# Properties

 Label 882.2.e.m Level $882$ Weight $2$ Character orbit 882.e Analytic conductor $7.043$ 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] = [882,2,Mod(373,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("882.373");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 882.e (of order $$3$$, degree $$2$$, not 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: $$7.04280545828$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 126) 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

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

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$785$$ $$\chi(n)$$ $$-1 + \beta_{2}$$ $$-1 + \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
373.1
 1.22474 + 0.707107i −1.22474 − 0.707107i 1.22474 − 0.707107i −1.22474 + 0.707107i
1.00000 −1.72474 0.158919i 1.00000 0.724745 + 1.25529i −1.72474 0.158919i 0 1.00000 2.94949 + 0.548188i 0.724745 + 1.25529i
373.2 1.00000 0.724745 1.57313i 1.00000 −1.72474 2.98735i 0.724745 1.57313i 0 1.00000 −1.94949 2.28024i −1.72474 2.98735i
655.1 1.00000 −1.72474 + 0.158919i 1.00000 0.724745 1.25529i −1.72474 + 0.158919i 0 1.00000 2.94949 0.548188i 0.724745 1.25529i
655.2 1.00000 0.724745 + 1.57313i 1.00000 −1.72474 + 2.98735i 0.724745 + 1.57313i 0 1.00000 −1.94949 + 2.28024i −1.72474 + 2.98735i
 $$n$$: 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
63.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.2.e.m 4
3.b odd 2 1 2646.2.e.l 4
7.b odd 2 1 882.2.e.n 4
7.c even 3 1 126.2.f.c 4
7.c even 3 1 882.2.h.k 4
7.d odd 6 1 882.2.f.j 4
7.d odd 6 1 882.2.h.l 4
9.c even 3 1 882.2.h.k 4
9.d odd 6 1 2646.2.h.m 4
21.c even 2 1 2646.2.e.k 4
21.g even 6 1 2646.2.f.k 4
21.g even 6 1 2646.2.h.n 4
21.h odd 6 1 378.2.f.d 4
21.h odd 6 1 2646.2.h.m 4
28.g odd 6 1 1008.2.r.e 4
63.g even 3 1 126.2.f.c 4
63.h even 3 1 inner 882.2.e.m 4
63.h even 3 1 1134.2.a.p 2
63.i even 6 1 2646.2.e.k 4
63.i even 6 1 7938.2.a.bm 2
63.j odd 6 1 1134.2.a.i 2
63.j odd 6 1 2646.2.e.l 4
63.k odd 6 1 882.2.f.j 4
63.l odd 6 1 882.2.h.l 4
63.n odd 6 1 378.2.f.d 4
63.o even 6 1 2646.2.h.n 4
63.s even 6 1 2646.2.f.k 4
63.t odd 6 1 882.2.e.n 4
63.t odd 6 1 7938.2.a.bn 2
84.n even 6 1 3024.2.r.e 4
252.o even 6 1 3024.2.r.e 4
252.u odd 6 1 9072.2.a.bk 2
252.bb even 6 1 9072.2.a.bd 2
252.bl odd 6 1 1008.2.r.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.c 4 7.c even 3 1
126.2.f.c 4 63.g even 3 1
378.2.f.d 4 21.h odd 6 1
378.2.f.d 4 63.n odd 6 1
882.2.e.m 4 1.a even 1 1 trivial
882.2.e.m 4 63.h even 3 1 inner
882.2.e.n 4 7.b odd 2 1
882.2.e.n 4 63.t odd 6 1
882.2.f.j 4 7.d odd 6 1
882.2.f.j 4 63.k odd 6 1
882.2.h.k 4 7.c even 3 1
882.2.h.k 4 9.c even 3 1
882.2.h.l 4 7.d odd 6 1
882.2.h.l 4 63.l odd 6 1
1008.2.r.e 4 28.g odd 6 1
1008.2.r.e 4 252.bl odd 6 1
1134.2.a.i 2 63.j odd 6 1
1134.2.a.p 2 63.h even 3 1
2646.2.e.k 4 21.c even 2 1
2646.2.e.k 4 63.i even 6 1
2646.2.e.l 4 3.b odd 2 1
2646.2.e.l 4 63.j odd 6 1
2646.2.f.k 4 21.g even 6 1
2646.2.f.k 4 63.s even 6 1
2646.2.h.m 4 9.d odd 6 1
2646.2.h.m 4 21.h odd 6 1
2646.2.h.n 4 21.g even 6 1
2646.2.h.n 4 63.o even 6 1
3024.2.r.e 4 84.n even 6 1
3024.2.r.e 4 252.o even 6 1
7938.2.a.bm 2 63.i even 6 1
7938.2.a.bn 2 63.t odd 6 1
9072.2.a.bd 2 252.bb even 6 1
9072.2.a.bk 2 252.u odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{4} + 2T_{5}^{3} + 9T_{5}^{2} - 10T_{5} + 25$$ T5^4 + 2*T5^3 + 9*T5^2 - 10*T5 + 25 $$T_{11}^{2} + 2T_{11} + 4$$ T11^2 + 2*T11 + 4 $$T_{13}^{4} + 24T_{13}^{2} + 576$$ T13^4 + 24*T13^2 + 576

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{4}$$
$3$ $$T^{4} + 2 T^{3} + T^{2} + 6 T + 9$$
$5$ $$T^{4} + 2 T^{3} + 9 T^{2} - 10 T + 25$$
$7$ $$T^{4}$$
$11$ $$(T^{2} + 2 T + 4)^{2}$$
$13$ $$T^{4} + 24T^{2} + 576$$
$17$ $$(T^{2} + 2 T + 4)^{2}$$
$19$ $$T^{4} + 10 T^{3} + 81 T^{2} + \cdots + 361$$
$23$ $$(T^{2} - T + 1)^{2}$$
$29$ $$T^{4} - 4 T^{3} + 36 T^{2} + 80 T + 400$$
$31$ $$(T - 6)^{4}$$
$37$ $$T^{4} + 4 T^{3} + 108 T^{2} + \cdots + 8464$$
$41$ $$T^{4} + 96T^{2} + 9216$$
$43$ $$T^{4} + 4 T^{3} + 36 T^{2} - 80 T + 400$$
$47$ $$(T^{2} - 96)^{2}$$
$53$ $$T^{4} - 12 T^{3} + 132 T^{2} + \cdots + 144$$
$59$ $$(T + 2)^{4}$$
$61$ $$(T^{2} - 18 T + 75)^{2}$$
$67$ $$(T^{2} + 16 T + 40)^{2}$$
$71$ $$(T^{2} - 10 T + 1)^{2}$$
$73$ $$T^{4} - 4 T^{3} + 36 T^{2} + 80 T + 400$$
$79$ $$(T^{2} - 6 T - 15)^{2}$$
$83$ $$(T^{2} + 2 T + 4)^{2}$$
$89$ $$T^{4} - 24 T^{3} + 456 T^{2} + \cdots + 14400$$
$97$ $$T^{4} - 4 T^{3} + 36 T^{2} + 80 T + 400$$