# Properties

 Label 504.2.t.c Level 504 Weight 2 Character orbit 504.t 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.t (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$22$$ Relative dimension: $$11$$ over $$\Q(\zeta_{3})$$ Coefficient ring index: multiple of None 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.

 $$\operatorname{Tr}(f)(q) =$$ $$22q - 2q^{3} - 2q^{5} - q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$22q - 2q^{3} - 2q^{5} - q^{7} - 6q^{11} + 7q^{13} - q^{15} - q^{17} + 13q^{19} + 33q^{21} + 44q^{25} - 2q^{27} - 7q^{29} + 6q^{31} + 9q^{33} + 2q^{35} + 6q^{37} - 4q^{39} + 4q^{41} + 2q^{43} + 17q^{47} + 29q^{49} - 25q^{51} + q^{53} + 2q^{55} - 21q^{57} - 21q^{59} + 31q^{61} - 7q^{63} - 3q^{65} - 26q^{67} - 40q^{69} - 32q^{71} + 17q^{73} - 16q^{75} - 4q^{77} - 16q^{79} - 36q^{83} + 28q^{85} + 7q^{87} - 2q^{89} + 15q^{91} - 56q^{93} - 24q^{95} + 19q^{97} + 24q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1 0 −1.72692 + 0.133195i 0 −4.22296 0 −2.37802 + 1.15974i 0 2.96452 0.460034i 0
193.2 0 −1.72608 0.143720i 0 2.77180 0 0.855737 + 2.50354i 0 2.95869 + 0.496145i 0
193.3 0 −1.07968 + 1.35436i 0 3.40736 0 −2.05842 1.66220i 0 −0.668594 2.92455i 0
193.4 0 −1.04208 1.38350i 0 −0.0619693 0 −1.63689 2.07860i 0 −0.828124 + 2.88344i 0
193.5 0 −0.816621 + 1.52746i 0 −1.78355 0 1.90167 1.83948i 0 −1.66626 2.49471i 0
193.6 0 −0.341725 1.69801i 0 0.526004 0 2.43963 + 1.02383i 0 −2.76645 + 1.16050i 0
193.7 0 0.666060 + 1.59886i 0 −0.468169 0 −2.39007 1.13471i 0 −2.11273 + 2.12988i 0
193.8 0 0.677409 1.59409i 0 −2.66851 0 −0.654882 + 2.56342i 0 −2.08224 2.15970i 0
193.9 0 1.13766 + 1.30604i 0 3.19500 0 2.61289 0.415693i 0 −0.411479 + 2.97165i 0
193.10 0 1.52946 + 0.812868i 0 −3.79940 0 2.59312 + 0.525101i 0 1.67849 + 2.48650i 0
193.11 0 1.72252 0.181425i 0 2.10440 0 −1.78475 + 1.95312i 0 2.93417 0.625017i 0
457.1 0 −1.72692 0.133195i 0 −4.22296 0 −2.37802 1.15974i 0 2.96452 + 0.460034i 0
457.2 0 −1.72608 + 0.143720i 0 2.77180 0 0.855737 2.50354i 0 2.95869 0.496145i 0
457.3 0 −1.07968 1.35436i 0 3.40736 0 −2.05842 + 1.66220i 0 −0.668594 + 2.92455i 0
457.4 0 −1.04208 + 1.38350i 0 −0.0619693 0 −1.63689 + 2.07860i 0 −0.828124 2.88344i 0
457.5 0 −0.816621 1.52746i 0 −1.78355 0 1.90167 + 1.83948i 0 −1.66626 + 2.49471i 0
457.6 0 −0.341725 + 1.69801i 0 0.526004 0 2.43963 1.02383i 0 −2.76645 1.16050i 0
457.7 0 0.666060 1.59886i 0 −0.468169 0 −2.39007 + 1.13471i 0 −2.11273 2.12988i 0
457.8 0 0.677409 + 1.59409i 0 −2.66851 0 −0.654882 2.56342i 0 −2.08224 + 2.15970i 0
457.9 0 1.13766 1.30604i 0 3.19500 0 2.61289 + 0.415693i 0 −0.411479 2.97165i 0
See all 22 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 457.11 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.g 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.t.c yes 22
3.b odd 2 1 1512.2.t.c 22
4.b odd 2 1 1008.2.t.l 22
7.c even 3 1 504.2.q.c 22
9.c even 3 1 504.2.q.c 22
9.d odd 6 1 1512.2.q.d 22
12.b even 2 1 3024.2.t.k 22
21.h odd 6 1 1512.2.q.d 22
28.g odd 6 1 1008.2.q.l 22
36.f odd 6 1 1008.2.q.l 22
36.h even 6 1 3024.2.q.l 22
63.g even 3 1 inner 504.2.t.c yes 22
63.n odd 6 1 1512.2.t.c 22
84.n even 6 1 3024.2.q.l 22
252.o even 6 1 3024.2.t.k 22
252.bl odd 6 1 1008.2.t.l 22

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{11} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database