# Properties

 Label 130.2.g.d Level $130$ Weight $2$ Character orbit 130.g Analytic conductor $1.038$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [130,2,Mod(57,130)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(130, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([1, 3]))

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

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

chi := DirichletCharacter("130.57");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$130 = 2 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 130.g (of order $$4$$, degree $$2$$, minimal)

## 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: no Analytic conductor: $$1.03805522628$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 5x^{2} + 9$$ x^4 - 5*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{3} + q^{4} + (\beta_{3} + 2) q^{5} + (\beta_{3} - \beta_{2} + \beta_1) q^{6} - 3 \beta_{2} q^{7} - q^{8} + ( - 2 \beta_{3} + 3 \beta_{2} - 1) q^{9}+O(q^{10})$$ q - q^2 + (-b3 + b2 - b1) * q^3 + q^4 + (b3 + 2) * q^5 + (b3 - b2 + b1) * q^6 - 3*b2 * q^7 - q^8 + (-2*b3 + 3*b2 - 1) * q^9 $$q - q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{3} + q^{4} + (\beta_{3} + 2) q^{5} + (\beta_{3} - \beta_{2} + \beta_1) q^{6} - 3 \beta_{2} q^{7} - q^{8} + ( - 2 \beta_{3} + 3 \beta_{2} - 1) q^{9} + ( - \beta_{3} - 2) q^{10} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{11} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{12} + ( - 3 \beta_{2} - 2) q^{13} + 3 \beta_{2} q^{14} + ( - \beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{15} + q^{16} + (\beta_{3} + \beta_{2} + \beta_1 + 2) q^{17} + (2 \beta_{3} - 3 \beta_{2} + 1) q^{18} + (2 \beta_{2} - 2) q^{19} + (\beta_{3} + 2) q^{20} + (3 \beta_{3} - 3 \beta_1 + 3) q^{21} + (2 \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{22} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{23} + (\beta_{3} - \beta_{2} + \beta_1) q^{24} + (3 \beta_{3} + 1) q^{25} + (3 \beta_{2} + 2) q^{26} + ( - \beta_{3} + 5 \beta_{2} + \beta_1 - 6) q^{27} - 3 \beta_{2} q^{28} + (2 \beta_{3} + 3 \beta_{2} + 1) q^{29} + (\beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{30} + (\beta_{2} + 1) q^{31} - q^{32} + ( - \beta_{2} + 2 \beta_1 - 11) q^{33} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{34} + ( - 6 \beta_{2} + 3 \beta_1) q^{35} + ( - 2 \beta_{3} + 3 \beta_{2} - 1) q^{36} - 3 \beta_{2} q^{37} + ( - 2 \beta_{2} + 2) q^{38} + (5 \beta_{3} - 2 \beta_{2} - \beta_1 + 3) q^{39} + ( - \beta_{3} - 2) q^{40} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 4) q^{41} + ( - 3 \beta_{3} + 3 \beta_1 - 3) q^{42} + (3 \beta_{3} - \beta_{2} - 3 \beta_1 + 4) q^{43} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{44} + ( - 3 \beta_{3} + 6 \beta_{2} - 3 \beta_1 + 4) q^{45} + (2 \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{46} + (2 \beta_{3} + 6 \beta_{2} + 1) q^{47} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{48} - 2 q^{49} + ( - 3 \beta_{3} - 1) q^{50} + ( - 2 \beta_{3} - 4 \beta_{2} - 1) q^{51} + ( - 3 \beta_{2} - 2) q^{52} + (4 \beta_{3} + \beta_{2} + 4 \beta_1 + 5) q^{53} + (\beta_{3} - 5 \beta_{2} - \beta_1 + 6) q^{54} + ( - 3 \beta_{3} + 4 \beta_{2} + 3 \beta_1 + 4) q^{55} + 3 \beta_{2} q^{56} + ( - 2 \beta_{2} + 4 \beta_1 - 2) q^{57} + ( - 2 \beta_{3} - 3 \beta_{2} - 1) q^{58} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 2) q^{59} + ( - \beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{60} + 2 q^{61} + ( - \beta_{2} - 1) q^{62} + (3 \beta_{2} - 6 \beta_1 + 9) q^{63} + q^{64} + ( - 2 \beta_{3} - 6 \beta_{2} + 3 \beta_1 - 4) q^{65} + (\beta_{2} - 2 \beta_1 + 11) q^{66} + (3 \beta_{2} - 6 \beta_1 + 5) q^{67} + (\beta_{3} + \beta_{2} + \beta_1 + 2) q^{68} + ( - \beta_{2} + 2 \beta_1 - 11) q^{69} + (6 \beta_{2} - 3 \beta_1) q^{70} + (\beta_{3} - 5 \beta_{2} + \beta_1 - 4) q^{71} + (2 \beta_{3} - 3 \beta_{2} + 1) q^{72} - 4 q^{73} + 3 \beta_{2} q^{74} + (2 \beta_{3} - 8 \beta_{2} - \beta_1 + 9) q^{75} + (2 \beta_{2} - 2) q^{76} + ( - 6 \beta_{3} + 3 \beta_{2} - 6 \beta_1 - 3) q^{77} + ( - 5 \beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{78} + ( - 6 \beta_{3} - 3 \beta_{2} - 3) q^{79} + (\beta_{3} + 2) q^{80} + ( - 3 \beta_{2} + 6 \beta_1 - 2) q^{81} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 4) q^{82} + (2 \beta_{3} + 3 \beta_{2} + 1) q^{83} + (3 \beta_{3} - 3 \beta_1 + 3) q^{84} + (3 \beta_{3} + 5 \beta_{2} + 1) q^{85} + ( - 3 \beta_{3} + \beta_{2} + 3 \beta_1 - 4) q^{86} + ( - 2 \beta_{3} - 5 \beta_{2} + 2 \beta_1 + 3) q^{87} + (2 \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{88} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 2) q^{89} + (3 \beta_{3} - 6 \beta_{2} + 3 \beta_1 - 4) q^{90} + (6 \beta_{2} - 9) q^{91} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{92} + ( - 2 \beta_{3} + \beta_{2} - 1) q^{93} + ( - 2 \beta_{3} - 6 \beta_{2} - 1) q^{94} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 4) q^{95} + (\beta_{3} - \beta_{2} + \beta_1) q^{96} + ( - 3 \beta_{2} + 6 \beta_1 - 1) q^{97} + 2 q^{98} + (6 \beta_{3} - 14 \beta_{2} + 6 \beta_1 - 8) q^{99}+O(q^{100})$$ q - q^2 + (-b3 + b2 - b1) * q^3 + q^4 + (b3 + 2) * q^5 + (b3 - b2 + b1) * q^6 - 3*b2 * q^7 - q^8 + (-2*b3 + 3*b2 - 1) * q^9 + (-b3 - 2) * q^10 + (-2*b3 - b2 + 2*b1 - 1) * q^11 + (-b3 + b2 - b1) * q^12 + (-3*b2 - 2) * q^13 + 3*b2 * q^14 + (-b3 - b2 - 2*b1 + 3) * q^15 + q^16 + (b3 + b2 + b1 + 2) * q^17 + (2*b3 - 3*b2 + 1) * q^18 + (2*b2 - 2) * q^19 + (b3 + 2) * q^20 + (3*b3 - 3*b1 + 3) * q^21 + (2*b3 + b2 - 2*b1 + 1) * q^22 + (-2*b3 - b2 + 2*b1 - 1) * q^23 + (b3 - b2 + b1) * q^24 + (3*b3 + 1) * q^25 + (3*b2 + 2) * q^26 + (-b3 + 5*b2 + b1 - 6) * q^27 - 3*b2 * q^28 + (2*b3 + 3*b2 + 1) * q^29 + (b3 + b2 + 2*b1 - 3) * q^30 + (b2 + 1) * q^31 - q^32 + (-b2 + 2*b1 - 11) * q^33 + (-b3 - b2 - b1 - 2) * q^34 + (-6*b2 + 3*b1) * q^35 + (-2*b3 + 3*b2 - 1) * q^36 - 3*b2 * q^37 + (-2*b2 + 2) * q^38 + (5*b3 - 2*b2 - b1 + 3) * q^39 + (-b3 - 2) * q^40 + (-2*b3 - 2*b2 - 2*b1 - 4) * q^41 + (-3*b3 + 3*b1 - 3) * q^42 + (3*b3 - b2 - 3*b1 + 4) * q^43 + (-2*b3 - b2 + 2*b1 - 1) * q^44 + (-3*b3 + 6*b2 - 3*b1 + 4) * q^45 + (2*b3 + b2 - 2*b1 + 1) * q^46 + (2*b3 + 6*b2 + 1) * q^47 + (-b3 + b2 - b1) * q^48 - 2 * q^49 + (-3*b3 - 1) * q^50 + (-2*b3 - 4*b2 - 1) * q^51 + (-3*b2 - 2) * q^52 + (4*b3 + b2 + 4*b1 + 5) * q^53 + (b3 - 5*b2 - b1 + 6) * q^54 + (-3*b3 + 4*b2 + 3*b1 + 4) * q^55 + 3*b2 * q^56 + (-2*b2 + 4*b1 - 2) * q^57 + (-2*b3 - 3*b2 - 1) * q^58 + (-2*b3 + 4*b2 - 2*b1 + 2) * q^59 + (-b3 - b2 - 2*b1 + 3) * q^60 + 2 * q^61 + (-b2 - 1) * q^62 + (3*b2 - 6*b1 + 9) * q^63 + q^64 + (-2*b3 - 6*b2 + 3*b1 - 4) * q^65 + (b2 - 2*b1 + 11) * q^66 + (3*b2 - 6*b1 + 5) * q^67 + (b3 + b2 + b1 + 2) * q^68 + (-b2 + 2*b1 - 11) * q^69 + (6*b2 - 3*b1) * q^70 + (b3 - 5*b2 + b1 - 4) * q^71 + (2*b3 - 3*b2 + 1) * q^72 - 4 * q^73 + 3*b2 * q^74 + (2*b3 - 8*b2 - b1 + 9) * q^75 + (2*b2 - 2) * q^76 + (-6*b3 + 3*b2 - 6*b1 - 3) * q^77 + (-5*b3 + 2*b2 + b1 - 3) * q^78 + (-6*b3 - 3*b2 - 3) * q^79 + (b3 + 2) * q^80 + (-3*b2 + 6*b1 - 2) * q^81 + (2*b3 + 2*b2 + 2*b1 + 4) * q^82 + (2*b3 + 3*b2 + 1) * q^83 + (3*b3 - 3*b1 + 3) * q^84 + (3*b3 + 5*b2 + 1) * q^85 + (-3*b3 + b2 + 3*b1 - 4) * q^86 + (-2*b3 - 5*b2 + 2*b1 + 3) * q^87 + (2*b3 + b2 - 2*b1 + 1) * q^88 + (-2*b3 + 4*b2 - 2*b1 + 2) * q^89 + (3*b3 - 6*b2 + 3*b1 - 4) * q^90 + (6*b2 - 9) * q^91 + (-2*b3 - b2 + 2*b1 - 1) * q^92 + (-2*b3 + b2 - 1) * q^93 + (-2*b3 - 6*b2 - 1) * q^94 + (-2*b3 + 4*b2 - 2*b1 - 4) * q^95 + (b3 - b2 + b1) * q^96 + (-3*b2 + 6*b1 - 1) * q^97 + 2 * q^98 + (6*b3 - 14*b2 + 6*b1 - 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} + 6 q^{5} - 2 q^{6} - 4 q^{8}+O(q^{10})$$ 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 + 6 * q^5 - 2 * q^6 - 4 * q^8 $$4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} + 6 q^{5} - 2 q^{6} - 4 q^{8} - 6 q^{10} + 2 q^{12} - 8 q^{13} + 14 q^{15} + 4 q^{16} + 6 q^{17} - 8 q^{19} + 6 q^{20} + 6 q^{21} - 2 q^{24} - 2 q^{25} + 8 q^{26} - 22 q^{27} - 14 q^{30} + 4 q^{31} - 4 q^{32} - 44 q^{33} - 6 q^{34} + 8 q^{38} + 2 q^{39} - 6 q^{40} - 12 q^{41} - 6 q^{42} + 10 q^{43} + 22 q^{45} + 2 q^{48} - 8 q^{49} + 2 q^{50} - 8 q^{52} + 12 q^{53} + 22 q^{54} + 22 q^{55} - 8 q^{57} + 12 q^{59} + 14 q^{60} + 8 q^{61} - 4 q^{62} + 36 q^{63} + 4 q^{64} - 12 q^{65} + 44 q^{66} + 20 q^{67} + 6 q^{68} - 44 q^{69} - 18 q^{71} - 16 q^{73} + 32 q^{75} - 8 q^{76} - 2 q^{78} + 6 q^{80} - 8 q^{81} + 12 q^{82} + 6 q^{84} - 2 q^{85} - 10 q^{86} + 16 q^{87} + 12 q^{89} - 22 q^{90} - 36 q^{91} - 12 q^{95} - 2 q^{96} - 4 q^{97} + 8 q^{98} - 44 q^{99}+O(q^{100})$$ 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 + 6 * q^5 - 2 * q^6 - 4 * q^8 - 6 * q^10 + 2 * q^12 - 8 * q^13 + 14 * q^15 + 4 * q^16 + 6 * q^17 - 8 * q^19 + 6 * q^20 + 6 * q^21 - 2 * q^24 - 2 * q^25 + 8 * q^26 - 22 * q^27 - 14 * q^30 + 4 * q^31 - 4 * q^32 - 44 * q^33 - 6 * q^34 + 8 * q^38 + 2 * q^39 - 6 * q^40 - 12 * q^41 - 6 * q^42 + 10 * q^43 + 22 * q^45 + 2 * q^48 - 8 * q^49 + 2 * q^50 - 8 * q^52 + 12 * q^53 + 22 * q^54 + 22 * q^55 - 8 * q^57 + 12 * q^59 + 14 * q^60 + 8 * q^61 - 4 * q^62 + 36 * q^63 + 4 * q^64 - 12 * q^65 + 44 * q^66 + 20 * q^67 + 6 * q^68 - 44 * q^69 - 18 * q^71 - 16 * q^73 + 32 * q^75 - 8 * q^76 - 2 * q^78 + 6 * q^80 - 8 * q^81 + 12 * q^82 + 6 * q^84 - 2 * q^85 - 10 * q^86 + 16 * q^87 + 12 * q^89 - 22 * q^90 - 36 * q^91 - 12 * q^95 - 2 * q^96 - 4 * q^97 + 8 * q^98 - 44 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 2\nu ) / 3$$ (v^3 - 2*v) / 3 $$\beta_{3}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3$$ b3 + 3 $$\nu^{3}$$ $$=$$ $$3\beta_{2} + 2\beta_1$$ 3*b2 + 2*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/130\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$41$$ $$\chi(n)$$ $$-\beta_{2}$$ $$\beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
57.1
 1.65831 − 0.500000i −1.65831 − 0.500000i 1.65831 + 0.500000i −1.65831 + 0.500000i
−1.00000 −1.15831 + 1.15831i 1.00000 1.50000 1.65831i 1.15831 1.15831i 3.00000i −1.00000 0.316625i −1.50000 + 1.65831i
57.2 −1.00000 2.15831 2.15831i 1.00000 1.50000 + 1.65831i −2.15831 + 2.15831i 3.00000i −1.00000 6.31662i −1.50000 1.65831i
73.1 −1.00000 −1.15831 1.15831i 1.00000 1.50000 + 1.65831i 1.15831 + 1.15831i 3.00000i −1.00000 0.316625i −1.50000 1.65831i
73.2 −1.00000 2.15831 + 2.15831i 1.00000 1.50000 1.65831i −2.15831 2.15831i 3.00000i −1.00000 6.31662i −1.50000 + 1.65831i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 130.2.g.d 4
3.b odd 2 1 1170.2.m.e 4
4.b odd 2 1 1040.2.bg.k 4
5.b even 2 1 650.2.g.g 4
5.c odd 4 1 130.2.j.d yes 4
5.c odd 4 1 650.2.j.f 4
13.d odd 4 1 130.2.j.d yes 4
15.e even 4 1 1170.2.w.e 4
20.e even 4 1 1040.2.cd.i 4
39.f even 4 1 1170.2.w.e 4
52.f even 4 1 1040.2.cd.i 4
65.f even 4 1 650.2.g.g 4
65.g odd 4 1 650.2.j.f 4
65.k even 4 1 inner 130.2.g.d 4
195.j odd 4 1 1170.2.m.e 4
260.s odd 4 1 1040.2.bg.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.g.d 4 1.a even 1 1 trivial
130.2.g.d 4 65.k even 4 1 inner
130.2.j.d yes 4 5.c odd 4 1
130.2.j.d yes 4 13.d odd 4 1
650.2.g.g 4 5.b even 2 1
650.2.g.g 4 65.f even 4 1
650.2.j.f 4 5.c odd 4 1
650.2.j.f 4 65.g odd 4 1
1040.2.bg.k 4 4.b odd 2 1
1040.2.bg.k 4 260.s odd 4 1
1040.2.cd.i 4 20.e even 4 1
1040.2.cd.i 4 52.f even 4 1
1170.2.m.e 4 3.b odd 2 1
1170.2.m.e 4 195.j odd 4 1
1170.2.w.e 4 15.e even 4 1
1170.2.w.e 4 39.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 2T_{3}^{3} + 2T_{3}^{2} + 10T_{3} + 25$$ acting on $$S_{2}^{\mathrm{new}}(130, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{4}$$
$3$ $$T^{4} - 2 T^{3} + 2 T^{2} + 10 T + 25$$
$5$ $$(T^{2} - 3 T + 5)^{2}$$
$7$ $$(T^{2} + 9)^{2}$$
$11$ $$T^{4} + 484$$
$13$ $$(T^{2} + 4 T + 13)^{2}$$
$17$ $$T^{4} - 6 T^{3} + 18 T^{2} + 6 T + 1$$
$19$ $$(T^{2} + 4 T + 8)^{2}$$
$23$ $$T^{4} + 484$$
$29$ $$T^{4} + 40T^{2} + 4$$
$31$ $$(T^{2} - 2 T + 2)^{2}$$
$37$ $$(T^{2} + 9)^{2}$$
$41$ $$T^{4} + 12 T^{3} + 72 T^{2} - 48 T + 16$$
$43$ $$T^{4} - 10 T^{3} + 50 T^{2} + \cdots + 1369$$
$47$ $$T^{4} + 94T^{2} + 625$$
$53$ $$T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 4900$$
$59$ $$T^{4} - 12 T^{3} + 72 T^{2} + 48 T + 16$$
$61$ $$(T - 2)^{4}$$
$67$ $$(T^{2} - 10 T - 74)^{2}$$
$71$ $$T^{4} + 18 T^{3} + 162 T^{2} + \cdots + 1225$$
$73$ $$(T + 4)^{4}$$
$79$ $$T^{4} + 216T^{2} + 8100$$
$83$ $$T^{4} + 40T^{2} + 4$$
$89$ $$T^{4} - 12 T^{3} + 72 T^{2} + 48 T + 16$$
$97$ $$(T^{2} + 2 T - 98)^{2}$$