Embedded newform 864.2.bt.a.11.14

• Label: 864.2.bt.a.11.14
• 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			11.14
Character	\(\chi\)	\(=\)	864.11
Dual form			864.2.bt.a.707.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.35346 - 0.410054i) q^{2} +(-0.153915 + 1.72520i) q^{3} +(1.66371 + 1.10998i) q^{4} +(2.53456 + 2.32250i) q^{5} +(0.915743 - 2.27187i) q^{6} +(0.366857 + 0.523926i) q^{7} +(-1.79661 - 2.18453i) q^{8} +(-2.95262 - 0.531068i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3408 q - 24 q^{2} - 24 q^{3} - 24 q^{4} - 24 q^{5} - 24 q^{6} - 24 q^{7} - 36 q^{8} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.35346	−	0.410054i	−0.957041	−	0.289952i
							\(3\)	−0.153915	+	1.72520i	−0.0888630	+	0.996044i
\(5\)	2.53456	+	2.32250i	1.13349	+	1.03865i								0.998859	+	0.0477569i	\(0.0152073\pi\)
							0.134631	+	0.990896i	\(0.457015\pi\)
\(7\)	0.366857	+	0.523926i	0.138659	+	0.198025i	0.882476	−	0.470358i	\(-0.155875\pi\)
							−0.743817	+	0.668384i	\(0.766986\pi\)
\(11\)	−0.283220	−	0.0123657i	−0.0853941	−	0.00372839i	0.000882471	−	1.00000i	\(-0.499719\pi\)
							−0.0862766	+	0.996271i	\(0.527497\pi\)
\(13\)	1.67021	+	0.869459i	0.463234	+	0.241144i	0.677346	−	0.735665i	\(-0.263130\pi\)
							−0.214112	+	0.976809i	\(0.568686\pi\)
\(17\)	3.70414	+	6.41576i	0.898387	+	1.55605i	0.829556	+	0.558423i	\(0.188593\pi\)
							0.0688304	+	0.997628i	\(0.478073\pi\)
\(19\)	0.332222	−	2.52348i	0.0762170	−	0.578926i	−0.910139	−	0.414302i	\(-0.864026\pi\)
							0.986356	−	0.164624i	\(-0.0526410\pi\)
\(23\)	−0.382238	−	0.267646i	−0.0797021	−	0.0558080i	0.533046	−	0.846086i	\(-0.321047\pi\)
							−0.612748	+	0.790278i	\(0.709936\pi\)
\(29\)	−2.13325	+	6.76581i	−0.396135	+	1.25638i	0.519535	+	0.854449i	\(0.326105\pi\)
							−0.915670	+	0.401931i	\(0.868339\pi\)
\(31\)	8.19279	−	1.44461i	1.47147	−	0.259460i	0.620305	−	0.784361i	\(-0.287009\pi\)
							0.851163	+	0.524901i	\(0.175898\pi\)
\(37\)	−8.99973	+	1.18484i	−1.47955	+	0.194786i	−0.826674	−	0.562681i	\(-0.809770\pi\)
							−0.652873	+	0.757467i	\(0.726437\pi\)
\(41\)	−2.89386	−	0.253180i	−0.451946	−	0.0395401i	−0.141089	−	0.989997i	\(-0.545060\pi\)
							−0.310857	+	0.950457i	\(0.600616\pi\)
\(43\)	0.244677	−	5.60402i	0.0373129	−	0.854605i	−0.884399	−	0.466731i	\(-0.845432\pi\)
							0.921712	−	0.387874i	\(-0.126790\pi\)
\(47\)	−1.49484	−	0.263581i	−0.218045	−	0.0384473i	0.0635581	−	0.997978i	\(-0.479755\pi\)
							−0.281604	+	0.959531i	\(0.590866\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.bt.a.11.14	✓	3408
27.5	odd	18	inner	864.2.bt.a.491.93	yes	3408
32.3	odd	8	inner	864.2.bt.a.227.93	yes	3408
864.707	even	72	inner	864.2.bt.a.707.14	yes	3408

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.2.bt.a.11.14	✓	3408	1.1	even	1	trivial
864.2.bt.a.227.93	yes	3408	32.3	odd	8	inner
864.2.bt.a.491.93	yes	3408	27.5	odd	18	inner
864.2.bt.a.707.14	yes	3408	864.707	even	72	inner