# Properties

 Label 225.2.e.d Level $225$ Weight $2$ Character orbit 225.e Analytic conductor $1.797$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,2,Mod(76,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 0]))

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

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

chi := DirichletCharacter("225.76");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 225.e (of order $$3$$, 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: $$1.79663404548$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{24})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 45) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{7} q^{2} + ( - \beta_{7} + \beta_{6}) q^{3} - \beta_{2} q^{4} + ( - \beta_{2} - \beta_1 - 1) q^{6} + (\beta_{7} - 2 \beta_{5} + 2 \beta_{3}) q^{7} - \beta_{6} q^{8} + ( - \beta_{4} - \beta_{2}) q^{9}+O(q^{10})$$ q - b7 * q^2 + (-b7 + b6) * q^3 - b2 * q^4 + (-b2 - b1 - 1) * q^6 + (b7 - 2*b5 + 2*b3) * q^7 - b6 * q^8 + (-b4 - b2) * q^9 $$q - \beta_{7} q^{2} + ( - \beta_{7} + \beta_{6}) q^{3} - \beta_{2} q^{4} + ( - \beta_{2} - \beta_1 - 1) q^{6} + (\beta_{7} - 2 \beta_{5} + 2 \beta_{3}) q^{7} - \beta_{6} q^{8} + ( - \beta_{4} - \beta_{2}) q^{9} + ( - \beta_{4} + \beta_{2} + 3 \beta_1 - 3) q^{11} + ( - \beta_{7} + 2 \beta_{6} + 3 \beta_{5} - 2 \beta_{3}) q^{12} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \beta_{3}) q^{13} - \beta_{2} q^{14} + (2 \beta_{4} - 2 \beta_{2} - \beta_1 + 1) q^{16} + ( - 2 \beta_{6} + \beta_{5}) q^{17} + (\beta_{7} + \beta_{6} + 3 \beta_{5} - \beta_{3}) q^{18} + (\beta_{4} + 1) q^{19} + ( - 2 \beta_{4} + \beta_{2} - 3 \beta_1 + 3) q^{21} + (4 \beta_{7} - 4 \beta_{6} - 4 \beta_{5} - \beta_{3}) q^{22} + ( - \beta_{7} + \beta_{6} + \beta_{5}) q^{23} + (\beta_{4} - \beta_1 - 1) q^{24} + (\beta_{4} + 3) q^{26} + (3 \beta_{6} + 3 \beta_{5} - 3 \beta_{3}) q^{27} + 3 \beta_{6} q^{28} + ( - 2 \beta_{4} + 2 \beta_{2} + 3 \beta_1 - 3) q^{29} + (\beta_{2} + \beta_1) q^{31} + ( - \beta_{7} + \beta_{6} + \beta_{5} + 4 \beta_{3}) q^{32} + (2 \beta_{7} - 4 \beta_{6} - 3 \beta_{5} - 2 \beta_{3}) q^{33} + ( - \beta_{4} + \beta_{2} - \beta_1 + 1) q^{34} + (6 \beta_1 - 3) q^{36} - 3 \beta_{5} q^{37} + ( - 2 \beta_{7} - \beta_{5} + \beta_{3}) q^{38} + (\beta_{4} + \beta_{2} - 3 \beta_1 + 3) q^{39} + (\beta_{2} - 6 \beta_1) q^{41} + ( - \beta_{7} + 2 \beta_{6} + 3 \beta_{5} - 2 \beta_{3}) q^{42} + ( - 4 \beta_{7} - \beta_{5} + \beta_{3}) q^{43} + (3 \beta_{4} + 3) q^{44} + ( - \beta_{4} - 2) q^{46} + ( - 3 \beta_{7} + \beta_{5} - \beta_{3}) q^{47} + ( - 4 \beta_{7} + 3 \beta_{6} + 6 \beta_{5} - \beta_{3}) q^{48} + (3 \beta_{2} + \beta_1) q^{49} + (2 \beta_{4} + \beta_{2} - \beta_1 - 4) q^{51} + ( - 3 \beta_{5} + 3 \beta_{3}) q^{52} + (2 \beta_{6} + 2 \beta_{5}) q^{53} + (3 \beta_{2} + 6 \beta_1 - 3) q^{54} + (2 \beta_{4} - 2 \beta_{2} + 3 \beta_1 - 3) q^{56} + ( - 2 \beta_{7} + \beta_{3}) q^{57} + (5 \beta_{7} - 5 \beta_{6} - 5 \beta_{5} - 2 \beta_{3}) q^{58} + (2 \beta_{2} - 6 \beta_1) q^{59} + ( - 5 \beta_{4} + 5 \beta_{2} + 2 \beta_1 - 2) q^{61} + ( - 2 \beta_{6} - 3 \beta_{5}) q^{62} + (3 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} + 3 \beta_{3}) q^{63} + ( - \beta_{4} - 4) q^{64} + (5 \beta_{4} - \beta_{2} - 3 \beta_1 + 9) q^{66} + (\beta_{7} - \beta_{6} - \beta_{5} + 5 \beta_{3}) q^{67} + (4 \beta_{7} - 4 \beta_{6} - 4 \beta_{5} + 5 \beta_{3}) q^{68} + ( - \beta_{4} + \beta_1 - 2) q^{69} + ( - 3 \beta_{4} + 9) q^{71} + (\beta_{7} - 2 \beta_{6} + 2 \beta_{3}) q^{72} + 6 \beta_{5} q^{73} + (3 \beta_{4} - 3 \beta_{2} - 3 \beta_1 + 3) q^{74} + ( - \beta_{2} - 3 \beta_1) q^{76} - 3 \beta_{3} q^{77} + ( - 4 \beta_{7} + 2 \beta_{6} + \beta_{3}) q^{78} + (2 \beta_{4} - 2 \beta_{2} + 4 \beta_1 - 4) q^{79} + 9 \beta_1 q^{81} + (5 \beta_{6} + 4 \beta_{5}) q^{82} + (\beta_{7} + 6 \beta_{5} - 6 \beta_{3}) q^{83} + ( - 3 \beta_{4} + 3 \beta_1 + 3) q^{84} + ( - 5 \beta_{2} - 9 \beta_1) q^{86} + (4 \beta_{7} - 5 \beta_{6} - 6 \beta_{5} - \beta_{3}) q^{87} + (2 \beta_{7} - \beta_{5} + \beta_{3}) q^{88} + ( - 6 \beta_{4} + 3) q^{89} + (3 \beta_{4} - 3) q^{91} + (\beta_{7} + \beta_{5} - \beta_{3}) q^{92} + ( - 2 \beta_{6} - 3 \beta_{5} + \beta_{3}) q^{93} + ( - 2 \beta_{2} - 5 \beta_1) q^{94} + ( - 5 \beta_{4} + 4 \beta_{2} - 3 \beta_1 - 6) q^{96} + ( - 2 \beta_{7} + 10 \beta_{5} - 10 \beta_{3}) q^{97} + ( - 4 \beta_{6} - 7 \beta_{5}) q^{98} + (6 \beta_{4} - 3 \beta_{2} - 3 \beta_1 + 6) q^{99}+O(q^{100})$$ q - b7 * q^2 + (-b7 + b6) * q^3 - b2 * q^4 + (-b2 - b1 - 1) * q^6 + (b7 - 2*b5 + 2*b3) * q^7 - b6 * q^8 + (-b4 - b2) * q^9 + (-b4 + b2 + 3*b1 - 3) * q^11 + (-b7 + 2*b6 + 3*b5 - 2*b3) * q^12 + (2*b7 - 2*b6 - 2*b5 + b3) * q^13 - b2 * q^14 + (2*b4 - 2*b2 - b1 + 1) * q^16 + (-2*b6 + b5) * q^17 + (b7 + b6 + 3*b5 - b3) * q^18 + (b4 + 1) * q^19 + (-2*b4 + b2 - 3*b1 + 3) * q^21 + (4*b7 - 4*b6 - 4*b5 - b3) * q^22 + (-b7 + b6 + b5) * q^23 + (b4 - b1 - 1) * q^24 + (b4 + 3) * q^26 + (3*b6 + 3*b5 - 3*b3) * q^27 + 3*b6 * q^28 + (-2*b4 + 2*b2 + 3*b1 - 3) * q^29 + (b2 + b1) * q^31 + (-b7 + b6 + b5 + 4*b3) * q^32 + (2*b7 - 4*b6 - 3*b5 - 2*b3) * q^33 + (-b4 + b2 - b1 + 1) * q^34 + (6*b1 - 3) * q^36 - 3*b5 * q^37 + (-2*b7 - b5 + b3) * q^38 + (b4 + b2 - 3*b1 + 3) * q^39 + (b2 - 6*b1) * q^41 + (-b7 + 2*b6 + 3*b5 - 2*b3) * q^42 + (-4*b7 - b5 + b3) * q^43 + (3*b4 + 3) * q^44 + (-b4 - 2) * q^46 + (-3*b7 + b5 - b3) * q^47 + (-4*b7 + 3*b6 + 6*b5 - b3) * q^48 + (3*b2 + b1) * q^49 + (2*b4 + b2 - b1 - 4) * q^51 + (-3*b5 + 3*b3) * q^52 + (2*b6 + 2*b5) * q^53 + (3*b2 + 6*b1 - 3) * q^54 + (2*b4 - 2*b2 + 3*b1 - 3) * q^56 + (-2*b7 + b3) * q^57 + (5*b7 - 5*b6 - 5*b5 - 2*b3) * q^58 + (2*b2 - 6*b1) * q^59 + (-5*b4 + 5*b2 + 2*b1 - 2) * q^61 + (-2*b6 - 3*b5) * q^62 + (3*b7 + 3*b6 - 3*b5 + 3*b3) * q^63 + (-b4 - 4) * q^64 + (5*b4 - b2 - 3*b1 + 9) * q^66 + (b7 - b6 - b5 + 5*b3) * q^67 + (4*b7 - 4*b6 - 4*b5 + 5*b3) * q^68 + (-b4 + b1 - 2) * q^69 + (-3*b4 + 9) * q^71 + (b7 - 2*b6 + 2*b3) * q^72 + 6*b5 * q^73 + (3*b4 - 3*b2 - 3*b1 + 3) * q^74 + (-b2 - 3*b1) * q^76 - 3*b3 * q^77 + (-4*b7 + 2*b6 + b3) * q^78 + (2*b4 - 2*b2 + 4*b1 - 4) * q^79 + 9*b1 * q^81 + (5*b6 + 4*b5) * q^82 + (b7 + 6*b5 - 6*b3) * q^83 + (-3*b4 + 3*b1 + 3) * q^84 + (-5*b2 - 9*b1) * q^86 + (4*b7 - 5*b6 - 6*b5 - b3) * q^87 + (2*b7 - b5 + b3) * q^88 + (-6*b4 + 3) * q^89 + (3*b4 - 3) * q^91 + (b7 + b5 - b3) * q^92 + (-2*b6 - 3*b5 + b3) * q^93 + (-2*b2 - 5*b1) * q^94 + (-5*b4 + 4*b2 - 3*b1 - 6) * q^96 + (-2*b7 + 10*b5 - 10*b3) * q^97 + (-4*b6 - 7*b5) * q^98 + (6*b4 - 3*b2 - 3*b1 + 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 12 q^{6}+O(q^{10})$$ 8 * q - 12 * q^6 $$8 q - 12 q^{6} - 12 q^{11} + 4 q^{16} + 8 q^{19} + 12 q^{21} - 12 q^{24} + 24 q^{26} - 12 q^{29} + 4 q^{31} + 4 q^{34} + 12 q^{39} - 24 q^{41} + 24 q^{44} - 16 q^{46} + 4 q^{49} - 36 q^{51} - 12 q^{56} - 24 q^{59} - 8 q^{61} - 32 q^{64} + 60 q^{66} - 12 q^{69} + 72 q^{71} + 12 q^{74} - 12 q^{76} - 16 q^{79} + 36 q^{81} + 36 q^{84} - 36 q^{86} + 24 q^{89} - 24 q^{91} - 20 q^{94} - 60 q^{96} + 36 q^{99}+O(q^{100})$$ 8 * q - 12 * q^6 - 12 * q^11 + 4 * q^16 + 8 * q^19 + 12 * q^21 - 12 * q^24 + 24 * q^26 - 12 * q^29 + 4 * q^31 + 4 * q^34 + 12 * q^39 - 24 * q^41 + 24 * q^44 - 16 * q^46 + 4 * q^49 - 36 * q^51 - 12 * q^56 - 24 * q^59 - 8 * q^61 - 32 * q^64 + 60 * q^66 - 12 * q^69 + 72 * q^71 + 12 * q^74 - 12 * q^76 - 16 * q^79 + 36 * q^81 + 36 * q^84 - 36 * q^86 + 24 * q^89 - 24 * q^91 - 20 * q^94 - 60 * q^96 + 36 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{24}^{4}$$ v^4 $$\beta_{2}$$ $$=$$ $$\zeta_{24}^{6} + \zeta_{24}^{2}$$ v^6 + v^2 $$\beta_{3}$$ $$=$$ $$\zeta_{24}^{7} + \zeta_{24}$$ v^7 + v $$\beta_{4}$$ $$=$$ $$-\zeta_{24}^{6} + 2\zeta_{24}^{2}$$ -v^6 + 2*v^2 $$\beta_{5}$$ $$=$$ $$-\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}$$ -v^5 + v^3 + v $$\beta_{6}$$ $$=$$ $$-\zeta_{24}^{7} + \zeta_{24}^{5}$$ -v^7 + v^5 $$\beta_{7}$$ $$=$$ $$-\zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}$$ -v^7 - v^5 + v
 $$\zeta_{24}$$ $$=$$ $$( \beta_{7} + \beta_{6} + 2\beta_{3} ) / 3$$ (b7 + b6 + 2*b3) / 3 $$\zeta_{24}^{2}$$ $$=$$ $$( \beta_{4} + \beta_{2} ) / 3$$ (b4 + b2) / 3 $$\zeta_{24}^{3}$$ $$=$$ $$( -2\beta_{7} + \beta_{6} + 3\beta_{5} - \beta_{3} ) / 3$$ (-2*b7 + b6 + 3*b5 - b3) / 3 $$\zeta_{24}^{4}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{24}^{5}$$ $$=$$ $$( -\beta_{7} + 2\beta_{6} + \beta_{3} ) / 3$$ (-b7 + 2*b6 + b3) / 3 $$\zeta_{24}^{6}$$ $$=$$ $$( -\beta_{4} + 2\beta_{2} ) / 3$$ (-b4 + 2*b2) / 3 $$\zeta_{24}^{7}$$ $$=$$ $$( -\beta_{7} - \beta_{6} + \beta_{3} ) / 3$$ (-b7 - b6 + b3) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-\beta_{1}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
76.1
 0.965926 − 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i −0.965926 + 0.258819i 0.965926 + 0.258819i 0.258819 − 0.965926i −0.258819 + 0.965926i −0.965926 − 0.258819i
−0.965926 1.67303i −0.448288 1.67303i −0.866025 + 1.50000i 0 −2.36603 + 2.36603i −0.448288 0.776457i −0.517638 −2.59808 + 1.50000i 0
76.2 −0.258819 0.448288i 1.67303 0.448288i 0.866025 1.50000i 0 −0.633975 0.633975i 1.67303 + 2.89778i −1.93185 2.59808 1.50000i 0
76.3 0.258819 + 0.448288i −1.67303 + 0.448288i 0.866025 1.50000i 0 −0.633975 0.633975i −1.67303 2.89778i 1.93185 2.59808 1.50000i 0
76.4 0.965926 + 1.67303i 0.448288 + 1.67303i −0.866025 + 1.50000i 0 −2.36603 + 2.36603i 0.448288 + 0.776457i 0.517638 −2.59808 + 1.50000i 0
151.1 −0.965926 + 1.67303i −0.448288 + 1.67303i −0.866025 1.50000i 0 −2.36603 2.36603i −0.448288 + 0.776457i −0.517638 −2.59808 1.50000i 0
151.2 −0.258819 + 0.448288i 1.67303 + 0.448288i 0.866025 + 1.50000i 0 −0.633975 + 0.633975i 1.67303 2.89778i −1.93185 2.59808 + 1.50000i 0
151.3 0.258819 0.448288i −1.67303 0.448288i 0.866025 + 1.50000i 0 −0.633975 + 0.633975i −1.67303 + 2.89778i 1.93185 2.59808 + 1.50000i 0
151.4 0.965926 1.67303i 0.448288 1.67303i −0.866025 1.50000i 0 −2.36603 2.36603i 0.448288 0.776457i 0.517638 −2.59808 1.50000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 76.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.2.e.d 8
3.b odd 2 1 675.2.e.d 8
5.b even 2 1 inner 225.2.e.d 8
5.c odd 4 2 45.2.j.a 8
9.c even 3 1 inner 225.2.e.d 8
9.c even 3 1 2025.2.a.t 4
9.d odd 6 1 675.2.e.d 8
9.d odd 6 1 2025.2.a.r 4
15.d odd 2 1 675.2.e.d 8
15.e even 4 2 135.2.j.a 8
20.e even 4 2 720.2.by.d 8
45.h odd 6 1 675.2.e.d 8
45.h odd 6 1 2025.2.a.r 4
45.j even 6 1 inner 225.2.e.d 8
45.j even 6 1 2025.2.a.t 4
45.k odd 12 2 45.2.j.a 8
45.k odd 12 2 405.2.b.d 4
45.l even 12 2 135.2.j.a 8
45.l even 12 2 405.2.b.c 4
60.l odd 4 2 2160.2.by.c 8
180.v odd 12 2 2160.2.by.c 8
180.x even 12 2 720.2.by.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.j.a 8 5.c odd 4 2
45.2.j.a 8 45.k odd 12 2
135.2.j.a 8 15.e even 4 2
135.2.j.a 8 45.l even 12 2
225.2.e.d 8 1.a even 1 1 trivial
225.2.e.d 8 5.b even 2 1 inner
225.2.e.d 8 9.c even 3 1 inner
225.2.e.d 8 45.j even 6 1 inner
405.2.b.c 4 45.l even 12 2
405.2.b.d 4 45.k odd 12 2
675.2.e.d 8 3.b odd 2 1
675.2.e.d 8 9.d odd 6 1
675.2.e.d 8 15.d odd 2 1
675.2.e.d 8 45.h odd 6 1
720.2.by.d 8 20.e even 4 2
720.2.by.d 8 180.x even 12 2
2025.2.a.r 4 9.d odd 6 1
2025.2.a.r 4 45.h odd 6 1
2025.2.a.t 4 9.c even 3 1
2025.2.a.t 4 45.j even 6 1
2160.2.by.c 8 60.l odd 4 2
2160.2.by.c 8 180.v odd 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 4T_{2}^{6} + 15T_{2}^{4} + 4T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(225, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 4 T^{6} + 15 T^{4} + 4 T^{2} + \cdots + 1$$
$3$ $$T^{8} - 9T^{4} + 81$$
$5$ $$T^{8}$$
$7$ $$T^{8} + 12 T^{6} + 135 T^{4} + \cdots + 81$$
$11$ $$(T^{4} + 6 T^{3} + 30 T^{2} + 36 T + 36)^{2}$$
$13$ $$(T^{4} + 6 T^{2} + 36)^{2}$$
$17$ $$(T^{4} - 28 T^{2} + 4)^{2}$$
$19$ $$(T^{2} - 2 T - 2)^{4}$$
$23$ $$T^{8} + 4 T^{6} + 15 T^{4} + 4 T^{2} + \cdots + 1$$
$29$ $$(T^{4} + 6 T^{3} + 39 T^{2} - 18 T + 9)^{2}$$
$31$ $$(T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4)^{2}$$
$37$ $$(T^{2} - 18)^{4}$$
$41$ $$(T^{4} + 12 T^{3} + 111 T^{2} + 396 T + 1089)^{2}$$
$43$ $$T^{8} + 84 T^{6} + 7020 T^{4} + \cdots + 1296$$
$47$ $$T^{8} + 28 T^{6} + 615 T^{4} + \cdots + 28561$$
$53$ $$(T^{4} - 16 T^{2} + 16)^{2}$$
$59$ $$(T^{4} + 12 T^{3} + 120 T^{2} + 288 T + 576)^{2}$$
$61$ $$(T^{4} + 4 T^{3} + 87 T^{2} - 284 T + 5041)^{2}$$
$67$ $$T^{8} + 84 T^{6} + 5535 T^{4} + \cdots + 2313441$$
$71$ $$(T^{2} - 18 T + 54)^{4}$$
$73$ $$(T^{2} - 72)^{4}$$
$79$ $$(T^{4} + 8 T^{3} + 60 T^{2} + 32 T + 16)^{2}$$
$83$ $$T^{8} + 172 T^{6} + \cdots + 47458321$$
$89$ $$(T^{2} - 6 T - 99)^{4}$$
$97$ $$T^{8} + 336 T^{6} + \cdots + 592240896$$