Embedded newform 324.2.p.a.203.51

• Label: 324.2.p.a.203.51
• Level: $324$
• Weight: $2$
• Character: 324.203
• Analytic conductor: $2.587$
• Analytic rank: $0$
• Dimension: $936$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	324.p (of order \(54\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.58715302549\)
Analytic rank:	\(0\)
Dimension:	\(936\)
Relative dimension:	\(52\) over \(\Q(\zeta_{54})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{54}]$

Embedding invariants

Embedding label			203.51
Character	\(\chi\)	\(=\)	324.203
Dual form			324.2.p.a.83.51

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.40911 + 0.120030i) q^{2} +(1.57425 - 0.722316i) q^{3} +(1.97119 + 0.338270i) q^{4} +(-0.187018 + 0.789092i) q^{5} +(2.30499 - 0.828867i) q^{6} +(-1.71428 - 1.27624i) q^{7} +(2.73702 + 0.713260i) q^{8} +(1.95652 - 2.27421i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 936 q - 18 q^{2} - 18 q^{4} - 36 q^{5} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\)	\(163\)	\(245\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{53}{54}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.40911	+	0.120030i	0.996392	+	0.0848737i
							\(3\)	1.57425	−	0.722316i	0.908893	−	0.417029i
\(5\)	−0.187018	+	0.789092i	−0.0836371	+	0.352893i								−0.998732	−	0.0503464i	\(-0.983967\pi\)
							0.915095	+	0.403239i	\(0.132116\pi\)
\(7\)	−1.71428	−	1.27624i	−0.647939	−	0.482372i	0.222316	−	0.974975i	\(-0.428638\pi\)
							−0.870255	+	0.492602i	\(0.836046\pi\)
\(11\)	−2.24280	−	2.37723i	−0.676231	−	0.716762i	0.294842	−	0.955546i	\(-0.404733\pi\)
							−0.971073	+	0.238783i	\(0.923251\pi\)
\(13\)	−3.56685	+	1.79134i	−0.989267	+	0.496828i	−0.868413	−	0.495842i	\(-0.834860\pi\)
							−0.120854	+	0.992670i	\(0.538563\pi\)
\(17\)	1.53626	+	4.22084i	0.372597	+	1.02370i	0.974353	+	0.225023i	\(0.0722458\pi\)
							−0.601756	+	0.798680i	\(0.705532\pi\)
\(19\)	−2.53116	+	6.95430i	−0.580687	+	1.59543i	0.206322	+	0.978484i	\(0.433851\pi\)
							−0.787009	+	0.616941i	\(0.788372\pi\)
\(23\)	−2.00970	−	2.69949i	−0.419050	−	0.562882i	0.541602	−	0.840635i	\(-0.317818\pi\)
							−0.960653	+	0.277752i	\(0.910410\pi\)
\(29\)	−3.77265	−	5.73604i	−0.700564	−	1.06516i	−0.994022	−	0.109184i	\(-0.965176\pi\)
							0.293458	−	0.955972i	\(-0.405194\pi\)
\(31\)	3.39428	+	1.46415i	0.609631	+	0.262969i	0.678434	−	0.734661i	\(-0.262659\pi\)
							−0.0688034	+	0.997630i	\(0.521918\pi\)
\(37\)	−6.90854	−	5.79695i	−1.13576	−	0.953013i	−0.136465	−	0.990645i	\(-0.543574\pi\)
							−0.999292	+	0.0376316i	\(0.988019\pi\)
\(41\)	−6.86170	−	0.399649i	−1.07162	−	0.0624146i	−0.486748	−	0.873542i	\(-0.661817\pi\)
							−0.584869	+	0.811128i	\(0.698854\pi\)
\(43\)	9.84330	+	2.94689i	1.50109	+	0.449396i	0.928777	−	0.370640i	\(-0.120861\pi\)
							0.572312	+	0.820036i	\(0.306047\pi\)
\(47\)	−0.646240	−	1.49815i	−0.0942638	−	0.218528i	0.864560	−	0.502530i	\(-0.167597\pi\)
							−0.958824	+	0.284001i	\(0.908338\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.2.p.a.203.51	yes	936
3.2	odd	2		972.2.p.a.395.2		936
4.3	odd	2	inner	324.2.p.a.203.48	yes	936
12.11	even	2		972.2.p.a.395.5		936
81.2	odd	54	inner	324.2.p.a.83.48	✓	936
81.79	even	27		972.2.p.a.251.5		936
324.79	odd	54		972.2.p.a.251.2		936
324.83	even	54	inner	324.2.p.a.83.51	yes	936

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.2.p.a.83.48	✓	936	81.2	odd	54	inner
324.2.p.a.83.51	yes	936	324.83	even	54	inner
324.2.p.a.203.48	yes	936	4.3	odd	2	inner
324.2.p.a.203.51	yes	936	1.1	even	1	trivial
972.2.p.a.251.2		936	324.79	odd	54
972.2.p.a.251.5		936	81.79	even	27
972.2.p.a.395.2		936	3.2	odd	2
972.2.p.a.395.5		936	12.11	even	2