Embedded newform 45.4.j.a.4.16

• Label: 45.4.j.a.4.16
• 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			4.16
Character	\(\chi\)	\(=\)	45.4
Dual form			45.4.j.a.34.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.37720 - 2.52718i) q^{2} +(-5.16592 - 0.559681i) q^{3} +(8.77324 - 15.1957i) q^{4} +(10.8464 - 2.71197i) q^{5} +(-24.0267 + 10.6054i) q^{6} +(-18.1967 + 10.5059i) q^{7} -48.2513i q^{8} +(26.3735 + 5.78253i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 54 q^{4} + 3 q^{5} - 12 q^{6} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.37720	−	2.52718i	1.54757	−	0.893492i	0.549247	−	0.835660i	\(-0.314915\pi\)
							0.998326	−	0.0578315i	\(-0.0184186\pi\)
\(3\)	−5.16592	−	0.559681i	−0.994182	−	0.107711i
							\(5\)	10.8464	−	2.71197i	0.970135	−	0.242566i
\(7\)	−18.1967	+	10.5059i	−0.982529	+	0.567263i								−0.903033	−	0.429572i	\(-0.858664\pi\)
							−0.0794959	+	0.996835i	\(0.525331\pi\)
\(11\)	14.2815	+	24.7362i	0.391457	+	0.678023i	0.992642	−	0.121087i	\(-0.0386379\pi\)
							−0.601185	+	0.799110i	\(0.705305\pi\)
\(13\)	8.72122	+	5.03520i	0.186064	+	0.107424i	0.590139	−	0.807302i	\(-0.299073\pi\)
							−0.404075	+	0.914726i	\(0.632406\pi\)
\(17\)			82.7541i			1.18064i	0.807171	+	0.590318i	\(0.200998\pi\)
							−0.807171	+	0.590318i	\(0.799002\pi\)
\(19\)	−1.91981			−0.0231807			−0.0115904	−	0.999933i	\(-0.503689\pi\)
							−0.0115904	+	0.999933i	\(0.503689\pi\)
\(23\)	−147.712	−	85.2818i	−1.33914	−	0.773151i	−0.352458	−	0.935828i	\(-0.614654\pi\)
							−0.986679	+	0.162677i	\(0.947987\pi\)
\(29\)	−128.214	−	222.074i	−0.820993	−	1.42200i	−0.904944	−	0.425531i	\(-0.860087\pi\)
							0.0839514	−	0.996470i	\(-0.473246\pi\)
\(31\)	−24.0262	+	41.6147i	−0.139201	+	0.241104i	−0.927195	−	0.374580i	\(-0.877787\pi\)
							0.787993	+	0.615684i	\(0.211120\pi\)
\(37\)		−	161.834i		−	0.719065i	−0.933132	−	0.359533i	\(-0.882936\pi\)
							0.933132	−	0.359533i	\(-0.117064\pi\)
\(41\)	−139.674	+	241.922i	−0.532034	+	0.921510i	0.467266	+	0.884117i	\(0.345239\pi\)
							−0.999301	+	0.0373937i	\(0.988094\pi\)
\(43\)	233.024	−	134.536i	0.826414	−	0.477130i	−0.0262093	−	0.999656i	\(-0.508344\pi\)
							0.852623	+	0.522526i	\(0.175010\pi\)
\(47\)	8.30380	−	4.79420i	0.0257709	−	0.0148789i	−0.487059	−	0.873369i	\(-0.661930\pi\)
							0.512830	+	0.858490i	\(0.328597\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.4.j.a.4.16	yes	32
3.2	odd	2		135.4.j.a.64.1		32
5.2	odd	4		225.4.e.g.76.16		32
5.3	odd	4		225.4.e.g.76.1		32
5.4	even	2	inner	45.4.j.a.4.1	✓	32
9.2	odd	6		135.4.j.a.19.16		32
9.4	even	3		405.4.b.e.244.16		16
9.5	odd	6		405.4.b.f.244.1		16
9.7	even	3	inner	45.4.j.a.34.1	yes	32
15.14	odd	2		135.4.j.a.64.16		32
45.4	even	6		405.4.b.e.244.1		16
45.7	odd	12		225.4.e.g.151.16		32
45.13	odd	12		2025.4.a.bk.1.16		16
45.14	odd	6		405.4.b.f.244.16		16
45.22	odd	12		2025.4.a.bk.1.1		16
45.23	even	12		2025.4.a.bl.1.1		16
45.29	odd	6		135.4.j.a.19.1		32
45.32	even	12		2025.4.a.bl.1.16		16
45.34	even	6	inner	45.4.j.a.34.16	yes	32
45.43	odd	12		225.4.e.g.151.1		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.j.a.4.1	✓	32	5.4	even	2	inner
45.4.j.a.4.16	yes	32	1.1	even	1	trivial
45.4.j.a.34.1	yes	32	9.7	even	3	inner
45.4.j.a.34.16	yes	32	45.34	even	6	inner
135.4.j.a.19.1		32	45.29	odd	6
135.4.j.a.19.16		32	9.2	odd	6
135.4.j.a.64.1		32	3.2	odd	2
135.4.j.a.64.16		32	15.14	odd	2
225.4.e.g.76.1		32	5.3	odd	4
225.4.e.g.76.16		32	5.2	odd	4
225.4.e.g.151.1		32	45.43	odd	12
225.4.e.g.151.16		32	45.7	odd	12
405.4.b.e.244.1		16	45.4	even	6
405.4.b.e.244.16		16	9.4	even	3
405.4.b.f.244.1		16	9.5	odd	6
405.4.b.f.244.16		16	45.14	odd	6
2025.4.a.bk.1.1		16	45.22	odd	12
2025.4.a.bk.1.16		16	45.13	odd	12
2025.4.a.bl.1.1		16	45.23	even	12
2025.4.a.bl.1.16		16	45.32	even	12