Embedded newform 27.8.e.a.7.2

• Label: 27.8.e.a.7.2
• Level: $27$
• Weight: $8$
• Character: 27.7
• Analytic conductor: $8.434$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	27.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.43439568807\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(20\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			7.2
Character	\(\chi\)	\(=\)	27.7
Dual form			27.8.e.a.4.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-19.1819 + 6.98163i) q^{2} +(-45.3621 + 11.3703i) q^{3} +(221.148 - 185.565i) q^{4} +(73.2555 + 415.452i) q^{5} +(790.746 - 534.805i) q^{6} +(1090.38 + 914.934i) q^{7} +(-1640.05 + 2840.66i) q^{8} +(1928.43 - 1031.56i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 219 q^{5} + 1386 q^{6} - 6 q^{7} - 4611 q^{8} - 1728 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−19.1819	+	6.98163i	−1.69545	+	0.617095i	−0.995295	−	0.0968913i	\(-0.969110\pi\)
							−0.700160	+	0.713986i	\(0.746888\pi\)
\(3\)	−45.3621	+	11.3703i	−0.969992	+	0.243135i
							\(5\)	73.2555	+	415.452i	0.262087	+	1.48637i	0.777204	+	0.629248i	\(0.216637\pi\)
−0.515117	+	0.857120i	\(0.672252\pi\)
\(7\)	1090.38	+	914.934i	1.20153	+	1.00820i	0.999585	+	0.0288014i	\(0.00916904\pi\)
							0.201940	+	0.979398i	\(0.435275\pi\)
\(11\)	−356.707	+	2022.99i	−0.0808049	+	0.458267i	0.917378	+	0.398016i	\(0.130301\pi\)
							−0.998183	+	0.0602510i	\(0.980810\pi\)
\(13\)	824.032	+	299.923i	0.104026	+	0.0378624i	0.393509	−	0.919321i	\(-0.371261\pi\)
							−0.289483	+	0.957183i	\(0.593483\pi\)
\(17\)	15072.9	+	26107.1i	0.744091	+	1.28880i	0.950618	+	0.310363i	\(0.100450\pi\)
							−0.206527	+	0.978441i	\(0.566216\pi\)
\(19\)	−7133.23	+	12355.1i	−0.238588	+	0.413246i	−0.960309	−	0.278937i	\(-0.910018\pi\)
							0.721721	+	0.692184i	\(0.243351\pi\)
\(23\)	27506.4	−	23080.6i	0.471396	−	0.395548i	−0.375907	−	0.926657i	\(-0.622669\pi\)
							0.847304	+	0.531109i	\(0.178224\pi\)
\(29\)	−18256.8	+	6644.93i	−0.139005	+	0.0505938i	−0.410586	−	0.911822i	\(-0.634676\pi\)
							0.271581	+	0.962416i	\(0.412454\pi\)
\(31\)	96577.6	−	81038.2i	0.582251	−	0.488567i	−0.303434	−	0.952852i	\(-0.598133\pi\)
							0.885686	+	0.464286i	\(0.153689\pi\)
\(37\)	80433.0	+	139314.i	0.261053	+	0.452157i	0.966522	−	0.256584i	\(-0.0825970\pi\)
							−0.705469	+	0.708741i	\(0.749264\pi\)
\(41\)	−748958.	−	272598.i	−1.69713	−	0.617703i	−0.701632	−	0.712539i	\(-0.747545\pi\)
							−0.995493	+	0.0948362i	\(0.969767\pi\)
\(43\)	−2452.85	+	13910.8i	−0.00470469	+	0.0266816i	−0.987070	−	0.160291i	\(-0.948757\pi\)
							0.982365	+	0.186973i	\(0.0598677\pi\)
\(47\)	−589117.	−	494328.i	−0.827673	−	0.694501i	0.127082	−	0.991892i	\(-0.459439\pi\)
							−0.954756	+	0.297392i	\(0.903883\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.e.a.7.2	yes	120
3.2	odd	2		81.8.e.a.19.19		120
27.4	even	9	inner	27.8.e.a.4.2	✓	120
27.23	odd	18		81.8.e.a.64.19		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.e.a.4.2	✓	120	27.4	even	9	inner
27.8.e.a.7.2	yes	120	1.1	even	1	trivial
81.8.e.a.19.19		120	3.2	odd	2
81.8.e.a.64.19		120	27.23	odd	18