# Properties

 Label 525.2.j.b Level 525 Weight 2 Character orbit 525.j Analytic conductor 4.192 Analytic rank 0 Dimension 24 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 525.j (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(i)$$ Coefficient ring index: multiple of None Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 4q^{3} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 4q^{3} - 16q^{12} + 8q^{13} - 16q^{16} + 20q^{18} + 4q^{21} - 8q^{22} + 16q^{27} - 28q^{33} + 16q^{36} + 16q^{37} + 20q^{42} + 40q^{43} - 64q^{46} - 16q^{48} - 20q^{51} - 4q^{57} - 40q^{58} + 32q^{61} + 8q^{63} - 16q^{66} - 24q^{67} + 8q^{72} - 32q^{73} + 32q^{76} - 60q^{78} + 52q^{81} + 80q^{82} - 4q^{87} - 96q^{88} - 24q^{91} + 76q^{93} - 96q^{96} - 24q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
218.1 −1.79963 1.79963i −0.491204 + 1.66094i 4.47734i 0 3.87306 2.10509i 0.707107 0.707107i 4.45829 4.45829i −2.51744 1.63172i 0
218.2 −1.54414 1.54414i 1.73204 + 0.00622252i 2.76875i 0 −2.66491 2.68412i −0.707107 + 0.707107i 1.18705 1.18705i 2.99992 + 0.0215553i 0
218.3 −1.24414 1.24414i −0.474620 1.66575i 1.09578i 0 −1.48194 + 2.66293i −0.707107 + 0.707107i −1.12498 + 1.12498i −2.54947 + 1.58120i 0
218.4 −0.800553 0.800553i 1.09397 + 1.34285i 0.718229i 0 0.199242 1.95080i 0.707107 0.707107i −2.17609 + 2.17609i −0.606476 + 2.93806i 0
218.5 −0.347054 0.347054i −0.176396 1.72305i 1.75911i 0 −0.536770 + 0.659208i 0.707107 0.707107i −1.30461 + 1.30461i −2.93777 + 0.607876i 0
218.6 −0.260263 0.260263i −0.826909 + 1.52191i 1.86453i 0 0.611312 0.180884i −0.707107 + 0.707107i −1.00579 + 1.00579i −1.63244 2.51697i 0
218.7 0.260263 + 0.260263i 1.52191 0.826909i 1.86453i 0 0.611312 + 0.180884i −0.707107 + 0.707107i 1.00579 1.00579i 1.63244 2.51697i 0
218.8 0.347054 + 0.347054i −1.72305 0.176396i 1.75911i 0 −0.536770 0.659208i 0.707107 0.707107i 1.30461 1.30461i 2.93777 + 0.607876i 0
218.9 0.800553 + 0.800553i 1.34285 + 1.09397i 0.718229i 0 0.199242 + 1.95080i 0.707107 0.707107i 2.17609 2.17609i 0.606476 + 2.93806i 0
218.10 1.24414 + 1.24414i −1.66575 0.474620i 1.09578i 0 −1.48194 2.66293i −0.707107 + 0.707107i 1.12498 1.12498i 2.54947 + 1.58120i 0
218.11 1.54414 + 1.54414i 0.00622252 + 1.73204i 2.76875i 0 −2.66491 + 2.68412i −0.707107 + 0.707107i −1.18705 + 1.18705i −2.99992 + 0.0215553i 0
218.12 1.79963 + 1.79963i 1.66094 0.491204i 4.47734i 0 3.87306 + 2.10509i 0.707107 0.707107i −4.45829 + 4.45829i 2.51744 1.63172i 0
407.1 −1.79963 + 1.79963i −0.491204 1.66094i 4.47734i 0 3.87306 + 2.10509i 0.707107 + 0.707107i 4.45829 + 4.45829i −2.51744 + 1.63172i 0
407.2 −1.54414 + 1.54414i 1.73204 0.00622252i 2.76875i 0 −2.66491 + 2.68412i −0.707107 0.707107i 1.18705 + 1.18705i 2.99992 0.0215553i 0
407.3 −1.24414 + 1.24414i −0.474620 + 1.66575i 1.09578i 0 −1.48194 2.66293i −0.707107 0.707107i −1.12498 1.12498i −2.54947 1.58120i 0
407.4 −0.800553 + 0.800553i 1.09397 1.34285i 0.718229i 0 0.199242 + 1.95080i 0.707107 + 0.707107i −2.17609 2.17609i −0.606476 2.93806i 0
407.5 −0.347054 + 0.347054i −0.176396 + 1.72305i 1.75911i 0 −0.536770 0.659208i 0.707107 + 0.707107i −1.30461 1.30461i −2.93777 0.607876i 0
407.6 −0.260263 + 0.260263i −0.826909 1.52191i 1.86453i 0 0.611312 + 0.180884i −0.707107 0.707107i −1.00579 1.00579i −1.63244 + 2.51697i 0
407.7 0.260263 0.260263i 1.52191 + 0.826909i 1.86453i 0 0.611312 0.180884i −0.707107 0.707107i 1.00579 + 1.00579i 1.63244 + 2.51697i 0
407.8 0.347054 0.347054i −1.72305 + 0.176396i 1.75911i 0 −0.536770 + 0.659208i 0.707107 + 0.707107i 1.30461 + 1.30461i 2.93777 0.607876i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 407.12 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.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.j.b 24
3.b odd 2 1 inner 525.2.j.b 24
5.b even 2 1 105.2.j.a 24
5.c odd 4 1 105.2.j.a 24
5.c odd 4 1 inner 525.2.j.b 24
15.d odd 2 1 105.2.j.a 24
15.e even 4 1 105.2.j.a 24
15.e even 4 1 inner 525.2.j.b 24
35.c odd 2 1 735.2.j.h 24
35.f even 4 1 735.2.j.h 24
35.i odd 6 2 735.2.y.g 48
35.j even 6 2 735.2.y.j 48
35.k even 12 2 735.2.y.g 48
35.l odd 12 2 735.2.y.j 48
105.g even 2 1 735.2.j.h 24
105.k odd 4 1 735.2.j.h 24
105.o odd 6 2 735.2.y.j 48
105.p even 6 2 735.2.y.g 48
105.w odd 12 2 735.2.y.g 48
105.x even 12 2 735.2.y.j 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.j.a 24 5.b even 2 1
105.2.j.a 24 5.c odd 4 1
105.2.j.a 24 15.d odd 2 1
105.2.j.a 24 15.e even 4 1
525.2.j.b 24 1.a even 1 1 trivial
525.2.j.b 24 3.b odd 2 1 inner
525.2.j.b 24 5.c odd 4 1 inner
525.2.j.b 24 15.e even 4 1 inner
735.2.j.h 24 35.c odd 2 1
735.2.j.h 24 35.f even 4 1
735.2.j.h 24 105.g even 2 1
735.2.j.h 24 105.k odd 4 1
735.2.y.g 48 35.i odd 6 2
735.2.y.g 48 35.k even 12 2
735.2.y.g 48 105.p even 6 2
735.2.y.g 48 105.w odd 12 2
735.2.y.j 48 35.j even 6 2
735.2.y.j 48 35.l odd 12 2
735.2.y.j 48 105.o odd 6 2
735.2.y.j 48 105.x even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{24} + 76 T_{2}^{20} + 1702 T_{2}^{16} + 11860 T_{2}^{12} + 15921 T_{2}^{8} + 1160 T_{2}^{4} + 16$$ acting on $$S_{2}^{\mathrm{new}}(525, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database