Newform orbit 45.4.j.a

• Label: 45.4.j.a
• Level: $45$
• Weight: $4$
• Character orbit: 45.j
• 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:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 54 q^{4} + 3 q^{5} - 12 q^{6} - 18 q^{9}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
4.1	−4.37720	+	2.52718i	5.16592	+	0.559681i	8.77324	−	15.1957i	−3.07459	−	10.7493i	−24.0267	+	10.6054i	18.1967	−	10.5059i			48.2513i	26.3735	+	5.78253i	40.6234	+	39.2817i
4.2	−4.35066	+	2.51186i	−1.99648	−	4.79730i	8.61884	−	14.9283i	5.42265	+	9.77726i	20.7361	+	15.8565i	−4.66092	+	2.69099i			46.4074i	−19.0281	+	19.1554i	−48.1512	−	28.9166i
4.3	−3.53014	+	2.03813i	−2.78227	+	4.38850i	4.30793	−	7.46156i	−8.75492	+	6.95352i	0.877481	−	21.1627i	11.5553	−	6.67148i			2.51043i	−11.5179	−	24.4200i	16.7339	−	42.3906i
4.4	−2.67377	+	1.54370i	2.05649	+	4.77188i	0.766022	−	1.32679i	11.1766	−	0.287878i	−12.8649	−	9.58430i	−27.1243	+	15.6602i		−	19.9692i	−18.5417	+	19.6266i	−29.4393	+	18.0231i
4.5	−2.31667	+	1.33753i	−5.14958	−	0.694136i	−0.422017	+	0.730954i	7.31972	−	8.45113i	12.8583	−	5.27964i	13.3501	−	7.70766i		−	23.6584i	26.0363	+	7.14902i	−5.65374	+	29.3689i
4.6	−2.13376	+	1.23192i	1.50557	−	4.97325i	−0.964722	+	1.67095i	−9.64354	−	5.65704i	2.91415	+	12.4665i	−16.6624	+	9.62007i		−	24.4647i	−22.4665	−	14.9752i	27.5460	+	0.190629i
4.7	−0.680118	+	0.392666i	3.65242	−	3.69592i	−3.69163	+	6.39408i	10.9698	+	2.15960i	−1.03281	+	3.94785i	18.1117	−	10.4568i		−	12.0810i	−0.319654	−	26.9981i	−8.30875	+	2.83868i
4.8	−0.410487	+	0.236995i	4.58061	+	2.45316i	−3.88767	+	6.73364i	−8.32653	+	7.46116i	−2.46167	+	0.0785933i	7.10710	−	4.10329i		−	7.47735i	14.9641	+	22.4739i	1.64968	−	5.03606i
4.9	0.410487	−	0.236995i	−4.58061	−	2.45316i	−3.88767	+	6.73364i	−2.29829	+	10.9416i	−2.46167	+	0.0785933i	−7.10710	+	4.10329i			7.47735i	14.9641	+	22.4739i	1.64968	+	5.03606i
4.10	0.680118	−	0.392666i	−3.65242	+	3.69592i	−3.69163	+	6.39408i	−7.35516	−	8.42031i	−1.03281	+	3.94785i	−18.1117	+	10.4568i			12.0810i	−0.319654	−	26.9981i	−8.30875	−	2.83868i
4.11	2.13376	−	1.23192i	−1.50557	+	4.97325i	−0.964722	+	1.67095i	9.72091	+	5.52303i	2.91415	+	12.4665i	16.6624	−	9.62007i			24.4647i	−22.4665	−	14.9752i	27.5460	−	0.190629i
4.12	2.31667	−	1.33753i	5.14958	+	0.694136i	−0.422017	+	0.730954i	3.65904	−	10.5646i	12.8583	−	5.27964i	−13.3501	+	7.70766i			23.6584i	26.0363	+	7.14902i	−5.65374	−	29.3689i
4.13	2.67377	−	1.54370i	−2.05649	−	4.77188i	0.766022	−	1.32679i	−5.33901	−	9.82319i	−12.8649	−	9.58430i	27.1243	−	15.6602i			19.9692i	−18.5417	+	19.6266i	−29.4393	−	18.0231i
4.14	3.53014	−	2.03813i	2.78227	−	4.38850i	4.30793	−	7.46156i	−1.64447	+	11.0587i	0.877481	−	21.1627i	−11.5553	+	6.67148i		−	2.51043i	−11.5179	−	24.4200i	16.7339	+	42.3906i
4.15	4.35066	−	2.51186i	1.99648	+	4.79730i	8.61884	−	14.9283i	−11.1787	+	0.192474i	20.7361	+	15.8565i	4.66092	−	2.69099i		−	46.4074i	−19.0281	+	19.1554i	−48.1512	+	28.9166i
4.16	4.37720	−	2.52718i	−5.16592	−	0.559681i	8.77324	−	15.1957i	10.8464	−	2.71197i	−24.0267	+	10.6054i	−18.1967	+	10.5059i		−	48.2513i	26.3735	+	5.78253i	40.6234	−	39.2817i
34.1	−4.37720	−	2.52718i	5.16592	−	0.559681i	8.77324	+	15.1957i	−3.07459	+	10.7493i	−24.0267	−	10.6054i	18.1967	+	10.5059i		−	48.2513i	26.3735	−	5.78253i	40.6234	−	39.2817i
34.2	−4.35066	−	2.51186i	−1.99648	+	4.79730i	8.61884	+	14.9283i	5.42265	−	9.77726i	20.7361	−	15.8565i	−4.66092	−	2.69099i		−	46.4074i	−19.0281	−	19.1554i	−48.1512	+	28.9166i
34.3	−3.53014	−	2.03813i	−2.78227	−	4.38850i	4.30793	+	7.46156i	−8.75492	−	6.95352i	0.877481	+	21.1627i	11.5553	+	6.67148i		−	2.51043i	−11.5179	+	24.4200i	16.7339	+	42.3906i
34.4	−2.67377	−	1.54370i	2.05649	−	4.77188i	0.766022	+	1.32679i	11.1766	+	0.287878i	−12.8649	+	9.58430i	−27.1243	−	15.6602i			19.9692i	−18.5417	−	19.6266i	−29.4393	−	18.0231i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
9.c	even	3	1	inner
45.j	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	45.4.j.a	✓	32
3.b	odd	2	1		135.4.j.a		32
5.b	even	2	1	inner	45.4.j.a	✓	32
5.c	odd	4	2		225.4.e.g		32
9.c	even	3	1	inner	45.4.j.a	✓	32
9.c	even	3	1		405.4.b.e		16
9.d	odd	6	1		135.4.j.a		32
9.d	odd	6	1		405.4.b.f		16
15.d	odd	2	1		135.4.j.a		32
45.h	odd	6	1		135.4.j.a		32
45.h	odd	6	1		405.4.b.f		16
45.j	even	6	1	inner	45.4.j.a	✓	32
45.j	even	6	1		405.4.b.e		16
45.k	odd	12	2		225.4.e.g		32
45.k	odd	12	2		2025.4.a.bk		16
45.l	even	12	2		2025.4.a.bl		16

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
45.4.j.a	✓	32	1.a	even	1	1	trivial
45.4.j.a	✓	32	5.b	even	2	1	inner
45.4.j.a	✓	32	9.c	even	3	1	inner
45.4.j.a	✓	32	45.j	even	6	1	inner
135.4.j.a		32	3.b	odd	2	1
135.4.j.a		32	9.d	odd	6	1
135.4.j.a		32	15.d	odd	2	1
135.4.j.a		32	45.h	odd	6	1
225.4.e.g		32	5.c	odd	4	2
225.4.e.g		32	45.k	odd	12	2
405.4.b.e		16	9.c	even	3	1
405.4.b.e		16	45.j	even	6	1
405.4.b.f		16	9.d	odd	6	1
405.4.b.f		16	45.h	odd	6	1
2025.4.a.bk		16	45.k	odd	12	2
2025.4.a.bl		16	45.l	even	12	2

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(45, [\chi])\).