# Properties

 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

# Related objects

## 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 $$\Q(\zeta_{6})$$ Coefficient ring index: multiple of None 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 12q^{4} + 6q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 12q^{4} + 6q^{9} - 12q^{16} - 6q^{21} - 18q^{24} + 84q^{36} + 12q^{39} + 36q^{46} + 12q^{49} - 12q^{51} + 36q^{54} + 36q^{61} - 24q^{64} - 72q^{66} - 48q^{79} - 6q^{81} - 48q^{84} - 96q^{91} + 72q^{94} - 90q^{96} + 48q^{99} + O(q^{100})$$

## 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
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 101.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(525, [\chi])$$:

 $$T_{2}^{12} - 9 T_{2}^{10} + 63 T_{2}^{8} - 150 T_{2}^{6} + 270 T_{2}^{4} - 108 T_{2}^{2} + 36$$ $$T_{13}^{6} + 30 T_{13}^{4} + 219 T_{13}^{2} + 28$$ $$T_{37}^{12} + 90 T_{37}^{10} + 6129 T_{37}^{8} + 153198 T_{37}^{6} + 2796201 T_{37}^{4} + 23841216 T_{37}^{2} + 146313216$$

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database