# Properties

 Label 525.2.bf.g Level 525 Weight 2 Character orbit 525.bf Analytic conductor 4.192 Analytic rank 0 Dimension 80 CM no Inner twists 16

# Related objects

## Newspace parameters

 Level: $$N$$ = $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 525.bf (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$20$$ over $$\Q(\zeta_{12})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$80q + 16q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80q + 16q^{6} + 72q^{16} + 44q^{21} + 72q^{31} - 240q^{36} - 92q^{51} - 24q^{61} - 216q^{66} - 208q^{76} - 20q^{81} - 40q^{91} - 156q^{96} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
32.1 −2.61504 0.700699i 0.131277 1.72707i 4.61542 + 2.66471i 0 −1.55345 + 4.42437i 1.33002 + 2.28715i −6.37367 6.37367i −2.96553 0.453449i 0
32.2 −2.61504 0.700699i 0.977224 + 1.43005i 4.61542 + 2.66471i 0 −1.55345 4.42437i −1.33002 2.28715i −6.37367 6.37367i −1.09007 + 2.79495i 0
32.3 −2.14072 0.573605i −1.69002 0.379239i 2.52162 + 1.45586i 0 3.40034 + 1.78125i −0.453654 + 2.60657i −1.42877 1.42877i 2.71236 + 1.28185i 0
32.4 −2.14072 0.573605i −1.27398 + 1.17344i 2.52162 + 1.45586i 0 3.40034 1.78125i 0.453654 2.60657i −1.42877 1.42877i 0.246067 2.98989i 0
32.5 −1.78672 0.478751i 0.293073 1.70708i 1.23113 + 0.710791i 0 −1.34091 + 2.90976i 0.585067 2.58025i 0.756555 + 0.756555i −2.82822 1.00060i 0
32.6 −1.78672 0.478751i 1.10735 + 1.33183i 1.23113 + 0.710791i 0 −1.34091 2.90976i −0.585067 + 2.58025i 0.756555 + 0.756555i −0.547566 + 2.94961i 0
32.7 −1.05572 0.282878i −1.36728 1.06328i −0.697534 0.402722i 0 1.14268 + 1.50929i −2.58881 + 0.545957i 2.16815 + 2.16815i 0.738892 + 2.90758i 0
32.8 −1.05572 0.282878i −0.652459 + 1.60446i −0.697534 0.402722i 0 1.14268 1.50929i 2.58881 0.545957i 2.16815 + 2.16815i −2.14860 2.09369i 0
32.9 −0.364909 0.0977772i 1.59960 0.664285i −1.60845 0.928640i 0 −0.648661 + 0.0859992i 2.42547 + 1.05692i 1.03040 + 1.03040i 2.11745 2.12518i 0
32.10 −0.364909 0.0977772i 1.71744 0.224513i −1.60845 0.928640i 0 −0.648661 0.0859992i −2.42547 1.05692i 1.03040 + 1.03040i 2.89919 0.771175i 0
32.11 0.364909 + 0.0977772i −1.71744 + 0.224513i −1.60845 0.928640i 0 −0.648661 0.0859992i 2.42547 + 1.05692i −1.03040 1.03040i 2.89919 0.771175i 0
32.12 0.364909 + 0.0977772i −1.59960 + 0.664285i −1.60845 0.928640i 0 −0.648661 + 0.0859992i −2.42547 1.05692i −1.03040 1.03040i 2.11745 2.12518i 0
32.13 1.05572 + 0.282878i 0.652459 1.60446i −0.697534 0.402722i 0 1.14268 1.50929i −2.58881 + 0.545957i −2.16815 2.16815i −2.14860 2.09369i 0
32.14 1.05572 + 0.282878i 1.36728 + 1.06328i −0.697534 0.402722i 0 1.14268 + 1.50929i 2.58881 0.545957i −2.16815 2.16815i 0.738892 + 2.90758i 0
32.15 1.78672 + 0.478751i −1.10735 1.33183i 1.23113 + 0.710791i 0 −1.34091 2.90976i 0.585067 2.58025i −0.756555 0.756555i −0.547566 + 2.94961i 0
32.16 1.78672 + 0.478751i −0.293073 + 1.70708i 1.23113 + 0.710791i 0 −1.34091 + 2.90976i −0.585067 + 2.58025i −0.756555 0.756555i −2.82822 1.00060i 0
32.17 2.14072 + 0.573605i 1.27398 1.17344i 2.52162 + 1.45586i 0 3.40034 1.78125i −0.453654 + 2.60657i 1.42877 + 1.42877i 0.246067 2.98989i 0
32.18 2.14072 + 0.573605i 1.69002 + 0.379239i 2.52162 + 1.45586i 0 3.40034 + 1.78125i 0.453654 2.60657i 1.42877 + 1.42877i 2.71236 + 1.28185i 0
32.19 2.61504 + 0.700699i −0.977224 1.43005i 4.61542 + 2.66471i 0 −1.55345 4.42437i 1.33002 + 2.28715i 6.37367 + 6.37367i −1.09007 + 2.79495i 0
32.20 2.61504 + 0.700699i −0.131277 + 1.72707i 4.61542 + 2.66471i 0 −1.55345 + 4.42437i −1.33002 2.28715i 6.37367 + 6.37367i −2.96553 0.453449i 0
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 443.20 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
5.c odd 4 2 inner
7.c even 3 1 inner
15.d odd 2 1 inner
15.e even 4 2 inner
21.h odd 6 1 inner
35.j even 6 1 inner
35.l odd 12 2 inner
105.o odd 6 1 inner
105.x even 12 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.bf.g 80
3.b odd 2 1 inner 525.2.bf.g 80
5.b even 2 1 inner 525.2.bf.g 80
5.c odd 4 2 inner 525.2.bf.g 80
7.c even 3 1 inner 525.2.bf.g 80
15.d odd 2 1 inner 525.2.bf.g 80
15.e even 4 2 inner 525.2.bf.g 80
21.h odd 6 1 inner 525.2.bf.g 80
35.j even 6 1 inner 525.2.bf.g 80
35.l odd 12 2 inner 525.2.bf.g 80
105.o odd 6 1 inner 525.2.bf.g 80
105.x even 12 2 inner 525.2.bf.g 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.bf.g 80 1.a even 1 1 trivial
525.2.bf.g 80 3.b odd 2 1 inner
525.2.bf.g 80 5.b even 2 1 inner
525.2.bf.g 80 5.c odd 4 2 inner
525.2.bf.g 80 7.c even 3 1 inner
525.2.bf.g 80 15.d odd 2 1 inner
525.2.bf.g 80 15.e even 4 2 inner
525.2.bf.g 80 21.h odd 6 1 inner
525.2.bf.g 80 35.j even 6 1 inner
525.2.bf.g 80 35.l odd 12 2 inner
525.2.bf.g 80 105.o odd 6 1 inner
525.2.bf.g 80 105.x even 12 2 inner

## Hecke kernels

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

 $$T_{2}^{40} - \cdots$$ $$T_{13}^{20} + 700 T_{13}^{16} + 76914 T_{13}^{12} + 675469 T_{13}^{8} + 1706287 T_{13}^{4} + 1172889$$

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database