Embedded newform 324.2.p.a.239.20

• Label: 324.2.p.a.239.20
• Level: $324$
• Weight: $2$
• Character: 324.239
• 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			239.20
Character	\(\chi\)	\(=\)	324.239
Dual form			324.2.p.a.263.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.478430 - 1.33083i) q^{2} +(1.39409 - 1.02787i) q^{3} +(-1.54221 + 1.27342i) q^{4} +(0.700373 + 1.39456i) q^{5} +(-2.03489 - 1.36354i) q^{6} +(3.95962 + 1.18543i) q^{7} +(2.43254 + 1.44318i) q^{8} +(0.886984 - 2.86588i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 936 q - 18 q^{2} - 18 q^{4} - 36 q^{5} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(163\)	\(245\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{29}{54}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.478430	−	1.33083i	−0.338301	−	0.941038i
							\(3\)	1.39409	−	1.02787i	0.804879	−	0.593439i
\(5\)	0.700373	+	1.39456i	0.313217	+	0.623665i								0.994069	−	0.108756i	\(-0.0346866\pi\)
							−0.680852	+	0.732421i	\(0.738390\pi\)
\(7\)	3.95962	+	1.18543i	1.49660	+	0.448052i	0.927375	−	0.374132i	\(-0.122059\pi\)
							0.569222	+	0.822184i	\(0.307245\pi\)
\(11\)	0.366190	+	6.28723i	0.110410	+	1.89567i	0.368616	+	0.929582i	\(0.379832\pi\)
							−0.258206	+	0.966090i	\(0.583131\pi\)
\(13\)	−1.88902	−	2.53739i	−0.523919	−	0.703746i	0.458864	−	0.888506i	\(-0.348256\pi\)
							−0.982784	+	0.184760i	\(0.940849\pi\)
\(17\)	−0.154501	+	0.184127i	−0.0374721	+	0.0446575i	−0.784455	−	0.620186i	\(-0.787057\pi\)
							0.746983	+	0.664843i	\(0.231502\pi\)
\(19\)	−0.281157	−	0.335070i	−0.0645018	−	0.0768703i	0.732829	−	0.680413i	\(-0.238200\pi\)
							−0.797331	+	0.603543i	\(0.793755\pi\)
\(23\)	−0.516923	−	1.72664i	−0.107786	−	0.360030i	0.887031	−	0.461709i	\(-0.152764\pi\)
							−0.994817	+	0.101679i	\(0.967578\pi\)
\(29\)	−8.08473	+	3.48741i	−1.50130	+	0.647596i	−0.978758	−	0.205019i	\(-0.934274\pi\)
							−0.522539	+	0.852615i	\(0.675015\pi\)
\(31\)	−6.47786	−	6.11155i	−1.16346	−	1.09767i	−0.993688	−	0.112182i	\(-0.964216\pi\)
							−0.169770	−	0.985484i	\(-0.554303\pi\)
\(37\)	−0.364098	+	2.06490i	−0.0598573	+	0.339468i	−0.999999	−	0.00131579i	\(-0.999581\pi\)
							0.940142	+	0.340783i	\(0.110692\pi\)
\(41\)	−1.29197	−	11.0535i	−0.201772	−	1.72627i	−0.592337	−	0.805691i	\(-0.701794\pi\)
							0.390565	−	0.920575i	\(-0.372280\pi\)
\(43\)	−0.947811	−	1.44108i	−0.144540	−	0.219762i	0.755961	−	0.654617i	\(-0.227170\pi\)
							−0.900501	+	0.434855i	\(0.856800\pi\)
\(47\)	5.93821	+	6.29414i	0.866177	+	0.918094i	0.997312	−	0.0732704i	\(-0.0233436\pi\)
							−0.131135	+	0.991365i	\(0.541862\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.2.p.a.239.20	✓	936
3.2	odd	2		972.2.p.a.179.33		936
4.3	odd	2	inner	324.2.p.a.239.37	yes	936
12.11	even	2		972.2.p.a.179.16		936
81.20	odd	54	inner	324.2.p.a.263.37	yes	936
81.61	even	27		972.2.p.a.467.16		936
324.223	odd	54		972.2.p.a.467.33		936
324.263	even	54	inner	324.2.p.a.263.20	yes	936

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.2.p.a.239.20	✓	936	1.1	even	1	trivial
324.2.p.a.239.37	yes	936	4.3	odd	2	inner
324.2.p.a.263.20	yes	936	324.263	even	54	inner
324.2.p.a.263.37	yes	936	81.20	odd	54	inner
972.2.p.a.179.16		936	12.11	even	2
972.2.p.a.179.33		936	3.2	odd	2
972.2.p.a.467.16		936	81.61	even	27
972.2.p.a.467.33		936	324.223	odd	54