# Properties

 Label 504.2.cx.a Level 504 Weight 2 Character orbit 504.cx Analytic conductor 4.024 Analytic rank 0 Dimension 48 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$48q + 2q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 2q^{9} + 8q^{15} - 10q^{21} + 48q^{25} + 18q^{27} + 18q^{29} + 18q^{31} + 12q^{33} - 4q^{39} - 6q^{41} - 6q^{43} - 18q^{45} + 18q^{47} - 12q^{49} + 6q^{51} - 12q^{53} + 4q^{57} + 18q^{61} - 32q^{63} - 36q^{65} - 12q^{77} + 6q^{79} + 6q^{81} - 54q^{87} - 18q^{89} + 6q^{91} + 4q^{93} - 54q^{95} - 64q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
185.1 0 −1.70958 0.278106i 0 −0.542075 0 −2.62378 + 0.340238i 0 2.84531 + 0.950888i 0
185.2 0 −1.69924 + 0.335557i 0 −3.64567 0 1.05524 + 2.42620i 0 2.77480 1.14038i 0
185.3 0 −1.67230 + 0.451026i 0 −0.203178 0 −1.27132 2.32029i 0 2.59315 1.50850i 0
185.4 0 −1.46327 + 0.926729i 0 4.09884 0 −0.146661 2.64168i 0 1.28235 2.71212i 0
185.5 0 −1.45594 0.938214i 0 −1.81173 0 1.69266 2.03345i 0 1.23951 + 2.73196i 0
185.6 0 −1.14647 1.29831i 0 −0.0525740 0 2.44149 1.01937i 0 −0.371197 + 2.97695i 0
185.7 0 −1.11231 1.32769i 0 3.53401 0 1.30083 + 2.30388i 0 −0.525517 + 2.95361i 0
185.8 0 −1.03050 + 1.39215i 0 2.20884 0 2.16520 + 1.52049i 0 −0.876153 2.86921i 0
185.9 0 −0.859090 + 1.50398i 0 −4.19811 0 1.67151 2.05087i 0 −1.52393 2.58411i 0
185.10 0 −0.601162 + 1.62438i 0 −0.207028 0 −1.37075 + 2.26297i 0 −2.27721 1.95303i 0
185.11 0 −0.498607 1.65873i 0 −0.623597 0 −0.996837 + 2.45078i 0 −2.50278 + 1.65411i 0
185.12 0 −0.210634 + 1.71920i 0 −1.07485 0 −1.26701 2.32265i 0 −2.91127 0.724244i 0
185.13 0 0.149059 1.72562i 0 −2.22094 0 −2.45091 0.996507i 0 −2.95556 0.514438i 0
185.14 0 0.310783 1.70394i 0 2.76937 0 −1.21939 2.34800i 0 −2.80683 1.05911i 0
185.15 0 0.549450 1.64259i 0 −1.05582 0 1.79851 + 1.94045i 0 −2.39621 1.80504i 0
185.16 0 0.930597 + 1.46082i 0 1.36828 0 2.64451 + 0.0810554i 0 −1.26798 + 2.71887i 0
185.17 0 0.958980 + 1.44234i 0 4.11484 0 −2.54819 + 0.711830i 0 −1.16071 + 2.76636i 0
185.18 0 0.994080 + 1.41838i 0 −1.58600 0 −1.06431 + 2.42224i 0 −1.02361 + 2.81997i 0
185.19 0 1.24547 1.20367i 0 2.04899 0 1.41312 2.23676i 0 0.102371 2.99825i 0
185.20 0 1.51239 0.844203i 0 −2.84763 0 −2.64481 + 0.0704652i 0 1.57464 2.55353i 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 425.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.s 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.cx.a yes 48
3.b odd 2 1 1512.2.cx.a 48
4.b odd 2 1 1008.2.df.e 48
7.d odd 6 1 504.2.bs.a 48
9.c even 3 1 1512.2.bs.a 48
9.d odd 6 1 504.2.bs.a 48
12.b even 2 1 3024.2.df.e 48
21.g even 6 1 1512.2.bs.a 48
28.f even 6 1 1008.2.ca.e 48
36.f odd 6 1 3024.2.ca.e 48
36.h even 6 1 1008.2.ca.e 48
63.k odd 6 1 1512.2.cx.a 48
63.s even 6 1 inner 504.2.cx.a yes 48
84.j odd 6 1 3024.2.ca.e 48
252.n even 6 1 3024.2.df.e 48
252.bn odd 6 1 1008.2.df.e 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.bs.a 48 7.d odd 6 1
504.2.bs.a 48 9.d odd 6 1
504.2.cx.a yes 48 1.a even 1 1 trivial
504.2.cx.a yes 48 63.s even 6 1 inner
1008.2.ca.e 48 28.f even 6 1
1008.2.ca.e 48 36.h even 6 1
1008.2.df.e 48 4.b odd 2 1
1008.2.df.e 48 252.bn odd 6 1
1512.2.bs.a 48 9.c even 3 1
1512.2.bs.a 48 21.g even 6 1
1512.2.cx.a 48 3.b odd 2 1
1512.2.cx.a 48 63.k odd 6 1
3024.2.ca.e 48 36.f odd 6 1
3024.2.ca.e 48 84.j odd 6 1
3024.2.df.e 48 12.b even 2 1
3024.2.df.e 48 252.n even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database