# Properties

 Label 504.6.a.a Level $504$ Weight $6$ Character orbit 504.a Self dual yes Analytic conductor $80.833$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [504,6,Mod(1,504)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(504, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("504.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 504.a (trivial)

## 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: yes Analytic conductor: $$80.8334451857$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 168) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 74 q^{5} + 49 q^{7}+O(q^{10})$$ q - 74 * q^5 + 49 * q^7 $$q - 74 q^{5} + 49 q^{7} - 216 q^{11} - 186 q^{13} - 1078 q^{17} - 908 q^{19} + 2236 q^{23} + 2351 q^{25} - 5366 q^{29} - 536 q^{31} - 3626 q^{35} + 3798 q^{37} - 18598 q^{41} + 15308 q^{43} - 23480 q^{47} + 2401 q^{49} - 9062 q^{53} + 15984 q^{55} + 49284 q^{59} + 17806 q^{61} + 13764 q^{65} + 24876 q^{67} - 3468 q^{71} - 32414 q^{73} - 10584 q^{77} + 25384 q^{79} + 67284 q^{83} + 79772 q^{85} + 698 q^{89} - 9114 q^{91} + 67192 q^{95} + 154906 q^{97}+O(q^{100})$$ q - 74 * q^5 + 49 * q^7 - 216 * q^11 - 186 * q^13 - 1078 * q^17 - 908 * q^19 + 2236 * q^23 + 2351 * q^25 - 5366 * q^29 - 536 * q^31 - 3626 * q^35 + 3798 * q^37 - 18598 * q^41 + 15308 * q^43 - 23480 * q^47 + 2401 * q^49 - 9062 * q^53 + 15984 * q^55 + 49284 * q^59 + 17806 * q^61 + 13764 * q^65 + 24876 * q^67 - 3468 * q^71 - 32414 * q^73 - 10584 * q^77 + 25384 * q^79 + 67284 * q^83 + 79772 * q^85 + 698 * q^89 - 9114 * q^91 + 67192 * q^95 + 154906 * q^97

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 −74.0000 0 49.0000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.6.a.a 1
3.b odd 2 1 168.6.a.c 1
4.b odd 2 1 1008.6.a.e 1
12.b even 2 1 336.6.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.6.a.c 1 3.b odd 2 1
336.6.a.p 1 12.b even 2 1
504.6.a.a 1 1.a even 1 1 trivial
1008.6.a.e 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(504))$$:

 $$T_{5} + 74$$ T5 + 74 $$T_{11} + 216$$ T11 + 216

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 74$$
$7$ $$T - 49$$
$11$ $$T + 216$$
$13$ $$T + 186$$
$17$ $$T + 1078$$
$19$ $$T + 908$$
$23$ $$T - 2236$$
$29$ $$T + 5366$$
$31$ $$T + 536$$
$37$ $$T - 3798$$
$41$ $$T + 18598$$
$43$ $$T - 15308$$
$47$ $$T + 23480$$
$53$ $$T + 9062$$
$59$ $$T - 49284$$
$61$ $$T - 17806$$
$67$ $$T - 24876$$
$71$ $$T + 3468$$
$73$ $$T + 32414$$
$79$ $$T - 25384$$
$83$ $$T - 67284$$
$89$ $$T - 698$$
$97$ $$T - 154906$$