Embedded newform 864.2.bt.a.11.7

• Label: 864.2.bt.a.11.7
• 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			11.7
Character	\(\chi\)	\(=\)	864.11
Dual form			864.2.bt.a.707.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40277 + 0.179519i) q^{2} +(1.46202 - 0.928701i) q^{3} +(1.93555 - 0.503650i) q^{4} +(0.162424 + 0.148834i) q^{5} +(-1.88417 + 1.56522i) q^{6} +(0.0602653 + 0.0860678i) q^{7} +(-2.62472 + 1.05398i) q^{8} +(1.27503 - 2.71557i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3408 q - 24 q^{2} - 24 q^{3} - 24 q^{4} - 24 q^{5} - 24 q^{6} - 24 q^{7} - 36 q^{8} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40277	+	0.179519i	−0.991910	+	0.126939i
							\(3\)	1.46202	−	0.928701i	0.844100	−	0.536186i
\(5\)	0.162424	+	0.148834i	0.0726382	+	0.0665607i								0.711089	−	0.703102i	\(-0.248202\pi\)
							−0.638451	+	0.769663i	\(0.720424\pi\)
\(7\)	0.0602653	+	0.0860678i	0.0227781	+	0.0325306i	0.830380	−	0.557198i	\(-0.188124\pi\)
							−0.807601	+	0.589729i	\(0.799235\pi\)
\(11\)	−1.63922	−	0.0715698i	−0.494243	−	0.0215791i	−0.204857	−	0.978792i	\(-0.565673\pi\)
							−0.289385	+	0.957213i	\(0.593451\pi\)
\(13\)	−3.16757	−	1.64893i	−0.878527	−	0.457332i	−0.0381104	−	0.999274i	\(-0.512134\pi\)
							−0.840416	+	0.541942i	\(0.817689\pi\)
\(17\)	−1.00470	−	1.74019i	−0.243676	−	0.422059i	0.718083	−	0.695958i	\(-0.245020\pi\)
							−0.961758	+	0.273899i	\(0.911687\pi\)
\(19\)	0.408303	−	3.10137i	0.0936711	−	0.711503i	−0.878585	−	0.477585i	\(-0.841512\pi\)
							0.972256	−	0.233917i	\(-0.0751545\pi\)
\(23\)	−4.11552	−	2.88172i	−0.858146	−	0.600880i	0.0595227	−	0.998227i	\(-0.481042\pi\)
							−0.917669	+	0.397347i	\(0.869931\pi\)
\(29\)	0.477873	−	1.51562i	0.0887388	−	0.281444i	−0.898909	−	0.438136i	\(-0.855639\pi\)
							0.987647	+	0.156693i	\(0.0500832\pi\)
\(31\)	9.86851	−	1.74009i	1.77244	−	0.312529i	0.810488	−	0.585756i	\(-0.199202\pi\)
							0.961950	+	0.273227i	\(0.0880911\pi\)
\(37\)	2.40035	−	0.316013i	0.394616	−	0.0519522i	0.0693926	−	0.997589i	\(-0.477894\pi\)
							0.325223	+	0.945637i	\(0.394561\pi\)
\(41\)	0.650814	+	0.0569388i	0.101640	+	0.00889235i	0.137862	−	0.990451i	\(-0.455977\pi\)
							−0.0362223	+	0.999344i	\(0.511532\pi\)
\(43\)	0.0143033	−	0.327599i	0.00218123	−	0.0499584i	−0.997645	−	0.0685850i	\(-0.978152\pi\)
							0.999827	+	0.0186265i	\(0.00592935\pi\)
\(47\)	2.18622	+	0.385490i	0.318893	+	0.0562294i	0.330804	−	0.943700i	\(-0.392680\pi\)
							−0.0119106	+	0.999929i	\(0.503791\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.bt.a.11.7	✓	3408
27.5	odd	18	inner	864.2.bt.a.491.72	yes	3408
32.3	odd	8	inner	864.2.bt.a.227.72	yes	3408
864.707	even	72	inner	864.2.bt.a.707.7	yes	3408

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.2.bt.a.11.7	✓	3408	1.1	even	1	trivial
864.2.bt.a.227.72	yes	3408	32.3	odd	8	inner
864.2.bt.a.491.72	yes	3408	27.5	odd	18	inner
864.2.bt.a.707.7	yes	3408	864.707	even	72	inner