# Properties

 Label 504.2.s.g Level $504$ Weight $2$ Character orbit 504.s Analytic conductor $4.024$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [504,2,Mod(289,504)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(504, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("504.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 168) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 2) q^{5} + ( - \zeta_{6} + 3) q^{7}+O(q^{10})$$ q + (-2*z + 2) * q^5 + (-z + 3) * q^7 $$q + ( - 2 \zeta_{6} + 2) q^{5} + ( - \zeta_{6} + 3) q^{7} - 6 \zeta_{6} q^{11} - 3 q^{13} + 4 \zeta_{6} q^{17} + ( - 5 \zeta_{6} + 5) q^{19} + (4 \zeta_{6} - 4) q^{23} + \zeta_{6} q^{25} + 4 q^{29} - 7 \zeta_{6} q^{31} + ( - 6 \zeta_{6} + 4) q^{35} + ( - 9 \zeta_{6} + 9) q^{37} + 2 q^{41} - q^{43} + ( - 2 \zeta_{6} + 2) q^{47} + ( - 5 \zeta_{6} + 8) q^{49} + 8 \zeta_{6} q^{53} - 12 q^{55} + (10 \zeta_{6} - 10) q^{61} + (6 \zeta_{6} - 6) q^{65} + 15 \zeta_{6} q^{67} + 6 q^{71} + 11 \zeta_{6} q^{73} + ( - 12 \zeta_{6} - 6) q^{77} + (\zeta_{6} - 1) q^{79} - 6 q^{83} + 8 q^{85} + (8 \zeta_{6} - 8) q^{89} + (3 \zeta_{6} - 9) q^{91} - 10 \zeta_{6} q^{95} - 14 q^{97} +O(q^{100})$$ q + (-2*z + 2) * q^5 + (-z + 3) * q^7 - 6*z * q^11 - 3 * q^13 + 4*z * q^17 + (-5*z + 5) * q^19 + (4*z - 4) * q^23 + z * q^25 + 4 * q^29 - 7*z * q^31 + (-6*z + 4) * q^35 + (-9*z + 9) * q^37 + 2 * q^41 - q^43 + (-2*z + 2) * q^47 + (-5*z + 8) * q^49 + 8*z * q^53 - 12 * q^55 + (10*z - 10) * q^61 + (6*z - 6) * q^65 + 15*z * q^67 + 6 * q^71 + 11*z * q^73 + (-12*z - 6) * q^77 + (z - 1) * q^79 - 6 * q^83 + 8 * q^85 + (8*z - 8) * q^89 + (3*z - 9) * q^91 - 10*z * q^95 - 14 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} + 5 q^{7}+O(q^{10})$$ 2 * q + 2 * q^5 + 5 * q^7 $$2 q + 2 q^{5} + 5 q^{7} - 6 q^{11} - 6 q^{13} + 4 q^{17} + 5 q^{19} - 4 q^{23} + q^{25} + 8 q^{29} - 7 q^{31} + 2 q^{35} + 9 q^{37} + 4 q^{41} - 2 q^{43} + 2 q^{47} + 11 q^{49} + 8 q^{53} - 24 q^{55} - 10 q^{61} - 6 q^{65} + 15 q^{67} + 12 q^{71} + 11 q^{73} - 24 q^{77} - q^{79} - 12 q^{83} + 16 q^{85} - 8 q^{89} - 15 q^{91} - 10 q^{95} - 28 q^{97}+O(q^{100})$$ 2 * q + 2 * q^5 + 5 * q^7 - 6 * q^11 - 6 * q^13 + 4 * q^17 + 5 * q^19 - 4 * q^23 + q^25 + 8 * q^29 - 7 * q^31 + 2 * q^35 + 9 * q^37 + 4 * q^41 - 2 * q^43 + 2 * q^47 + 11 * q^49 + 8 * q^53 - 24 * q^55 - 10 * q^61 - 6 * q^65 + 15 * q^67 + 12 * q^71 + 11 * q^73 - 24 * q^77 - q^79 - 12 * q^83 + 16 * q^85 - 8 * q^89 - 15 * q^91 - 10 * q^95 - 28 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/504\mathbb{Z}\right)^\times$$.

 $$n$$ $$73$$ $$127$$ $$253$$ $$281$$ $$\chi(n)$$ $$-1 + \zeta_{6}$$ $$1$$ $$1$$ $$1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 1.00000 1.73205i 0 2.50000 0.866025i 0 0 0
361.1 0 0 0 1.00000 + 1.73205i 0 2.50000 + 0.866025i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c 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.s.g 2
3.b odd 2 1 168.2.q.b 2
4.b odd 2 1 1008.2.s.m 2
7.b odd 2 1 3528.2.s.d 2
7.c even 3 1 inner 504.2.s.g 2
7.c even 3 1 3528.2.a.f 1
7.d odd 6 1 3528.2.a.y 1
7.d odd 6 1 3528.2.s.d 2
12.b even 2 1 336.2.q.a 2
21.c even 2 1 1176.2.q.e 2
21.g even 6 1 1176.2.a.e 1
21.g even 6 1 1176.2.q.e 2
21.h odd 6 1 168.2.q.b 2
21.h odd 6 1 1176.2.a.d 1
24.f even 2 1 1344.2.q.t 2
24.h odd 2 1 1344.2.q.i 2
28.f even 6 1 7056.2.a.bn 1
28.g odd 6 1 1008.2.s.m 2
28.g odd 6 1 7056.2.a.i 1
84.h odd 2 1 2352.2.q.v 2
84.j odd 6 1 2352.2.a.e 1
84.j odd 6 1 2352.2.q.v 2
84.n even 6 1 336.2.q.a 2
84.n even 6 1 2352.2.a.x 1
168.s odd 6 1 1344.2.q.i 2
168.s odd 6 1 9408.2.a.cd 1
168.v even 6 1 1344.2.q.t 2
168.v even 6 1 9408.2.a.f 1
168.ba even 6 1 9408.2.a.bk 1
168.be odd 6 1 9408.2.a.cs 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.q.b 2 3.b odd 2 1
168.2.q.b 2 21.h odd 6 1
336.2.q.a 2 12.b even 2 1
336.2.q.a 2 84.n even 6 1
504.2.s.g 2 1.a even 1 1 trivial
504.2.s.g 2 7.c even 3 1 inner
1008.2.s.m 2 4.b odd 2 1
1008.2.s.m 2 28.g odd 6 1
1176.2.a.d 1 21.h odd 6 1
1176.2.a.e 1 21.g even 6 1
1176.2.q.e 2 21.c even 2 1
1176.2.q.e 2 21.g even 6 1
1344.2.q.i 2 24.h odd 2 1
1344.2.q.i 2 168.s odd 6 1
1344.2.q.t 2 24.f even 2 1
1344.2.q.t 2 168.v even 6 1
2352.2.a.e 1 84.j odd 6 1
2352.2.a.x 1 84.n even 6 1
2352.2.q.v 2 84.h odd 2 1
2352.2.q.v 2 84.j odd 6 1
3528.2.a.f 1 7.c even 3 1
3528.2.a.y 1 7.d odd 6 1
3528.2.s.d 2 7.b odd 2 1
3528.2.s.d 2 7.d odd 6 1
7056.2.a.i 1 28.g odd 6 1
7056.2.a.bn 1 28.f even 6 1
9408.2.a.f 1 168.v even 6 1
9408.2.a.bk 1 168.ba even 6 1
9408.2.a.cd 1 168.s odd 6 1
9408.2.a.cs 1 168.be odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$:

 $$T_{5}^{2} - 2T_{5} + 4$$ T5^2 - 2*T5 + 4 $$T_{11}^{2} + 6T_{11} + 36$$ T11^2 + 6*T11 + 36 $$T_{13} + 3$$ T13 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 2T + 4$$
$7$ $$T^{2} - 5T + 7$$
$11$ $$T^{2} + 6T + 36$$
$13$ $$(T + 3)^{2}$$
$17$ $$T^{2} - 4T + 16$$
$19$ $$T^{2} - 5T + 25$$
$23$ $$T^{2} + 4T + 16$$
$29$ $$(T - 4)^{2}$$
$31$ $$T^{2} + 7T + 49$$
$37$ $$T^{2} - 9T + 81$$
$41$ $$(T - 2)^{2}$$
$43$ $$(T + 1)^{2}$$
$47$ $$T^{2} - 2T + 4$$
$53$ $$T^{2} - 8T + 64$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 10T + 100$$
$67$ $$T^{2} - 15T + 225$$
$71$ $$(T - 6)^{2}$$
$73$ $$T^{2} - 11T + 121$$
$79$ $$T^{2} + T + 1$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} + 8T + 64$$
$97$ $$(T + 14)^{2}$$