Embedded newform 45.2.l.a.32.1

• Label: 45.2.l.a.32.1
• 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			32.1
Root			\(2.24352 - 0.601150i\) of defining polynomial
Character	\(\chi\)	\(=\)	45.32
Dual form			45.2.l.a.38.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.24352 - 0.601150i) q^{2} +(0.173261 + 1.72336i) q^{3} +(2.93996 + 1.69739i) q^{4} +(2.10323 + 0.759216i) q^{5} +(0.647285 - 3.97056i) q^{6} +(-0.201351 + 0.751454i) q^{7} +(-2.29074 - 2.29074i) q^{8} +(-2.93996 + 0.597183i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{2} - 6 q^{3} - 6 q^{5} - 2 q^{7}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.24352	−	0.601150i	−1.58641	−	0.425077i	−0.645507	−	0.763754i	\(-0.723354\pi\)
							−0.940903	+	0.338677i	\(0.890021\pi\)
\(3\)	0.173261	+	1.72336i	0.100032	+	0.994984i
							\(5\)	2.10323	+	0.759216i	0.940595	+	0.339532i
\(7\)	−0.201351	+	0.751454i	−0.0761037	+	0.284023i								−0.993481	−	0.113995i	\(-0.963635\pi\)
							0.917378	+	0.398018i	\(0.130302\pi\)
\(11\)	−0.220188	+	0.127126i	−0.0663893	+	0.0383299i	−0.532827	−	0.846224i	\(-0.678870\pi\)
							0.466438	+	0.884554i	\(0.345537\pi\)
\(13\)	−0.992714	−	3.70486i	−0.275329	−	1.02754i	−0.955620	−	0.294601i	\(-0.904813\pi\)
							0.680291	−	0.732942i	\(-0.261853\pi\)
\(17\)	3.93311	−	3.93311i	0.953920	−	0.953920i	−0.0450642	−	0.998984i	\(-0.514349\pi\)
							0.998984	+	0.0450642i	\(0.0143492\pi\)
\(19\)			0.440377i			0.101029i	0.998723	+	0.0505147i	\(0.0160862\pi\)
							−0.998723	+	0.0505147i	\(0.983914\pi\)
\(23\)	−3.42258	+	0.917076i	−0.713656	+	0.191224i	−0.597339	−	0.801989i	\(-0.703775\pi\)
							−0.116317	+	0.993212i	\(0.537109\pi\)
\(29\)	−2.76265	−	4.78505i	−0.513011	−	0.888561i	−0.999886	−	0.0150897i	\(-0.995197\pi\)
							0.486875	−	0.873472i	\(-0.338137\pi\)
\(31\)	−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	0.696924	−	0.717145i	\(-0.254552\pi\)
							−0.717145	+	0.696924i	\(0.754552\pi\)
\(41\)	−3.88223	−	2.24141i	−0.606303	−	0.350049i	0.165214	−	0.986258i	\(-0.447168\pi\)
							−0.771517	+	0.636209i	\(0.780502\pi\)
\(43\)	1.33488	+	0.357680i	0.203567	+	0.0545456i	0.359162	−	0.933275i	\(-0.383063\pi\)
							−0.155595	+	0.987821i	\(0.549729\pi\)
\(47\)	4.17348	+	1.11828i	0.608765	+	0.163118i	0.550015	−	0.835155i	\(-0.314622\pi\)
							0.0587499	+	0.998273i	\(0.481289\pi\)