# Properties

 Label 504.2.ch.b Level 504 Weight 2 Character orbit 504.ch Analytic conductor 4.024 Analytic rank 0 Dimension 56 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 504.ch (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$56$$ Relative dimension: $$28$$ over $$\Q(\zeta_{6})$$ Coefficient ring index: multiple of None Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$56q - 8q^{4} - 20q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$56q - 8q^{4} - 20q^{7} + 20q^{16} - 16q^{22} + 8q^{25} + 36q^{28} - 36q^{31} + 60q^{40} - 8q^{46} - 28q^{49} + 36q^{52} - 44q^{58} + 40q^{64} - 60q^{70} + 72q^{73} - 12q^{79} - 36q^{82} + 4q^{88} - 180q^{94} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
269.1 −1.41410 + 0.0182389i 0 1.99933 0.0515832i −0.785247 + 0.453362i 0 −2.47043 0.947077i −2.82631 + 0.109409i 0 1.10215 0.655420i
269.2 −1.40843 0.127777i 0 1.96735 + 0.359931i 1.87230 1.08097i 0 2.55958 + 0.669737i −2.72488 0.758320i 0 −2.77512 + 1.28324i
269.3 −1.39540 + 0.229930i 0 1.89426 0.641688i −3.16007 + 1.82447i 0 −1.64838 + 2.06951i −2.49571 + 1.33096i 0 3.99006 3.27246i
269.4 −1.30255 0.550794i 0 1.39325 + 1.43487i −1.00441 + 0.579896i 0 1.24394 + 2.33508i −1.02446 2.63638i 0 1.62769 0.202118i
269.5 −1.13135 0.848557i 0 0.559903 + 1.92003i −1.53798 + 0.887954i 0 0.843933 2.50754i 0.995807 2.64733i 0 2.49347 + 0.300478i
269.6 −1.11071 0.875401i 0 0.467346 + 1.94463i 3.18706 1.84005i 0 −0.998380 2.45015i 1.18325 2.56903i 0 −5.15068 0.746198i
269.7 −0.937852 1.05851i 0 −0.240867 + 1.98544i −0.331990 + 0.191675i 0 −2.03027 + 1.69647i 2.32750 1.60709i 0 0.514246 + 0.171651i
269.8 −0.896824 + 1.09348i 0 −0.391414 1.96132i 3.16007 1.82447i 0 −1.64838 + 2.06951i 2.49571 + 1.33096i 0 −0.839002 + 5.09172i
269.9 −0.722843 + 1.21552i 0 −0.954995 1.75727i 0.785247 0.453362i 0 −2.47043 0.947077i 2.82631 + 0.109409i 0 −0.0165377 + 1.28220i
269.10 −0.593556 + 1.28362i 0 −1.29538 1.52381i −1.87230 + 1.08097i 0 2.55958 + 0.669737i 2.72488 0.758320i 0 −0.276248 3.04495i
269.11 −0.447766 1.34146i 0 −1.59901 + 1.20132i −0.331990 + 0.191675i 0 −2.03027 + 1.69647i 2.32750 + 1.60709i 0 0.405777 + 0.359525i
269.12 −0.202765 1.39960i 0 −1.91777 + 0.567582i 3.18706 1.84005i 0 −0.998380 2.45015i 1.18325 + 2.56903i 0 −3.22157 4.08752i
269.13 −0.174271 + 1.40343i 0 −1.93926 0.489157i 1.00441 0.579896i 0 1.24394 + 2.33508i 1.02446 2.63638i 0 0.638806 + 1.51068i
269.14 −0.169197 1.40406i 0 −1.94274 + 0.475124i −1.53798 + 0.887954i 0 0.843933 2.50754i 0.995807 + 2.64733i 0 1.50696 + 2.00917i
269.15 0.169197 + 1.40406i 0 −1.94274 + 0.475124i 1.53798 0.887954i 0 0.843933 2.50754i −0.995807 2.64733i 0 1.50696 + 2.00917i
269.16 0.174271 1.40343i 0 −1.93926 0.489157i −1.00441 + 0.579896i 0 1.24394 + 2.33508i −1.02446 + 2.63638i 0 0.638806 + 1.51068i
269.17 0.202765 + 1.39960i 0 −1.91777 + 0.567582i −3.18706 + 1.84005i 0 −0.998380 2.45015i −1.18325 2.56903i 0 −3.22157 4.08752i
269.18 0.447766 + 1.34146i 0 −1.59901 + 1.20132i 0.331990 0.191675i 0 −2.03027 + 1.69647i −2.32750 1.60709i 0 0.405777 + 0.359525i
269.19 0.593556 1.28362i 0 −1.29538 1.52381i 1.87230 1.08097i 0 2.55958 + 0.669737i −2.72488 + 0.758320i 0 −0.276248 3.04495i
269.20 0.722843 1.21552i 0 −0.954995 1.75727i −0.785247 + 0.453362i 0 −2.47043 0.947077i −2.82631 0.109409i 0 −0.0165377 + 1.28220i
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 341.28 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
7.d odd 6 1 inner
8.b even 2 1 inner
21.g even 6 1 inner
24.h odd 2 1 inner
56.j odd 6 1 inner
168.ba even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.ch.b 56
3.b odd 2 1 inner 504.2.ch.b 56
4.b odd 2 1 2016.2.cp.b 56
7.d odd 6 1 inner 504.2.ch.b 56
8.b even 2 1 inner 504.2.ch.b 56
8.d odd 2 1 2016.2.cp.b 56
12.b even 2 1 2016.2.cp.b 56
21.g even 6 1 inner 504.2.ch.b 56
24.f even 2 1 2016.2.cp.b 56
24.h odd 2 1 inner 504.2.ch.b 56
28.f even 6 1 2016.2.cp.b 56
56.j odd 6 1 inner 504.2.ch.b 56
56.m even 6 1 2016.2.cp.b 56
84.j odd 6 1 2016.2.cp.b 56
168.ba even 6 1 inner 504.2.ch.b 56
168.be odd 6 1 2016.2.cp.b 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.ch.b 56 1.a even 1 1 trivial
504.2.ch.b 56 3.b odd 2 1 inner
504.2.ch.b 56 7.d odd 6 1 inner
504.2.ch.b 56 8.b even 2 1 inner
504.2.ch.b 56 21.g even 6 1 inner
504.2.ch.b 56 24.h odd 2 1 inner
504.2.ch.b 56 56.j odd 6 1 inner
504.2.ch.b 56 168.ba even 6 1 inner
2016.2.cp.b 56 4.b odd 2 1
2016.2.cp.b 56 8.d odd 2 1
2016.2.cp.b 56 12.b even 2 1
2016.2.cp.b 56 24.f even 2 1
2016.2.cp.b 56 28.f even 6 1
2016.2.cp.b 56 56.m even 6 1
2016.2.cp.b 56 84.j odd 6 1
2016.2.cp.b 56 168.be odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{28} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database