# Properties

 Label 5225.2.a.z Level $5225$ Weight $2$ Character orbit 5225.a Self dual yes Analytic conductor $41.722$ Analytic rank $1$ Dimension $16$ Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5225,2,Mod(1,5225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5225, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5225 = 5^{2} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5225.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: $$41.7218350561$$ Analytic rank: $$1$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - 19x^{14} + 144x^{12} - 552x^{10} + 1119x^{8} - 1146x^{6} + 524x^{4} - 83x^{2} + 4$$ x^16 - 19*x^14 + 144*x^12 - 552*x^10 + 1119*x^8 - 1146*x^6 + 524*x^4 - 83*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 1045) 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - \beta_{11} q^{3} + \beta_{2} q^{4} + (\beta_{9} - \beta_{8} + \beta_{6} - 1) q^{6} - \beta_{5} q^{7} + \beta_{3} q^{8} + (\beta_{8} + 1) q^{9}+O(q^{10})$$ q + b1 * q^2 - b11 * q^3 + b2 * q^4 + (b9 - b8 + b6 - 1) * q^6 - b5 * q^7 + b3 * q^8 + (b8 + 1) * q^9 $$q + \beta_1 q^{2} - \beta_{11} q^{3} + \beta_{2} q^{4} + (\beta_{9} - \beta_{8} + \beta_{6} - 1) q^{6} - \beta_{5} q^{7} + \beta_{3} q^{8} + (\beta_{8} + 1) q^{9} - q^{11} + (\beta_{15} + \beta_{12} + \cdots - \beta_1) q^{12}+ \cdots + ( - \beta_{8} - 1) q^{99}+O(q^{100})$$ q + b1 * q^2 - b11 * q^3 + b2 * q^4 + (b9 - b8 + b6 - 1) * q^6 - b5 * q^7 + b3 * q^8 + (b8 + 1) * q^9 - q^11 + (b15 + b12 - b11 + b3 - b1) * q^12 + (b14 - b12 + b11 + b5 - b1) * q^13 + (-b6 - b4) * q^14 + (b4 - b2 - 1) * q^16 + (-b15 + b11 - b10 + b5 - b3) * q^17 + (-b15 + b11 + b10 - b3 + b1) * q^18 + q^19 + (-b9 - b4 - b2) * q^21 - b1 * q^22 + (b15 - 2*b14 + b10 - b5 + b1) * q^23 + (-b9 + b7 - 1) * q^24 + (-b9 + b8 + b7 - b6 - b2 - 1) * q^26 + (-b14 + b12 + b11 - b10 - b3 - b1) * q^27 + (-b12 + b11 - 2*b3) * q^28 + (-b9 - b7 + b4) * q^29 + (-b8 - b7 - b6 + b4 - b2 - 2) * q^31 + (b5 - 2*b3 - 3*b1) * q^32 + b11 * q^33 + (-b13 - b9 + b8 - 2*b7 - b2 + 3) * q^34 + (b13 - b9 + b8 - b6 + b4 + 1) * q^36 + (-2*b15 + b14 - b12 + b11 - b10 + b5) * q^37 + b1 * q^38 + (-b13 - 2*b6 + 2*b4 + b2 - 3) * q^39 + (b13 - b9 - b4 - 2*b2 + 1) * q^41 + (b11 - b10 - b3 - 2*b1) * q^42 + (2*b15 - b14 + b12 - b11 + b5 + b3) * q^43 - b2 * q^44 + (b13 - b6 + b4 + b2 - 1) * q^46 + (2*b14 - b12 + b11 + 2*b10 - 2*b1) * q^47 + (-b15 + b14 - 2*b12 + 2*b11 + b5 + b1) * q^48 + (-b8 + b7 - b4 - 3) * q^49 + (b13 + 2*b9 - b2 - 1) * q^51 + (-b14 + b12 + b10 - 2*b5 - b3 - b1) * q^52 + (b14 + b12 + b11 + 2*b10 - b5 - 2*b3 - b1) * q^53 + (-b13 - b9 - 2*b7 - b4 - 2*b2 + 1) * q^54 + (-b9 + b8 - 2*b2 + 2) * q^56 - b11 * q^57 + (-b15 - b14 + 2*b11 - 2*b10 + 2*b5 + b3) * q^58 + (b9 + 3*b7 - b6 + b4 + 3*b2 - 4) * q^59 + (-2*b9 + b8 + 2*b7 + b6 - 2*b4 - b2 - 2) * q^61 + (-b14 - b12 + b11 - 2*b10 - b3 - 4*b1) * q^62 + (-b14 + b11 + b10 + b5) * q^63 + (b6 - 3*b4 - 3*b2 - 2) * q^64 + (-b9 + b8 - b6 + 1) * q^66 + (2*b15 - 2*b14 + 3*b12 - b11 + b10 + b3) * q^67 + (-b15 - 2*b14 + 2*b11 - 2*b10) * q^68 + (-b13 - 2*b9 + b7 + b6 - 2*b4 - b2 - 1) * q^69 + (-2*b13 + 2*b9 - b7 + 2*b6 + b4 - 2) * q^71 + (b15 - b12 + b11 + b3) * q^72 + (-b15 + b14 + 3*b12 + b10 + 2*b3) * q^73 + (-b13 - b9 + 2*b8 - 2*b7 - b6 + b4 + 2) * q^74 + b2 * q^76 + b5 * q^77 + (-2*b12 + 2*b11 - 2*b10 + b5 + 2*b3 - 2*b1) * q^78 + (-2*b13 + b9 + 2*b8 + 3*b7 - b6 - 4) * q^79 + (2*b13 - 2*b8 - b6 - b4 - 5) * q^81 + (b11 + b10 - b5 - 3*b3 - 2*b1) * q^82 + (b15 + 2*b14 + 2*b12 - b11 - b10 + b5 + 5*b3 - 2*b1) * q^83 + (-b13 + b9 - b7 - b6 - b2 - 1) * q^84 + (b9 - 3*b8 + b7 + 3*b6 + b4 + b2 - 2) * q^86 + (b15 + b14 + 2*b10 - b5 + b3 - 2*b1) * q^87 - b3 * q^88 + (2*b13 - 2*b8 + 2*b6 - b4 - b2 - 2) * q^89 + (b13 + b9 - b8 - b7 + b6 + b4 + 2*b2 - 3) * q^91 + (-2*b15 + 4*b14 - b12 + b11 + b5) * q^92 + (-b15 + 3*b14 - 3*b12 + 5*b11 + b10 + b5 - b3 - b1) * q^93 + (2*b13 - b9 + 3*b8 + 4*b7 - 2*b6 - 2*b2 - 4) * q^94 + (3*b8 - 2*b7 - 3*b6 + b4 + b2 + 5) * q^96 + (-b15 - 4*b14 + 2*b12 + b10 - b5 + b3 + 2*b1) * q^97 + (2*b15 + b14 - 2*b11 - b5 + b3 - 3*b1) * q^98 + (-b8 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 6 q^{4} - 8 q^{6} + 8 q^{9}+O(q^{10})$$ 16 * q + 6 * q^4 - 8 * q^6 + 8 * q^9 $$16 q + 6 q^{4} - 8 q^{6} + 8 q^{9} - 16 q^{11} - 4 q^{14} - 18 q^{16} + 16 q^{19} - 10 q^{21} - 10 q^{24} - 24 q^{26} - 2 q^{29} - 32 q^{31} + 16 q^{34} + 18 q^{36} - 40 q^{39} + 6 q^{41} - 6 q^{44} - 38 q^{49} - 16 q^{51} - 18 q^{54} + 12 q^{56} - 24 q^{59} - 42 q^{61} - 62 q^{64} + 8 q^{66} - 30 q^{69} - 46 q^{71} + 2 q^{74} + 6 q^{76} - 74 q^{79} - 56 q^{81} - 34 q^{84} + 8 q^{86} - 14 q^{89} - 24 q^{91} - 64 q^{94} + 54 q^{96} - 8 q^{99}+O(q^{100})$$ 16 * q + 6 * q^4 - 8 * q^6 + 8 * q^9 - 16 * q^11 - 4 * q^14 - 18 * q^16 + 16 * q^19 - 10 * q^21 - 10 * q^24 - 24 * q^26 - 2 * q^29 - 32 * q^31 + 16 * q^34 + 18 * q^36 - 40 * q^39 + 6 * q^41 - 6 * q^44 - 38 * q^49 - 16 * q^51 - 18 * q^54 + 12 * q^56 - 24 * q^59 - 42 * q^61 - 62 * q^64 + 8 * q^66 - 30 * q^69 - 46 * q^71 + 2 * q^74 + 6 * q^76 - 74 * q^79 - 56 * q^81 - 34 * q^84 + 8 * q^86 - 14 * q^89 - 24 * q^91 - 64 * q^94 + 54 * q^96 - 8 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 19x^{14} + 144x^{12} - 552x^{10} + 1119x^{8} - 1146x^{6} + 524x^{4} - 83x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 4\nu$$ v^3 - 4*v $$\beta_{4}$$ $$=$$ $$\nu^{4} - 5\nu^{2} + 3$$ v^4 - 5*v^2 + 3 $$\beta_{5}$$ $$=$$ $$\nu^{5} - 6\nu^{3} + 7\nu$$ v^5 - 6*v^3 + 7*v $$\beta_{6}$$ $$=$$ $$\nu^{6} - 7\nu^{4} + 12\nu^{2} - 3$$ v^6 - 7*v^4 + 12*v^2 - 3 $$\beta_{7}$$ $$=$$ $$\nu^{12} - 14\nu^{10} + 72\nu^{8} - 165\nu^{6} + 163\nu^{4} - 60\nu^{2} + 6$$ v^12 - 14*v^10 + 72*v^8 - 165*v^6 + 163*v^4 - 60*v^2 + 6 $$\beta_{8}$$ $$=$$ $$\nu^{12} - 15\nu^{10} + 84\nu^{8} - 215\nu^{6} + 246\nu^{4} - 104\nu^{2} + 7$$ v^12 - 15*v^10 + 84*v^8 - 215*v^6 + 246*v^4 - 104*v^2 + 7 $$\beta_{9}$$ $$=$$ $$\nu^{12} - 15\nu^{10} + 85\nu^{8} - 225\nu^{6} + 277\nu^{4} - 134\nu^{2} + 13$$ v^12 - 15*v^10 + 85*v^8 - 225*v^6 + 277*v^4 - 134*v^2 + 13 $$\beta_{10}$$ $$=$$ $$( \nu^{15} - 17\nu^{13} + 114\nu^{11} - 382\nu^{9} + 667\nu^{7} - 572\nu^{5} + 198\nu^{3} - 15\nu ) / 2$$ (v^15 - 17*v^13 + 114*v^11 - 382*v^9 + 667*v^7 - 572*v^5 + 198*v^3 - 15*v) / 2 $$\beta_{11}$$ $$=$$ $$( \nu^{15} - 19\nu^{13} + 144\nu^{11} - 552\nu^{9} + 1117\nu^{7} - 1128\nu^{5} + 476\nu^{3} - 47\nu ) / 2$$ (v^15 - 19*v^13 + 144*v^11 - 552*v^9 + 1117*v^7 - 1128*v^5 + 476*v^3 - 47*v) / 2 $$\beta_{12}$$ $$=$$ $$( \nu^{15} - 19\nu^{13} + 144\nu^{11} - 552\nu^{9} + 1119\nu^{7} - 1144\nu^{5} + 510\nu^{3} - 59\nu ) / 2$$ (v^15 - 19*v^13 + 144*v^11 - 552*v^9 + 1119*v^7 - 1144*v^5 + 510*v^3 - 59*v) / 2 $$\beta_{13}$$ $$=$$ $$\nu^{14} - 17\nu^{12} + 114\nu^{10} - 382\nu^{8} + 667\nu^{6} - 573\nu^{4} + 203\nu^{2} - 17$$ v^14 - 17*v^12 + 114*v^10 - 382*v^8 + 667*v^6 - 573*v^4 + 203*v^2 - 17 $$\beta_{14}$$ $$=$$ $$-\nu^{15} + 19\nu^{13} - 143\nu^{11} + 538\nu^{9} - 1047\nu^{7} + 981\nu^{5} - 362\nu^{3} + 28\nu$$ -v^15 + 19*v^13 - 143*v^11 + 538*v^9 - 1047*v^7 + 981*v^5 - 362*v^3 + 28*v $$\beta_{15}$$ $$=$$ $$\nu^{15} - 19\nu^{13} + 144\nu^{11} - 551\nu^{9} + 1107\nu^{7} - 1096\nu^{5} + 440\nu^{3} - 34\nu$$ v^15 - 19*v^13 + 144*v^11 - 551*v^9 + 1107*v^7 - 1096*v^5 + 440*v^3 - 34*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 4\beta_1$$ b3 + 4*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + 5\beta_{2} + 7$$ b4 + 5*b2 + 7 $$\nu^{5}$$ $$=$$ $$\beta_{5} + 6\beta_{3} + 17\beta_1$$ b5 + 6*b3 + 17*b1 $$\nu^{6}$$ $$=$$ $$\beta_{6} + 7\beta_{4} + 23\beta_{2} + 28$$ b6 + 7*b4 + 23*b2 + 28 $$\nu^{7}$$ $$=$$ $$\beta_{12} - \beta_{11} + 8\beta_{5} + 31\beta_{3} + 74\beta_1$$ b12 - b11 + 8*b5 + 31*b3 + 74*b1 $$\nu^{8}$$ $$=$$ $$\beta_{9} - \beta_{8} + 10\beta_{6} + 39\beta_{4} + 105\beta_{2} + 117$$ b9 - b8 + 10*b6 + 39*b4 + 105*b2 + 117 $$\nu^{9}$$ $$=$$ $$\beta_{15} + 10\beta_{12} - 12\beta_{11} + 48\beta_{5} + 154\beta_{3} + 327\beta_1$$ b15 + 10*b12 - 12*b11 + 48*b5 + 154*b3 + 327*b1 $$\nu^{10}$$ $$=$$ $$12\beta_{9} - 13\beta_{8} + \beta_{7} + 70\beta_{6} + 201\beta_{4} + 481\beta_{2} + 498$$ 12*b9 - 13*b8 + b7 + 70*b6 + 201*b4 + 481*b2 + 498 $$\nu^{11}$$ $$=$$ $$14\beta_{15} + \beta_{14} + 70\beta_{12} - 96\beta_{11} + 259\beta_{5} + 754\beta_{3} + 1460\beta_1$$ 14*b15 + b14 + 70*b12 - 96*b11 + 259*b5 + 754*b3 + 1460*b1 $$\nu^{12}$$ $$=$$ $$96\beta_{9} - 110\beta_{8} + 15\beta_{7} + 425\beta_{6} + 998\beta_{4} + 2214\beta_{2} + 2141$$ 96*b9 - 110*b8 + 15*b7 + 425*b6 + 998*b4 + 2214*b2 + 2141 $$\nu^{13}$$ $$=$$ $$125 \beta_{15} + 15 \beta_{14} + 425 \beta_{12} - 646 \beta_{11} + \beta_{10} + 1327 \beta_{5} + \cdots + 6569 \beta_1$$ 125*b15 + 15*b14 + 425*b12 - 646*b11 + b10 + 1327*b5 + 3666*b3 + 6569*b1 $$\nu^{14}$$ $$=$$ $$\beta_{13} + 646\beta_{9} - 770\beta_{8} + 141\beta_{7} + 2398\beta_{6} + 4854\beta_{4} + 10235\beta_{2} + 9265$$ b13 + 646*b9 - 770*b8 + 141*b7 + 2398*b6 + 4854*b4 + 10235*b2 + 9265 $$\nu^{15}$$ $$=$$ $$911 \beta_{15} + 141 \beta_{14} + 2398 \beta_{12} - 3955 \beta_{11} + 19 \beta_{10} + 6605 \beta_{5} + \cdots + 29736 \beta_1$$ 911*b15 + 141*b14 + 2398*b12 - 3955*b11 + 19*b10 + 6605*b5 + 17751*b3 + 29736*b1

## 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
 −2.18657 −2.10564 −2.05535 −1.78099 −1.19140 −0.853553 −0.387488 −0.301154 0.301154 0.387488 0.853553 1.19140 1.78099 2.05535 2.10564 2.18657
−2.18657 −1.58567 2.78109 0 3.46718 2.56330 −1.70792 −0.485645 0
1.2 −2.10564 1.90142 2.43371 0 −4.00370 0.116917 −0.913237 0.615392 0
1.3 −2.05535 2.80376 2.22445 0 −5.76269 −1.02917 −0.461316 4.86106 0
1.4 −1.78099 −0.213501 1.17191 0 0.380243 −3.50934 1.47481 −2.95442 0
1.5 −1.19140 −2.62369 −0.580557 0 3.12588 0.593512 3.07449 3.88377 0
1.6 −0.853553 1.84364 −1.27145 0 −1.57364 2.69678 2.79235 0.399003 0
1.7 −0.387488 0.494325 −1.84985 0 −0.191545 2.37207 1.49177 −2.75564 0
1.8 −0.301154 −1.85377 −1.90931 0 0.558272 1.94668 1.17730 0.436480 0
1.9 0.301154 1.85377 −1.90931 0 0.558272 −1.94668 −1.17730 0.436480 0
1.10 0.387488 −0.494325 −1.84985 0 −0.191545 −2.37207 −1.49177 −2.75564 0
1.11 0.853553 −1.84364 −1.27145 0 −1.57364 −2.69678 −2.79235 0.399003 0
1.12 1.19140 2.62369 −0.580557 0 3.12588 −0.593512 −3.07449 3.88377 0
1.13 1.78099 0.213501 1.17191 0 0.380243 3.50934 −1.47481 −2.95442 0
1.14 2.05535 −2.80376 2.22445 0 −5.76269 1.02917 0.461316 4.86106 0
1.15 2.10564 −1.90142 2.43371 0 −4.00370 −0.116917 0.913237 0.615392 0
1.16 2.18657 1.58567 2.78109 0 3.46718 −2.56330 1.70792 −0.485645 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$11$$ $$+1$$
$$19$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5225.2.a.z 16
5.b even 2 1 inner 5225.2.a.z 16
5.c odd 4 2 1045.2.b.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.b.b 16 5.c odd 4 2
5225.2.a.z 16 1.a even 1 1 trivial
5225.2.a.z 16 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5225))$$:

 $$T_{2}^{16} - 19T_{2}^{14} + 144T_{2}^{12} - 552T_{2}^{10} + 1119T_{2}^{8} - 1146T_{2}^{6} + 524T_{2}^{4} - 83T_{2}^{2} + 4$$ T2^16 - 19*T2^14 + 144*T2^12 - 552*T2^10 + 1119*T2^8 - 1146*T2^6 + 524*T2^4 - 83*T2^2 + 4 $$T_{7}^{16} - 37T_{7}^{14} + 537T_{7}^{12} - 3908T_{7}^{10} + 14949T_{7}^{8} - 28337T_{7}^{6} + 21900T_{7}^{4} - 4976T_{7}^{2} + 64$$ T7^16 - 37*T7^14 + 537*T7^12 - 3908*T7^10 + 14949*T7^8 - 28337*T7^6 + 21900*T7^4 - 4976*T7^2 + 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} - 19 T^{14} + \cdots + 4$$
$3$ $$T^{16} - 28 T^{14} + \cdots + 64$$
$5$ $$T^{16}$$
$7$ $$T^{16} - 37 T^{14} + \cdots + 64$$
$11$ $$(T + 1)^{16}$$
$13$ $$T^{16} - 81 T^{14} + \cdots + 65536$$
$17$ $$T^{16} - 113 T^{14} + \cdots + 6390784$$
$19$ $$(T - 1)^{16}$$
$23$ $$T^{16} - 181 T^{14} + \cdots + 1936$$
$29$ $$(T^{8} + T^{7} - 59 T^{6} + \cdots - 784)^{2}$$
$31$ $$(T^{8} + 16 T^{7} + \cdots - 3104)^{2}$$
$37$ $$T^{16} + \cdots + 476636224$$
$41$ $$(T^{8} - 3 T^{7} + \cdots - 3152)^{2}$$
$43$ $$T^{16} + \cdots + 277688896$$
$47$ $$T^{16} + \cdots + 37937690176$$
$53$ $$T^{16} + \cdots + 32319410176$$
$59$ $$(T^{8} + 12 T^{7} + \cdots + 9973312)^{2}$$
$61$ $$(T^{8} + 21 T^{7} + \cdots + 269488)^{2}$$
$67$ $$T^{16} + \cdots + 3129530826304$$
$71$ $$(T^{8} + 23 T^{7} + \cdots + 1022368)^{2}$$
$73$ $$T^{16} + \cdots + 3803560873984$$
$79$ $$(T^{8} + 37 T^{7} + \cdots + 143552)^{2}$$
$83$ $$T^{16} + \cdots + 28036855560256$$
$89$ $$(T^{8} + 7 T^{7} + \cdots - 10084)^{2}$$
$97$ $$T^{16} + \cdots + 574639907835904$$
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