Properties

 Label 504.6.a.k 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: $$\Q(\sqrt{106})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 106$$ x^2 - 106 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 $$\beta = 6\sqrt{106}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 38) q^{5} + 49 q^{7}+O(q^{10})$$ q + (b - 38) * q^5 + 49 * q^7 $$q + (\beta - 38) q^{5} + 49 q^{7} + ( - 5 \beta - 282) q^{11} + (4 \beta + 258) q^{13} + ( - 9 \beta + 38) q^{17} + ( - 4 \beta + 1180) q^{19} + (41 \beta + 1018) q^{23} + ( - 76 \beta + 2135) q^{25} + ( - 20 \beta - 4160) q^{29} + ( - 84 \beta - 3140) q^{31} + (49 \beta - 1862) q^{35} + ( - 60 \beta - 1230) q^{37} + ( - 211 \beta + 7154) q^{41} + (32 \beta + 11564) q^{43} + (222 \beta - 6356) q^{47} + 2401 q^{49} + ( - 26 \beta + 4948) q^{53} + ( - 92 \beta - 8364) q^{55} + ( - 198 \beta - 30444) q^{59} + (700 \beta + 8086) q^{61} + (106 \beta + 5460) q^{65} + ( - 40 \beta - 31284) q^{67} + (283 \beta + 366) q^{71} + ( - 344 \beta + 622) q^{73} + ( - 245 \beta - 13818) q^{77} + (1512 \beta + 5800) q^{79} + (80 \beta - 66144) q^{83} + (380 \beta - 35788) q^{85} + ( - 703 \beta - 46726) q^{89} + (196 \beta + 12642) q^{91} + (1332 \beta - 60104) q^{95} + ( - 312 \beta - 136466) q^{97}+O(q^{100})$$ q + (b - 38) * q^5 + 49 * q^7 + (-5*b - 282) * q^11 + (4*b + 258) * q^13 + (-9*b + 38) * q^17 + (-4*b + 1180) * q^19 + (41*b + 1018) * q^23 + (-76*b + 2135) * q^25 + (-20*b - 4160) * q^29 + (-84*b - 3140) * q^31 + (49*b - 1862) * q^35 + (-60*b - 1230) * q^37 + (-211*b + 7154) * q^41 + (32*b + 11564) * q^43 + (222*b - 6356) * q^47 + 2401 * q^49 + (-26*b + 4948) * q^53 + (-92*b - 8364) * q^55 + (-198*b - 30444) * q^59 + (700*b + 8086) * q^61 + (106*b + 5460) * q^65 + (-40*b - 31284) * q^67 + (283*b + 366) * q^71 + (-344*b + 622) * q^73 + (-245*b - 13818) * q^77 + (1512*b + 5800) * q^79 + (80*b - 66144) * q^83 + (380*b - 35788) * q^85 + (-703*b - 46726) * q^89 + (196*b + 12642) * q^91 + (1332*b - 60104) * q^95 + (-312*b - 136466) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 76 q^{5} + 98 q^{7}+O(q^{10})$$ 2 * q - 76 * q^5 + 98 * q^7 $$2 q - 76 q^{5} + 98 q^{7} - 564 q^{11} + 516 q^{13} + 76 q^{17} + 2360 q^{19} + 2036 q^{23} + 4270 q^{25} - 8320 q^{29} - 6280 q^{31} - 3724 q^{35} - 2460 q^{37} + 14308 q^{41} + 23128 q^{43} - 12712 q^{47} + 4802 q^{49} + 9896 q^{53} - 16728 q^{55} - 60888 q^{59} + 16172 q^{61} + 10920 q^{65} - 62568 q^{67} + 732 q^{71} + 1244 q^{73} - 27636 q^{77} + 11600 q^{79} - 132288 q^{83} - 71576 q^{85} - 93452 q^{89} + 25284 q^{91} - 120208 q^{95} - 272932 q^{97}+O(q^{100})$$ 2 * q - 76 * q^5 + 98 * q^7 - 564 * q^11 + 516 * q^13 + 76 * q^17 + 2360 * q^19 + 2036 * q^23 + 4270 * q^25 - 8320 * q^29 - 6280 * q^31 - 3724 * q^35 - 2460 * q^37 + 14308 * q^41 + 23128 * q^43 - 12712 * q^47 + 4802 * q^49 + 9896 * q^53 - 16728 * q^55 - 60888 * q^59 + 16172 * q^61 + 10920 * q^65 - 62568 * q^67 + 732 * q^71 + 1244 * q^73 - 27636 * q^77 + 11600 * q^79 - 132288 * q^83 - 71576 * q^85 - 93452 * q^89 + 25284 * q^91 - 120208 * q^95 - 272932 * 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
 −10.2956 10.2956
0 0 0 −99.7738 0 49.0000 0 0 0
1.2 0 0 0 23.7738 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.k 2
3.b odd 2 1 504.6.a.u yes 2
4.b odd 2 1 1008.6.a.bg 2
12.b even 2 1 1008.6.a.bv 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.6.a.k 2 1.a even 1 1 trivial
504.6.a.u yes 2 3.b odd 2 1
1008.6.a.bg 2 4.b odd 2 1
1008.6.a.bv 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(504))$$:

 $$T_{5}^{2} + 76T_{5} - 2372$$ T5^2 + 76*T5 - 2372 $$T_{11}^{2} + 564T_{11} - 15876$$ T11^2 + 564*T11 - 15876

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 76T - 2372$$
$7$ $$(T - 49)^{2}$$
$11$ $$T^{2} + 564T - 15876$$
$13$ $$T^{2} - 516T + 5508$$
$17$ $$T^{2} - 76T - 307652$$
$19$ $$T^{2} - 2360 T + 1331344$$
$23$ $$T^{2} - 2036 T - 5378372$$
$29$ $$T^{2} + 8320 T + 15779200$$
$31$ $$T^{2} + 6280 T - 17066096$$
$37$ $$T^{2} + 2460 T - 12224700$$
$41$ $$T^{2} - 14308 T - 118712420$$
$43$ $$T^{2} - 23128 T + 129818512$$
$47$ $$T^{2} + 12712 T - 147669008$$
$53$ $$T^{2} - 9896 T + 21903088$$
$59$ $$T^{2} + 60888 T + 777234672$$
$61$ $$T^{2} + \cdots - 1804456604$$
$67$ $$T^{2} + 62568 T + 972583056$$
$71$ $$T^{2} - 732 T - 305485668$$
$73$ $$T^{2} - 1244 T - 451183292$$
$79$ $$T^{2} + \cdots - 8690285504$$
$83$ $$T^{2} + \cdots + 4350606336$$
$89$ $$T^{2} + 93452 T + 297417532$$
$97$ $$T^{2} + \cdots + 18251504452$$