Embedded newform 45.2.l.a.2.2

• Label: 45.2.l.a.2.2
• Level: $45$
• Weight: $2$
• Character: 45.2
• 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			2.2
Root			\(0.430324 - 1.60599i\) of defining polynomial
Character	\(\chi\)	\(=\)	45.2
Dual form			45.2.l.a.23.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.430324 - 1.60599i) q^{2} +(1.35314 + 1.08121i) q^{3} +(-0.661975 + 0.382191i) q^{4} +(-2.23073 + 0.154373i) q^{5} +(1.15412 - 2.63840i) q^{6} +(-1.73749 + 0.465559i) q^{7} +(-1.45267 - 1.45267i) q^{8} +(0.661975 + 2.92605i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{2} - 6 q^{3} - 6 q^{5} - 2 q^{7}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.430324	−	1.60599i	−0.304285	−	1.13561i	−0.933559	−	0.358423i	\(-0.883314\pi\)
							0.629274	−	0.777183i	\(-0.283352\pi\)
\(3\)	1.35314	+	1.08121i	0.781236	+	0.624236i
							\(5\)	−2.23073	+	0.154373i	−0.997614	+	0.0690379i
\(7\)	−1.73749	+	0.465559i	−0.656710	+	0.175965i								−0.571761	−	0.820421i	\(-0.693739\pi\)
							−0.0849489	+	0.996385i	\(0.527073\pi\)
\(11\)	3.12636	+	1.80501i	0.942634	+	0.544230i	0.890785	−	0.454425i	\(-0.150155\pi\)
							0.0518493	+	0.998655i	\(0.483488\pi\)
\(13\)	−1.27850	−	0.342574i	−0.354593	−	0.0950128i	0.0771255	−	0.997021i	\(-0.475426\pi\)
							−0.431718	+	0.902009i	\(0.642092\pi\)
\(17\)	0.277007	−	0.277007i	0.0671841	−	0.0671841i	−0.672716	−	0.739900i	\(-0.734873\pi\)
							0.739900	+	0.672716i	\(0.234873\pi\)
\(19\)		−	6.25273i		−	1.43447i	−0.696829	−	0.717237i	\(-0.745406\pi\)
							0.696829	−	0.717237i	\(-0.254594\pi\)
\(23\)	0.579699	−	2.16347i	0.120876	−	0.451114i	−0.878784	−	0.477221i	\(-0.841644\pi\)
							0.999659	+	0.0261067i	\(0.00831095\pi\)
\(29\)	−1.56832	+	2.71642i	−0.291230	+	0.504426i	−0.974101	−	0.226114i	\(-0.927398\pi\)
							0.682871	+	0.730539i	\(0.260731\pi\)
\(31\)	−2.42605	−	4.20205i	−0.435732	−	0.754710i	0.561623	−	0.827393i	\(-0.310177\pi\)
							−0.997355	+	0.0726832i	\(0.976844\pi\)
\(37\)	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	−0.395844	−	0.918318i	\(-0.629548\pi\)
							0.597365	+	0.801970i	\(0.296215\pi\)
\(43\)	−1.10121	−	4.10976i	−0.167933	−	0.626733i	−0.997648	−	0.0685463i	\(-0.978164\pi\)
							0.829715	−	0.558187i	\(-0.188503\pi\)
\(47\)	−1.02368	−	3.82042i	−0.149319	−	0.557266i	−0.999525	−	0.0308158i	\(-0.990189\pi\)
							0.850206	−	0.526450i	\(-0.176477\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.2.l.a.2.2	✓	16	1.1	even	1	trivial
45.2.l.a.23.2	yes	16	45.23	even	12	inner
45.2.l.a.32.2	yes	16	9.5	odd	6	inner
45.2.l.a.38.2	yes	16	5.3	odd	4	inner
135.2.m.a.8.3		16	15.8	even	4
135.2.m.a.17.3		16	9.4	even	3
135.2.m.a.62.3		16	3.2	odd	2
135.2.m.a.98.3		16	45.13	odd	12
225.2.p.b.32.3		16	45.14	odd	6
225.2.p.b.68.3		16	45.32	even	12
225.2.p.b.182.3		16	5.4	even	2
225.2.p.b.218.3		16	5.2	odd	4
405.2.f.a.242.2		16	9.7	even	3
405.2.f.a.242.7		16	9.2	odd	6
405.2.f.a.323.2		16	45.38	even	12
405.2.f.a.323.7		16	45.43	odd	12
675.2.q.a.143.2		16	15.2	even	4
675.2.q.a.332.2		16	15.14	odd	2
675.2.q.a.368.2		16	45.22	odd	12
675.2.q.a.557.2		16	45.4	even	6
720.2.cu.c.113.1		16	180.23	odd	12
720.2.cu.c.257.3		16	36.23	even	6
720.2.cu.c.353.3		16	20.3	even	4
720.2.cu.c.497.1		16	4.3	odd	2