# Properties

 Label 504.6.a.i Level $504$ Weight $6$ Character orbit 504.a Self dual yes Analytic conductor $80.833$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{345})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 86$$ x^2 - x - 86 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 56) 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 $$\beta = 3\sqrt{345}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 41) q^{5} - 49 q^{7}+O(q^{10})$$ q + (-b - 41) * q^5 - 49 * q^7 $$q + ( - \beta - 41) q^{5} - 49 q^{7} + ( - 2 \beta - 170) q^{11} + ( - 13 \beta + 455) q^{13} + ( - 2 \beta - 1608) q^{17} + ( - 15 \beta - 337) q^{19} + ( - 24 \beta + 552) q^{23} + (82 \beta + 1661) q^{25} + ( - 38 \beta - 4032) q^{29} + ( - 70 \beta - 3106) q^{31} + (49 \beta + 2009) q^{35} + ( - 130 \beta - 4256) q^{37} + (226 \beta + 652) q^{41} + ( - 266 \beta - 5002) q^{43} + ( - 238 \beta + 6374) q^{47} + 2401 q^{49} + ( - 256 \beta + 5610) q^{53} + (252 \beta + 13180) q^{55} + (751 \beta + 6009) q^{59} + (25 \beta + 51369) q^{61} + (78 \beta + 21710) q^{65} + ( - 648 \beta + 12068) q^{67} + (580 \beta - 44860) q^{71} + (572 \beta - 27794) q^{73} + (98 \beta + 8330) q^{77} + ( - 652 \beta + 24412) q^{79} + ( - 1149 \beta - 17891) q^{83} + (1690 \beta + 72138) q^{85} + (1576 \beta + 9150) q^{89} + (637 \beta - 22295) q^{91} + (952 \beta + 60392) q^{95} + (154 \beta - 34992) q^{97}+O(q^{100})$$ q + (-b - 41) * q^5 - 49 * q^7 + (-2*b - 170) * q^11 + (-13*b + 455) * q^13 + (-2*b - 1608) * q^17 + (-15*b - 337) * q^19 + (-24*b + 552) * q^23 + (82*b + 1661) * q^25 + (-38*b - 4032) * q^29 + (-70*b - 3106) * q^31 + (49*b + 2009) * q^35 + (-130*b - 4256) * q^37 + (226*b + 652) * q^41 + (-266*b - 5002) * q^43 + (-238*b + 6374) * q^47 + 2401 * q^49 + (-256*b + 5610) * q^53 + (252*b + 13180) * q^55 + (751*b + 6009) * q^59 + (25*b + 51369) * q^61 + (78*b + 21710) * q^65 + (-648*b + 12068) * q^67 + (580*b - 44860) * q^71 + (572*b - 27794) * q^73 + (98*b + 8330) * q^77 + (-652*b + 24412) * q^79 + (-1149*b - 17891) * q^83 + (1690*b + 72138) * q^85 + (1576*b + 9150) * q^89 + (637*b - 22295) * q^91 + (952*b + 60392) * q^95 + (154*b - 34992) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 82 q^{5} - 98 q^{7}+O(q^{10})$$ 2 * q - 82 * q^5 - 98 * q^7 $$2 q - 82 q^{5} - 98 q^{7} - 340 q^{11} + 910 q^{13} - 3216 q^{17} - 674 q^{19} + 1104 q^{23} + 3322 q^{25} - 8064 q^{29} - 6212 q^{31} + 4018 q^{35} - 8512 q^{37} + 1304 q^{41} - 10004 q^{43} + 12748 q^{47} + 4802 q^{49} + 11220 q^{53} + 26360 q^{55} + 12018 q^{59} + 102738 q^{61} + 43420 q^{65} + 24136 q^{67} - 89720 q^{71} - 55588 q^{73} + 16660 q^{77} + 48824 q^{79} - 35782 q^{83} + 144276 q^{85} + 18300 q^{89} - 44590 q^{91} + 120784 q^{95} - 69984 q^{97}+O(q^{100})$$ 2 * q - 82 * q^5 - 98 * q^7 - 340 * q^11 + 910 * q^13 - 3216 * q^17 - 674 * q^19 + 1104 * q^23 + 3322 * q^25 - 8064 * q^29 - 6212 * q^31 + 4018 * q^35 - 8512 * q^37 + 1304 * q^41 - 10004 * q^43 + 12748 * q^47 + 4802 * q^49 + 11220 * q^53 + 26360 * q^55 + 12018 * q^59 + 102738 * q^61 + 43420 * q^65 + 24136 * q^67 - 89720 * q^71 - 55588 * q^73 + 16660 * q^77 + 48824 * q^79 - 35782 * q^83 + 144276 * q^85 + 18300 * q^89 - 44590 * q^91 + 120784 * q^95 - 69984 * 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
 9.78709 −8.78709
0 0 0 −96.7225 0 −49.0000 0 0 0
1.2 0 0 0 14.7225 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.i 2
3.b odd 2 1 56.6.a.e 2
4.b odd 2 1 1008.6.a.bd 2
12.b even 2 1 112.6.a.i 2
21.c even 2 1 392.6.a.d 2
21.g even 6 2 392.6.i.i 4
21.h odd 6 2 392.6.i.j 4
24.f even 2 1 448.6.a.t 2
24.h odd 2 1 448.6.a.v 2
84.h odd 2 1 784.6.a.u 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.a.e 2 3.b odd 2 1
112.6.a.i 2 12.b even 2 1
392.6.a.d 2 21.c even 2 1
392.6.i.i 4 21.g even 6 2
392.6.i.j 4 21.h odd 6 2
448.6.a.t 2 24.f even 2 1
448.6.a.v 2 24.h odd 2 1
504.6.a.i 2 1.a even 1 1 trivial
784.6.a.u 2 84.h odd 2 1
1008.6.a.bd 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(504))$$:

 $$T_{5}^{2} + 82T_{5} - 1424$$ T5^2 + 82*T5 - 1424 $$T_{11}^{2} + 340T_{11} + 16480$$ T11^2 + 340*T11 + 16480

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 82T - 1424$$
$7$ $$(T + 49)^{2}$$
$11$ $$T^{2} + 340T + 16480$$
$13$ $$T^{2} - 910T - 317720$$
$17$ $$T^{2} + 3216 T + 2573244$$
$19$ $$T^{2} + 674T - 585056$$
$23$ $$T^{2} - 1104 T - 1483776$$
$29$ $$T^{2} + 8064 T + 11773404$$
$31$ $$T^{2} + 6212 T - 5567264$$
$37$ $$T^{2} + 8512 T - 34360964$$
$41$ $$T^{2} - 1304 T - 158165876$$
$43$ $$T^{2} + 10004 T - 194677376$$
$47$ $$T^{2} - 12748 T - 135251744$$
$53$ $$T^{2} - 11220 T - 172017180$$
$59$ $$T^{2} + \cdots - 1715115024$$
$61$ $$T^{2} + \cdots + 2636833536$$
$67$ $$T^{2} + \cdots - 1158165296$$
$71$ $$T^{2} + 89720 T + 967897600$$
$73$ $$T^{2} + 55588 T - 243399884$$
$79$ $$T^{2} - 48824 T - 724002176$$
$83$ $$T^{2} + \cdots - 3779136224$$
$89$ $$T^{2} + \cdots - 7628401980$$
$97$ $$T^{2} + \cdots + 1150801884$$