Embedded newform 54.2.e.b.49.1

• Label: 54.2.e.b.49.1
• Level: $54$
• Weight: $2$
• Character: 54.49
• Analytic conductor: $0.431$
• Analytic rank: $0$
• Dimension: $12$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	54.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.431192170915\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{9})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	x^12 - 6*x^11 + 33*x^10 - 110*x^9 + 318*x^8 - 678*x^7 + 1225*x^6 - 1698*x^5 + 1905*x^4 - 1584*x^3 + 936*x^2 - 342*x + 57
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			49.1
Root			\(0.500000 - 1.80139i\) of defining polynomial
Character	\(\chi\)	\(=\)	54.49
Dual form			54.2.e.b.43.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.766044 - 0.642788i) q^{2} +(-1.36085 - 1.07149i) q^{3} +(0.173648 - 0.984808i) q^{4} +(0.696050 - 0.253341i) q^{5} +(-1.73121 + 0.0539310i) q^{6} +(0.717657 + 4.07003i) q^{7} +(-0.500000 - 0.866025i) q^{8} +(0.703829 + 2.91627i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{5} + 3 q^{6} - 3 q^{7} - 6 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/54\mathbb{Z}\right)^\times\).

\(n\)	\(29\)
\(\chi(n)\)	\(e\left(\frac{7}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.766044	−	0.642788i	0.541675	−	0.454519i
							\(3\)	−1.36085	−	1.07149i	−0.785688	−	0.618624i
\(5\)	0.696050	−	0.253341i	0.311283	−	0.113298i								−0.181655	−	0.983362i	\(-0.558145\pi\)
							0.492938	+	0.870065i	\(0.335923\pi\)
\(7\)	0.717657	+	4.07003i	0.271249	+	1.53833i	0.750631	+	0.660722i	\(0.229750\pi\)
							−0.479382	+	0.877606i	\(0.659139\pi\)
\(11\)	−4.27215	−	1.55493i	−1.28810	−	0.468830i	−0.394998	−	0.918682i	\(-0.629255\pi\)
							−0.893103	+	0.449852i	\(0.851477\pi\)
\(13\)	0.662744	+	0.556108i	0.183812	+	0.154237i	0.730051	−	0.683393i	\(-0.239496\pi\)
							−0.546239	+	0.837630i	\(0.683941\pi\)
\(17\)	2.17975	−	3.77544i	0.528667	−	0.915678i	−0.470774	−	0.882254i	\(-0.656025\pi\)
							0.999441	−	0.0334246i	\(-0.0106414\pi\)
\(19\)	−0.777964	−	1.34747i	−0.178477	−	0.309132i	0.762882	−	0.646538i	\(-0.223784\pi\)
							−0.941359	+	0.337406i	\(0.890450\pi\)
\(23\)	−0.608539	+	3.45119i	−0.126889	+	0.719624i	0.853279	+	0.521455i	\(0.174611\pi\)
							−0.980168	+	0.198169i	\(0.936501\pi\)
\(29\)	−2.50318	+	2.10042i	−0.464829	+	0.390038i	−0.844904	−	0.534918i	\(-0.820343\pi\)
							0.380075	+	0.924956i	\(0.375898\pi\)
\(31\)	1.85778	−	10.5360i	0.333667	−	1.89232i	−0.106350	−	0.994329i	\(-0.533916\pi\)
							0.440016	−	0.897990i	\(-0.354973\pi\)
\(37\)	0.880842	−	1.52566i	0.144810	−	0.250817i	−0.784492	−	0.620139i	\(-0.787076\pi\)
							0.929302	+	0.369321i	\(0.120410\pi\)
\(41\)	−1.97401	−	1.65639i	−0.308289	−	0.258685i	0.475495	−	0.879718i	\(-0.342269\pi\)
							−0.783784	+	0.621033i	\(0.786713\pi\)
\(43\)	2.58757	+	0.941797i	0.394600	+	0.143623i	0.531696	−	0.846935i	\(-0.321555\pi\)
							−0.137096	+	0.990558i	\(0.543777\pi\)
\(47\)	1.68378	+	9.54918i	0.245604	+	1.39289i	0.819085	+	0.573672i	\(0.194481\pi\)
							−0.573481	+	0.819219i	\(0.694407\pi\)