# Properties

 Label 504.2.bk.b Level 504 Weight 2 Character orbit 504.bk Analytic conductor 4.024 Analytic rank 0 Dimension 32 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.bk (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 2q^{4} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 2q^{4} + 18q^{10} - 10q^{16} - 12q^{22} - 16q^{25} - 6q^{28} - 30q^{40} + 16q^{43} + 16q^{46} + 8q^{49} - 72q^{52} - 38q^{58} + 44q^{64} + 16q^{67} - 18q^{70} - 24q^{73} - 96q^{82} - 30q^{88} - 8q^{91} - 72q^{94} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 −1.40711 0.141583i 0 1.95991 + 0.398446i −1.91923 + 3.32420i 0 1.55724 + 2.13893i −2.70139 0.838146i 0 3.17121 4.40578i
19.2 −1.35459 0.406311i 0 1.66982 + 1.10077i 1.09169 1.89086i 0 1.40064 2.24459i −1.81467 2.16955i 0 −2.24706 + 2.11777i
19.3 −1.28822 + 0.583515i 0 1.31902 1.50339i −0.245316 + 0.424900i 0 −2.09582 1.61479i −0.821939 + 2.70637i 0 0.0680858 0.690511i
19.4 −1.02917 0.969953i 0 0.118381 + 1.99649i 1.09169 1.89086i 0 −1.40064 + 2.24459i 1.81467 2.16955i 0 −2.95757 + 0.887128i
19.5 −0.826169 1.14780i 0 −0.634890 + 1.89655i −1.91923 + 3.32420i 0 −1.55724 2.13893i 2.70139 0.838146i 0 5.40112 0.543460i
19.6 −0.809233 + 1.15980i 0 −0.690284 1.87710i −1.03180 + 1.78713i 0 2.39181 1.13104i 2.73567 + 0.718419i 0 −1.23775 2.64289i
19.7 −0.599802 + 1.28072i 0 −1.28048 1.53635i 1.03180 1.78713i 0 −2.39181 + 1.13104i 2.73567 0.718419i 0 1.66993 + 2.39337i
19.8 −0.138771 1.40739i 0 −1.96148 + 0.390611i −0.245316 + 0.424900i 0 2.09582 + 1.61479i 0.821939 + 2.70637i 0 0.632043 + 0.286291i
19.9 0.138771 + 1.40739i 0 −1.96148 + 0.390611i 0.245316 0.424900i 0 2.09582 + 1.61479i −0.821939 2.70637i 0 0.632043 + 0.286291i
19.10 0.599802 1.28072i 0 −1.28048 1.53635i −1.03180 + 1.78713i 0 −2.39181 + 1.13104i −2.73567 + 0.718419i 0 1.66993 + 2.39337i
19.11 0.809233 1.15980i 0 −0.690284 1.87710i 1.03180 1.78713i 0 2.39181 1.13104i −2.73567 0.718419i 0 −1.23775 2.64289i
19.12 0.826169 + 1.14780i 0 −0.634890 + 1.89655i 1.91923 3.32420i 0 −1.55724 2.13893i −2.70139 + 0.838146i 0 5.40112 0.543460i
19.13 1.02917 + 0.969953i 0 0.118381 + 1.99649i −1.09169 + 1.89086i 0 −1.40064 + 2.24459i −1.81467 + 2.16955i 0 −2.95757 + 0.887128i
19.14 1.28822 0.583515i 0 1.31902 1.50339i 0.245316 0.424900i 0 −2.09582 1.61479i 0.821939 2.70637i 0 0.0680858 0.690511i
19.15 1.35459 + 0.406311i 0 1.66982 + 1.10077i −1.09169 + 1.89086i 0 1.40064 2.24459i 1.81467 + 2.16955i 0 −2.24706 + 2.11777i
19.16 1.40711 + 0.141583i 0 1.95991 + 0.398446i 1.91923 3.32420i 0 1.55724 + 2.13893i 2.70139 + 0.838146i 0 3.17121 4.40578i
451.1 −1.40711 + 0.141583i 0 1.95991 0.398446i −1.91923 3.32420i 0 1.55724 2.13893i −2.70139 + 0.838146i 0 3.17121 + 4.40578i
451.2 −1.35459 + 0.406311i 0 1.66982 1.10077i 1.09169 + 1.89086i 0 1.40064 + 2.24459i −1.81467 + 2.16955i 0 −2.24706 2.11777i
451.3 −1.28822 0.583515i 0 1.31902 + 1.50339i −0.245316 0.424900i 0 −2.09582 + 1.61479i −0.821939 2.70637i 0 0.0680858 + 0.690511i
451.4 −1.02917 + 0.969953i 0 0.118381 1.99649i 1.09169 + 1.89086i 0 −1.40064 2.24459i 1.81467 + 2.16955i 0 −2.95757 0.887128i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 451.16 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.d odd 2 1 inner
21.g even 6 1 inner
24.f even 2 1 inner
56.m even 6 1 inner
168.be odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.bk.b 32
3.b odd 2 1 inner 504.2.bk.b 32
4.b odd 2 1 2016.2.bs.b 32
7.d odd 6 1 inner 504.2.bk.b 32
8.b even 2 1 2016.2.bs.b 32
8.d odd 2 1 inner 504.2.bk.b 32
12.b even 2 1 2016.2.bs.b 32
21.g even 6 1 inner 504.2.bk.b 32
24.f even 2 1 inner 504.2.bk.b 32
24.h odd 2 1 2016.2.bs.b 32
28.f even 6 1 2016.2.bs.b 32
56.j odd 6 1 2016.2.bs.b 32
56.m even 6 1 inner 504.2.bk.b 32
84.j odd 6 1 2016.2.bs.b 32
168.ba even 6 1 2016.2.bs.b 32
168.be odd 6 1 inner 504.2.bk.b 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.bk.b 32 1.a even 1 1 trivial
504.2.bk.b 32 3.b odd 2 1 inner
504.2.bk.b 32 7.d odd 6 1 inner
504.2.bk.b 32 8.d odd 2 1 inner
504.2.bk.b 32 21.g even 6 1 inner
504.2.bk.b 32 24.f even 2 1 inner
504.2.bk.b 32 56.m even 6 1 inner
504.2.bk.b 32 168.be odd 6 1 inner
2016.2.bs.b 32 4.b odd 2 1
2016.2.bs.b 32 8.b even 2 1
2016.2.bs.b 32 12.b even 2 1
2016.2.bs.b 32 24.h odd 2 1
2016.2.bs.b 32 28.f even 6 1
2016.2.bs.b 32 56.j odd 6 1
2016.2.bs.b 32 84.j odd 6 1
2016.2.bs.b 32 168.ba even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{16} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database