Embedded newform 45.2.l.a.2.4

• Label: 45.2.l.a.2.4
• 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			2.4
Root			\(-0.347596 + 1.29724i\) of defining polynomial
Character	\(\chi\)	\(=\)	45.2
Dual form			45.2.l.a.23.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.347596 + 1.29724i) q^{2} +(-1.18953 + 1.25897i) q^{3} +(0.170031 - 0.0981673i) q^{4} +(-1.59371 - 1.56847i) q^{5} +(-2.04667 - 1.10550i) q^{6} +(1.97869 - 0.530190i) q^{7} +(2.08575 + 2.08575i) q^{8} +(-0.170031 - 2.99518i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{2} - 6 q^{3} - 6 q^{5} - 2 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.347596	+	1.29724i	0.245787	+	0.917290i	0.972986	+	0.230863i	\(0.0741551\pi\)
							−0.727199	+	0.686427i	\(0.759178\pi\)
\(3\)	−1.18953	+	1.25897i	−0.686776	+	0.726869i
							\(5\)	−1.59371	−	1.56847i	−0.712727	−	0.701442i
\(7\)	1.97869	−	0.530190i	0.747876	−	0.200393i								0.135300	−	0.990805i	\(-0.456800\pi\)
							0.612576	+	0.790412i	\(0.290133\pi\)
\(11\)	−0.762281	−	0.440103i	−0.229836	−	0.132696i	0.380660	−	0.924715i	\(-0.375697\pi\)
							−0.610497	+	0.792019i	\(0.709030\pi\)
\(13\)	−5.36743	−	1.43820i	−1.48866	−	0.398885i	−0.579374	−	0.815062i	\(-0.696703\pi\)
							−0.909283	+	0.416177i	\(0.863370\pi\)
\(17\)	−1.13610	+	1.13610i	−0.275544	+	0.275544i	−0.831327	−	0.555783i	\(-0.812418\pi\)
							0.555783	+	0.831327i	\(0.312418\pi\)
\(19\)			1.52456i			0.349758i	0.984590	+	0.174879i	\(0.0559535\pi\)
							−0.984590	+	0.174879i	\(0.944047\pi\)
\(23\)	0.410850	−	1.53331i	0.0856682	−	0.319718i	−0.909772	−	0.415109i	\(-0.863743\pi\)
							0.995440	+	0.0953909i	\(0.0304101\pi\)
\(29\)	0.796583	−	1.37972i	0.147922	−	0.256208i	−0.782537	−	0.622603i	\(-0.786075\pi\)
							0.930459	+	0.366396i	\(0.119408\pi\)
\(31\)	3.49518	+	6.05383i	0.627752	+	1.08730i	0.988002	+	0.154443i	\(0.0493583\pi\)
							−0.360249	+	0.932856i	\(0.617308\pi\)
\(37\)	−4.25746	−	4.25746i	−0.699922	−	0.699922i	0.264472	−	0.964393i	\(-0.414802\pi\)
							−0.964393	+	0.264472i	\(0.914802\pi\)
\(41\)	3.11546	−	1.79871i	0.486552	−	0.280911i	−0.236591	−	0.971609i	\(-0.576030\pi\)
							0.723143	+	0.690698i	\(0.242697\pi\)
\(43\)	0.497959	+	1.85841i	0.0759380	+	0.283404i	0.993444	−	0.114317i	\(-0.0364679\pi\)
							−0.917506	+	0.397721i	\(0.869801\pi\)
\(47\)	−2.14344	−	7.99942i	−0.312652	−	1.16683i	−0.926155	−	0.377142i	\(-0.876907\pi\)
							0.613503	−	0.789693i	\(-0.289760\pi\)