# Properties

 Label 338.3.d.d Level $338$ Weight $3$ Character orbit 338.d Analytic conductor $9.210$ 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] = [338,3,Mod(99,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("338.99");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 338.d (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: $$9.20983293538$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 26) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - 1) q^{2} + (\beta_{3} - \beta_{2} + \beta_1) q^{3} + 2 \beta_{2} q^{4} + (\beta_{2} - 2 \beta_1 + 1) q^{5} + (\beta_{2} - 2 \beta_1 + 1) q^{6} + (7 \beta_{3} - 4 \beta_{2} + 4) q^{7} + ( - 2 \beta_{2} + 2) q^{8} - 6 q^{9}+O(q^{10})$$ q + (-b2 - 1) * q^2 + (b3 - b2 + b1) * q^3 + 2*b2 * q^4 + (b2 - 2*b1 + 1) * q^5 + (b2 - 2*b1 + 1) * q^6 + (7*b3 - 4*b2 + 4) * q^7 + (-2*b2 + 2) * q^8 - 6 * q^9 $$q + ( - \beta_{2} - 1) q^{2} + (\beta_{3} - \beta_{2} + \beta_1) q^{3} + 2 \beta_{2} q^{4} + (\beta_{2} - 2 \beta_1 + 1) q^{5} + (\beta_{2} - 2 \beta_1 + 1) q^{6} + (7 \beta_{3} - 4 \beta_{2} + 4) q^{7} + ( - 2 \beta_{2} + 2) q^{8} - 6 q^{9} + ( - 2 \beta_{3} + 2 \beta_1 - 2) q^{10} + (\beta_{3} - 5 \beta_{2} + 5) q^{11} + ( - 2 \beta_{3} + 2 \beta_1 - 2) q^{12} + ( - 7 \beta_{3} + 7 \beta_{2} - 7 \beta_1 - 1) q^{14} + ( - 3 \beta_{2} - 3) q^{15} - 4 q^{16} + ( - 13 \beta_{3} - 6 \beta_{2} + 13 \beta_1 - 13) q^{17} + (6 \beta_{2} + 6) q^{18} + (12 \beta_{2} - 17 \beta_1 + 12) q^{19} + (4 \beta_{3} - 2 \beta_{2} + 2) q^{20} + (\beta_{3} - 11 \beta_{2} + 11) q^{21} + ( - \beta_{3} + \beta_{2} - \beta_1 - 9) q^{22} + (2 \beta_{3} - 21 \beta_{2} - 2 \beta_1 + 2) q^{23} + (4 \beta_{3} - 2 \beta_{2} + 2) q^{24} - 19 \beta_{2} q^{25} + ( - 15 \beta_{3} + 15 \beta_{2} - 15 \beta_1) q^{27} + ( - 6 \beta_{2} + 14 \beta_1 - 6) q^{28} + (16 \beta_{3} - 16 \beta_{2} + 16 \beta_1 + 27) q^{29} + 6 \beta_{2} q^{30} + ( - 27 \beta_{2} - 10 \beta_1 - 27) q^{31} + (4 \beta_{2} + 4) q^{32} + (9 \beta_{3} - 6 \beta_{2} + 6) q^{33} + (26 \beta_{3} - 7 \beta_{2} + 7) q^{34} + ( - \beta_{3} + \beta_{2} - \beta_1 - 21) q^{35} - 12 \beta_{2} q^{36} + (7 \beta_{3} + 6 \beta_{2} - 6) q^{37} + ( - 17 \beta_{3} - 7 \beta_{2} + 17 \beta_1 - 17) q^{38} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{40} + ( - 26 \beta_{2} + 13 \beta_1 - 26) q^{41} + ( - \beta_{3} + \beta_{2} - \beta_1 - 21) q^{42} + ( - 20 \beta_{3} - 39 \beta_{2} + 20 \beta_1 - 20) q^{43} + (8 \beta_{2} + 2 \beta_1 + 8) q^{44} + ( - 6 \beta_{2} + 12 \beta_1 - 6) q^{45} + ( - 4 \beta_{3} + 23 \beta_{2} - 23) q^{46} + ( - 33 \beta_{2} + 33) q^{47} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{48} + (7 \beta_{3} - 25 \beta_{2} - 7 \beta_1 + 7) q^{49} + (19 \beta_{2} - 19) q^{50} + (6 \beta_{3} + 39 \beta_{2} - 6 \beta_1 + 6) q^{51} + (22 \beta_{3} - 22 \beta_{2} + 22 \beta_1 - 42) q^{53} + ( - 15 \beta_{2} + 30 \beta_1 - 15) q^{54} + ( - 9 \beta_{3} + 9 \beta_{2} - 9 \beta_1 - 3) q^{55} + (14 \beta_{3} - 2 \beta_{2} - 14 \beta_1 + 14) q^{56} + ( - 29 \beta_{2} + 7 \beta_1 - 29) q^{57} + ( - 11 \beta_{2} - 32 \beta_1 - 11) q^{58} + ( - 13 \beta_{3} - 10 \beta_{2} + 10) q^{59} + ( - 6 \beta_{2} + 6) q^{60} + ( - 6 \beta_{3} + 6 \beta_{2} - 6 \beta_1 - 39) q^{61} + ( - 10 \beta_{3} + 64 \beta_{2} + 10 \beta_1 - 10) q^{62} + ( - 42 \beta_{3} + 24 \beta_{2} - 24) q^{63} - 8 \beta_{2} q^{64} + ( - 9 \beta_{3} + 9 \beta_{2} - 9 \beta_1 - 3) q^{66} + (23 \beta_{2} - 33 \beta_1 + 23) q^{67} + ( - 26 \beta_{3} + 26 \beta_{2} - 26 \beta_1 + 12) q^{68} + (21 \beta_{3} - 6 \beta_{2} - 21 \beta_1 + 21) q^{69} + (20 \beta_{2} + 2 \beta_1 + 20) q^{70} + (4 \beta_{2} + 25 \beta_1 + 4) q^{71} + (12 \beta_{2} - 12) q^{72} + ( - 54 \beta_{3} + 61 \beta_{2} - 61) q^{73} + ( - 7 \beta_{3} + 7 \beta_{2} - 7 \beta_1 + 19) q^{74} + (19 \beta_{3} - 19 \beta_1 + 19) q^{75} + (34 \beta_{3} - 10 \beta_{2} + 10) q^{76} + (32 \beta_{3} - 15 \beta_{2} - 32 \beta_1 + 32) q^{77} + (36 \beta_{3} - 36 \beta_{2} + 36 \beta_1 + 48) q^{79} + ( - 4 \beta_{2} + 8 \beta_1 - 4) q^{80} + 9 q^{81} + (13 \beta_{3} + 39 \beta_{2} - 13 \beta_1 + 13) q^{82} + (71 \beta_{2} - 46 \beta_1 + 71) q^{83} + (20 \beta_{2} + 2 \beta_1 + 20) q^{84} + ( - 12 \beta_{3} - 33 \beta_{2} + 33) q^{85} + (40 \beta_{3} + 19 \beta_{2} - 19) q^{86} + (27 \beta_{3} - 27 \beta_{2} + 27 \beta_1 + 48) q^{87} + (2 \beta_{3} - 18 \beta_{2} - 2 \beta_1 + 2) q^{88} + (13 \beta_{3} - 56 \beta_{2} + 56) q^{89} + (12 \beta_{3} - 12 \beta_1 + 12) q^{90} + (4 \beta_{3} - 4 \beta_{2} + 4 \beta_1 + 42) q^{92} + (17 \beta_{2} - 64 \beta_1 + 17) q^{93} - 66 q^{94} + (7 \beta_{3} + 51 \beta_{2} - 7 \beta_1 + 7) q^{95} + ( - 4 \beta_{2} + 8 \beta_1 - 4) q^{96} + ( - 71 \beta_{2} + 69 \beta_1 - 71) q^{97} + ( - 14 \beta_{3} + 32 \beta_{2} - 32) q^{98} + ( - 6 \beta_{3} + 30 \beta_{2} - 30) q^{99}+O(q^{100})$$ q + (-b2 - 1) * q^2 + (b3 - b2 + b1) * q^3 + 2*b2 * q^4 + (b2 - 2*b1 + 1) * q^5 + (b2 - 2*b1 + 1) * q^6 + (7*b3 - 4*b2 + 4) * q^7 + (-2*b2 + 2) * q^8 - 6 * q^9 + (-2*b3 + 2*b1 - 2) * q^10 + (b3 - 5*b2 + 5) * q^11 + (-2*b3 + 2*b1 - 2) * q^12 + (-7*b3 + 7*b2 - 7*b1 - 1) * q^14 + (-3*b2 - 3) * q^15 - 4 * q^16 + (-13*b3 - 6*b2 + 13*b1 - 13) * q^17 + (6*b2 + 6) * q^18 + (12*b2 - 17*b1 + 12) * q^19 + (4*b3 - 2*b2 + 2) * q^20 + (b3 - 11*b2 + 11) * q^21 + (-b3 + b2 - b1 - 9) * q^22 + (2*b3 - 21*b2 - 2*b1 + 2) * q^23 + (4*b3 - 2*b2 + 2) * q^24 - 19*b2 * q^25 + (-15*b3 + 15*b2 - 15*b1) * q^27 + (-6*b2 + 14*b1 - 6) * q^28 + (16*b3 - 16*b2 + 16*b1 + 27) * q^29 + 6*b2 * q^30 + (-27*b2 - 10*b1 - 27) * q^31 + (4*b2 + 4) * q^32 + (9*b3 - 6*b2 + 6) * q^33 + (26*b3 - 7*b2 + 7) * q^34 + (-b3 + b2 - b1 - 21) * q^35 - 12*b2 * q^36 + (7*b3 + 6*b2 - 6) * q^37 + (-17*b3 - 7*b2 + 17*b1 - 17) * q^38 + (-4*b3 + 4*b2 - 4*b1) * q^40 + (-26*b2 + 13*b1 - 26) * q^41 + (-b3 + b2 - b1 - 21) * q^42 + (-20*b3 - 39*b2 + 20*b1 - 20) * q^43 + (8*b2 + 2*b1 + 8) * q^44 + (-6*b2 + 12*b1 - 6) * q^45 + (-4*b3 + 23*b2 - 23) * q^46 + (-33*b2 + 33) * q^47 + (-4*b3 + 4*b2 - 4*b1) * q^48 + (7*b3 - 25*b2 - 7*b1 + 7) * q^49 + (19*b2 - 19) * q^50 + (6*b3 + 39*b2 - 6*b1 + 6) * q^51 + (22*b3 - 22*b2 + 22*b1 - 42) * q^53 + (-15*b2 + 30*b1 - 15) * q^54 + (-9*b3 + 9*b2 - 9*b1 - 3) * q^55 + (14*b3 - 2*b2 - 14*b1 + 14) * q^56 + (-29*b2 + 7*b1 - 29) * q^57 + (-11*b2 - 32*b1 - 11) * q^58 + (-13*b3 - 10*b2 + 10) * q^59 + (-6*b2 + 6) * q^60 + (-6*b3 + 6*b2 - 6*b1 - 39) * q^61 + (-10*b3 + 64*b2 + 10*b1 - 10) * q^62 + (-42*b3 + 24*b2 - 24) * q^63 - 8*b2 * q^64 + (-9*b3 + 9*b2 - 9*b1 - 3) * q^66 + (23*b2 - 33*b1 + 23) * q^67 + (-26*b3 + 26*b2 - 26*b1 + 12) * q^68 + (21*b3 - 6*b2 - 21*b1 + 21) * q^69 + (20*b2 + 2*b1 + 20) * q^70 + (4*b2 + 25*b1 + 4) * q^71 + (12*b2 - 12) * q^72 + (-54*b3 + 61*b2 - 61) * q^73 + (-7*b3 + 7*b2 - 7*b1 + 19) * q^74 + (19*b3 - 19*b1 + 19) * q^75 + (34*b3 - 10*b2 + 10) * q^76 + (32*b3 - 15*b2 - 32*b1 + 32) * q^77 + (36*b3 - 36*b2 + 36*b1 + 48) * q^79 + (-4*b2 + 8*b1 - 4) * q^80 + 9 * q^81 + (13*b3 + 39*b2 - 13*b1 + 13) * q^82 + (71*b2 - 46*b1 + 71) * q^83 + (20*b2 + 2*b1 + 20) * q^84 + (-12*b3 - 33*b2 + 33) * q^85 + (40*b3 + 19*b2 - 19) * q^86 + (27*b3 - 27*b2 + 27*b1 + 48) * q^87 + (2*b3 - 18*b2 - 2*b1 + 2) * q^88 + (13*b3 - 56*b2 + 56) * q^89 + (12*b3 - 12*b1 + 12) * q^90 + (4*b3 - 4*b2 + 4*b1 + 42) * q^92 + (17*b2 - 64*b1 + 17) * q^93 - 66 * q^94 + (7*b3 + 51*b2 - 7*b1 + 7) * q^95 + (-4*b2 + 8*b1 - 4) * q^96 + (-71*b2 + 69*b1 - 71) * q^97 + (-14*b3 + 32*b2 - 32) * q^98 + (-6*b3 + 30*b2 - 30) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} + 2 q^{7} + 8 q^{8} - 24 q^{9}+O(q^{10})$$ 4 * q - 4 * q^2 + 2 * q^7 + 8 * q^8 - 24 * q^9 $$4 q - 4 q^{2} + 2 q^{7} + 8 q^{8} - 24 q^{9} + 18 q^{11} - 4 q^{14} - 12 q^{15} - 16 q^{16} + 24 q^{18} + 14 q^{19} + 42 q^{21} - 36 q^{22} + 4 q^{28} + 108 q^{29} - 128 q^{31} + 16 q^{32} + 6 q^{33} - 24 q^{34} - 84 q^{35} - 38 q^{37} - 78 q^{41} - 84 q^{42} + 36 q^{44} - 84 q^{46} + 132 q^{47} - 76 q^{50} - 168 q^{53} - 12 q^{55} - 102 q^{57} - 108 q^{58} + 66 q^{59} + 24 q^{60} - 156 q^{61} - 12 q^{63} - 12 q^{66} + 26 q^{67} + 48 q^{68} + 84 q^{70} + 66 q^{71} - 48 q^{72} - 136 q^{73} + 76 q^{74} - 28 q^{76} + 192 q^{79} + 36 q^{81} + 192 q^{83} + 84 q^{84} + 156 q^{85} - 156 q^{86} + 192 q^{87} + 198 q^{89} + 168 q^{92} - 60 q^{93} - 264 q^{94} - 146 q^{97} - 100 q^{98} - 108 q^{99}+O(q^{100})$$ 4 * q - 4 * q^2 + 2 * q^7 + 8 * q^8 - 24 * q^9 + 18 * q^11 - 4 * q^14 - 12 * q^15 - 16 * q^16 + 24 * q^18 + 14 * q^19 + 42 * q^21 - 36 * q^22 + 4 * q^28 + 108 * q^29 - 128 * q^31 + 16 * q^32 + 6 * q^33 - 24 * q^34 - 84 * q^35 - 38 * q^37 - 78 * q^41 - 84 * q^42 + 36 * q^44 - 84 * q^46 + 132 * q^47 - 76 * q^50 - 168 * q^53 - 12 * q^55 - 102 * q^57 - 108 * q^58 + 66 * q^59 + 24 * q^60 - 156 * q^61 - 12 * q^63 - 12 * q^66 + 26 * q^67 + 48 * q^68 + 84 * q^70 + 66 * q^71 - 48 * q^72 - 136 * q^73 + 76 * q^74 - 28 * q^76 + 192 * q^79 + 36 * q^81 + 192 * q^83 + 84 * q^84 + 156 * q^85 - 156 * q^86 + 192 * q^87 + 198 * q^89 + 168 * q^92 - 60 * q^93 - 264 * q^94 - 146 * q^97 - 100 * q^98 - 108 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{2} + \zeta_{12}$$ v^2 + v $$\beta_{2}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{2} + \zeta_{12}$$ -v^2 + v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$( -\beta_{3} + \beta_1 ) / 2$$ (-b3 + b1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_{2}$$ b2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/338\mathbb{Z}\right)^\times$$.

 $$n$$ $$171$$ $$\chi(n)$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i
−1.00000 1.00000i −1.73205 2.00000i 1.73205 + 1.73205i 1.73205 + 1.73205i −5.56218 + 5.56218i 2.00000 2.00000i −6.00000 3.46410i
99.2 −1.00000 1.00000i 1.73205 2.00000i −1.73205 1.73205i −1.73205 1.73205i 6.56218 6.56218i 2.00000 2.00000i −6.00000 3.46410i
239.1 −1.00000 + 1.00000i −1.73205 2.00000i 1.73205 1.73205i 1.73205 1.73205i −5.56218 5.56218i 2.00000 + 2.00000i −6.00000 3.46410i
239.2 −1.00000 + 1.00000i 1.73205 2.00000i −1.73205 + 1.73205i −1.73205 + 1.73205i 6.56218 + 6.56218i 2.00000 + 2.00000i −6.00000 3.46410i
 $$n$$: 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
13.d odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.3.d.d 4
13.b even 2 1 338.3.d.e 4
13.c even 3 1 26.3.f.a 4
13.c even 3 1 338.3.f.f 4
13.d odd 4 1 inner 338.3.d.d 4
13.d odd 4 1 338.3.d.e 4
13.e even 6 1 338.3.f.c 4
13.e even 6 1 338.3.f.d 4
13.f odd 12 1 26.3.f.a 4
13.f odd 12 1 338.3.f.c 4
13.f odd 12 1 338.3.f.d 4
13.f odd 12 1 338.3.f.f 4
39.i odd 6 1 234.3.bb.b 4
39.k even 12 1 234.3.bb.b 4
52.j odd 6 1 208.3.bd.c 4
52.l even 12 1 208.3.bd.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.3.f.a 4 13.c even 3 1
26.3.f.a 4 13.f odd 12 1
208.3.bd.c 4 52.j odd 6 1
208.3.bd.c 4 52.l even 12 1
234.3.bb.b 4 39.i odd 6 1
234.3.bb.b 4 39.k even 12 1
338.3.d.d 4 1.a even 1 1 trivial
338.3.d.d 4 13.d odd 4 1 inner
338.3.d.e 4 13.b even 2 1
338.3.d.e 4 13.d odd 4 1
338.3.f.c 4 13.e even 6 1
338.3.f.c 4 13.f odd 12 1
338.3.f.d 4 13.e even 6 1
338.3.f.d 4 13.f odd 12 1
338.3.f.f 4 13.c even 3 1
338.3.f.f 4 13.f odd 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(338, [\chi])$$:

 $$T_{3}^{2} - 3$$ T3^2 - 3 $$T_{5}^{4} + 36$$ T5^4 + 36 $$T_{7}^{4} - 2T_{7}^{3} + 2T_{7}^{2} + 146T_{7} + 5329$$ T7^4 - 2*T7^3 + 2*T7^2 + 146*T7 + 5329

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2 T + 2)^{2}$$
$3$ $$(T^{2} - 3)^{2}$$
$5$ $$T^{4} + 36$$
$7$ $$T^{4} - 2 T^{3} + 2 T^{2} + 146 T + 5329$$
$11$ $$T^{4} - 18 T^{3} + 162 T^{2} + \cdots + 1521$$
$13$ $$T^{4}$$
$17$ $$T^{4} + 1086 T^{2} + 221841$$
$19$ $$T^{4} - 14 T^{3} + 98 T^{2} + \cdots + 167281$$
$23$ $$T^{4} + 906 T^{2} + 184041$$
$29$ $$(T^{2} - 54 T - 39)^{2}$$
$31$ $$T^{4} + 128 T^{3} + 8192 T^{2} + \cdots + 3602404$$
$37$ $$T^{4} + 38 T^{3} + 722 T^{2} + \cdots + 11449$$
$41$ $$T^{4} + 78 T^{3} + 3042 T^{2} + \cdots + 257049$$
$43$ $$T^{4} + 5442 T^{2} + 103041$$
$47$ $$(T^{2} - 66 T + 2178)^{2}$$
$53$ $$(T^{2} + 84 T + 312)^{2}$$
$59$ $$T^{4} - 66 T^{3} + 2178 T^{2} + \cdots + 84681$$
$61$ $$(T^{2} + 78 T + 1413)^{2}$$
$67$ $$T^{4} - 26 T^{3} + 338 T^{2} + \cdots + 2399401$$
$71$ $$T^{4} - 66 T^{3} + 2178 T^{2} + \cdots + 154449$$
$73$ $$T^{4} + 136 T^{3} + 9248 T^{2} + \cdots + 4251844$$
$79$ $$(T^{2} - 96 T - 1584)^{2}$$
$83$ $$T^{4} - 192 T^{3} + 18432 T^{2} + \cdots + 2056356$$
$89$ $$T^{4} - 198 T^{3} + \cdots + 21594609$$
$97$ $$T^{4} + 146 T^{3} + \cdots + 20043529$$