# Properties

 Label 504.2.q.c Level $504$ Weight $2$ Character orbit 504.q Analytic conductor $4.024$ Analytic rank $0$ Dimension $22$ 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.q (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$22$$ Relative dimension: $$11$$ over $$\Q(\zeta_{3})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$22q - 2q^{3} + q^{5} + 5q^{7} + 6q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$22q - 2q^{3} + q^{5} + 5q^{7} + 6q^{9} + 3q^{11} + 7q^{13} - q^{15} - q^{17} + 13q^{19} - 22q^{25} - 2q^{27} - 7q^{29} - 12q^{31} - 3q^{33} + 2q^{35} + 6q^{37} - 4q^{39} + 4q^{41} + 2q^{43} - 3q^{45} - 34q^{47} - 25q^{49} + 53q^{51} + q^{53} + 2q^{55} - 21q^{57} + 42q^{59} - 62q^{61} - 22q^{63} + 6q^{65} + 52q^{67} - 40q^{69} - 32q^{71} + 17q^{73} + 53q^{75} - q^{77} + 32q^{79} - 6q^{81} - 36q^{83} + 28q^{85} - 5q^{87} - 2q^{89} + 15q^{91} - 11q^{93} + 48q^{95} + 19q^{97} + 24q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1 0 −1.71769 + 0.222607i 0 0.234085 0.405446i 0 0.212345 + 2.63722i 0 2.90089 0.764737i 0
25.2 0 −1.69989 0.332219i 0 −1.59750 + 2.76695i 0 −1.66645 2.05498i 0 2.77926 + 1.12947i 0
25.3 0 −1.46869 0.918117i 0 1.89970 3.29038i 0 −0.841809 2.50826i 0 1.31412 + 2.69686i 0
25.4 0 −0.914508 + 1.47094i 0 0.891774 1.54460i 0 −2.54386 0.727153i 0 −1.32735 2.69038i 0
25.5 0 −0.704143 1.58246i 0 −1.05220 + 1.82246i 0 2.58382 + 0.569079i 0 −2.00837 + 2.22856i 0
25.6 0 −0.633073 + 1.61221i 0 −1.70368 + 2.95086i 0 −0.410295 + 2.61374i 0 −2.19844 2.04129i 0
25.7 0 0.748111 + 1.56216i 0 2.11148 3.65719i 0 2.19338 + 1.47956i 0 −1.88066 + 2.33733i 0
25.8 0 0.987504 + 1.42297i 0 −1.38590 + 2.40045i 0 1.74026 1.99286i 0 −1.04967 + 2.81037i 0
25.9 0 1.04182 1.38370i 0 1.33425 2.31099i 0 2.54743 0.714566i 0 −0.829236 2.88312i 0
25.10 0 1.64138 0.553060i 0 −0.263002 + 0.455533i 0 −0.333150 2.62469i 0 2.38825 1.81556i 0
25.11 0 1.71918 + 0.210723i 0 0.0309846 0.0536670i 0 −0.981674 + 2.45689i 0 2.91119 + 0.724543i 0
121.1 0 −1.71769 0.222607i 0 0.234085 + 0.405446i 0 0.212345 2.63722i 0 2.90089 + 0.764737i 0
121.2 0 −1.69989 + 0.332219i 0 −1.59750 2.76695i 0 −1.66645 + 2.05498i 0 2.77926 1.12947i 0
121.3 0 −1.46869 + 0.918117i 0 1.89970 + 3.29038i 0 −0.841809 + 2.50826i 0 1.31412 2.69686i 0
121.4 0 −0.914508 1.47094i 0 0.891774 + 1.54460i 0 −2.54386 + 0.727153i 0 −1.32735 + 2.69038i 0
121.5 0 −0.704143 + 1.58246i 0 −1.05220 1.82246i 0 2.58382 0.569079i 0 −2.00837 2.22856i 0
121.6 0 −0.633073 1.61221i 0 −1.70368 2.95086i 0 −0.410295 2.61374i 0 −2.19844 + 2.04129i 0
121.7 0 0.748111 1.56216i 0 2.11148 + 3.65719i 0 2.19338 1.47956i 0 −1.88066 2.33733i 0
121.8 0 0.987504 1.42297i 0 −1.38590 2.40045i 0 1.74026 + 1.99286i 0 −1.04967 2.81037i 0
121.9 0 1.04182 + 1.38370i 0 1.33425 + 2.31099i 0 2.54743 + 0.714566i 0 −0.829236 + 2.88312i 0
See all 22 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 121.11 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.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.q.c 22
3.b odd 2 1 1512.2.q.d 22
4.b odd 2 1 1008.2.q.l 22
7.c even 3 1 504.2.t.c yes 22
9.c even 3 1 504.2.t.c yes 22
9.d odd 6 1 1512.2.t.c 22
12.b even 2 1 3024.2.q.l 22
21.h odd 6 1 1512.2.t.c 22
28.g odd 6 1 1008.2.t.l 22
36.f odd 6 1 1008.2.t.l 22
36.h even 6 1 3024.2.t.k 22
63.h even 3 1 inner 504.2.q.c 22
63.j odd 6 1 1512.2.q.d 22
84.n even 6 1 3024.2.t.k 22
252.u odd 6 1 1008.2.q.l 22
252.bb even 6 1 3024.2.q.l 22

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.q.c 22 1.a even 1 1 trivial
504.2.q.c 22 63.h even 3 1 inner
504.2.t.c yes 22 7.c even 3 1
504.2.t.c yes 22 9.c even 3 1
1008.2.q.l 22 4.b odd 2 1
1008.2.q.l 22 252.u odd 6 1
1008.2.t.l 22 28.g odd 6 1
1008.2.t.l 22 36.f odd 6 1
1512.2.q.d 22 3.b odd 2 1
1512.2.q.d 22 63.j odd 6 1
1512.2.t.c 22 9.d odd 6 1
1512.2.t.c 22 21.h odd 6 1
3024.2.q.l 22 12.b even 2 1
3024.2.q.l 22 252.bb even 6 1
3024.2.t.k 22 36.h even 6 1
3024.2.t.k 22 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{22} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database