Embedded newform 45.4.j.a.4.14

• Label: 45.4.j.a.4.14
• Level: $45$
• Weight: $4$
• Character: 45.4
• 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			4.14
Character	\(\chi\)	\(=\)	45.4
Dual form			45.4.j.a.34.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.53014 - 2.03813i) q^{2} +(2.78227 - 4.38850i) q^{3} +(4.30793 - 7.46156i) q^{4} +(-1.64447 + 11.0587i) q^{5} +(0.877481 - 21.1627i) q^{6} +(-11.5553 + 6.67148i) q^{7} -2.51043i q^{8} +(-11.5179 - 24.4200i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 54 q^{4} + 3 q^{5} - 12 q^{6} - 18 q^{9}+O(q^{10}) \)

Character values

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

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

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	\(-0.410539\pi\)
							0.970729	+	0.240178i	\(0.0772057\pi\)
\(3\)	2.78227	−	4.38850i	0.535448	−	0.844568i
							\(5\)	−1.64447	+	11.0587i	−0.147086	+	0.989124i
\(7\)	−11.5553	+	6.67148i	−0.623930	+	0.360226i								−0.778397	−	0.627772i	\(-0.783967\pi\)
							0.154468	+	0.987998i	\(0.450634\pi\)
\(11\)	−5.78085	−	10.0127i	−0.158454	−	0.274450i	0.775858	−	0.630908i	\(-0.217318\pi\)
							−0.934311	+	0.356458i	\(0.883984\pi\)
\(13\)	34.7099	+	20.0398i	0.740523	+	0.427541i	0.822260	−	0.569113i	\(-0.192713\pi\)
							−0.0817363	+	0.996654i	\(0.526047\pi\)
\(17\)		−	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	\(-0.110741\pi\)
							0.174792	+	0.984605i	\(0.444075\pi\)
\(29\)	−87.0236	−	150.729i	−0.557237	−	0.965163i	−0.997726	−	0.0674045i	\(-0.978528\pi\)
							0.440489	−	0.897758i	\(-0.354805\pi\)
\(31\)	−124.434	+	215.526i	−0.720937	+	1.24870i	0.239688	+	0.970850i	\(0.422955\pi\)
							−0.960625	+	0.277849i	\(0.910378\pi\)
\(37\)		−	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	−0.0196924	−	0.999806i	\(-0.506269\pi\)
							0.875704	−	0.482849i	\(-0.160398\pi\)
\(43\)	242.357	−	139.925i	0.859514	−	0.496241i	−0.00433558	−	0.999991i	\(-0.501380\pi\)
							0.863849	+	0.503750i	\(0.168047\pi\)
\(47\)	−29.3378	+	16.9382i	−0.0910501	+	0.0525678i	−0.544834	−	0.838544i	\(-0.683407\pi\)
							0.453784	+	0.891112i	\(0.350074\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.j.a.4.3	✓	32	5.4	even	2	inner
45.4.j.a.4.14	yes	32	1.1	even	1	trivial
45.4.j.a.34.3	yes	32	9.7	even	3	inner
45.4.j.a.34.14	yes	32	45.34	even	6	inner
135.4.j.a.19.3		32	45.29	odd	6
135.4.j.a.19.14		32	9.2	odd	6
135.4.j.a.64.3		32	3.2	odd	2
135.4.j.a.64.14		32	15.14	odd	2
225.4.e.g.76.3		32	5.3	odd	4
225.4.e.g.76.14		32	5.2	odd	4
225.4.e.g.151.3		32	45.43	odd	12
225.4.e.g.151.14		32	45.7	odd	12
405.4.b.e.244.3		16	45.4	even	6
405.4.b.e.244.14		16	9.4	even	3
405.4.b.f.244.3		16	9.5	odd	6
405.4.b.f.244.14		16	45.14	odd	6
2025.4.a.bk.1.3		16	45.22	odd	12
2025.4.a.bk.1.14		16	45.13	odd	12
2025.4.a.bl.1.3		16	45.23	even	12
2025.4.a.bl.1.14		16	45.32	even	12