Embedded newform 45.4.j.a.4.9

• Label: 45.4.j.a.4.9
• 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			4.9
Character	\(\chi\)	\(=\)	45.4
Dual form			45.4.j.a.34.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.410487 - 0.236995i) q^{2} +(-4.58061 - 2.45316i) q^{3} +(-3.88767 + 6.73364i) q^{4} +(-2.29829 + 10.9416i) q^{5} +(-2.46167 + 0.0785933i) q^{6} +(-7.10710 + 4.10329i) q^{7} +7.47735i q^{8} +(14.9641 + 22.4739i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 54 q^{4} + 3 q^{5} - 12 q^{6} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.410487	−	0.236995i	0.145129	−	0.0837904i	−0.425677	−	0.904875i	\(-0.639964\pi\)
							0.570806	+	0.821085i	\(0.306631\pi\)
\(3\)	−4.58061	−	2.45316i	−0.881540	−	0.472110i
							\(5\)	−2.29829	+	10.9416i	−0.205565	+	0.978643i
\(7\)	−7.10710	+	4.10329i	−0.383747	+	0.221557i								−0.679447	−	0.733724i	\(-0.737780\pi\)
							0.295700	+	0.955281i	\(0.404447\pi\)
\(11\)	−2.63291	−	4.56034i	−0.0721685	−	0.125000i	0.827683	−	0.561196i	\(-0.189659\pi\)
							−0.899851	+	0.436196i	\(0.856325\pi\)
\(13\)	−66.7136	−	38.5171i	−1.42331	−	0.821748i	−0.426730	−	0.904379i	\(-0.640334\pi\)
							−0.996580	+	0.0826310i	\(0.973668\pi\)
\(17\)			88.9397i			1.26888i	0.772970	+	0.634442i	\(0.218770\pi\)
							−0.772970	+	0.634442i	\(0.781230\pi\)
\(19\)	91.7358			1.10766			0.553832	−	0.832628i	\(-0.313165\pi\)
							0.553832	+	0.832628i	\(0.313165\pi\)
\(23\)	133.750	+	77.2204i	1.21255	+	0.700068i	0.963315	−	0.268374i	\(-0.0864864\pi\)
							0.249238	+	0.968442i	\(0.419820\pi\)
\(29\)	84.1517	+	145.755i	0.538847	+	0.933311i	0.998966	+	0.0454539i	\(0.0144734\pi\)
							−0.460119	+	0.887857i	\(0.652193\pi\)
\(31\)	−36.3943	+	63.0367i	−0.210858	+	0.365217i	−0.951983	−	0.306150i	\(-0.900959\pi\)
							0.741125	+	0.671367i	\(0.234292\pi\)
\(37\)			154.836i			0.687971i	0.938975	+	0.343986i	\(0.111777\pi\)
							−0.938975	+	0.343986i	\(0.888223\pi\)
\(41\)	−3.97579	+	6.88627i	−0.0151442	+	0.0262306i	−0.873498	−	0.486827i	\(-0.838154\pi\)
							0.858354	+	0.513058i	\(0.171487\pi\)
\(43\)	−20.0930	+	11.6007i	−0.0712594	+	0.0411416i	−0.535206	−	0.844721i	\(-0.679766\pi\)
							0.463947	+	0.885863i	\(0.346433\pi\)
\(47\)	−261.018	+	150.699i	−0.810073	+	0.467696i	−0.846981	−	0.531623i	\(-0.821582\pi\)
							0.0369081	+	0.999319i	\(0.488249\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.4.j.a.4.9	yes	32
3.2	odd	2		135.4.j.a.64.8		32
5.2	odd	4		225.4.e.g.76.9		32
5.3	odd	4		225.4.e.g.76.8		32
5.4	even	2	inner	45.4.j.a.4.8	✓	32
9.2	odd	6		135.4.j.a.19.9		32
9.4	even	3		405.4.b.e.244.9		16
9.5	odd	6		405.4.b.f.244.8		16
9.7	even	3	inner	45.4.j.a.34.8	yes	32
15.14	odd	2		135.4.j.a.64.9		32
45.4	even	6		405.4.b.e.244.8		16
45.7	odd	12		225.4.e.g.151.9		32
45.13	odd	12		2025.4.a.bk.1.9		16
45.14	odd	6		405.4.b.f.244.9		16
45.22	odd	12		2025.4.a.bk.1.8		16
45.23	even	12		2025.4.a.bl.1.8		16
45.29	odd	6		135.4.j.a.19.8		32
45.32	even	12		2025.4.a.bl.1.9		16
45.34	even	6	inner	45.4.j.a.34.9	yes	32
45.43	odd	12		225.4.e.g.151.8		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.j.a.4.8	✓	32	5.4	even	2	inner
45.4.j.a.4.9	yes	32	1.1	even	1	trivial
45.4.j.a.34.8	yes	32	9.7	even	3	inner
45.4.j.a.34.9	yes	32	45.34	even	6	inner
135.4.j.a.19.8		32	45.29	odd	6
135.4.j.a.19.9		32	9.2	odd	6
135.4.j.a.64.8		32	3.2	odd	2
135.4.j.a.64.9		32	15.14	odd	2
225.4.e.g.76.8		32	5.3	odd	4
225.4.e.g.76.9		32	5.2	odd	4
225.4.e.g.151.8		32	45.43	odd	12
225.4.e.g.151.9		32	45.7	odd	12
405.4.b.e.244.8		16	45.4	even	6
405.4.b.e.244.9		16	9.4	even	3
405.4.b.f.244.8		16	9.5	odd	6
405.4.b.f.244.9		16	45.14	odd	6
2025.4.a.bk.1.8		16	45.22	odd	12
2025.4.a.bk.1.9		16	45.13	odd	12
2025.4.a.bl.1.8		16	45.23	even	12
2025.4.a.bl.1.9		16	45.32	even	12