Embedded newform 45.2.l.a.2.3

• Label: 45.2.l.a.2.3
• Level: $45$
• Weight: $2$
• Character: 45.2
• Analytic conductor: $0.359$
• Analytic rank: $0$
• Dimension: $16$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	45.l (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.359326809096\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{12})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - 6*x^15 + 18*x^14 - 36*x^13 + 34*x^12 + 18*x^11 - 72*x^10 + 132*x^9 - 93*x^8 - 102*x^7 + 144*x^6 - 432*x^5 + 502*x^4 + 288*x^3 + 72*x^2 + 12*x + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			2.3
Root			\(-0.0499037 + 0.186243i\) of defining polynomial
Character	\(\chi\)	\(=\)	45.2
Dual form			45.2.l.a.23.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0499037 + 0.186243i) q^{2} +(-0.806271 - 1.53295i) q^{3} +(1.69985 - 0.981412i) q^{4} +(-0.250705 + 2.22197i) q^{5} +(0.245265 - 0.226662i) q^{6} +(-2.35868 + 0.632007i) q^{7} +(0.540289 + 0.540289i) q^{8} +(-1.69985 + 2.47194i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{2} - 6 q^{3} - 6 q^{5} - 2 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\)	\(11\)	\(37\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0499037	+	0.186243i	0.0352872	+	0.131694i	0.981322	−	0.192370i	\(-0.0616174\pi\)
							−0.946035	+	0.324064i	\(0.894951\pi\)
\(3\)	−0.806271	−	1.53295i	−0.465501	−	0.885048i
							\(5\)	−0.250705	+	2.22197i	−0.112119	+	0.993695i
\(7\)	−2.35868	+	0.632007i	−0.891499	+	0.238876i								−0.675362	−	0.737487i	\(-0.736012\pi\)
							−0.216137	+	0.976363i	\(0.569346\pi\)
\(11\)	−2.14390	−	1.23778i	−0.646409	−	0.373204i	0.140670	−	0.990057i	\(-0.455074\pi\)
							−0.787079	+	0.616852i	\(0.788408\pi\)
\(13\)	1.57505	+	0.422032i	0.436839	+	0.117051i	0.470535	−	0.882381i	\(-0.344061\pi\)
							−0.0336956	+	0.999432i	\(0.510728\pi\)
\(17\)	−0.403949	+	0.403949i	−0.0979721	+	0.0979721i	−0.754394	−	0.656422i	\(-0.772069\pi\)
							0.656422	+	0.754394i	\(0.272069\pi\)
\(19\)			4.28779i			0.983687i	0.870684	+	0.491843i	\(0.163677\pi\)
							−0.870684	+	0.491843i	\(0.836323\pi\)
\(23\)	1.82845	−	6.82387i	0.381258	−	1.42288i	−0.462723	−	0.886503i	\(-0.653128\pi\)
							0.843981	−	0.536373i	\(-0.180206\pi\)
\(29\)	3.20524	−	5.55164i	0.595199	−	1.03091i	−0.398320	−	0.917247i	\(-0.630407\pi\)
							0.993519	−	0.113668i	\(-0.0362600\pi\)
\(31\)	−1.97194	−	3.41550i	−0.354171	−	0.613442i	0.632805	−	0.774312i	\(-0.281904\pi\)
							−0.986976	+	0.160869i	\(0.948570\pi\)
\(37\)	−0.171954	−	0.171954i	−0.0282691	−	0.0282691i	0.692831	−	0.721100i	\(-0.256363\pi\)
							−0.721100	+	0.692831i	\(0.756363\pi\)
\(41\)	−6.52359	+	3.76639i	−1.01881	+	0.588212i	−0.913760	−	0.406255i	\(-0.866835\pi\)
							−0.105053	+	0.994467i	\(0.533501\pi\)
\(43\)	1.32695	+	4.95226i	0.202359	+	0.755213i	0.990238	+	0.139384i	\(0.0445122\pi\)
							−0.787880	+	0.615829i	\(0.788821\pi\)
\(47\)	0.780885	+	2.91430i	0.113904	+	0.425095i	0.999203	−	0.0399279i	\(-0.0127128\pi\)
							−0.885299	+	0.465023i	\(0.846046\pi\)