Embedded newform 45.2.l.a.32.2

• Label: 45.2.l.a.32.2
• Level: $45$
• Weight: $2$
• Character: 45.32
• Analytic conductor: $0.359$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• 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			32.2
Root			\(1.60599 - 0.430324i\) of defining polynomial
Character	\(\chi\)	\(=\)	45.32
Dual form			45.2.l.a.38.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.60599 - 0.430324i) q^{2} +(-1.08121 - 1.35314i) q^{3} +(0.661975 + 0.382191i) q^{4} +(-1.24906 - 1.85468i) q^{5} +(1.15412 + 2.63840i) q^{6} +(0.465559 - 1.73749i) q^{7} +(1.45267 + 1.45267i) q^{8} +(-0.661975 + 2.92605i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{2} - 6 q^{3} - 6 q^{5} - 2 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.60599	−	0.430324i	−1.13561	−	0.304285i	−0.358423	−	0.933559i	\(-0.616686\pi\)
							−0.777183	+	0.629274i	\(0.783352\pi\)
\(3\)	−1.08121	−	1.35314i	−0.624236	−	0.781236i
							\(5\)	−1.24906	−	1.85468i	−0.558596	−	0.829440i
\(7\)	0.465559	−	1.73749i	0.175965	−	0.656710i								−0.820421	−	0.571761i	\(-0.806261\pi\)
							0.996385	−	0.0849489i	\(-0.0270727\pi\)
\(11\)	3.12636	−	1.80501i	0.942634	−	0.544230i	0.0518493	−	0.998655i	\(-0.483488\pi\)
							0.890785	+	0.454425i	\(0.150155\pi\)
\(13\)	0.342574	+	1.27850i	0.0950128	+	0.354593i	0.997021	−	0.0771255i	\(-0.0245742\pi\)
							−0.902009	+	0.431718i	\(0.857908\pi\)
\(17\)	−0.277007	+	0.277007i	−0.0671841	+	0.0671841i	−0.739900	−	0.672716i	\(-0.765127\pi\)
							0.672716	+	0.739900i	\(0.265127\pi\)
\(19\)		−	6.25273i		−	1.43447i	−0.696829	−	0.717237i	\(-0.745406\pi\)
							0.696829	−	0.717237i	\(-0.254594\pi\)
\(23\)	2.16347	−	0.579699i	0.451114	−	0.120876i	−0.0261067	−	0.999659i	\(-0.508311\pi\)
							0.477221	+	0.878784i	\(0.341644\pi\)
\(29\)	1.56832	+	2.71642i	0.291230	+	0.504426i	0.974101	−	0.226114i	\(-0.0726021\pi\)
							−0.682871	+	0.730539i	\(0.739269\pi\)
\(31\)	−2.42605	+	4.20205i	−0.435732	+	0.754710i	−0.997355	−	0.0726832i	\(-0.976844\pi\)
							0.561623	+	0.827393i	\(0.310177\pi\)
\(37\)	5.55242	+	5.55242i	0.912812	+	0.912812i	0.996493	−	0.0836807i	\(-0.0266676\pi\)
							−0.0836807	+	0.996493i	\(0.526668\pi\)
\(41\)	1.29036	+	0.744991i	0.201521	+	0.116348i	0.597365	−	0.801970i	\(-0.296215\pi\)
							−0.395844	+	0.918318i	\(0.629548\pi\)
\(43\)	4.10976	+	1.10121i	0.626733	+	0.167933i	0.558187	−	0.829715i	\(-0.311497\pi\)
							0.0685463	+	0.997648i	\(0.478164\pi\)
\(47\)	−3.82042	−	1.02368i	−0.557266	−	0.149319i	−0.0308158	−	0.999525i	\(-0.509811\pi\)
							−0.526450	+	0.850206i	\(0.676477\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.2.l.a.32.2	yes	16
3.2	odd	2		135.2.m.a.17.3		16
4.3	odd	2		720.2.cu.c.257.3		16
5.2	odd	4		225.2.p.b.68.3		16
5.3	odd	4	inner	45.2.l.a.23.2	yes	16
5.4	even	2		225.2.p.b.32.3		16
9.2	odd	6	inner	45.2.l.a.2.2	✓	16
9.4	even	3		405.2.f.a.242.7		16
9.5	odd	6		405.2.f.a.242.2		16
9.7	even	3		135.2.m.a.62.3		16
15.2	even	4		675.2.q.a.368.2		16
15.8	even	4		135.2.m.a.98.3		16
15.14	odd	2		675.2.q.a.557.2		16
20.3	even	4		720.2.cu.c.113.1		16
36.11	even	6		720.2.cu.c.497.1		16
45.2	even	12		225.2.p.b.218.3		16
45.7	odd	12		675.2.q.a.143.2		16
45.13	odd	12		405.2.f.a.323.2		16
45.23	even	12		405.2.f.a.323.7		16
45.29	odd	6		225.2.p.b.182.3		16
45.34	even	6		675.2.q.a.332.2		16
45.38	even	12	inner	45.2.l.a.38.2	yes	16
45.43	odd	12		135.2.m.a.8.3		16
180.83	odd	12		720.2.cu.c.353.3		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.2.l.a.2.2	✓	16	9.2	odd	6	inner
45.2.l.a.23.2	yes	16	5.3	odd	4	inner
45.2.l.a.32.2	yes	16	1.1	even	1	trivial
45.2.l.a.38.2	yes	16	45.38	even	12	inner
135.2.m.a.8.3		16	45.43	odd	12
135.2.m.a.17.3		16	3.2	odd	2
135.2.m.a.62.3		16	9.7	even	3
135.2.m.a.98.3		16	15.8	even	4
225.2.p.b.32.3		16	5.4	even	2
225.2.p.b.68.3		16	5.2	odd	4
225.2.p.b.182.3		16	45.29	odd	6
225.2.p.b.218.3		16	45.2	even	12
405.2.f.a.242.2		16	9.5	odd	6
405.2.f.a.242.7		16	9.4	even	3
405.2.f.a.323.2		16	45.13	odd	12
405.2.f.a.323.7		16	45.23	even	12
675.2.q.a.143.2		16	45.7	odd	12
675.2.q.a.332.2		16	45.34	even	6
675.2.q.a.368.2		16	15.2	even	4
675.2.q.a.557.2		16	15.14	odd	2
720.2.cu.c.113.1		16	20.3	even	4
720.2.cu.c.257.3		16	4.3	odd	2
720.2.cu.c.353.3		16	180.83	odd	12
720.2.cu.c.497.1		16	36.11	even	6