Embedded newform 45.4.j.a.34.3

• Label: 45.4.j.a.34.3
• Level: $45$
• Weight: $4$
• Character: 45.34
• Analytic conductor: $2.655$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	45.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.65508595026\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			34.3
Character	\(\chi\)	\(=\)	45.34
Dual form			45.4.j.a.4.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.53014 - 2.03813i) q^{2} +(-2.78227 - 4.38850i) q^{3} +(4.30793 + 7.46156i) q^{4} +(-8.75492 - 6.95352i) q^{5} +(0.877481 + 21.1627i) q^{6} +(11.5553 + 6.67148i) q^{7} -2.51043i q^{8} +(-11.5179 + 24.4200i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 54 q^{4} + 3 q^{5} - 12 q^{6} - 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(11\)	\(37\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.53014	−	2.03813i	−1.24809	−	0.720587i	−0.277365	−	0.960765i	\(-0.589461\pi\)
							−0.970729	+	0.240178i	\(0.922794\pi\)
\(3\)	−2.78227	−	4.38850i	−0.535448	−	0.844568i
							\(5\)	−8.75492	−	6.95352i	−0.783064	−	0.621942i
\(7\)	11.5553	+	6.67148i	0.623930	+	0.360226i								0.778397	−	0.627772i	\(-0.216033\pi\)
							−0.154468	+	0.987998i	\(0.549366\pi\)
\(11\)	−5.78085	+	10.0127i	−0.158454	+	0.274450i	−0.934311	−	0.356458i	\(-0.883984\pi\)
							0.775858	+	0.630908i	\(0.217318\pi\)
\(13\)	−34.7099	+	20.0398i	−0.740523	+	0.427541i	−0.822260	−	0.569113i	\(-0.807287\pi\)
							0.0817363	+	0.996654i	\(0.473953\pi\)
\(17\)		−	93.3392i		−	1.33165i	−0.746107	−	0.665826i	\(-0.768079\pi\)
							0.746107	−	0.665826i	\(-0.231921\pi\)
\(19\)	−75.1002			−0.906799			−0.453399	−	0.891307i	\(-0.649789\pi\)
							−0.453399	+	0.891307i	\(0.649789\pi\)
\(23\)	−122.976	+	71.0003i	−1.11488	+	0.643677i	−0.940089	−	0.340928i	\(-0.889259\pi\)
							−0.174792	+	0.984605i	\(0.555925\pi\)
\(29\)	−87.0236	+	150.729i	−0.557237	+	0.965163i	0.440489	+	0.897758i	\(0.354805\pi\)
							−0.997726	+	0.0674045i	\(0.978528\pi\)
\(31\)	−124.434	−	215.526i	−0.720937	−	1.24870i	−0.960625	−	0.277849i	\(-0.910378\pi\)
							0.239688	−	0.970850i	\(-0.422955\pi\)
\(37\)		−	82.9086i		−	0.368381i	−0.982891	−	0.184190i	\(-0.941034\pi\)
							0.982891	−	0.184190i	\(-0.0589663\pi\)
\(41\)	224.727	+	389.239i	0.856011	+	1.48266i	0.875704	+	0.482849i	\(0.160398\pi\)
							−0.0196924	+	0.999806i	\(0.506269\pi\)
\(43\)	−242.357	−	139.925i	−0.859514	−	0.496241i	0.00433558	−	0.999991i	\(-0.498620\pi\)
							−0.863849	+	0.503750i	\(0.831953\pi\)
\(47\)	29.3378	+	16.9382i	0.0910501	+	0.0525678i	0.544834	−	0.838544i	\(-0.316593\pi\)
							−0.453784	+	0.891112i	\(0.649926\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.4.j.a.34.3	yes	32
3.2	odd	2		135.4.j.a.19.14		32
5.2	odd	4		225.4.e.g.151.14		32
5.3	odd	4		225.4.e.g.151.3		32
5.4	even	2	inner	45.4.j.a.34.14	yes	32
9.2	odd	6		405.4.b.f.244.3		16
9.4	even	3	inner	45.4.j.a.4.14	yes	32
9.5	odd	6		135.4.j.a.64.3		32
9.7	even	3		405.4.b.e.244.14		16
15.14	odd	2		135.4.j.a.19.3		32
45.2	even	12		2025.4.a.bl.1.14		16
45.4	even	6	inner	45.4.j.a.4.3	✓	32
45.7	odd	12		2025.4.a.bk.1.3		16
45.13	odd	12		225.4.e.g.76.3		32
45.14	odd	6		135.4.j.a.64.14		32
45.22	odd	12		225.4.e.g.76.14		32
45.29	odd	6		405.4.b.f.244.14		16
45.34	even	6		405.4.b.e.244.3		16
45.38	even	12		2025.4.a.bl.1.3		16
45.43	odd	12		2025.4.a.bk.1.14		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.j.a.4.3	✓	32	45.4	even	6	inner
45.4.j.a.4.14	yes	32	9.4	even	3	inner
45.4.j.a.34.3	yes	32	1.1	even	1	trivial
45.4.j.a.34.14	yes	32	5.4	even	2	inner
135.4.j.a.19.3		32	15.14	odd	2
135.4.j.a.19.14		32	3.2	odd	2
135.4.j.a.64.3		32	9.5	odd	6
135.4.j.a.64.14		32	45.14	odd	6
225.4.e.g.76.3		32	45.13	odd	12
225.4.e.g.76.14		32	45.22	odd	12
225.4.e.g.151.3		32	5.3	odd	4
225.4.e.g.151.14		32	5.2	odd	4
405.4.b.e.244.3		16	45.34	even	6
405.4.b.e.244.14		16	9.7	even	3
405.4.b.f.244.3		16	9.2	odd	6
405.4.b.f.244.14		16	45.29	odd	6
2025.4.a.bk.1.3		16	45.7	odd	12
2025.4.a.bk.1.14		16	45.43	odd	12
2025.4.a.bl.1.3		16	45.38	even	12
2025.4.a.bl.1.14		16	45.2	even	12