# Properties

 Label 504.2.i.b Level 504 Weight 2 Character orbit 504.i Analytic conductor 4.024 Analytic rank 0 Dimension 24 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.i (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$24$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 48q^{22} - 72q^{25} - 24q^{28} - 72q^{49} + 48q^{58} + 96q^{64} - 24q^{70} + 48q^{79} - 144q^{88} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
125.1 −1.39273 0.245576i 0 1.87939 + 0.684040i 2.37285i 0 −1.53209 + 2.15701i −2.44949 1.41421i 0 −0.582714 + 3.30474i
125.2 −1.39273 0.245576i 0 1.87939 + 0.684040i 2.37285i 0 −1.53209 2.15701i −2.44949 1.41421i 0 0.582714 3.30474i
125.3 −1.39273 + 0.245576i 0 1.87939 0.684040i 2.37285i 0 −1.53209 + 2.15701i −2.44949 + 1.41421i 0 0.582714 + 3.30474i
125.4 −1.39273 + 0.245576i 0 1.87939 0.684040i 2.37285i 0 −1.53209 2.15701i −2.44949 + 1.41421i 0 −0.582714 3.30474i
125.5 −0.909039 1.08335i 0 −0.347296 + 1.96962i 3.85006i 0 1.87939 1.86223i 2.44949 1.41421i 0 −4.17096 + 3.49985i
125.6 −0.909039 1.08335i 0 −0.347296 + 1.96962i 3.85006i 0 1.87939 + 1.86223i 2.44949 1.41421i 0 4.17096 3.49985i
125.7 −0.909039 + 1.08335i 0 −0.347296 1.96962i 3.85006i 0 1.87939 1.86223i 2.44949 + 1.41421i 0 4.17096 + 3.49985i
125.8 −0.909039 + 1.08335i 0 −0.347296 1.96962i 3.85006i 0 1.87939 + 1.86223i 2.44949 + 1.41421i 0 −4.17096 3.49985i
125.9 −0.483690 1.32893i 0 −1.53209 + 1.28558i 1.88325i 0 −0.347296 + 2.62286i 2.44949 + 1.41421i 0 −2.50270 + 0.910909i
125.10 −0.483690 1.32893i 0 −1.53209 + 1.28558i 1.88325i 0 −0.347296 2.62286i 2.44949 + 1.41421i 0 2.50270 0.910909i
125.11 −0.483690 + 1.32893i 0 −1.53209 1.28558i 1.88325i 0 −0.347296 + 2.62286i 2.44949 1.41421i 0 2.50270 + 0.910909i
125.12 −0.483690 + 1.32893i 0 −1.53209 1.28558i 1.88325i 0 −0.347296 2.62286i 2.44949 1.41421i 0 −2.50270 0.910909i
125.13 0.483690 1.32893i 0 −1.53209 1.28558i 1.88325i 0 −0.347296 2.62286i −2.44949 + 1.41421i 0 −2.50270 0.910909i
125.14 0.483690 1.32893i 0 −1.53209 1.28558i 1.88325i 0 −0.347296 + 2.62286i −2.44949 + 1.41421i 0 2.50270 + 0.910909i
125.15 0.483690 + 1.32893i 0 −1.53209 + 1.28558i 1.88325i 0 −0.347296 2.62286i −2.44949 1.41421i 0 2.50270 0.910909i
125.16 0.483690 + 1.32893i 0 −1.53209 + 1.28558i 1.88325i 0 −0.347296 + 2.62286i −2.44949 1.41421i 0 −2.50270 + 0.910909i
125.17 0.909039 1.08335i 0 −0.347296 1.96962i 3.85006i 0 1.87939 + 1.86223i −2.44949 1.41421i 0 −4.17096 3.49985i
125.18 0.909039 1.08335i 0 −0.347296 1.96962i 3.85006i 0 1.87939 1.86223i −2.44949 1.41421i 0 4.17096 + 3.49985i
125.19 0.909039 + 1.08335i 0 −0.347296 + 1.96962i 3.85006i 0 1.87939 + 1.86223i −2.44949 + 1.41421i 0 4.17096 3.49985i
125.20 0.909039 + 1.08335i 0 −0.347296 + 1.96962i 3.85006i 0 1.87939 1.86223i −2.44949 + 1.41421i 0 −4.17096 + 3.49985i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 125.24 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.b odd 2 1 inner
8.b even 2 1 inner
21.c even 2 1 inner
24.h odd 2 1 inner
56.h odd 2 1 inner
168.i even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.i.b 24
3.b odd 2 1 inner 504.2.i.b 24
4.b odd 2 1 2016.2.i.b 24
7.b odd 2 1 inner 504.2.i.b 24
8.b even 2 1 inner 504.2.i.b 24
8.d odd 2 1 2016.2.i.b 24
12.b even 2 1 2016.2.i.b 24
21.c even 2 1 inner 504.2.i.b 24
24.f even 2 1 2016.2.i.b 24
24.h odd 2 1 inner 504.2.i.b 24
28.d even 2 1 2016.2.i.b 24
56.e even 2 1 2016.2.i.b 24
56.h odd 2 1 inner 504.2.i.b 24
84.h odd 2 1 2016.2.i.b 24
168.e odd 2 1 2016.2.i.b 24
168.i even 2 1 inner 504.2.i.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.i.b 24 1.a even 1 1 trivial
504.2.i.b 24 3.b odd 2 1 inner
504.2.i.b 24 7.b odd 2 1 inner
504.2.i.b 24 8.b even 2 1 inner
504.2.i.b 24 21.c even 2 1 inner
504.2.i.b 24 24.h odd 2 1 inner
504.2.i.b 24 56.h odd 2 1 inner
504.2.i.b 24 168.i even 2 1 inner
2016.2.i.b 24 4.b odd 2 1
2016.2.i.b 24 8.d odd 2 1
2016.2.i.b 24 12.b even 2 1
2016.2.i.b 24 24.f even 2 1
2016.2.i.b 24 28.d even 2 1
2016.2.i.b 24 56.e even 2 1
2016.2.i.b 24 84.h odd 2 1
2016.2.i.b 24 168.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} + 24 T_{5}^{4} + 156 T_{5}^{2} + 296$$ acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database