# Properties

 Label 504.6.a.w Level $504$ Weight $6$ Character orbit 504.a Self dual yes Analytic conductor $80.833$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{429})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 107$$ x^2 - x - 107 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{429}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta + 40) q^{5} + 49 q^{7}+O(q^{10})$$ q + (-b + 40) * q^5 + 49 * q^7 $$q + ( - \beta + 40) q^{5} + 49 q^{7} + ( - 3 \beta + 216) q^{11} + (16 \beta - 438) q^{13} + ( - 15 \beta - 424) q^{17} + (16 \beta - 1676) q^{19} + ( - \beta - 2648) q^{23} + ( - 80 \beta + 191) q^{25} + (188 \beta - 128) q^{29} + (208 \beta - 428) q^{31} + ( - 49 \beta + 1960) q^{35} + (176 \beta + 1818) q^{37} + ( - 405 \beta - 1528) q^{41} + ( - 224 \beta - 13108) q^{43} + (82 \beta + 1456) q^{47} + 2401 q^{49} + (306 \beta - 13616) q^{53} + ( - 336 \beta + 13788) q^{55} + ( - 394 \beta + 29520) q^{59} + (560 \beta - 20210) q^{61} + (1078 \beta - 44976) q^{65} + (256 \beta + 26460) q^{67} + ( - 1219 \beta - 26376) q^{71} + ( - 1440 \beta + 9406) q^{73} + ( - 147 \beta + 10584) q^{77} + ( - 864 \beta + 25144) q^{79} + (464 \beta - 33024) q^{83} + ( - 176 \beta + 8780) q^{85} + (839 \beta - 65752) q^{89} + (784 \beta - 21462) q^{91} + (2316 \beta - 94496) q^{95} + (416 \beta + 82174) q^{97}+O(q^{100})$$ q + (-b + 40) * q^5 + 49 * q^7 + (-3*b + 216) * q^11 + (16*b - 438) * q^13 + (-15*b - 424) * q^17 + (16*b - 1676) * q^19 + (-b - 2648) * q^23 + (-80*b + 191) * q^25 + (188*b - 128) * q^29 + (208*b - 428) * q^31 + (-49*b + 1960) * q^35 + (176*b + 1818) * q^37 + (-405*b - 1528) * q^41 + (-224*b - 13108) * q^43 + (82*b + 1456) * q^47 + 2401 * q^49 + (306*b - 13616) * q^53 + (-336*b + 13788) * q^55 + (-394*b + 29520) * q^59 + (560*b - 20210) * q^61 + (1078*b - 44976) * q^65 + (256*b + 26460) * q^67 + (-1219*b - 26376) * q^71 + (-1440*b + 9406) * q^73 + (-147*b + 10584) * q^77 + (-864*b + 25144) * q^79 + (464*b - 33024) * q^83 + (-176*b + 8780) * q^85 + (839*b - 65752) * q^89 + (784*b - 21462) * q^91 + (2316*b - 94496) * q^95 + (416*b + 82174) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 80 q^{5} + 98 q^{7}+O(q^{10})$$ 2 * q + 80 * q^5 + 98 * q^7 $$2 q + 80 q^{5} + 98 q^{7} + 432 q^{11} - 876 q^{13} - 848 q^{17} - 3352 q^{19} - 5296 q^{23} + 382 q^{25} - 256 q^{29} - 856 q^{31} + 3920 q^{35} + 3636 q^{37} - 3056 q^{41} - 26216 q^{43} + 2912 q^{47} + 4802 q^{49} - 27232 q^{53} + 27576 q^{55} + 59040 q^{59} - 40420 q^{61} - 89952 q^{65} + 52920 q^{67} - 52752 q^{71} + 18812 q^{73} + 21168 q^{77} + 50288 q^{79} - 66048 q^{83} + 17560 q^{85} - 131504 q^{89} - 42924 q^{91} - 188992 q^{95} + 164348 q^{97}+O(q^{100})$$ 2 * q + 80 * q^5 + 98 * q^7 + 432 * q^11 - 876 * q^13 - 848 * q^17 - 3352 * q^19 - 5296 * q^23 + 382 * q^25 - 256 * q^29 - 856 * q^31 + 3920 * q^35 + 3636 * q^37 - 3056 * q^41 - 26216 * q^43 + 2912 * q^47 + 4802 * q^49 - 27232 * q^53 + 27576 * q^55 + 59040 * q^59 - 40420 * q^61 - 89952 * q^65 + 52920 * q^67 - 52752 * q^71 + 18812 * q^73 + 21168 * q^77 + 50288 * q^79 - 66048 * q^83 + 17560 * q^85 - 131504 * q^89 - 42924 * q^91 - 188992 * q^95 + 164348 * 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
 10.8562 −9.85616
0 0 0 −1.42463 0 49.0000 0 0 0
1.2 0 0 0 81.4246 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.w yes 2
3.b odd 2 1 504.6.a.j 2
4.b odd 2 1 1008.6.a.bx 2
12.b even 2 1 1008.6.a.be 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.6.a.j 2 3.b odd 2 1
504.6.a.w yes 2 1.a even 1 1 trivial
1008.6.a.be 2 12.b even 2 1
1008.6.a.bx 2 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(504))$$:

 $$T_{5}^{2} - 80T_{5} - 116$$ T5^2 - 80*T5 - 116 $$T_{11}^{2} - 432T_{11} + 31212$$ T11^2 - 432*T11 + 31212

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 80T - 116$$
$7$ $$(T - 49)^{2}$$
$11$ $$T^{2} - 432T + 31212$$
$13$ $$T^{2} + 876T - 247452$$
$17$ $$T^{2} + 848T - 206324$$
$19$ $$T^{2} + 3352 T + 2369680$$
$23$ $$T^{2} + 5296 T + 7010188$$
$29$ $$T^{2} + 256 T - 60633920$$
$31$ $$T^{2} + 856 T - 74057840$$
$37$ $$T^{2} - 3636 T - 49849692$$
$41$ $$T^{2} + 3056 T - 279132116$$
$43$ $$T^{2} + 26216 T + 85717648$$
$47$ $$T^{2} - 2912 T - 9418448$$
$53$ $$T^{2} + 27232 T + 24716080$$
$59$ $$T^{2} - 59040 T + 605045424$$
$61$ $$T^{2} + 40420 T - 129693500$$
$67$ $$T^{2} - 52920 T + 587671824$$
$71$ $$T^{2} + \cdots - 1854215700$$
$73$ $$T^{2} + \cdots - 3469824764$$
$79$ $$T^{2} - 50288 T - 648766400$$
$83$ $$T^{2} + 66048 T + 721136640$$
$89$ $$T^{2} + \cdots + 3115397068$$
$97$ $$T^{2} + \cdots + 6455602180$$