# Properties

 Label 504.6.a.r 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: $$\Q(\sqrt{37})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 9$$ x^2 - x - 9 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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 $$\beta = 2\sqrt{37}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (5 \beta + 24) q^{5} - 49 q^{7}+O(q^{10})$$ q + (5*b + 24) * q^5 - 49 * q^7 $$q + (5 \beta + 24) q^{5} - 49 q^{7} + (37 \beta + 184) q^{11} + (80 \beta - 78) q^{13} + ( - 33 \beta - 1656) q^{17} + (48 \beta + 1868) q^{19} + ( - 17 \beta + 776) q^{23} + (240 \beta + 1151) q^{25} + ( - 192 \beta + 864) q^{29} + ( - 16 \beta - 1812) q^{31} + ( - 245 \beta - 1176) q^{35} + (112 \beta + 3498) q^{37} + (341 \beta + 13080) q^{41} + ( - 32 \beta - 15092) q^{43} + (58 \beta - 5712) q^{47} + 2401 q^{49} + (2322 \beta - 8688) q^{53} + (1808 \beta + 31796) q^{55} + ( - 1810 \beta + 7504) q^{59} + ( - 1168 \beta + 17782) q^{61} + (1530 \beta + 57328) q^{65} + ( - 512 \beta - 35252) q^{67} + ( - 2435 \beta + 20376) q^{71} + ( - 1184 \beta - 26946) q^{73} + ( - 1813 \beta - 9016) q^{77} + ( - 2720 \beta - 4872) q^{79} + ( - 7800 \beta + 15680) q^{83} + ( - 9072 \beta - 64164) q^{85} + (6129 \beta + 17976) q^{89} + ( - 3920 \beta + 3822) q^{91} + (10492 \beta + 80352) q^{95} + (11680 \beta + 33326) q^{97}+O(q^{100})$$ q + (5*b + 24) * q^5 - 49 * q^7 + (37*b + 184) * q^11 + (80*b - 78) * q^13 + (-33*b - 1656) * q^17 + (48*b + 1868) * q^19 + (-17*b + 776) * q^23 + (240*b + 1151) * q^25 + (-192*b + 864) * q^29 + (-16*b - 1812) * q^31 + (-245*b - 1176) * q^35 + (112*b + 3498) * q^37 + (341*b + 13080) * q^41 + (-32*b - 15092) * q^43 + (58*b - 5712) * q^47 + 2401 * q^49 + (2322*b - 8688) * q^53 + (1808*b + 31796) * q^55 + (-1810*b + 7504) * q^59 + (-1168*b + 17782) * q^61 + (1530*b + 57328) * q^65 + (-512*b - 35252) * q^67 + (-2435*b + 20376) * q^71 + (-1184*b - 26946) * q^73 + (-1813*b - 9016) * q^77 + (-2720*b - 4872) * q^79 + (-7800*b + 15680) * q^83 + (-9072*b - 64164) * q^85 + (6129*b + 17976) * q^89 + (-3920*b + 3822) * q^91 + (10492*b + 80352) * q^95 + (11680*b + 33326) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 48 q^{5} - 98 q^{7}+O(q^{10})$$ 2 * q + 48 * q^5 - 98 * q^7 $$2 q + 48 q^{5} - 98 q^{7} + 368 q^{11} - 156 q^{13} - 3312 q^{17} + 3736 q^{19} + 1552 q^{23} + 2302 q^{25} + 1728 q^{29} - 3624 q^{31} - 2352 q^{35} + 6996 q^{37} + 26160 q^{41} - 30184 q^{43} - 11424 q^{47} + 4802 q^{49} - 17376 q^{53} + 63592 q^{55} + 15008 q^{59} + 35564 q^{61} + 114656 q^{65} - 70504 q^{67} + 40752 q^{71} - 53892 q^{73} - 18032 q^{77} - 9744 q^{79} + 31360 q^{83} - 128328 q^{85} + 35952 q^{89} + 7644 q^{91} + 160704 q^{95} + 66652 q^{97}+O(q^{100})$$ 2 * q + 48 * q^5 - 98 * q^7 + 368 * q^11 - 156 * q^13 - 3312 * q^17 + 3736 * q^19 + 1552 * q^23 + 2302 * q^25 + 1728 * q^29 - 3624 * q^31 - 2352 * q^35 + 6996 * q^37 + 26160 * q^41 - 30184 * q^43 - 11424 * q^47 + 4802 * q^49 - 17376 * q^53 + 63592 * q^55 + 15008 * q^59 + 35564 * q^61 + 114656 * q^65 - 70504 * q^67 + 40752 * q^71 - 53892 * q^73 - 18032 * q^77 - 9744 * q^79 + 31360 * q^83 - 128328 * q^85 + 35952 * q^89 + 7644 * q^91 + 160704 * q^95 + 66652 * 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
 −2.54138 3.54138
0 0 0 −36.8276 0 −49.0000 0 0 0
1.2 0 0 0 84.8276 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.r yes 2
3.b odd 2 1 504.6.a.l 2
4.b odd 2 1 1008.6.a.bs 2
12.b even 2 1 1008.6.a.bh 2

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

 $$T_{5}^{2} - 48T_{5} - 3124$$ T5^2 - 48*T5 - 3124 $$T_{11}^{2} - 368T_{11} - 168756$$ T11^2 - 368*T11 - 168756

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 48T - 3124$$
$7$ $$(T + 49)^{2}$$
$11$ $$T^{2} - 368T - 168756$$
$13$ $$T^{2} + 156T - 941116$$
$17$ $$T^{2} + 3312 T + 2581164$$
$19$ $$T^{2} - 3736 T + 3148432$$
$23$ $$T^{2} - 1552 T + 559404$$
$29$ $$T^{2} - 1728 T - 4709376$$
$31$ $$T^{2} + 3624 T + 3245456$$
$37$ $$T^{2} - 6996 T + 10379492$$
$41$ $$T^{2} - 26160 T + 153876812$$
$43$ $$T^{2} + 30184 T + 227616912$$
$47$ $$T^{2} + 11424 T + 32129072$$
$53$ $$T^{2} + 17376 T - 722487888$$
$59$ $$T^{2} - 15008 T - 428552784$$
$61$ $$T^{2} - 35564 T + 114294372$$
$67$ $$T^{2} + \cdots + 1203906192$$
$71$ $$T^{2} - 40752 T - 462343924$$
$73$ $$T^{2} + 53892 T + 518612228$$
$79$ $$T^{2} + \cdots - 1071226816$$
$83$ $$T^{2} + \cdots - 8758457600$$
$89$ $$T^{2} + \cdots - 5236430292$$
$97$ $$T^{2} + \cdots - 19079892924$$