# Properties

 Label 7728.2.a.bx Level $7728$ Weight $2$ Character orbit 7728.a Self dual yes Analytic conductor $61.708$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7728,2,Mod(1,7728)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7728.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: $$61.7083906820$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.837.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 6x - 1$$ x^3 - 6*x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3864) 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + \beta_1 q^{5} - q^{7} + q^{9}+O(q^{10})$$ q + q^3 + b1 * q^5 - q^7 + q^9 $$q + q^{3} + \beta_1 q^{5} - q^{7} + q^{9} + ( - \beta_{2} + \beta_1 - 2) q^{11} + (\beta_{2} - 2 \beta_1) q^{13} + \beta_1 q^{15} + ( - \beta_{2} - \beta_1) q^{17} + (\beta_{2} - \beta_1 - 2) q^{19} - q^{21} + q^{23} + (\beta_{2} - 1) q^{25} + q^{27} + 5 q^{29} + ( - \beta_{2} + \beta_1 - 4) q^{31} + ( - \beta_{2} + \beta_1 - 2) q^{33} - \beta_1 q^{35} + (2 \beta_1 - 3) q^{37} + (\beta_{2} - 2 \beta_1) q^{39} + (2 \beta_{2} + 3) q^{41} + (\beta_{2} - 4 \beta_1 + 2) q^{43} + \beta_1 q^{45} + ( - \beta_{2} - \beta_1 - 1) q^{47} + q^{49} + ( - \beta_{2} - \beta_1) q^{51} + ( - \beta_{2} - 2 \beta_1 - 1) q^{53} + (\beta_{2} - 4 \beta_1 + 3) q^{55} + (\beta_{2} - \beta_1 - 2) q^{57} + ( - 3 \beta_{2} - 7) q^{59} + ( - 2 \beta_{2} + \beta_1 + 1) q^{61} - q^{63} + ( - 2 \beta_{2} + 2 \beta_1 - 7) q^{65} + (3 \beta_{2} + 2 \beta_1 - 1) q^{67} + q^{69} + (2 \beta_{2} - \beta_1 - 1) q^{71} + (3 \beta_{2} + 3 \beta_1) q^{73} + (\beta_{2} - 1) q^{75} + (\beta_{2} - \beta_1 + 2) q^{77} + ( - \beta_{2} - 3 \beta_1 - 6) q^{79} + q^{81} + (3 \beta_{2} + 5 \beta_1 - 2) q^{83} + ( - \beta_{2} - 2 \beta_1 - 5) q^{85} + 5 q^{87} + ( - 2 \beta_{2} + \beta_1 + 1) q^{89} + ( - \beta_{2} + 2 \beta_1) q^{91} + ( - \beta_{2} + \beta_1 - 4) q^{93} + ( - \beta_{2} - 3) q^{95} + (2 \beta_{2} - 4 \beta_1 - 7) q^{97} + ( - \beta_{2} + \beta_1 - 2) q^{99}+O(q^{100})$$ q + q^3 + b1 * q^5 - q^7 + q^9 + (-b2 + b1 - 2) * q^11 + (b2 - 2*b1) * q^13 + b1 * q^15 + (-b2 - b1) * q^17 + (b2 - b1 - 2) * q^19 - q^21 + q^23 + (b2 - 1) * q^25 + q^27 + 5 * q^29 + (-b2 + b1 - 4) * q^31 + (-b2 + b1 - 2) * q^33 - b1 * q^35 + (2*b1 - 3) * q^37 + (b2 - 2*b1) * q^39 + (2*b2 + 3) * q^41 + (b2 - 4*b1 + 2) * q^43 + b1 * q^45 + (-b2 - b1 - 1) * q^47 + q^49 + (-b2 - b1) * q^51 + (-b2 - 2*b1 - 1) * q^53 + (b2 - 4*b1 + 3) * q^55 + (b2 - b1 - 2) * q^57 + (-3*b2 - 7) * q^59 + (-2*b2 + b1 + 1) * q^61 - q^63 + (-2*b2 + 2*b1 - 7) * q^65 + (3*b2 + 2*b1 - 1) * q^67 + q^69 + (2*b2 - b1 - 1) * q^71 + (3*b2 + 3*b1) * q^73 + (b2 - 1) * q^75 + (b2 - b1 + 2) * q^77 + (-b2 - 3*b1 - 6) * q^79 + q^81 + (3*b2 + 5*b1 - 2) * q^83 + (-b2 - 2*b1 - 5) * q^85 + 5 * q^87 + (-2*b2 + b1 + 1) * q^89 + (-b2 + 2*b1) * q^91 + (-b2 + b1 - 4) * q^93 + (-b2 - 3) * q^95 + (2*b2 - 4*b1 - 7) * q^97 + (-b2 + b1 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 - 3 * q^7 + 3 * q^9 $$3 q + 3 q^{3} - 3 q^{7} + 3 q^{9} - 6 q^{11} - 6 q^{19} - 3 q^{21} + 3 q^{23} - 3 q^{25} + 3 q^{27} + 15 q^{29} - 12 q^{31} - 6 q^{33} - 9 q^{37} + 9 q^{41} + 6 q^{43} - 3 q^{47} + 3 q^{49} - 3 q^{53} + 9 q^{55} - 6 q^{57} - 21 q^{59} + 3 q^{61} - 3 q^{63} - 21 q^{65} - 3 q^{67} + 3 q^{69} - 3 q^{71} - 3 q^{75} + 6 q^{77} - 18 q^{79} + 3 q^{81} - 6 q^{83} - 15 q^{85} + 15 q^{87} + 3 q^{89} - 12 q^{93} - 9 q^{95} - 21 q^{97} - 6 q^{99}+O(q^{100})$$ 3 * q + 3 * q^3 - 3 * q^7 + 3 * q^9 - 6 * q^11 - 6 * q^19 - 3 * q^21 + 3 * q^23 - 3 * q^25 + 3 * q^27 + 15 * q^29 - 12 * q^31 - 6 * q^33 - 9 * q^37 + 9 * q^41 + 6 * q^43 - 3 * q^47 + 3 * q^49 - 3 * q^53 + 9 * q^55 - 6 * q^57 - 21 * q^59 + 3 * q^61 - 3 * q^63 - 21 * q^65 - 3 * q^67 + 3 * q^69 - 3 * q^71 - 3 * q^75 + 6 * q^77 - 18 * q^79 + 3 * q^81 - 6 * q^83 - 15 * q^85 + 15 * q^87 + 3 * q^89 - 12 * q^93 - 9 * q^95 - 21 * q^97 - 6 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 6x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ b2 + 4

## 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
 −2.36147 −0.167449 2.52892
0 1.00000 0 −2.36147 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 −0.167449 0 −1.00000 0 1.00000 0
1.3 0 1.00000 0 2.52892 0 −1.00000 0 1.00000 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$7$$ $$1$$
$$23$$ $$-1$$

## 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 7728.2.a.bx 3
4.b odd 2 1 3864.2.a.m 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3864.2.a.m 3 4.b odd 2 1
7728.2.a.bx 3 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7728))$$:

 $$T_{5}^{3} - 6T_{5} - 1$$ T5^3 - 6*T5 - 1 $$T_{11}^{3} + 6T_{11}^{2} - 3T_{11} - 20$$ T11^3 + 6*T11^2 - 3*T11 - 20 $$T_{13}^{3} - 30T_{13} - 61$$ T13^3 - 30*T13 - 61 $$T_{17}^{3} - 21T_{17} + 16$$ T17^3 - 21*T17 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T - 1)^{3}$$
$5$ $$T^{3} - 6T - 1$$
$7$ $$(T + 1)^{3}$$
$11$ $$T^{3} + 6 T^{2} - 3 T - 20$$
$13$ $$T^{3} - 30T - 61$$
$17$ $$T^{3} - 21T + 16$$
$19$ $$T^{3} + 6 T^{2} - 3 T - 24$$
$23$ $$(T - 1)^{3}$$
$29$ $$(T - 5)^{3}$$
$31$ $$T^{3} + 12 T^{2} + 33 T + 6$$
$37$ $$T^{3} + 9 T^{2} + 3 T - 53$$
$41$ $$T^{3} - 9 T^{2} - 21 T + 237$$
$43$ $$T^{3} - 6 T^{2} - 84 T - 97$$
$47$ $$T^{3} + 3 T^{2} - 18 T - 4$$
$53$ $$T^{3} + 3 T^{2} - 39 T + 60$$
$59$ $$T^{3} + 21 T^{2} + 39 T - 818$$
$61$ $$T^{3} - 3 T^{2} - 45 T - 50$$
$67$ $$T^{3} + 3 T^{2} - 147 T - 148$$
$71$ $$T^{3} + 3 T^{2} - 45 T + 50$$
$73$ $$T^{3} - 189T - 432$$
$79$ $$T^{3} + 18 T^{2} + 33 T + 12$$
$83$ $$T^{3} + 6 T^{2} - 291 T - 2388$$
$89$ $$T^{3} - 3 T^{2} - 45 T - 50$$
$97$ $$T^{3} + 21 T^{2} + 27 T - 985$$