Embedded newform 45.2.l.a.23.1

• Label: 45.2.l.a.23.1
• Level: $45$
• Weight: $2$
• Character: 45.23
• 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			23.1
Root			\(0.601150 + 2.24352i\) of defining polynomial
Character	\(\chi\)	\(=\)	45.23
Dual form			45.2.l.a.2.1

$q$-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 \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\)	\(11\)	\(37\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{3}{4}\right)\)

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