Embedded newform 45.2.l.a.23.2

• Label: 45.2.l.a.23.2
• Level: $45$
• Weight: $2$
• Character: 45.23
• 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			23.2
Root			\(0.430324 + 1.60599i\) of defining polynomial
Character	\(\chi\)	\(=\)	45.23
Dual form			45.2.l.a.2.2

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.430324	+	1.60599i	−0.304285	+	1.13561i	0.629274	+	0.777183i	\(0.283352\pi\)
							−0.933559	+	0.358423i	\(0.883314\pi\)
\(3\)	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								−0.0849489	−	0.996385i	\(-0.527073\pi\)
							−0.571761	+	0.820421i	\(0.693739\pi\)
\(11\)	3.12636	−	1.80501i	0.942634	−	0.544230i	0.0518493	−	0.998655i	\(-0.483488\pi\)
							0.890785	+	0.454425i	\(0.150155\pi\)
\(13\)	−1.27850	+	0.342574i	−0.354593	+	0.0950128i	−0.431718	−	0.902009i	\(-0.642092\pi\)
							0.0771255	+	0.997021i	\(0.475426\pi\)
\(17\)	0.277007	+	0.277007i	0.0671841	+	0.0671841i	0.739900	−	0.672716i	\(-0.234873\pi\)
							−0.672716	+	0.739900i	\(0.734873\pi\)
\(19\)			6.25273i			1.43447i	0.696829	+	0.717237i	\(0.254594\pi\)
							−0.696829	+	0.717237i	\(0.745406\pi\)
\(23\)	0.579699	+	2.16347i	0.120876	+	0.451114i	0.999659	−	0.0261067i	\(-0.00831095\pi\)
							−0.878784	+	0.477221i	\(0.841644\pi\)
\(29\)	−1.56832	−	2.71642i	−0.291230	−	0.504426i	0.682871	−	0.730539i	\(-0.260731\pi\)
							−0.974101	+	0.226114i	\(0.927398\pi\)
\(31\)	−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	−0.0836807	−	0.996493i	\(-0.526668\pi\)
							0.996493	+	0.0836807i	\(0.0266676\pi\)
\(41\)	1.29036	+	0.744991i	0.201521	+	0.116348i	0.597365	−	0.801970i	\(-0.296215\pi\)
							−0.395844	+	0.918318i	\(0.629548\pi\)
\(43\)	−1.10121	+	4.10976i	−0.167933	+	0.626733i	0.829715	+	0.558187i	\(0.188503\pi\)
							−0.997648	+	0.0685463i	\(0.978164\pi\)
\(47\)	−1.02368	+	3.82042i	−0.149319	+	0.557266i	0.850206	+	0.526450i	\(0.176477\pi\)
							−0.999525	+	0.0308158i	\(0.990189\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.2.l.a.2.2	✓	16	45.2	even	12	inner
45.2.l.a.23.2	yes	16	1.1	even	1	trivial
45.2.l.a.32.2	yes	16	5.2	odd	4	inner
45.2.l.a.38.2	yes	16	9.2	odd	6	inner
135.2.m.a.8.3		16	9.7	even	3
135.2.m.a.17.3		16	15.2	even	4
135.2.m.a.62.3		16	45.7	odd	12
135.2.m.a.98.3		16	3.2	odd	2
225.2.p.b.32.3		16	5.3	odd	4
225.2.p.b.68.3		16	5.4	even	2
225.2.p.b.182.3		16	45.38	even	12
225.2.p.b.218.3		16	45.29	odd	6
405.2.f.a.242.2		16	45.32	even	12
405.2.f.a.242.7		16	45.22	odd	12
405.2.f.a.323.2		16	9.4	even	3
405.2.f.a.323.7		16	9.5	odd	6
675.2.q.a.143.2		16	45.34	even	6
675.2.q.a.332.2		16	45.43	odd	12
675.2.q.a.368.2		16	15.14	odd	2
675.2.q.a.557.2		16	15.8	even	4
720.2.cu.c.113.1		16	4.3	odd	2
720.2.cu.c.257.3		16	20.7	even	4
720.2.cu.c.353.3		16	36.11	even	6
720.2.cu.c.497.1		16	180.47	odd	12