Newform orbit 135.2.p.a

• Label: 135.2.p.a
• Level: $135$
• Weight: $2$
• Character orbit: 135.p
• Analytic conductor: $1.078$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	135.p (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.07798042729\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(16\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 12 q^{4} - 9 q^{5} - 6 q^{6} - 18 q^{9}+O(q^{10}) \)

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	−0.911584	+	2.50456i	−1.32331	−	1.11752i	−3.90973	−	3.28066i	1.78103	−	1.35201i	4.00520	−	2.29560i	−1.98995	−	2.37153i	7.16422	−	4.13626i	0.502310	+	2.95765i	1.76262	+	5.69317i
4.2	−0.846914	+	2.32688i	0.666374	+	1.59873i	−3.16501	−	2.65576i	−2.21912	−	0.274758i	−4.28442	+	0.196582i	−1.41496	−	1.68629i	4.57119	−	2.63918i	−2.11189	+	2.13071i	2.51874	−	4.93093i
4.3	−0.691324	+	1.89940i	1.47769	−	0.903565i	−1.59769	−	1.34062i	0.122356	−	2.23272i	0.694666	+	3.43138i	2.82126	+	3.36225i	0.149906	−	0.0865484i	1.36714	−	2.67038i	4.15623	+	1.77593i
4.4	−0.613751	+	1.68627i	−1.05122	+	1.37657i	−0.934715	−	0.784319i	1.43461	+	1.71520i	−1.67607	−	2.61750i	0.337711	+	0.402468i	−1.21189	+	0.699685i	−0.789881	−	2.89415i	−3.77277	+	1.36642i
4.5	−0.557159	+	1.53078i	−0.844941	−	1.51198i	−0.500777	−	0.420202i	−1.06369	+	1.96687i	2.78527	−	0.451009i	1.77805	+	2.11900i	−1.89930	+	1.09656i	−1.57215	+	2.55506i	−2.41819	−	2.72414i
4.6	−0.289634	+	0.795763i	1.07411	−	1.35878i	0.982738	+	0.824615i	2.08094	+	0.818343i	0.770166	+	1.24829i	−3.00798	−	3.58478i	−2.40759	+	1.39002i	−0.692560	−	2.91897i	−1.25392	+	1.41892i
4.7	−0.228876	+	0.628833i	1.71312	+	0.255373i	1.18904	+	0.997725i	−2.05600	+	0.879131i	−0.552680	+	1.01882i	−0.158590	−	0.189000i	−2.05862	+	1.18854i	2.86957	+	0.874971i	−0.0822570	−	1.49409i
4.8	−0.141924	+	0.389932i	0.332960	+	1.69975i	1.40018	+	1.17489i	1.17994	−	1.89940i	−0.710041	−	0.111402i	−0.408725	−	0.487099i	−1.37557	+	0.794189i	−2.77828	+	1.13189i	0.573177	+	0.729668i
4.9	0.141924	−	0.389932i	−0.332960	−	1.69975i	1.40018	+	1.17489i	−0.459147	−	2.18842i	−0.710041	−	0.111402i	0.408725	+	0.487099i	1.37557	−	0.794189i	−2.77828	+	1.13189i	−0.918499	−	0.131552i
4.10	0.228876	−	0.628833i	−1.71312	−	0.255373i	1.18904	+	0.997725i	1.63133	+	1.52931i	−0.552680	+	1.01882i	0.158590	+	0.189000i	2.05862	−	1.18854i	2.86957	+	0.874971i	1.33505	−	0.675809i
4.11	0.289634	−	0.795763i	−1.07411	+	1.35878i	0.982738	+	0.824615i	−2.23533	+	0.0572668i	0.770166	+	1.24829i	3.00798	+	3.58478i	2.40759	−	1.39002i	−0.692560	−	2.91897i	−0.601858	+	1.79538i
4.12	0.557159	−	1.53078i	0.844941	+	1.51198i	−0.500777	−	0.420202i	0.326837	+	2.21205i	2.78527	−	0.451009i	−1.77805	−	2.11900i	1.89930	−	1.09656i	−1.57215	+	2.55506i	3.56827	+	0.732149i
4.13	0.613751	−	1.68627i	1.05122	−	1.37657i	−0.934715	−	0.784319i	−1.93472	+	1.12109i	−1.67607	−	2.61750i	−0.337711	−	0.402468i	1.21189	−	0.699685i	−0.789881	−	2.89415i	0.703028	+	3.95053i
4.14	0.691324	−	1.89940i	−1.47769	+	0.903565i	−1.59769	−	1.34062i	0.648657	−	2.13992i	0.694666	+	3.43138i	−2.82126	−	3.36225i	−0.149906	+	0.0865484i	1.36714	−	2.67038i	−3.61612	−	2.71143i
4.15	0.846914	−	2.32688i	−0.666374	−	1.59873i	−3.16501	−	2.65576i	2.17927	+	0.500797i	−4.28442	+	0.196582i	1.41496	+	1.68629i	−4.57119	+	2.63918i	−2.11189	+	2.13071i	3.01095	−	4.64676i
4.16	0.911584	−	2.50456i	1.32331	+	1.11752i	−3.90973	−	3.28066i	−1.21121	−	1.87962i	4.00520	−	2.29560i	1.98995	+	2.37153i	−7.16422	+	4.13626i	0.502310	+	2.95765i	−5.81174	+	1.32011i
34.1	−0.911584	−	2.50456i	−1.32331	+	1.11752i	−3.90973	+	3.28066i	1.78103	+	1.35201i	4.00520	+	2.29560i	−1.98995	+	2.37153i	7.16422	+	4.13626i	0.502310	−	2.95765i	1.76262	−	5.69317i
34.2	−0.846914	−	2.32688i	0.666374	−	1.59873i	−3.16501	+	2.65576i	−2.21912	+	0.274758i	−4.28442	−	0.196582i	−1.41496	+	1.68629i	4.57119	+	2.63918i	−2.11189	−	2.13071i	2.51874	+	4.93093i
34.3	−0.691324	−	1.89940i	1.47769	+	0.903565i	−1.59769	+	1.34062i	0.122356	+	2.23272i	0.694666	−	3.43138i	2.82126	−	3.36225i	0.149906	+	0.0865484i	1.36714	+	2.67038i	4.15623	−	1.77593i
34.4	−0.613751	−	1.68627i	−1.05122	−	1.37657i	−0.934715	+	0.784319i	1.43461	−	1.71520i	−1.67607	+	2.61750i	0.337711	−	0.402468i	−1.21189	−	0.699685i	−0.789881	+	2.89415i	−3.77277	−	1.36642i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
27.e	even	9	1	inner
135.p	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	135.2.p.a	✓	96
3.b	odd	2	1		405.2.p.a		96
5.b	even	2	1	inner	135.2.p.a	✓	96
5.c	odd	4	2		675.2.l.h		96
15.d	odd	2	1		405.2.p.a		96
27.e	even	9	1	inner	135.2.p.a	✓	96
27.f	odd	18	1		405.2.p.a		96
135.n	odd	18	1		405.2.p.a		96
135.p	even	18	1	inner	135.2.p.a	✓	96
135.r	odd	36	2		675.2.l.h		96

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
135.2.p.a	✓	96	1.a	even	1	1	trivial
135.2.p.a	✓	96	5.b	even	2	1	inner
135.2.p.a	✓	96	27.e	even	9	1	inner
135.2.p.a	✓	96	135.p	even	18	1	inner
405.2.p.a		96	3.b	odd	2	1
405.2.p.a		96	15.d	odd	2	1
405.2.p.a		96	27.f	odd	18	1
405.2.p.a		96	135.n	odd	18	1
675.2.l.h		96	5.c	odd	4	2
675.2.l.h		96	135.r	odd	36	2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(135, [\chi])\).