# Properties

 Label 504.6.a.n 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: $$\Q(\sqrt{114})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 114$$ x^2 - 114 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 3$$ 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 $$\beta = 6\sqrt{114}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 14) q^{5} - 49 q^{7}+O(q^{10})$$ q + (b - 14) * q^5 - 49 * q^7 $$q + (\beta - 14) q^{5} - 49 q^{7} + ( - 7 \beta + 298) q^{11} + (4 \beta + 266) q^{13} + (11 \beta - 1050) q^{17} + ( - 12 \beta - 1372) q^{19} + ( - 21 \beta + 1830) q^{23} + ( - 28 \beta + 1175) q^{25} + (56 \beta + 360) q^{29} + ( - 92 \beta + 1988) q^{31} + ( - 49 \beta + 686) q^{35} + ( - 140 \beta + 3394) q^{37} + (89 \beta + 2002) q^{41} + ( - 112 \beta + 9740) q^{43} + ( - 158 \beta - 10780) q^{47} + 2401 q^{49} + ( - 14 \beta - 28788) q^{53} + (396 \beta - 32900) q^{55} + ( - 58 \beta - 11172) q^{59} + (380 \beta - 19362) q^{61} + (210 \beta + 12692) q^{65} + (504 \beta + 3644) q^{67} + ( - 175 \beta - 1966) q^{71} + (616 \beta + 9646) q^{73} + (343 \beta - 14602) q^{77} + ( - 392 \beta + 7816) q^{79} + (456 \beta - 11312) q^{83} + ( - 1204 \beta + 59844) q^{85} + ( - 235 \beta - 56406) q^{89} + ( - 196 \beta - 13034) q^{91} + ( - 1204 \beta - 30040) q^{95} + (1448 \beta - 18018) q^{97}+O(q^{100})$$ q + (b - 14) * q^5 - 49 * q^7 + (-7*b + 298) * q^11 + (4*b + 266) * q^13 + (11*b - 1050) * q^17 + (-12*b - 1372) * q^19 + (-21*b + 1830) * q^23 + (-28*b + 1175) * q^25 + (56*b + 360) * q^29 + (-92*b + 1988) * q^31 + (-49*b + 686) * q^35 + (-140*b + 3394) * q^37 + (89*b + 2002) * q^41 + (-112*b + 9740) * q^43 + (-158*b - 10780) * q^47 + 2401 * q^49 + (-14*b - 28788) * q^53 + (396*b - 32900) * q^55 + (-58*b - 11172) * q^59 + (380*b - 19362) * q^61 + (210*b + 12692) * q^65 + (504*b + 3644) * q^67 + (-175*b - 1966) * q^71 + (616*b + 9646) * q^73 + (343*b - 14602) * q^77 + (-392*b + 7816) * q^79 + (456*b - 11312) * q^83 + (-1204*b + 59844) * q^85 + (-235*b - 56406) * q^89 + (-196*b - 13034) * q^91 + (-1204*b - 30040) * q^95 + (1448*b - 18018) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 28 q^{5} - 98 q^{7}+O(q^{10})$$ 2 * q - 28 * q^5 - 98 * q^7 $$2 q - 28 q^{5} - 98 q^{7} + 596 q^{11} + 532 q^{13} - 2100 q^{17} - 2744 q^{19} + 3660 q^{23} + 2350 q^{25} + 720 q^{29} + 3976 q^{31} + 1372 q^{35} + 6788 q^{37} + 4004 q^{41} + 19480 q^{43} - 21560 q^{47} + 4802 q^{49} - 57576 q^{53} - 65800 q^{55} - 22344 q^{59} - 38724 q^{61} + 25384 q^{65} + 7288 q^{67} - 3932 q^{71} + 19292 q^{73} - 29204 q^{77} + 15632 q^{79} - 22624 q^{83} + 119688 q^{85} - 112812 q^{89} - 26068 q^{91} - 60080 q^{95} - 36036 q^{97}+O(q^{100})$$ 2 * q - 28 * q^5 - 98 * q^7 + 596 * q^11 + 532 * q^13 - 2100 * q^17 - 2744 * q^19 + 3660 * q^23 + 2350 * q^25 + 720 * q^29 + 3976 * q^31 + 1372 * q^35 + 6788 * q^37 + 4004 * q^41 + 19480 * q^43 - 21560 * q^47 + 4802 * q^49 - 57576 * q^53 - 65800 * q^55 - 22344 * q^59 - 38724 * q^61 + 25384 * q^65 + 7288 * q^67 - 3932 * q^71 + 19292 * q^73 - 29204 * q^77 + 15632 * q^79 - 22624 * q^83 + 119688 * q^85 - 112812 * q^89 - 26068 * q^91 - 60080 * q^95 - 36036 * 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
 −10.6771 10.6771
0 0 0 −78.0625 0 −49.0000 0 0 0
1.2 0 0 0 50.0625 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.n 2
3.b odd 2 1 504.6.a.q yes 2
4.b odd 2 1 1008.6.a.bj 2
12.b even 2 1 1008.6.a.br 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.6.a.n 2 1.a even 1 1 trivial
504.6.a.q yes 2 3.b odd 2 1
1008.6.a.bj 2 4.b odd 2 1
1008.6.a.br 2 12.b even 2 1

## Hecke kernels

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

 $$T_{5}^{2} + 28T_{5} - 3908$$ T5^2 + 28*T5 - 3908 $$T_{11}^{2} - 596T_{11} - 112292$$ T11^2 - 596*T11 - 112292

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 28T - 3908$$
$7$ $$(T + 49)^{2}$$
$11$ $$T^{2} - 596T - 112292$$
$13$ $$T^{2} - 532T + 5092$$
$17$ $$T^{2} + 2100 T + 605916$$
$19$ $$T^{2} + 2744 T + 1291408$$
$23$ $$T^{2} - 3660 T + 1539036$$
$29$ $$T^{2} - 720 T - 12740544$$
$31$ $$T^{2} - 3976 T - 30784112$$
$37$ $$T^{2} - 6788 T - 68919164$$
$41$ $$T^{2} - 4004 T - 28499780$$
$43$ $$T^{2} - 19480 T + 43387024$$
$47$ $$T^{2} + 21560 T + 13756144$$
$53$ $$T^{2} + 57576 T + 827944560$$
$59$ $$T^{2} + 22344 T + 111007728$$
$61$ $$T^{2} + 38724 T - 217730556$$
$67$ $$T^{2} + \cdots - 1029202928$$
$71$ $$T^{2} + 3932 T - 121819844$$
$73$ $$T^{2} + \cdots - 1464242108$$
$79$ $$T^{2} - 15632 T - 569547200$$
$83$ $$T^{2} + 22624 T - 725408000$$
$89$ $$T^{2} + \cdots + 2954993436$$
$97$ $$T^{2} + \cdots - 8280224892$$