Newform orbit 525.2.t.j

• Label: 525.2.t.j
• Level: $525$
• Weight: $2$
• Character orbit: 525.t
• Analytic conductor: $4.192$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.19214610612\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(\zeta_{6})\)
Twist minimal:	no (minimal twist has level 105)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{4} + 6 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
26.1	−2.17197	−	1.25399i	−1.73159	−	0.0401158i	2.14497	+	3.71520i		0		3.71065	+	2.25852i	2.06025	+	1.65993i		−	5.74313i	2.99678	+	0.138928i		0
26.2	−2.17197	−	1.25399i	0.831052	−	1.51966i	2.14497	+	3.71520i		0		−3.71065	+	2.25852i	−2.06025	−	1.65993i		−	5.74313i	−1.61871	−	2.52582i		0
26.3	−1.31176	−	0.757344i	−1.20362	+	1.24551i	0.147140	+	0.254854i		0		2.52214	−	0.722254i	−2.64468	−	0.0753638i			2.58363i	−0.102593	−	2.99825i		0
26.4	−1.31176	−	0.757344i	1.68045	−	0.419611i	0.147140	+	0.254854i		0		−2.52214	−	0.722254i	2.64468	+	0.0753638i			2.58363i	2.64785	−	1.41027i		0
26.5	−0.558418	−	0.322403i	−1.28838	−	1.15761i	−0.792113	−	1.37198i		0		0.346239	+	1.06181i	0.105130	+	2.64366i			2.31113i	0.319861	+	2.98290i		0
26.6	−0.558418	−	0.322403i	−0.358331	−	1.69458i	−0.792113	−	1.37198i		0		−0.346239	+	1.06181i	−0.105130	−	2.64366i			2.31113i	−2.74320	+	1.21444i		0
26.7	0.558418	+	0.322403i	0.358331	+	1.69458i	−0.792113	−	1.37198i		0		−0.346239	+	1.06181i	0.105130	+	2.64366i		−	2.31113i	−2.74320	+	1.21444i		0
26.8	0.558418	+	0.322403i	1.28838	+	1.15761i	−0.792113	−	1.37198i		0		0.346239	+	1.06181i	−0.105130	−	2.64366i		−	2.31113i	0.319861	+	2.98290i		0
26.9	1.31176	+	0.757344i	−1.68045	+	0.419611i	0.147140	+	0.254854i		0		−2.52214	−	0.722254i	−2.64468	−	0.0753638i		−	2.58363i	2.64785	−	1.41027i		0
26.10	1.31176	+	0.757344i	1.20362	−	1.24551i	0.147140	+	0.254854i		0		2.52214	−	0.722254i	2.64468	+	0.0753638i		−	2.58363i	−0.102593	−	2.99825i		0
26.11	2.17197	+	1.25399i	−0.831052	+	1.51966i	2.14497	+	3.71520i		0		−3.71065	+	2.25852i	2.06025	+	1.65993i			5.74313i	−1.61871	−	2.52582i		0
26.12	2.17197	+	1.25399i	1.73159	+	0.0401158i	2.14497	+	3.71520i		0		3.71065	+	2.25852i	−2.06025	−	1.65993i			5.74313i	2.99678	+	0.138928i		0
101.1	−2.17197	+	1.25399i	−1.73159	+	0.0401158i	2.14497	−	3.71520i		0		3.71065	−	2.25852i	2.06025	−	1.65993i			5.74313i	2.99678	−	0.138928i		0
101.2	−2.17197	+	1.25399i	0.831052	+	1.51966i	2.14497	−	3.71520i		0		−3.71065	−	2.25852i	−2.06025	+	1.65993i			5.74313i	−1.61871	+	2.52582i		0
101.3	−1.31176	+	0.757344i	−1.20362	−	1.24551i	0.147140	−	0.254854i		0		2.52214	+	0.722254i	−2.64468	+	0.0753638i		−	2.58363i	−0.102593	+	2.99825i		0
101.4	−1.31176	+	0.757344i	1.68045	+	0.419611i	0.147140	−	0.254854i		0		−2.52214	+	0.722254i	2.64468	−	0.0753638i		−	2.58363i	2.64785	+	1.41027i		0
101.5	−0.558418	+	0.322403i	−1.28838	+	1.15761i	−0.792113	+	1.37198i		0		0.346239	−	1.06181i	0.105130	−	2.64366i		−	2.31113i	0.319861	−	2.98290i		0
101.6	−0.558418	+	0.322403i	−0.358331	+	1.69458i	−0.792113	+	1.37198i		0		−0.346239	−	1.06181i	−0.105130	+	2.64366i		−	2.31113i	−2.74320	−	1.21444i		0
101.7	0.558418	−	0.322403i	0.358331	−	1.69458i	−0.792113	+	1.37198i		0		−0.346239	−	1.06181i	0.105130	−	2.64366i			2.31113i	−2.74320	−	1.21444i		0
101.8	0.558418	−	0.322403i	1.28838	−	1.15761i	−0.792113	+	1.37198i		0		0.346239	−	1.06181i	−0.105130	+	2.64366i			2.31113i	0.319861	−	2.98290i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
7.d	odd	6	1	inner
15.d	odd	2	1	inner
21.g	even	6	1	inner
35.i	odd	6	1	inner
105.p	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	525.2.t.j		24
3.b	odd	2	1	inner	525.2.t.j		24
5.b	even	2	1	inner	525.2.t.j		24
5.c	odd	4	2		105.2.p.a	✓	24
7.d	odd	6	1	inner	525.2.t.j		24
15.d	odd	2	1	inner	525.2.t.j		24
15.e	even	4	2		105.2.p.a	✓	24
21.g	even	6	1	inner	525.2.t.j		24
35.f	even	4	2		735.2.p.f		24
35.i	odd	6	1	inner	525.2.t.j		24
35.k	even	12	2		105.2.p.a	✓	24
35.k	even	12	2		735.2.g.b		24
35.l	odd	12	2		735.2.g.b		24
35.l	odd	12	2		735.2.p.f		24
105.k	odd	4	2		735.2.p.f		24
105.p	even	6	1	inner	525.2.t.j		24
105.w	odd	12	2		105.2.p.a	✓	24
105.w	odd	12	2		735.2.g.b		24
105.x	even	12	2		735.2.g.b		24
105.x	even	12	2		735.2.p.f		24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
105.2.p.a	✓	24	5.c	odd	4	2
105.2.p.a	✓	24	15.e	even	4	2
105.2.p.a	✓	24	35.k	even	12	2
105.2.p.a	✓	24	105.w	odd	12	2
525.2.t.j		24	1.a	even	1	1	trivial
525.2.t.j		24	3.b	odd	2	1	inner
525.2.t.j		24	5.b	even	2	1	inner
525.2.t.j		24	7.d	odd	6	1	inner
525.2.t.j		24	15.d	odd	2	1	inner
525.2.t.j		24	21.g	even	6	1	inner
525.2.t.j		24	35.i	odd	6	1	inner
525.2.t.j		24	105.p	even	6	1	inner
735.2.g.b		24	35.k	even	12	2
735.2.g.b		24	35.l	odd	12	2
735.2.g.b		24	105.w	odd	12	2
735.2.g.b		24	105.x	even	12	2
735.2.p.f		24	35.f	even	4	2
735.2.p.f		24	35.l	odd	12	2
735.2.p.f		24	105.k	odd	4	2
735.2.p.f		24	105.x	even	12	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{12} - 9T_{2}^{10} + 63T_{2}^{8} - 150T_{2}^{6} + 270T_{2}^{4} - 108T_{2}^{2} + 36 \)

\( T_{13}^{6} + 30T_{13}^{4} + 219T_{13}^{2} + 28 \)

\( T_{37}^{12} + 90T_{37}^{10} + 6129T_{37}^{8} + 153198T_{37}^{6} + 2796201T_{37}^{4} + 23841216T_{37}^{2} + 146313216 \)