Embedded newform 45.4.j.a.4.4

• Label: 45.4.j.a.4.4
• 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			4.4
Character	\(\chi\)	\(=\)	45.4
Dual form			45.4.j.a.34.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.67377 + 1.54370i) q^{2} +(2.05649 + 4.77188i) q^{3} +(0.766022 - 1.32679i) q^{4} +(11.1766 - 0.287878i) q^{5} +(-12.8649 - 9.58430i) q^{6} +(-27.1243 + 15.6602i) q^{7} -19.9692i q^{8} +(-18.5417 + 19.6266i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 54 q^{4} + 3 q^{5} - 12 q^{6} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.67377	+	1.54370i	−0.945320	+	0.545781i	−0.891624	−	0.452777i	\(-0.850433\pi\)
							−0.0536957	+	0.998557i	\(0.517100\pi\)
\(3\)	2.05649	+	4.77188i	0.395771	+	0.918349i
							\(5\)	11.1766	−	0.287878i	0.999668	−	0.0257486i
\(7\)	−27.1243	+	15.6602i	−1.46457	+	0.845571i								−0.999217	−	0.0395535i	\(-0.987406\pi\)
							−0.465354	+	0.885124i	\(0.654073\pi\)
\(11\)	9.59684	+	16.6222i	0.263051	+	0.455617i	0.967051	−	0.254583i	\(-0.0819381\pi\)
							−0.704000	+	0.710199i	\(0.748605\pi\)
\(13\)	−18.1787	−	10.4955i	−0.387835	−	0.223917i	0.293387	−	0.955994i	\(-0.405218\pi\)
							−0.681222	+	0.732077i	\(0.738551\pi\)
\(17\)		−	6.19137i		−	0.0883311i	−0.999024	−	0.0441656i	\(-0.985937\pi\)
							0.999024	−	0.0441656i	\(-0.0140629\pi\)
\(19\)	96.6654			1.16719			0.583594	−	0.812046i	\(-0.301646\pi\)
							0.583594	+	0.812046i	\(0.301646\pi\)
\(23\)	141.150	+	81.4928i	1.27964	+	0.738801i	0.976782	−	0.214234i	\(-0.0687255\pi\)
							0.302859	+	0.953035i	\(0.402059\pi\)
\(29\)	3.82461	+	6.62442i	0.0244901	+	0.0424181i	0.878011	−	0.478641i	\(-0.158870\pi\)
							−0.853521	+	0.521059i	\(0.825537\pi\)
\(31\)	−112.512	+	194.877i	−0.651865	+	1.12906i	0.330805	+	0.943699i	\(0.392680\pi\)
							−0.982670	+	0.185364i	\(0.940654\pi\)
\(37\)			155.911i			0.692748i	0.938097	+	0.346374i	\(0.112587\pi\)
							−0.938097	+	0.346374i	\(0.887413\pi\)
\(41\)	157.611	−	272.989i	0.600357	−	1.03985i	−0.392410	−	0.919790i	\(-0.628359\pi\)
							0.992767	−	0.120058i	\(-0.0383081\pi\)
\(43\)	166.934	−	96.3793i	0.592027	−	0.341807i	−0.173871	−	0.984768i	\(-0.555628\pi\)
							0.765899	+	0.642961i	\(0.222294\pi\)
\(47\)	275.666	−	159.156i	0.855533	−	0.493942i	−0.00698057	−	0.999976i	\(-0.502222\pi\)
							0.862514	+	0.506033i	\(0.168889\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.4.j.a.4.4	✓	32
3.2	odd	2		135.4.j.a.64.13		32
5.2	odd	4		225.4.e.g.76.4		32
5.3	odd	4		225.4.e.g.76.13		32
5.4	even	2	inner	45.4.j.a.4.13	yes	32
9.2	odd	6		135.4.j.a.19.4		32
9.4	even	3		405.4.b.e.244.4		16
9.5	odd	6		405.4.b.f.244.13		16
9.7	even	3	inner	45.4.j.a.34.13	yes	32
15.14	odd	2		135.4.j.a.64.4		32
45.4	even	6		405.4.b.e.244.13		16
45.7	odd	12		225.4.e.g.151.4		32
45.13	odd	12		2025.4.a.bk.1.4		16
45.14	odd	6		405.4.b.f.244.4		16
45.22	odd	12		2025.4.a.bk.1.13		16
45.23	even	12		2025.4.a.bl.1.13		16
45.29	odd	6		135.4.j.a.19.13		32
45.32	even	12		2025.4.a.bl.1.4		16
45.34	even	6	inner	45.4.j.a.34.4	yes	32
45.43	odd	12		225.4.e.g.151.13		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.j.a.4.4	✓	32	1.1	even	1	trivial
45.4.j.a.4.13	yes	32	5.4	even	2	inner
45.4.j.a.34.4	yes	32	45.34	even	6	inner
45.4.j.a.34.13	yes	32	9.7	even	3	inner
135.4.j.a.19.4		32	9.2	odd	6
135.4.j.a.19.13		32	45.29	odd	6
135.4.j.a.64.4		32	15.14	odd	2
135.4.j.a.64.13		32	3.2	odd	2
225.4.e.g.76.4		32	5.2	odd	4
225.4.e.g.76.13		32	5.3	odd	4
225.4.e.g.151.4		32	45.7	odd	12
225.4.e.g.151.13		32	45.43	odd	12
405.4.b.e.244.4		16	9.4	even	3
405.4.b.e.244.13		16	45.4	even	6
405.4.b.f.244.4		16	45.14	odd	6
405.4.b.f.244.13		16	9.5	odd	6
2025.4.a.bk.1.4		16	45.13	odd	12
2025.4.a.bk.1.13		16	45.22	odd	12
2025.4.a.bl.1.4		16	45.32	even	12
2025.4.a.bl.1.13		16	45.23	even	12