Newform orbit 135.4.j.a

• Label: 135.4.j.a
• Level: $135$
• Weight: $4$
• Character orbit: 135.j
• Analytic conductor: $7.965$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	135.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.96525785077\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	no (minimal twist has level 45)
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.

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

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
19.1	−4.37720	−	2.52718i		0		8.77324	+	15.1957i	−10.8464	−	2.71197i		0		−18.1967	−	10.5059i		−	48.2513i		0		40.6234	+	39.2817i
19.2	−4.35066	−	2.51186i		0		8.61884	+	14.9283i	11.1787	+	0.192474i		0		4.66092	+	2.69099i		−	46.4074i		0		−48.1512	−	28.9166i
19.3	−3.53014	−	2.03813i		0		4.30793	+	7.46156i	1.64447	+	11.0587i		0		−11.5553	−	6.67148i		−	2.51043i		0		16.7339	−	42.3906i
19.4	−2.67377	−	1.54370i		0		0.766022	+	1.32679i	5.33901	−	9.82319i		0		27.1243	+	15.6602i			19.9692i		0		−29.4393	+	18.0231i
19.5	−2.31667	−	1.33753i		0		−0.422017	−	0.730954i	−3.65904	−	10.5646i		0		−13.3501	−	7.70766i			23.6584i		0		−5.65374	+	29.3689i
19.6	−2.13376	−	1.23192i		0		−0.964722	−	1.67095i	−9.72091	+	5.52303i		0		16.6624	+	9.62007i			24.4647i		0		27.5460	+	0.190629i
19.7	−0.680118	−	0.392666i		0		−3.69163	−	6.39408i	7.35516	−	8.42031i		0		−18.1117	−	10.4568i			12.0810i		0		−8.30875	+	2.83868i
19.8	−0.410487	−	0.236995i		0		−3.88767	−	6.73364i	2.29829	+	10.9416i		0		−7.10710	−	4.10329i			7.47735i		0		1.64968	−	5.03606i
19.9	0.410487	+	0.236995i		0		−3.88767	−	6.73364i	8.32653	+	7.46116i		0		7.10710	+	4.10329i		−	7.47735i		0		1.64968	+	5.03606i
19.10	0.680118	+	0.392666i		0		−3.69163	−	6.39408i	−10.9698	+	2.15960i		0		18.1117	+	10.4568i		−	12.0810i		0		−8.30875	−	2.83868i
19.11	2.13376	+	1.23192i		0		−0.964722	−	1.67095i	9.64354	−	5.65704i		0		−16.6624	−	9.62007i		−	24.4647i		0		27.5460	−	0.190629i
19.12	2.31667	+	1.33753i		0		−0.422017	−	0.730954i	−7.31972	−	8.45113i		0		13.3501	+	7.70766i		−	23.6584i		0		−5.65374	−	29.3689i
19.13	2.67377	+	1.54370i		0		0.766022	+	1.32679i	−11.1766	−	0.287878i		0		−27.1243	−	15.6602i		−	19.9692i		0		−29.4393	−	18.0231i
19.14	3.53014	+	2.03813i		0		4.30793	+	7.46156i	8.75492	+	6.95352i		0		11.5553	+	6.67148i			2.51043i		0		16.7339	+	42.3906i
19.15	4.35066	+	2.51186i		0		8.61884	+	14.9283i	−5.42265	+	9.77726i		0		−4.66092	−	2.69099i			46.4074i		0		−48.1512	+	28.9166i
19.16	4.37720	+	2.52718i		0		8.77324	+	15.1957i	3.07459	−	10.7493i		0		18.1967	+	10.5059i			48.2513i		0		40.6234	−	39.2817i
64.1	−4.37720	+	2.52718i		0		8.77324	−	15.1957i	−10.8464	+	2.71197i		0		−18.1967	+	10.5059i			48.2513i		0		40.6234	−	39.2817i
64.2	−4.35066	+	2.51186i		0		8.61884	−	14.9283i	11.1787	−	0.192474i		0		4.66092	−	2.69099i			46.4074i		0		−48.1512	+	28.9166i
64.3	−3.53014	+	2.03813i		0		4.30793	−	7.46156i	1.64447	−	11.0587i		0		−11.5553	+	6.67148i			2.51043i		0		16.7339	+	42.3906i
64.4	−2.67377	+	1.54370i		0		0.766022	−	1.32679i	5.33901	+	9.82319i		0		27.1243	−	15.6602i		−	19.9692i		0		−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	135.4.j.a		32
3.b	odd	2	1		45.4.j.a	✓	32
5.b	even	2	1	inner	135.4.j.a		32
9.c	even	3	1	inner	135.4.j.a		32
9.c	even	3	1		405.4.b.f		16
9.d	odd	6	1		45.4.j.a	✓	32
9.d	odd	6	1		405.4.b.e		16
15.d	odd	2	1		45.4.j.a	✓	32
15.e	even	4	2		225.4.e.g		32
45.h	odd	6	1		45.4.j.a	✓	32
45.h	odd	6	1		405.4.b.e		16
45.j	even	6	1	inner	135.4.j.a		32
45.j	even	6	1		405.4.b.f		16
45.k	odd	12	2		2025.4.a.bl		16
45.l	even	12	2		225.4.e.g		32
45.l	even	12	2		2025.4.a.bk		16

    

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

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(135, [\chi])\).