# Properties

 Label 7728.2.a.bd Level $7728$ Weight $2$ Character orbit 7728.a Self dual yes Analytic conductor $61.708$ 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] = [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: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 966) 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 $$\beta = 2\sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + 2 q^{5} - q^{7} + q^{9}+O(q^{10})$$ q - q^3 + 2 * q^5 - q^7 + q^9 $$q - q^{3} + 2 q^{5} - q^{7} + q^{9} - \beta q^{13} - 2 q^{15} + \beta q^{17} + ( - \beta + 2) q^{19} + q^{21} + q^{23} - q^{25} - q^{27} - 2 q^{29} + (\beta - 2) q^{31} - 2 q^{35} + (2 \beta - 2) q^{37} + \beta q^{39} - 6 q^{41} + (2 \beta - 4) q^{43} + 2 q^{45} + (\beta - 2) q^{47} + q^{49} - \beta q^{51} + ( - 2 \beta - 2) q^{53} + (\beta - 2) q^{57} - 4 q^{59} - 6 q^{61} - q^{63} - 2 \beta q^{65} + ( - 2 \beta + 4) q^{67} - q^{69} + ( - 2 \beta + 4) q^{71} + (2 \beta + 6) q^{73} + q^{75} + (2 \beta - 4) q^{79} + q^{81} + ( - \beta + 2) q^{83} + 2 \beta q^{85} + 2 q^{87} + (3 \beta - 4) q^{89} + \beta q^{91} + ( - \beta + 2) q^{93} + ( - 2 \beta + 4) q^{95} + ( - \beta + 4) q^{97} +O(q^{100})$$ q - q^3 + 2 * q^5 - q^7 + q^9 - b * q^13 - 2 * q^15 + b * q^17 + (-b + 2) * q^19 + q^21 + q^23 - q^25 - q^27 - 2 * q^29 + (b - 2) * q^31 - 2 * q^35 + (2*b - 2) * q^37 + b * q^39 - 6 * q^41 + (2*b - 4) * q^43 + 2 * q^45 + (b - 2) * q^47 + q^49 - b * q^51 + (-2*b - 2) * q^53 + (b - 2) * q^57 - 4 * q^59 - 6 * q^61 - q^63 - 2*b * q^65 + (-2*b + 4) * q^67 - q^69 + (-2*b + 4) * q^71 + (2*b + 6) * q^73 + q^75 + (2*b - 4) * q^79 + q^81 + (-b + 2) * q^83 + 2*b * q^85 + 2 * q^87 + (3*b - 4) * q^89 + b * q^91 + (-b + 2) * q^93 + (-2*b + 4) * q^95 + (-b + 4) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 4 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 4 * q^5 - 2 * q^7 + 2 * q^9 $$2 q - 2 q^{3} + 4 q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{15} + 4 q^{19} + 2 q^{21} + 2 q^{23} - 2 q^{25} - 2 q^{27} - 4 q^{29} - 4 q^{31} - 4 q^{35} - 4 q^{37} - 12 q^{41} - 8 q^{43} + 4 q^{45} - 4 q^{47} + 2 q^{49} - 4 q^{53} - 4 q^{57} - 8 q^{59} - 12 q^{61} - 2 q^{63} + 8 q^{67} - 2 q^{69} + 8 q^{71} + 12 q^{73} + 2 q^{75} - 8 q^{79} + 2 q^{81} + 4 q^{83} + 4 q^{87} - 8 q^{89} + 4 q^{93} + 8 q^{95} + 8 q^{97}+O(q^{100})$$ 2 * q - 2 * q^3 + 4 * q^5 - 2 * q^7 + 2 * q^9 - 4 * q^15 + 4 * q^19 + 2 * q^21 + 2 * q^23 - 2 * q^25 - 2 * q^27 - 4 * q^29 - 4 * q^31 - 4 * q^35 - 4 * q^37 - 12 * q^41 - 8 * q^43 + 4 * q^45 - 4 * q^47 + 2 * q^49 - 4 * q^53 - 4 * q^57 - 8 * q^59 - 12 * q^61 - 2 * q^63 + 8 * q^67 - 2 * q^69 + 8 * q^71 + 12 * q^73 + 2 * q^75 - 8 * q^79 + 2 * q^81 + 4 * q^83 + 4 * q^87 - 8 * q^89 + 4 * q^93 + 8 * q^95 + 8 * 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
 1.61803 −0.618034
0 −1.00000 0 2.00000 0 −1.00000 0 1.00000 0
1.2 0 −1.00000 0 2.00000 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.bd 2
4.b odd 2 1 966.2.a.p 2
12.b even 2 1 2898.2.a.v 2
28.d even 2 1 6762.2.a.bz 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.a.p 2 4.b odd 2 1
2898.2.a.v 2 12.b even 2 1
6762.2.a.bz 2 28.d even 2 1
7728.2.a.bd 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7728))$$:

 $$T_{5} - 2$$ T5 - 2 $$T_{11}$$ T11 $$T_{13}^{2} - 20$$ T13^2 - 20 $$T_{17}^{2} - 20$$ T17^2 - 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$(T - 2)^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 20$$
$17$ $$T^{2} - 20$$
$19$ $$T^{2} - 4T - 16$$
$23$ $$(T - 1)^{2}$$
$29$ $$(T + 2)^{2}$$
$31$ $$T^{2} + 4T - 16$$
$37$ $$T^{2} + 4T - 76$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} + 8T - 64$$
$47$ $$T^{2} + 4T - 16$$
$53$ $$T^{2} + 4T - 76$$
$59$ $$(T + 4)^{2}$$
$61$ $$(T + 6)^{2}$$
$67$ $$T^{2} - 8T - 64$$
$71$ $$T^{2} - 8T - 64$$
$73$ $$T^{2} - 12T - 44$$
$79$ $$T^{2} + 8T - 64$$
$83$ $$T^{2} - 4T - 16$$
$89$ $$T^{2} + 8T - 164$$
$97$ $$T^{2} - 8T - 4$$