Embedded newform 45.4.j.a.4.6

• Label: 45.4.j.a.4.6
• 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			4.6
Character	\(\chi\)	\(=\)	45.4
Dual form			45.4.j.a.34.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.13376 + 1.23192i) q^{2} +(1.50557 - 4.97325i) q^{3} +(-0.964722 + 1.67095i) q^{4} +(-9.64354 - 5.65704i) q^{5} +(2.91415 + 12.4665i) q^{6} +(-16.6624 + 9.62007i) q^{7} -24.4647i q^{8} +(-22.4665 - 14.9752i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 54 q^{4} + 3 q^{5} - 12 q^{6} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.13376	+	1.23192i	−0.754397	+	0.435551i	−0.827280	−	0.561789i	\(-0.810113\pi\)
							0.0728836	+	0.997340i	\(0.476780\pi\)
\(3\)	1.50557	−	4.97325i	0.289747	−	0.957103i
							\(5\)	−9.64354	−	5.65704i	−0.862545	−	0.505981i
\(7\)	−16.6624	+	9.62007i	−0.899688	+	0.519435i								−0.877099	−	0.480310i	\(-0.840524\pi\)
							−0.0225888	+	0.999745i	\(0.507191\pi\)
\(11\)	−19.9274	−	34.5153i	−0.546213	−	0.946069i	−0.998529	−	0.0542113i	\(-0.982736\pi\)
							0.452316	−	0.891858i	\(-0.350598\pi\)
\(13\)	0.873903	+	0.504548i	0.0186444	+	0.0107643i	0.509293	−	0.860593i	\(-0.329907\pi\)
							−0.490649	+	0.871357i	\(0.663240\pi\)
\(17\)			52.6170i			0.750676i	0.926888	+	0.375338i	\(0.122473\pi\)
							−0.926888	+	0.375338i	\(0.877527\pi\)
\(19\)	49.5371			0.598136			0.299068	−	0.954232i	\(-0.403324\pi\)
							0.299068	+	0.954232i	\(0.403324\pi\)
\(23\)	23.7307	+	13.7009i	0.215139	+	0.124210i	0.603697	−	0.797214i	\(-0.293694\pi\)
							−0.388559	+	0.921424i	\(0.627027\pi\)
\(29\)	−127.477	−	220.796i	−0.816269	−	1.41382i	−0.908413	−	0.418074i	\(-0.862705\pi\)
							0.0921441	−	0.995746i	\(-0.470628\pi\)
\(31\)	84.3282	−	146.061i	0.488574	−	0.846234i	−0.511340	−	0.859379i	\(-0.670851\pi\)
							0.999914	+	0.0131441i	\(0.00418403\pi\)
\(37\)		−	419.938i		−	1.86588i	−0.360038	−	0.932938i	\(-0.617236\pi\)
							0.360038	−	0.932938i	\(-0.382764\pi\)
\(41\)	−199.285	+	345.172i	−0.759100	+	1.31480i	0.184210	+	0.982887i	\(0.441027\pi\)
							−0.943310	+	0.331913i	\(0.892306\pi\)
\(43\)	−310.754	+	179.414i	−1.10208	+	0.636288i	−0.936767	−	0.349953i	\(-0.886198\pi\)
							−0.165316	+	0.986241i	\(0.552864\pi\)
\(47\)	122.371	−	70.6510i	0.379780	−	0.219266i	−0.297943	−	0.954584i	\(-0.596300\pi\)
							0.677723	+	0.735318i	\(0.262967\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.4.j.a.4.6	✓	32
3.2	odd	2		135.4.j.a.64.11		32
5.2	odd	4		225.4.e.g.76.6		32
5.3	odd	4		225.4.e.g.76.11		32
5.4	even	2	inner	45.4.j.a.4.11	yes	32
9.2	odd	6		135.4.j.a.19.6		32
9.4	even	3		405.4.b.e.244.6		16
9.5	odd	6		405.4.b.f.244.11		16
9.7	even	3	inner	45.4.j.a.34.11	yes	32
15.14	odd	2		135.4.j.a.64.6		32
45.4	even	6		405.4.b.e.244.11		16
45.7	odd	12		225.4.e.g.151.6		32
45.13	odd	12		2025.4.a.bk.1.6		16
45.14	odd	6		405.4.b.f.244.6		16
45.22	odd	12		2025.4.a.bk.1.11		16
45.23	even	12		2025.4.a.bl.1.11		16
45.29	odd	6		135.4.j.a.19.11		32
45.32	even	12		2025.4.a.bl.1.6		16
45.34	even	6	inner	45.4.j.a.34.6	yes	32
45.43	odd	12		225.4.e.g.151.11		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.j.a.4.6	✓	32	1.1	even	1	trivial
45.4.j.a.4.11	yes	32	5.4	even	2	inner
45.4.j.a.34.6	yes	32	45.34	even	6	inner
45.4.j.a.34.11	yes	32	9.7	even	3	inner
135.4.j.a.19.6		32	9.2	odd	6
135.4.j.a.19.11		32	45.29	odd	6
135.4.j.a.64.6		32	15.14	odd	2
135.4.j.a.64.11		32	3.2	odd	2
225.4.e.g.76.6		32	5.2	odd	4
225.4.e.g.76.11		32	5.3	odd	4
225.4.e.g.151.6		32	45.7	odd	12
225.4.e.g.151.11		32	45.43	odd	12
405.4.b.e.244.6		16	9.4	even	3
405.4.b.e.244.11		16	45.4	even	6
405.4.b.f.244.6		16	45.14	odd	6
405.4.b.f.244.11		16	9.5	odd	6
2025.4.a.bk.1.6		16	45.13	odd	12
2025.4.a.bk.1.11		16	45.22	odd	12
2025.4.a.bl.1.6		16	45.32	even	12
2025.4.a.bl.1.11		16	45.23	even	12