# Properties

 Label 841.2.a.i Level $841$ Weight $2$ Character orbit 841.a Self dual yes Analytic conductor $6.715$ Analytic rank $1$ Dimension $8$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [841,2,Mod(1,841)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(841, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("841.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$841 = 29^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 841.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: $$6.71541880999$$ Analytic rank: $$1$$ Dimension: $$8$$ Coefficient field: 8.8.2841328125.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 4x^{7} - 3x^{6} + 23x^{5} - 43x^{3} + 2x^{2} + 24x + 1$$ x^8 - 4*x^7 - 3*x^6 + 23*x^5 - 43*x^3 + 2*x^2 + 24*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{5} - 1) q^{3} + (\beta_{2} + \beta_1) q^{4} + (\beta_{6} - \beta_{3}) q^{5} + ( - \beta_{6} - \beta_{4} + \cdots + \beta_1) q^{6}+ \cdots + ( - \beta_{7} - \beta_{5} + \beta_{4} + \cdots + 2) q^{9}+O(q^{10})$$ q - b1 * q^2 + (b5 - 1) * q^3 + (b2 + b1) * q^4 + (b6 - b3) * q^5 + (-b6 - b4 + b3 + b1) * q^6 + (b7 + b6 - b5 + b1) * q^7 + (-b3 - b2 - 2) * q^8 + (-b7 - b5 + b4 - b2 - b1 + 2) * q^9 $$q - \beta_1 q^{2} + (\beta_{5} - 1) q^{3} + (\beta_{2} + \beta_1) q^{4} + (\beta_{6} - \beta_{3}) q^{5} + ( - \beta_{6} - \beta_{4} + \cdots + \beta_1) q^{6}+ \cdots + (\beta_{7} + 4 \beta_{6} - 3 \beta_{5} + \cdots + 2) q^{99}+O(q^{100})$$ q - b1 * q^2 + (b5 - 1) * q^3 + (b2 + b1) * q^4 + (b6 - b3) * q^5 + (-b6 - b4 + b3 + b1) * q^6 + (b7 + b6 - b5 + b1) * q^7 + (-b3 - b2 - 2) * q^8 + (-b7 - b5 + b4 - b2 - b1 + 2) * q^9 + (-b7 - b6 + b4 + b3 + b2) * q^10 + (b7 + b3 - b1) * q^11 + (b7 + b6 - b5 - 2*b2 - b1) * q^12 + (-b7 - 2*b6 - b5 - 1) * q^13 + (-b7 - 3*b6 - b3 - b1 - 3) * q^14 + (-2*b6 + b3 - b2 + b1) * q^15 + (b4 + 2*b3 - b2 + b1) * q^16 + (-2*b7 + b5 + b3 - b1) * q^17 + (4*b6 - b5 + b4 - b3 + 3) * q^18 + (b6 - b5 - b4 + b3 - b2 + b1 - 1) * q^19 + (b7 + 2*b6 - b5 - b4 - b3 - 2*b2 - b1 + 1) * q^20 + (-b6 - b5 - b4 - b3 + 2*b2 - 4) * q^21 + (-3*b6 - 2*b4 - b3 + b2 + b1 + 1) * q^22 + (b7 - 2*b6 + b5 + 2*b4 + b3 + b1 - 3) * q^23 + (-b7 - b6 + 2*b4 - b3 + 2*b2 + b1 + 1) * q^24 + (-b5 - b4 - b3 + b1 - 2) * q^25 + (2*b7 + 6*b6 + 2*b4 - b3 - b2 + b1 + 1) * q^26 + (b7 - b6 + b5 - b4 + 2*b2 + 3*b1 - 5) * q^27 + (b7 + 4*b6 + 2*b5 + 2*b4 + b3 + b2 + 2*b1 + 3) * q^28 + (2*b7 + 2*b6 - b4 - 2*b2 - 2) * q^30 + (b7 - 3*b6 - b4 - b3 + 2*b2 + b1 - 3) * q^31 + (-b5 - 3*b4 - b2 + 2) * q^32 + (-b7 - b6 - b5 - 2*b4 + 2*b2) * q^33 + (5*b6 - 2*b2 + b1 + 4) * q^34 + (-b5 - 3*b4 - b3) * q^35 + (-2*b7 - 3*b6 + b5 - b4 - b3 + 3*b2 - b1 - 4) * q^36 + (-b7 - 2*b6 - b5 + b4 + 2*b3 - b2 + b1 - 1) * q^37 + (-b7 + b5 + b4 - 2*b2 + b1 - 2) * q^38 + (2*b7 + 3*b6 + b5 + b4 - b3 + b2 + b1 - 4) * q^39 + (-2*b6 + b5 + b3 + b2 + 2*b1 + 1) * q^40 + (-2*b7 - 3*b6 + b5 - 2*b4 - 2*b3 + b2 + b1 - 3) * q^41 + (b7 + 2*b6 + b5 + 3*b4 - b3 + b2 + 2*b1) * q^42 + (3*b6 - b5 + 2*b4 + b3 + b2 + 2) * q^43 + (b7 + 3*b6 + 2*b5 + 3*b4 - b1 - 2) * q^44 + (-b7 + b6 + 2*b5 + 3*b4 - b3 + 4*b2 - 2*b1 - 1) * q^45 + (2*b7 - 2*b6 - 2*b5 - 5*b4 - 2*b3 - b2 + 2*b1 - 3) * q^46 + (2*b6 + 2*b5 + b3 - 3) * q^47 + (-b7 + 2*b6 - 3*b3 + 3*b2 - 2*b1 - 1) * q^48 + (4*b6 + 3*b5 + 2*b4 + 2*b3 - 2*b2 + b1 + 2) * q^49 + (b6 + b5 + 3*b4 + b3 + b1 - 2) * q^50 + (b7 + 2*b6 - b5 - b4 + 3*b3 - 5*b2 - b1 + 7) * q^51 + (-4*b7 - 8*b6 - 3*b4 + 2*b2 - b1 - 2) * q^52 + (-5*b6 - b5 - 4*b4 + b3 - 2*b2 - 2*b1 + 3) * q^53 + (b7 - 3*b6 + b5 - b4 - 2*b2 - 7) * q^54 + (-2*b7 - 3*b6 + 2*b3 + b2 - 3) * q^55 + (-2*b7 - 3*b6 - 2*b5 - 6*b4 - 2*b2 - 4*b1 + 1) * q^56 + (2*b6 - 3*b5 - b4 + b2 - 2*b1 + 2) * q^57 + (-2*b7 - 5*b6 + 2*b5 - b4 - b3 + b2 - 3) * q^59 + (-2*b7 - 4*b6 + b5 - b4 + b3 + 4*b2 + 2*b1 - 2) * q^60 + (b7 + 3*b6 + 3*b3 - 5*b2 + 1) * q^61 + (3*b7 + b5 + b4 + b2 - 3) * q^62 + (-2*b6 - 2*b5 - b4 + 2*b3 - 3*b2 - 2*b1 + 1) * q^63 + (b6 + 3*b5 + 2*b4 - b3 + 2*b2 - 3*b1) * q^64 + (b7 + 3*b6 + 2*b5 + 3*b4 + b3 + b2 - 2*b1 - 1) * q^65 + (b7 + 5*b6 + 2*b5 + 4*b4 - b3 - b2 - 2*b1 + 1) * q^66 + (4*b7 + 2*b6 + 2*b5 + 2*b4 - 2*b3 + 2*b2 + 2*b1 - 3) * q^67 + (-b7 - 5*b6 - 2*b5 - b2 - b1 - 2) * q^68 + (-2*b7 + b6 + 3*b4 - 6*b3 + 5*b2 - b1 - 1) * q^69 + (b6 + 3*b5 + 5*b4 + 3*b3 + b2) * q^70 + (b7 + 3*b6 + 2*b5 + 3*b4 + 2*b3 + b2 - 4*b1 + 1) * q^71 + (3*b7 + 3*b5 + b4 + 2*b3 + 2*b1 - 2) * q^72 + (-b7 - b6 - 4*b5 - 3*b4 - 4*b3 - b2 - b1 + 5) * q^73 + (2*b7 + 6*b6 - b5 - b4 - 3*b3 - 4*b2 + b1 - 1) * q^74 + (b7 + b6 - 2*b5 + b4) * q^75 + (b5 + b4 + b1 + 1) * q^76 + (-b7 - 2*b6 + 2*b5 + 4*b4 - 1) * q^77 + (-3*b7 - 10*b6 - b5 - 3*b4 + 2*b2 + 2*b1 - 4) * q^78 + (-b6 + b5 - b3 + 3*b2 - 3*b1 - 4) * q^79 + (-3*b6 + 2*b5 + b3 + b2 - 2*b1 - 6) * q^80 + (3*b7 + 4*b6 - 4*b5 - 2*b3 - 2*b1 + 4) * q^81 + (3*b7 + 8*b6 + 2*b5 + 5*b4 + 4*b3 - b2 + b1) * q^82 + (-2*b7 + b6 + 2*b5 + b3 + 4*b2 - 3*b1 + 1) * q^83 + (-2*b7 - 4*b6 - b5 - 2*b4 - 4*b2 - 3*b1 + 3) * q^84 + (b7 - b6 + 3*b4 + 2*b3 - 2*b1) * q^85 + (-3*b7 - 2*b6 - 2*b5 - 2*b4 - 5*b3 - b2 - 3*b1) * q^86 + (-3*b7 - 2*b6 - 3*b5 - 2*b4 + b3 + b1 - 1) * q^88 + (-b7 + b6 - 2*b5 - 5*b2 - 2*b1 - 1) * q^89 + (-b7 - 3*b5 - 3*b4 - 4*b3 + 2*b2 - b1 + 5) * q^90 + (-2*b7 - 2*b6 + b5 - b4 + 2*b3 - 3*b2 - 4*b1 + 3) * q^91 + (2*b6 + 3*b5 + 3*b4 + 4*b3 + 2*b2) * q^92 + (b7 + 3*b6 - 2*b5 + 3*b4 - 2*b3 - b1 + 1) * q^93 + (-2*b7 - 4*b6 - 3*b4 + b3 - b2 + 3*b1) * q^94 + (2*b7 - b6 - b5 - 3*b4 + b3 - 1) * q^95 + (3*b6 + 2*b3 - 2*b1 + 3) * q^96 + (b6 - 3*b4 - 2*b3 + b2 - 3*b1 + 4) * q^97 + (-4*b7 - 7*b6 - 2*b5 - 7*b4 + b3 - 3*b2 - b1 - 2) * q^98 + (b7 + 4*b6 - 3*b5 - 2*b4 + b3 - 6*b2 + 2*b1 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 4 q^{2} - 6 q^{3} + 6 q^{4} - q^{5} + 3 q^{6} - 15 q^{8} + 8 q^{9}+O(q^{10})$$ 8 * q - 4 * q^2 - 6 * q^3 + 6 * q^4 - q^5 + 3 * q^6 - 15 * q^8 + 8 * q^9 $$8 q - 4 q^{2} - 6 q^{3} + 6 q^{4} - q^{5} + 3 q^{6} - 15 q^{8} + 8 q^{9} + 3 q^{10} - 5 q^{11} - 12 q^{12} - 4 q^{13} - 15 q^{14} + 7 q^{15} - 2 q^{16} - 9 q^{17} + 11 q^{18} - 17 q^{19} - 7 q^{20} - 25 q^{21} + 25 q^{22} - 7 q^{23} + 25 q^{24} - 13 q^{25} - 3 q^{26} - 18 q^{27} + 25 q^{28} - 26 q^{30} - q^{31} + 6 q^{32} + 12 q^{34} - 5 q^{35} - 19 q^{36} - 6 q^{37} - 14 q^{38} - 27 q^{39} + 25 q^{40} - 6 q^{41} + 15 q^{42} + 5 q^{43} - 20 q^{44} - q^{45} - 14 q^{46} - 31 q^{47} - 11 q^{48} + 4 q^{49} - 11 q^{50} + 23 q^{51} + 2 q^{52} + 19 q^{53} - 46 q^{54} - 20 q^{55} - 20 q^{56} - 6 q^{57} - q^{59} + 9 q^{60} - 21 q^{61} - 12 q^{62} - 10 q^{63} + q^{64} - 17 q^{65} - 5 q^{66} + 2 q^{67} - 8 q^{68} + 14 q^{69} + 5 q^{70} - 12 q^{71} + 34 q^{73} - 27 q^{74} - 4 q^{75} + 16 q^{76} + 10 q^{77} + 6 q^{78} - 29 q^{79} - 41 q^{80} + 12 q^{81} - 22 q^{82} - 3 q^{83} + 10 q^{84} - 2 q^{85} - 5 q^{86} - 15 q^{88} - 36 q^{89} + 38 q^{90} - 4 q^{92} + 2 q^{93} + 13 q^{94} - 11 q^{95} - 2 q^{96} + 18 q^{97} - 27 q^{98} - 15 q^{99}+O(q^{100})$$ 8 * q - 4 * q^2 - 6 * q^3 + 6 * q^4 - q^5 + 3 * q^6 - 15 * q^8 + 8 * q^9 + 3 * q^10 - 5 * q^11 - 12 * q^12 - 4 * q^13 - 15 * q^14 + 7 * q^15 - 2 * q^16 - 9 * q^17 + 11 * q^18 - 17 * q^19 - 7 * q^20 - 25 * q^21 + 25 * q^22 - 7 * q^23 + 25 * q^24 - 13 * q^25 - 3 * q^26 - 18 * q^27 + 25 * q^28 - 26 * q^30 - q^31 + 6 * q^32 + 12 * q^34 - 5 * q^35 - 19 * q^36 - 6 * q^37 - 14 * q^38 - 27 * q^39 + 25 * q^40 - 6 * q^41 + 15 * q^42 + 5 * q^43 - 20 * q^44 - q^45 - 14 * q^46 - 31 * q^47 - 11 * q^48 + 4 * q^49 - 11 * q^50 + 23 * q^51 + 2 * q^52 + 19 * q^53 - 46 * q^54 - 20 * q^55 - 20 * q^56 - 6 * q^57 - q^59 + 9 * q^60 - 21 * q^61 - 12 * q^62 - 10 * q^63 + q^64 - 17 * q^65 - 5 * q^66 + 2 * q^67 - 8 * q^68 + 14 * q^69 + 5 * q^70 - 12 * q^71 + 34 * q^73 - 27 * q^74 - 4 * q^75 + 16 * q^76 + 10 * q^77 + 6 * q^78 - 29 * q^79 - 41 * q^80 + 12 * q^81 - 22 * q^82 - 3 * q^83 + 10 * q^84 - 2 * q^85 - 5 * q^86 - 15 * q^88 - 36 * q^89 + 38 * q^90 - 4 * q^92 + 2 * q^93 + 13 * q^94 - 11 * q^95 - 2 * q^96 + 18 * q^97 - 27 * q^98 - 15 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} - 3x^{6} + 23x^{5} - 43x^{3} + 2x^{2} + 24x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 3\nu$$ v^3 - v^2 - 3*v $$\beta_{4}$$ $$=$$ $$\nu^{4} - 2\nu^{3} - 3\nu^{2} + 4\nu + 2$$ v^4 - 2*v^3 - 3*v^2 + 4*v + 2 $$\beta_{5}$$ $$=$$ $$\nu^{5} - 3\nu^{4} - 2\nu^{3} + 8\nu^{2} + \nu - 2$$ v^5 - 3*v^4 - 2*v^3 + 8*v^2 + v - 2 $$\beta_{6}$$ $$=$$ $$\nu^{6} - 3\nu^{5} - 3\nu^{4} + 11\nu^{3} + 3\nu^{2} - 9\nu - 2$$ v^6 - 3*v^5 - 3*v^4 + 11*v^3 + 3*v^2 - 9*v - 2 $$\beta_{7}$$ $$=$$ $$\nu^{7} - 4\nu^{6} + 14\nu^{4} - 8\nu^{3} - 12\nu^{2} + 7\nu + 2$$ v^7 - 4*v^6 + 14*v^4 - 8*v^3 - 12*v^2 + 7*v + 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4\beta _1 + 2$$ b3 + b2 + 4*b1 + 2 $$\nu^{4}$$ $$=$$ $$\beta_{4} + 2\beta_{3} + 5\beta_{2} + 7\beta _1 + 8$$ b4 + 2*b3 + 5*b2 + 7*b1 + 8 $$\nu^{5}$$ $$=$$ $$\beta_{5} + 3\beta_{4} + 8\beta_{3} + 9\beta_{2} + 20\beta _1 + 14$$ b5 + 3*b4 + 8*b3 + 9*b2 + 20*b1 + 14 $$\nu^{6}$$ $$=$$ $$\beta_{6} + 3\beta_{5} + 12\beta_{4} + 19\beta_{3} + 28\beta_{2} + 43\beta _1 + 40$$ b6 + 3*b5 + 12*b4 + 19*b3 + 28*b2 + 43*b1 + 40 $$\nu^{7}$$ $$=$$ $$\beta_{7} + 4\beta_{6} + 12\beta_{5} + 34\beta_{4} + 56\beta_{3} + 62\beta_{2} + 111\beta _1 + 86$$ b7 + 4*b6 + 12*b5 + 34*b4 + 56*b3 + 62*b2 + 111*b1 + 86

## 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
 2.51918 2.39427 1.92862 1.04195 −0.0419454 −0.928616 −1.39427 −1.51918
−2.51918 −1.05053 4.34627 −1.46556 2.64648 4.12665 −5.91068 −1.89638 3.69200
1.2 −2.39427 −2.10137 3.73253 −2.42792 5.03125 −0.378370 −4.14815 1.41576 5.81310
1.3 −1.92862 −0.484840 1.71956 2.94984 0.935071 2.60541 0.540863 −2.76493 −5.68911
1.4 −1.04195 2.15694 −0.914350 1.46226 −2.24741 −3.80089 3.03659 1.65237 −1.52360
1.5 0.0419454 −3.02773 −1.99824 −1.74204 −0.126999 2.05366 −0.167708 6.16716 −0.0730704
1.6 0.928616 1.65019 −1.13767 −0.504715 1.53239 −4.15272 −2.91369 −0.276865 −0.468687
1.7 1.39427 −0.0278335 −0.0560094 −1.14641 −0.0388074 −0.110460 −2.86663 −2.99923 −1.59840
1.8 1.51918 −3.11482 0.307910 1.87453 −4.73198 −0.343265 −2.57059 6.70211 2.84776
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$29$$ $$+1$$

## 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 841.2.a.i 8
3.b odd 2 1 7569.2.a.bi 8
29.b even 2 1 841.2.a.j yes 8
29.c odd 4 2 841.2.b.f 16
29.d even 7 6 841.2.d.q 48
29.e even 14 6 841.2.d.p 48
29.f odd 28 12 841.2.e.m 96
87.d odd 2 1 7569.2.a.bd 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
841.2.a.i 8 1.a even 1 1 trivial
841.2.a.j yes 8 29.b even 2 1
841.2.b.f 16 29.c odd 4 2
841.2.d.p 48 29.e even 14 6
841.2.d.q 48 29.d even 7 6
841.2.e.m 96 29.f odd 28 12
7569.2.a.bd 8 87.d odd 2 1
7569.2.a.bi 8 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 4T_{2}^{7} - 3T_{2}^{6} - 23T_{2}^{5} + 43T_{2}^{3} + 2T_{2}^{2} - 24T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(841))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 4 T^{7} + \cdots + 1$$
$3$ $$T^{8} + 6 T^{7} + \cdots + 1$$
$5$ $$T^{8} + T^{7} + \cdots - 29$$
$7$ $$T^{8} - 30 T^{6} + \cdots - 5$$
$11$ $$T^{8} + 5 T^{7} + \cdots + 295$$
$13$ $$T^{8} + 4 T^{7} + \cdots - 1769$$
$17$ $$T^{8} + 9 T^{7} + \cdots - 4799$$
$19$ $$T^{8} + 17 T^{7} + \cdots - 29$$
$23$ $$T^{8} + 7 T^{7} + \cdots + 25681$$
$29$ $$T^{8}$$
$31$ $$T^{8} + T^{7} + \cdots + 631$$
$37$ $$T^{8} + 6 T^{7} + \cdots - 229709$$
$41$ $$T^{8} + 6 T^{7} + \cdots + 210811$$
$43$ $$T^{8} - 5 T^{7} + \cdots - 305$$
$47$ $$T^{8} + 31 T^{7} + \cdots + 25111$$
$53$ $$T^{8} - 19 T^{7} + \cdots - 322349$$
$59$ $$T^{8} + T^{7} + \cdots + 179131$$
$61$ $$T^{8} + 21 T^{7} + \cdots - 365879$$
$67$ $$T^{8} - 2 T^{7} + \cdots + 14622931$$
$71$ $$T^{8} + 12 T^{7} + \cdots + 2574301$$
$73$ $$T^{8} - 34 T^{7} + \cdots + 1223191$$
$79$ $$T^{8} + 29 T^{7} + \cdots + 878851$$
$83$ $$T^{8} + 3 T^{7} + \cdots - 343139$$
$89$ $$T^{8} + 36 T^{7} + \cdots - 108359$$
$97$ $$T^{8} - 18 T^{7} + \cdots + 3862591$$