Embedded newform 45.4.j.a.34.2

• Label: 45.4.j.a.34.2
• 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			34.2
Character	\(\chi\)	\(=\)	45.34
Dual form			45.4.j.a.4.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.35066 - 2.51186i) q^{2} +(-1.99648 + 4.79730i) q^{3} +(8.61884 + 14.9283i) q^{4} +(5.42265 - 9.77726i) q^{5} +(20.7361 - 15.8565i) q^{6} +(-4.66092 - 2.69099i) q^{7} -46.4074i q^{8} +(-19.0281 - 19.1554i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 54 q^{4} + 3 q^{5} - 12 q^{6} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.35066	−	2.51186i	−1.53819	−	0.888075i	−0.998945	−	0.0459269i	\(-0.985376\pi\)
							−0.539246	−	0.842148i	\(-0.681291\pi\)
\(3\)	−1.99648	+	4.79730i	−0.384223	+	0.923240i
							\(5\)	5.42265	−	9.77726i	0.485017	−	0.874505i
\(7\)	−4.66092	−	2.69099i	−0.251666	−	0.145300i								0.368861	−	0.929485i	\(-0.379748\pi\)
							−0.620527	+	0.784185i	\(0.713081\pi\)
\(11\)	19.5509	−	33.8631i	0.535892	−	0.928192i	−0.463227	−	0.886239i	\(-0.653309\pi\)
							0.999120	−	0.0419529i	\(-0.0133580\pi\)
\(13\)	75.0212	−	43.3135i	1.60055	−	0.924078i	0.609173	−	0.793037i	\(-0.291502\pi\)
							0.991377	−	0.131040i	\(-0.0418318\pi\)
\(17\)		−	15.4194i		−	0.219985i	−0.993932	−	0.109993i	\(-0.964917\pi\)
							0.993932	−	0.109993i	\(-0.0350827\pi\)
\(19\)	26.8412			0.324094			0.162047	−	0.986783i	\(-0.448190\pi\)
							0.162047	+	0.986783i	\(0.448190\pi\)
\(23\)	−96.2486	+	55.5692i	−0.872575	+	0.503781i	−0.868203	−	0.496209i	\(-0.834725\pi\)
							−0.00437188	+	0.999990i	\(0.501392\pi\)
\(29\)	24.6174	−	42.6385i	0.157632	−	0.273027i	−0.776382	−	0.630262i	\(-0.782947\pi\)
							0.934014	+	0.357236i	\(0.116281\pi\)
\(31\)	89.9383	+	155.778i	0.521077	+	0.902532i	0.999700	+	0.0245112i	\(0.00780295\pi\)
							−0.478622	+	0.878021i	\(0.658864\pi\)
\(37\)		−	293.496i		−	1.30407i	−0.758191	−	0.652033i	\(-0.773916\pi\)
							0.758191	−	0.652033i	\(-0.226084\pi\)
\(41\)	−13.8016	−	23.9051i	−0.0525719	−	0.0910571i	0.838542	−	0.544837i	\(-0.183409\pi\)
							−0.891114	+	0.453780i	\(0.850075\pi\)
\(43\)	−52.3971	−	30.2515i	−0.185825	−	0.107286i	0.404201	−	0.914670i	\(-0.367549\pi\)
							−0.590027	+	0.807384i	\(0.700883\pi\)
\(47\)	83.3420	+	48.1175i	0.258653	+	0.149333i	0.623720	−	0.781648i	\(-0.285621\pi\)
							−0.365067	+	0.930981i	\(0.618954\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.4.j.a.34.2	yes	32
3.2	odd	2		135.4.j.a.19.15		32
5.2	odd	4		225.4.e.g.151.15		32
5.3	odd	4		225.4.e.g.151.2		32
5.4	even	2	inner	45.4.j.a.34.15	yes	32
9.2	odd	6		405.4.b.f.244.2		16
9.4	even	3	inner	45.4.j.a.4.15	yes	32
9.5	odd	6		135.4.j.a.64.2		32
9.7	even	3		405.4.b.e.244.15		16
15.14	odd	2		135.4.j.a.19.2		32
45.2	even	12		2025.4.a.bl.1.15		16
45.4	even	6	inner	45.4.j.a.4.2	✓	32
45.7	odd	12		2025.4.a.bk.1.2		16
45.13	odd	12		225.4.e.g.76.2		32
45.14	odd	6		135.4.j.a.64.15		32
45.22	odd	12		225.4.e.g.76.15		32
45.29	odd	6		405.4.b.f.244.15		16
45.34	even	6		405.4.b.e.244.2		16
45.38	even	12		2025.4.a.bl.1.2		16
45.43	odd	12		2025.4.a.bk.1.15		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.j.a.4.2	✓	32	45.4	even	6	inner
45.4.j.a.4.15	yes	32	9.4	even	3	inner
45.4.j.a.34.2	yes	32	1.1	even	1	trivial
45.4.j.a.34.15	yes	32	5.4	even	2	inner
135.4.j.a.19.2		32	15.14	odd	2
135.4.j.a.19.15		32	3.2	odd	2
135.4.j.a.64.2		32	9.5	odd	6
135.4.j.a.64.15		32	45.14	odd	6
225.4.e.g.76.2		32	45.13	odd	12
225.4.e.g.76.15		32	45.22	odd	12
225.4.e.g.151.2		32	5.3	odd	4
225.4.e.g.151.15		32	5.2	odd	4
405.4.b.e.244.2		16	45.34	even	6
405.4.b.e.244.15		16	9.7	even	3
405.4.b.f.244.2		16	9.2	odd	6
405.4.b.f.244.15		16	45.29	odd	6
2025.4.a.bk.1.2		16	45.7	odd	12
2025.4.a.bk.1.15		16	45.43	odd	12
2025.4.a.bl.1.2		16	45.38	even	12
2025.4.a.bl.1.15		16	45.2	even	12