Embedded newform 63.4.f.b.22.2

• Label: 63.4.f.b.22.2
• Level: $63$
• Weight: $4$
• Character: 63.22
• Analytic conductor: $3.717$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	63.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.71712033036\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	\(x^{16} - 3 x^{15} + 58 x^{14} - 129 x^{13} + 2107 x^{12} - 4455 x^{11} + 42901 x^{10} - 76404 x^{9} + 599392 x^{8} - 1089732 x^{7} + 4808401 x^{6} - 7939134 x^{5} + 26225236 x^{4} - 39450864 x^{3} + 62254768 x^{2} - 39660672 x + 21307456\)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 2^{2}\cdot 3^{7} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			22.2
Root			\(2.28179 + 3.95218i\) of defining polynomial
Character	\(\chi\)	\(=\)	63.22
Dual form			63.4.f.b.43.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.28179 - 3.95218i) q^{2} +(1.09409 - 5.07966i) q^{3} +(-6.41313 + 11.1079i) q^{4} +(-10.3955 + 18.0055i) q^{5} +(-22.5722 + 7.26670i) q^{6} +(3.50000 + 6.06218i) q^{7} +22.0250 q^{8} +(-24.6060 - 11.1152i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16q - 3q^{2} + 2q^{3} - 43q^{4} - 30q^{5} + 19q^{6} + 56q^{7} + 12q^{8} - 124q^{9} + O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\)	\(10\)	\(29\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.28179	−	3.95218i	−0.806734	−	1.39730i	−0.915114	−	0.403195i	\(-0.867900\pi\)
							0.108380	−	0.994110i	\(-0.465434\pi\)
\(3\)	1.09409	−	5.07966i	0.210557	−	0.977582i
							\(5\)	−10.3955	+	18.0055i	−0.929801	+	1.61046i	−0.146149	+	0.989263i	\(0.546688\pi\)
−0.783652	+	0.621200i	\(0.786645\pi\)
\(7\)	3.50000	+	6.06218i	0.188982	+	0.327327i
							\(11\)	−22.6993	−	39.3163i	−0.622190	−	1.07766i	−0.989077	−	0.147399i	\(-0.952910\pi\)
0.366887	−	0.930265i	\(-0.380423\pi\)
\(13\)	−14.6521	+	25.3782i	−0.312597	+	0.541434i	−0.978924	−	0.204226i	\(-0.934532\pi\)
							0.666327	+	0.745660i	\(0.267866\pi\)
\(17\)	−98.0555			−1.39894			−0.699469	−	0.714663i	\(-0.746580\pi\)
							−0.699469	+	0.714663i	\(0.746580\pi\)
\(19\)	31.1554			0.376186			0.188093	−	0.982151i	\(-0.439769\pi\)
							0.188093	+	0.982151i	\(0.439769\pi\)
\(23\)	−4.19596	+	7.26761i	−0.0380399	+	0.0658870i	−0.884419	−	0.466695i	\(-0.845445\pi\)
							0.846379	+	0.532582i	\(0.178778\pi\)
\(29\)	36.1332	+	62.5845i	0.231371	+	0.400747i	0.958212	−	0.286059i	\(-0.0923455\pi\)
							−0.726841	+	0.686806i	\(0.759012\pi\)
\(31\)	15.4299	−	26.7254i	0.0893965	−	0.154839i	−0.817860	−	0.575418i	\(-0.804840\pi\)
							0.907256	+	0.420578i	\(0.138173\pi\)
\(37\)	−196.369			−0.872511			−0.436255	−	0.899823i	\(-0.643696\pi\)
							−0.436255	+	0.899823i	\(0.643696\pi\)
\(41\)	−106.336	+	184.180i	−0.405048	+	0.701564i	−0.994327	−	0.106366i	\(-0.966079\pi\)
							0.589279	+	0.807930i	\(0.299412\pi\)
\(43\)	−118.438	−	205.141i	−0.420039	−	0.727529i	0.575904	−	0.817517i	\(-0.304650\pi\)
							−0.995943	+	0.0899887i	\(0.971317\pi\)
\(47\)	110.018	+	190.557i	0.341442	+	0.591395i	0.984701	−	0.174254i	\(-0.0557513\pi\)
							−0.643259	+	0.765649i	\(0.722418\pi\)