Embedded newform 45.2.l.a.2.1

• Label: 45.2.l.a.2.1
• 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			2.1
Root			\(0.601150 - 2.24352i\) of defining polynomial
Character	\(\chi\)	\(=\)	45.2
Dual form			45.2.l.a.23.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.601150 - 2.24352i) q^{2} +(-1.72336 - 0.173261i) q^{3} +(-2.93996 + 1.69739i) q^{4} +(1.70912 - 1.44185i) q^{5} +(0.647285 + 3.97056i) q^{6} +(0.751454 - 0.201351i) q^{7} +(2.29074 + 2.29074i) q^{8} +(2.93996 + 0.597183i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{2} - 6 q^{3} - 6 q^{5} - 2 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.601150	−	2.24352i	−0.425077	−	1.58641i	−0.763754	−	0.645507i	\(-0.776646\pi\)
							0.338677	−	0.940903i	\(-0.390021\pi\)
\(3\)	−1.72336	−	0.173261i	−0.994984	−	0.100032i
							\(5\)	1.70912	−	1.44185i	0.764341	−	0.644813i
\(7\)	0.751454	−	0.201351i	0.284023	−	0.0761037i								−0.113995	−	0.993481i	\(-0.536365\pi\)
							0.398018	+	0.917378i	\(0.369698\pi\)
\(11\)	−0.220188	−	0.127126i	−0.0663893	−	0.0383299i	0.466438	−	0.884554i	\(-0.345537\pi\)
							−0.532827	+	0.846224i	\(0.678870\pi\)
\(13\)	3.70486	+	0.992714i	1.02754	+	0.275329i	0.732942	−	0.680291i	\(-0.238147\pi\)
							0.294601	+	0.955620i	\(0.404813\pi\)
\(17\)	−3.93311	+	3.93311i	−0.953920	+	0.953920i	−0.998984	−	0.0450642i	\(-0.985651\pi\)
							0.0450642	+	0.998984i	\(0.485651\pi\)
\(19\)			0.440377i			0.101029i	0.998723	+	0.0505147i	\(0.0160862\pi\)
							−0.998723	+	0.0505147i	\(0.983914\pi\)
\(23\)	−0.917076	+	3.42258i	−0.191224	+	0.713656i	0.801989	+	0.597339i	\(0.203775\pi\)
							−0.993212	+	0.116317i	\(0.962891\pi\)
\(29\)	2.76265	−	4.78505i	0.513011	−	0.888561i	−0.486875	−	0.873472i	\(-0.661863\pi\)
							0.999886	−	0.0150897i	\(-0.00480338\pi\)
\(31\)	−0.0971829	−	0.168326i	−0.0174546	−	0.0302322i	0.857166	−	0.515040i	\(-0.172223\pi\)
							−0.874621	+	0.484808i	\(0.838890\pi\)
\(37\)	−0.123005	−	0.123005i	−0.0202220	−	0.0202220i	0.696924	−	0.717145i	\(-0.254552\pi\)
							−0.717145	+	0.696924i	\(0.754552\pi\)
\(41\)	−3.88223	+	2.24141i	−0.606303	+	0.350049i	−0.771517	−	0.636209i	\(-0.780502\pi\)
							0.165214	+	0.986258i	\(0.447168\pi\)
\(43\)	−0.357680	−	1.33488i	−0.0545456	−	0.203567i	0.933275	−	0.359162i	\(-0.116937\pi\)
							−0.987821	+	0.155595i	\(0.950271\pi\)
\(47\)	1.11828	+	4.17348i	0.163118	+	0.608765i	0.998273	+	0.0587499i	\(0.0187115\pi\)
							−0.835155	+	0.550015i	\(0.814622\pi\)