Embedded newform 324.2.p.a.227.34

• Label: 324.2.p.a.227.34
• Level: $324$
• Weight: $2$
• Character: 324.227
• 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			227.34
Character	\(\chi\)	\(=\)	324.227
Dual form			324.2.p.a.167.34

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.708680 - 1.22384i) q^{2} +(0.722052 - 1.57437i) q^{3} +(-0.995546 - 1.73461i) q^{4} +(-0.582254 - 0.433472i) q^{5} +(-1.41507 - 1.99940i) q^{6} +(-0.981557 + 1.49238i) q^{7} +(-2.82841 - 0.0109009i) q^{8} +(-1.95728 - 2.27355i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 936 q - 18 q^{2} - 18 q^{4} - 36 q^{5} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(163\)	\(245\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{31}{54}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.708680	−	1.22384i	0.501112	−	0.865382i
							\(3\)	0.722052	−	1.57437i	0.416877	−	0.908963i
\(5\)	−0.582254	−	0.433472i	−0.260392	−	0.193855i								0.459044	−	0.888414i	\(-0.348192\pi\)
							−0.719436	+	0.694559i	\(0.755600\pi\)
\(7\)	−0.981557	+	1.49238i	−0.370994	+	0.564068i	−0.971942	−	0.235219i	\(-0.924419\pi\)
							0.600949	+	0.799288i	\(0.294790\pi\)
\(11\)	−2.76409	+	0.323076i	−0.833405	+	0.0974111i	−0.522085	−	0.852894i	\(-0.674845\pi\)
							−0.311321	+	0.950305i	\(0.600771\pi\)
\(13\)	1.62701	−	5.43458i	0.451250	−	1.50728i	−0.365788	−	0.930698i	\(-0.619201\pi\)
							0.817039	−	0.576583i	\(-0.195614\pi\)
\(17\)	5.89010	−	1.03858i	1.42856	−	0.251893i	0.594734	−	0.803923i	\(-0.297257\pi\)
							0.833825	+	0.552029i	\(0.186146\pi\)
\(19\)	7.99765	+	1.41020i	1.83479	+	0.323523i	0.980536	−	0.196338i	\(-0.0629049\pi\)
							0.854251	+	0.519860i	\(0.174016\pi\)
\(23\)	−2.66287	+	1.75140i	−0.555247	+	0.365192i	−0.795908	−	0.605418i	\(-0.793006\pi\)
							0.240661	+	0.970609i	\(0.422636\pi\)
\(29\)	2.70720	+	2.55411i	0.502714	+	0.474286i	0.895322	−	0.445420i	\(-0.146946\pi\)
							−0.392608	+	0.919706i	\(0.628427\pi\)
\(31\)	−0.406280	+	0.0236631i	−0.0729701	+	0.00425002i	−0.0945910	−	0.995516i	\(-0.530154\pi\)
							0.0216210	+	0.999766i	\(0.493117\pi\)
\(37\)	3.55379	+	1.29347i	0.584239	+	0.212646i	0.617194	−	0.786811i	\(-0.288270\pi\)
							−0.0329548	+	0.999457i	\(0.510492\pi\)
\(41\)	−0.930104	−	3.92442i	−0.145258	−	0.612891i	−0.996159	−	0.0875649i	\(-0.972091\pi\)
							0.850901	−	0.525326i	\(-0.176057\pi\)
\(43\)	6.36503	+	2.74560i	0.970658	+	0.418701i	0.821560	−	0.570122i	\(-0.193104\pi\)
							0.149097	+	0.988823i	\(0.452363\pi\)
\(47\)	0.517357	−	8.88268i	0.0754643	−	1.29567i	−0.721825	−	0.692075i	\(-0.756697\pi\)
							0.797290	−	0.603597i	\(-0.206266\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.2.p.a.227.34	yes	936
3.2	odd	2		972.2.p.a.35.19		936
4.3	odd	2	inner	324.2.p.a.227.27	yes	936
12.11	even	2		972.2.p.a.35.26		936
81.5	odd	54	inner	324.2.p.a.167.27	✓	936
81.76	even	27		972.2.p.a.611.26		936
324.167	even	54	inner	324.2.p.a.167.34	yes	936
324.319	odd	54		972.2.p.a.611.19		936

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.2.p.a.167.27	✓	936	81.5	odd	54	inner
324.2.p.a.167.34	yes	936	324.167	even	54	inner
324.2.p.a.227.27	yes	936	4.3	odd	2	inner
324.2.p.a.227.34	yes	936	1.1	even	1	trivial
972.2.p.a.35.19		936	3.2	odd	2
972.2.p.a.35.26		936	12.11	even	2
972.2.p.a.611.19		936	324.319	odd	54
972.2.p.a.611.26		936	81.76	even	27