Embedded newform 45.2.l.a.32.3

• Label: 45.2.l.a.32.3
• Level: $45$
• Weight: $2$
• Character: 45.32
• 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			32.3
Root			\(-0.186243 + 0.0499037i\) of defining polynomial
Character	\(\chi\)	\(=\)	45.32
Dual form			45.2.l.a.38.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.186243 + 0.0499037i) q^{2} +(1.53295 + 0.806271i) q^{3} +(-1.69985 - 0.981412i) q^{4} +(-2.04963 + 0.893868i) q^{5} +(0.245265 + 0.226662i) q^{6} +(0.632007 - 2.35868i) q^{7} +(-0.540289 - 0.540289i) q^{8} +(1.69985 + 2.47194i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{2} - 6 q^{3} - 6 q^{5} - 2 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(11\)	\(37\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.186243	+	0.0499037i	0.131694	+	0.0352872i	0.324064	−	0.946035i	\(-0.394951\pi\)
							−0.192370	+	0.981322i	\(0.561617\pi\)
\(3\)	1.53295	+	0.806271i	0.885048	+	0.465501i
							\(5\)	−2.04963	+	0.893868i	−0.916624	+	0.399750i
\(7\)	0.632007	−	2.35868i	0.238876	−	0.891499i								−0.737487	−	0.675362i	\(-0.763988\pi\)
							0.976363	−	0.216137i	\(-0.0693458\pi\)
\(11\)	−2.14390	+	1.23778i	−0.646409	+	0.373204i	−0.787079	−	0.616852i	\(-0.788408\pi\)
							0.140670	+	0.990057i	\(0.455074\pi\)
\(13\)	−0.422032	−	1.57505i	−0.117051	−	0.436839i	0.882381	−	0.470535i	\(-0.155939\pi\)
							−0.999432	+	0.0336956i	\(0.989272\pi\)
\(17\)	0.403949	−	0.403949i	0.0979721	−	0.0979721i	−0.656422	−	0.754394i	\(-0.727931\pi\)
							0.754394	+	0.656422i	\(0.227931\pi\)
\(19\)			4.28779i			0.983687i	0.870684	+	0.491843i	\(0.163677\pi\)
							−0.870684	+	0.491843i	\(0.836323\pi\)
\(23\)	6.82387	−	1.82845i	1.42288	−	0.381258i	0.536373	−	0.843981i	\(-0.319794\pi\)
							0.886503	+	0.462723i	\(0.153128\pi\)
\(29\)	−3.20524	−	5.55164i	−0.595199	−	1.03091i	−0.993519	−	0.113668i	\(-0.963740\pi\)
							0.398320	−	0.917247i	\(-0.369593\pi\)
\(31\)	−1.97194	+	3.41550i	−0.354171	+	0.613442i	−0.986976	−	0.160869i	\(-0.948570\pi\)
							0.632805	+	0.774312i	\(0.281904\pi\)
\(37\)	−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	−0.105053	−	0.994467i	\(-0.533501\pi\)
							−0.913760	+	0.406255i	\(0.866835\pi\)
\(43\)	−4.95226	−	1.32695i	−0.755213	−	0.202359i	−0.139384	−	0.990238i	\(-0.544512\pi\)
							−0.615829	+	0.787880i	\(0.711179\pi\)
\(47\)	2.91430	+	0.780885i	0.425095	+	0.113904i	0.465023	−	0.885299i	\(-0.346046\pi\)
							−0.0399279	+	0.999203i	\(0.512713\pi\)