0.987334\pi\)
							0.841321	+	0.540536i	\(0.181778\pi\)
\(31\)	2.09802	−	0.369937i	0.376815	−	0.0664427i	0.0179669	−	0.999839i	\(-0.494281\pi\)
							0.358848	+	0.933396i	\(0.383170\pi\)
\(37\)	10.1069	−	1.33060i	1.66156	−	0.218749i	0.759547	−	0.650452i	\(-0.225421\pi\)
							0.902016	+	0.431703i	\(0.142087\pi\)
\(41\)	3.25060	+	0.284390i	0.507658	+	0.0444143i	0.338108	−	0.941107i	\(-0.390213\pi\)
							0.169550	+	0.985522i	\(0.445769\pi\)
\(43\)	0.235571	−	5.39547i	0.0359243	−	0.822801i	−0.892448	−	0.451150i	\(-0.851014\pi\)
							0.928372	−	0.371651i	\(-0.121208\pi\)
\(47\)	9.02468	+	1.59129i	1.31638	+	0.232114i	0.787360	−	0.616493i	\(-0.211447\pi\)
							0.529024	+	0.848607i	\(0.322558\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.bt.a.11.15	✓	3408
27.5	odd	18	inner	864.2.bt.a.491.94	yes	3408
32.3	odd	8	inner	864.2.bt.a.227.94	yes	3408
864.707	even	72	inner	864.2.bt.a.707.15	yes	3408

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.2.bt.a.11.15	✓	3408	1.1	even	1	trivial
864.2.bt.a.227.94	yes	3408	32.3	odd	8	inner
864.2.bt.a.491.94	yes	3408	27.5	odd	18	inner
864.2.bt.a.707.15	yes	3408	864.707	even	72	inner