# Properties

 Label 1305.2.c.i Level $1305$ Weight $2$ Character orbit 1305.c Analytic conductor $10.420$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,2,Mod(784,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("1305.784");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1305 = 3^{2} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1305.c (of order $$2$$, degree $$1$$, 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: $$10.4204774638$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: 10.0.7025129046016.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} - 2x^{9} + 2x^{8} + 2x^{7} + 23x^{6} - 36x^{5} + 28x^{4} - 2x^{3} + x^{2} - 2x + 2$$ x^10 - 2*x^9 + 2*x^8 + 2*x^7 + 23*x^6 - 36*x^5 + 28*x^4 - 2*x^3 + x^2 - 2*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 435) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{9} + \beta_{2} + \beta_1) q^{2} + ( - \beta_{7} + \beta_{5} + \beta_{2} + \cdots - 1) q^{4}+ \cdots + ( - 2 \beta_{9} - \beta_{8} + \cdots - \beta_1) q^{8}+O(q^{10})$$ q + (b9 + b2 + b1) * q^2 + (-b7 + b5 + b2 - b1 - 1) * q^4 + (-b6 + b5 + b4 + b2) * q^5 + (2*b9 + b8 + b4) * q^7 + (-2*b9 - b8 + 3*b4 - b2 - b1) * q^8 $$q + (\beta_{9} + \beta_{2} + \beta_1) q^{2} + ( - \beta_{7} + \beta_{5} + \beta_{2} + \cdots - 1) q^{4}+ \cdots + ( - 6 \beta_{9} - 2 \beta_{4} + \cdots - 2 \beta_1) q^{98}+O(q^{100})$$ q + (b9 + b2 + b1) * q^2 + (-b7 + b5 + b2 - b1 - 1) * q^4 + (-b6 + b5 + b4 + b2) * q^5 + (2*b9 + b8 + b4) * q^7 + (-2*b9 - b8 + 3*b4 - b2 - b1) * q^8 + (b9 + b7 - b6 - b5 + b4 + b3 - b1) * q^10 + (2*b7 + b2 - b1 + 1) * q^11 + (2*b9 + b8 - b4) * q^13 + (-2*b7 + 2*b6 - b5 - 2*b3 - 2*b2 + 2*b1 - 2) * q^14 + (-b6 - 3*b5 + b3 - 2*b2 + 2*b1 + 2) * q^16 + (b8 - 3*b4 - b2 - b1) * q^17 + (-2*b7 + b6 - b3) * q^19 + (2*b9 + 3*b8 - b7 + 2*b6 - b5 + b3 - 3*b2 + b1) * q^20 + (3*b9 + b6 + 2*b4 + b3 + b2 + b1) * q^22 + (b8 + 2*b4 - 2*b2 - 2*b1) * q^23 + (b9 + b8 + b7 + b6 + b5 + b4 - b3 - 2*b1) * q^25 + (-2*b7 + 2*b6 + b5 - 2*b3 - 2) * q^26 + (-3*b9 - b8 - 3*b6 + b4 - 3*b3 + b2 + b1) * q^28 - q^29 + (b6 + 2*b5 - b3 - b2 + b1) * q^31 + (b9 + 3*b8 + 2*b6 - 3*b4 + 2*b3) * q^32 + (b6 + 3*b5 - b3 + 2*b2 - 2*b1 + 2) * q^34 + (2*b9 - b8 + 2*b7 - 3*b5 + 3*b4 - b3 + b2 + 4*b1) * q^35 + (-b8 - 2*b6 + 4*b4 - 2*b3 + 2*b2 + 2*b1) * q^37 + (-6*b9 - 2*b8 - b6 + 2*b4 - b3 - b2 - b1) * q^38 + (2*b9 - 3*b7 + b6 + 3*b5 - 2*b4 - 4*b3 + 3*b1 - 3) * q^40 + (b7 + b6 + 2*b5 - b3 + b2 - b1 + 4) * q^41 + (-2*b9 + b8 + 3*b6 - 2*b4 + 3*b3 - 2*b2 - 2*b1) * q^43 + (-b7 + 3*b6 - b5 - 3*b3 - 2*b2 + 2*b1 - 5) * q^44 + (2*b6 - 4*b5 - 2*b3 - 5*b2 + 5*b1 + 4) * q^46 + (b8 - 2*b6 - 3*b4 - 2*b3 + b2 + b1) * q^47 + (-b7 + b6 - 2*b5 - b3 - b2 + b1 - 2) * q^49 + (-b9 - 3*b8 - b7 + b6 - 2*b5 + 5*b4 - 3*b3 - 2*b2 + 4*b1 + 1) * q^50 + (-5*b9 - 3*b8 - 3*b6 + 3*b4 - 3*b3 - b2 - b1) * q^52 + (-b8 + 2*b6 + 2*b3 - b2 - b1) * q^53 + (-2*b9 - 3*b8 + b7 + 3*b5 + b3 - b1 + 1) * q^55 + (5*b7 - 3*b6 - 6*b5 + 3*b3 + 2*b2 - 2*b1 + 3) * q^56 + (-b9 - b2 - b1) * q^58 + (-4*b7 - b2 + b1 - 4) * q^59 + (2*b7 + 3*b6 - 4*b5 - 3*b3 - 2) * q^61 + (2*b9 - 4*b8 - 4*b6 + 2*b4 - 4*b3 + 5*b2 + 5*b1) * q^62 + (-5*b7 + b6 + 6*b5 - b3 - b2 + b1 - 1) * q^64 + (-b8 + 2*b7 - 3*b5 + 3*b4 + b3 + b2 + 2*b1 + 2) * q^65 + (-4*b9 + 3*b8 + 2*b6 + b4 + 2*b3 - 4*b2 - 4*b1) * q^67 + (-b9 - 3*b8 - 2*b6 + 3*b4 - 2*b3) * q^68 + (3*b9 + 2*b8 - b7 - 3*b5 - 5*b4 + 2*b3 + 2*b2 + 3*b1 - 6) * q^70 + (-2*b7 + b6 - 2*b5 - b3 + 4) * q^71 + (-5*b8 - 4*b4 + 2*b2 + 2*b1) * q^73 + (4*b7 - 4*b6 - 6*b5 + 4*b3 + b2 - b1) * q^74 + (4*b7 - 4*b6 - 4*b5 + 4*b3 + b2 - b1 + 10) * q^76 + (6*b9 - 3*b8 - b4 + 3*b2 + 3*b1) * q^77 + (4*b7 - b6 - 4*b5 + b3 - 4) * q^79 + (-4*b9 - 2*b8 - b7 - 4*b6 - 2*b5 + 5*b4 - 4*b3 + b2 - 2) * q^80 + (4*b9 - 4*b8 - 2*b6 + 6*b4 - 2*b3 + 6*b2 + 6*b1) * q^82 + (3*b8 - 3*b6 - 3*b3) * q^83 + (-4*b9 + b7 + b6 - b5 + 2*b3 - b1 + 3) * q^85 + (-4*b7 + 3*b6 + 8*b5 - 3*b3 - 2*b2 + 2*b1) * q^86 + (-4*b9 - 5*b8 - 2*b6 + 5*b4 - 2*b3 + 2*b2 + 2*b1) * q^88 + (-b7 + b6 + 6*b5 - b3 + 6*b2 - 6*b1 + 3) * q^89 + (-3*b7 + b6 + 2*b5 - b3 - b2 + b1 - 7) * q^91 + (6*b9 + 2*b8 - 3*b6 - 6*b4 - 3*b3 + 8*b2 + 8*b1) * q^92 + (4*b7 - 3*b6 + 3*b5 + 3*b3 + 8*b2 - 8*b1 + 2) * q^94 + (3*b9 + 4*b8 - 2*b7 - b5 + 2*b3 - b2 + 3*b1 - 4) * q^95 + (6*b9 + 3*b8 + 2*b4 + b2 + b1) * q^97 + (-6*b9 - 2*b4 - 2*b2 - 2*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 16 q^{4} - 2 q^{5}+O(q^{10})$$ 10 * q - 16 * q^4 - 2 * q^5 $$10 q - 16 q^{4} - 2 q^{5} + 10 q^{11} - 16 q^{14} + 28 q^{16} - 4 q^{19} + 6 q^{20} - 2 q^{25} - 24 q^{26} - 10 q^{29} + 4 q^{31} + 12 q^{34} + 10 q^{35} - 30 q^{40} + 38 q^{41} - 44 q^{44} + 60 q^{46} - 18 q^{49} + 20 q^{50} + 10 q^{55} + 32 q^{56} - 44 q^{59} - 16 q^{61} - 16 q^{64} + 26 q^{65} - 60 q^{70} + 36 q^{71} + 4 q^{74} + 104 q^{76} - 32 q^{79} - 24 q^{80} + 30 q^{85} + 4 q^{89} - 72 q^{91} - 4 q^{94} - 36 q^{95}+O(q^{100})$$ 10 * q - 16 * q^4 - 2 * q^5 + 10 * q^11 - 16 * q^14 + 28 * q^16 - 4 * q^19 + 6 * q^20 - 2 * q^25 - 24 * q^26 - 10 * q^29 + 4 * q^31 + 12 * q^34 + 10 * q^35 - 30 * q^40 + 38 * q^41 - 44 * q^44 + 60 * q^46 - 18 * q^49 + 20 * q^50 + 10 * q^55 + 32 * q^56 - 44 * q^59 - 16 * q^61 - 16 * q^64 + 26 * q^65 - 60 * q^70 + 36 * q^71 + 4 * q^74 + 104 * q^76 - 32 * q^79 - 24 * q^80 + 30 * q^85 + 4 * q^89 - 72 * q^91 - 4 * q^94 - 36 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2x^{9} + 2x^{8} + 2x^{7} + 23x^{6} - 36x^{5} + 28x^{4} - 2x^{3} + x^{2} - 2x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 8190 \nu^{9} - 16246 \nu^{8} + 17565 \nu^{7} + 16143 \nu^{6} + 188739 \nu^{5} - 288291 \nu^{4} + \cdots - 18260 ) / 71579$$ (8190*v^9 - 16246*v^8 + 17565*v^7 + 16143*v^6 + 188739*v^5 - 288291*v^4 + 277881*v^3 - 7953*v^2 + 7955*v - 18260) / 71579 $$\beta_{3}$$ $$=$$ $$( - 8852 \nu^{9} + 1845 \nu^{8} - 369 \nu^{7} - 21171 \nu^{6} - 255280 \nu^{5} - 76611 \nu^{4} + \cdots + 17988 ) / 71579$$ (-8852*v^9 + 1845*v^8 - 369*v^7 - 21171*v^6 - 255280*v^5 - 76611*v^4 - 8625*v^3 + 83898*v^2 - 116185*v + 17988) / 71579 $$\beta_{4}$$ $$=$$ $$( - 9130 \nu^{9} + 10070 \nu^{8} - 2014 \nu^{7} - 35825 \nu^{6} - 226133 \nu^{5} + 139941 \nu^{4} + \cdots + 10305 ) / 71579$$ (-9130*v^9 + 10070*v^8 - 2014*v^7 - 35825*v^6 - 226133*v^5 + 139941*v^4 + 32651*v^3 - 259621*v^2 - 1177*v + 10305) / 71579 $$\beta_{5}$$ $$=$$ $$( - 21643 \nu^{9} + 61268 \nu^{8} - 55201 \nu^{7} - 40903 \nu^{6} - 440793 \nu^{5} + 1260561 \nu^{4} + \cdots + 160648 ) / 71579$$ (-21643*v^9 + 61268*v^8 - 55201*v^7 - 40903*v^6 - 440793*v^5 + 1260561*v^4 - 669334*v^3 + 19260*v^2 + 154080*v + 160648) / 71579 $$\beta_{6}$$ $$=$$ $$( - 22439 \nu^{9} + 45682 \nu^{8} - 37768 \nu^{7} - 46300 \nu^{6} - 513883 \nu^{5} + 847098 \nu^{4} + \cdots + 33598 ) / 71579$$ (-22439*v^9 + 45682*v^8 - 37768*v^7 - 46300*v^6 - 513883*v^5 + 847098*v^4 - 408505*v^3 + 95440*v^2 - 23849*v + 33598) / 71579 $$\beta_{7}$$ $$=$$ $$( - 30637 \nu^{9} + 70404 \nu^{8} - 71344 \nu^{7} - 59260 \nu^{6} - 668826 \nu^{5} + 1329065 \nu^{4} + \cdots + 62451 ) / 71579$$ (-30637*v^9 + 70404*v^8 - 71344*v^7 - 59260*v^6 - 668826*v^5 + 1329065*v^4 - 997777*v^3 + 28623*v^2 + 228984*v + 62451) / 71579 $$\beta_{8}$$ $$=$$ $$( 40942 \nu^{9} - 72754 \nu^{8} + 71814 \nu^{7} + 83898 \nu^{6} + 977491 \nu^{5} - 1247779 \nu^{4} + \cdots - 80707 ) / 71579$$ (40942*v^9 - 72754*v^8 + 71814*v^7 + 83898*v^6 + 977491*v^5 - 1247779*v^4 + 1006435*v^3 - 114535*v^2 + 300563*v - 80707) / 71579 $$\beta_{9}$$ $$=$$ $$( - 57741 \nu^{9} + 83913 \nu^{8} - 59730 \nu^{7} - 155264 \nu^{6} - 1410168 \nu^{5} + 1338660 \nu^{4} + \cdots + 90456 ) / 71579$$ (-57741*v^9 + 83913*v^8 - 59730*v^7 - 155264*v^6 - 1410168*v^5 + 1338660*v^4 - 629709*v^3 - 260372*v^2 - 221922*v + 90456) / 71579
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{9} - \beta_{6} - 2\beta_{4} - \beta_{3} + \beta_{2} + \beta_1$$ b9 - b6 - 2*b4 - b3 + b2 + b1 $$\nu^{3}$$ $$=$$ $$-\beta_{8} + \beta_{7} - \beta_{6} + 6\beta_{2}$$ -b8 + b7 - b6 + 6*b2 $$\nu^{4}$$ $$=$$ $$\beta_{7} - 6\beta_{6} + 5\beta_{5} + 6\beta_{3} + 7\beta_{2} - 7\beta _1 - 9$$ b7 - 6*b6 + 5*b5 + 6*b3 + 7*b2 - 7*b1 - 9 $$\nu^{5}$$ $$=$$ $$-\beta_{9} + 6\beta_{8} + 6\beta_{7} + \beta_{5} + 2\beta_{4} + 9\beta_{3} - 35\beta _1 - 2$$ -b9 + 6*b8 + 6*b7 + b5 + 2*b4 + 9*b3 - 35*b1 - 2 $$\nu^{6}$$ $$=$$ $$-26\beta_{9} + 9\beta_{8} + 34\beta_{6} + 47\beta_{4} + 34\beta_{3} - 46\beta_{2} - 46\beta_1$$ -26*b9 + 9*b8 + 34*b6 + 47*b4 + 34*b3 - 46*b2 - 46*b1 $$\nu^{7}$$ $$=$$ $$-12\beta_{9} + 34\beta_{8} - 34\beta_{7} + 66\beta_{6} - 12\beta_{5} + 23\beta_{4} - 207\beta_{2} + 23$$ -12*b9 + 34*b8 - 34*b7 + 66*b6 - 12*b5 + 23*b4 - 207*b2 + 23 $$\nu^{8}$$ $$=$$ $$-66\beta_{7} + 195\beta_{6} - 141\beta_{5} - 195\beta_{3} - 296\beta_{2} + 296\beta _1 + 256$$ -66*b7 + 195*b6 - 141*b5 - 195*b3 - 296*b2 + 296*b1 + 256 $$\nu^{9}$$ $$=$$ $$101\beta_{9} - 195\beta_{8} - 195\beta_{7} - 101\beta_{5} - 190\beta_{4} - 451\beta_{3} + 1238\beta _1 + 190$$ 101*b9 - 195*b8 - 195*b7 - 101*b5 - 190*b4 - 451*b3 + 1238*b1 + 190

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1305\mathbb{Z}\right)^\times$$.

 $$n$$ $$146$$ $$262$$ $$901$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
784.1
 0.664979 − 0.664979i 1.75525 − 1.75525i 0.410556 + 0.410556i −1.47774 − 1.47774i −0.353040 − 0.353040i −0.353040 + 0.353040i −1.47774 + 1.47774i 0.410556 − 0.410556i 1.75525 + 1.75525i 0.664979 + 0.664979i
2.68032i 0 −5.18413 −0.117379 + 2.23299i 0 3.20454i 8.53449i 0 5.98512 + 0.314615i
784.2 2.45441i 0 −4.02413 −1.55285 1.60893i 0 2.54244i 4.96805i 0 −3.94898 + 3.81133i
784.3 1.88448i 0 −1.55125 2.18424 0.478651i 0 3.97545i 0.845662i 0 −0.902006 4.11614i
784.4 1.10415i 0 0.780857 −2.10939 0.741948i 0 2.02596i 3.07048i 0 −0.819220 + 2.32908i
784.5 0.146109i 0 1.97865 0.595378 2.15535i 0 2.71260i 0.581318i 0 −0.314916 0.0869902i
784.6 0.146109i 0 1.97865 0.595378 + 2.15535i 0 2.71260i 0.581318i 0 −0.314916 + 0.0869902i
784.7 1.10415i 0 0.780857 −2.10939 + 0.741948i 0 2.02596i 3.07048i 0 −0.819220 2.32908i
784.8 1.88448i 0 −1.55125 2.18424 + 0.478651i 0 3.97545i 0.845662i 0 −0.902006 + 4.11614i
784.9 2.45441i 0 −4.02413 −1.55285 + 1.60893i 0 2.54244i 4.96805i 0 −3.94898 3.81133i
784.10 2.68032i 0 −5.18413 −0.117379 2.23299i 0 3.20454i 8.53449i 0 5.98512 0.314615i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 784.10 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1305.2.c.i 10
3.b odd 2 1 435.2.c.d 10
5.b even 2 1 inner 1305.2.c.i 10
5.c odd 4 1 6525.2.a.bn 5
5.c odd 4 1 6525.2.a.br 5
15.d odd 2 1 435.2.c.d 10
15.e even 4 1 2175.2.a.x 5
15.e even 4 1 2175.2.a.y 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.c.d 10 3.b odd 2 1
435.2.c.d 10 15.d odd 2 1
1305.2.c.i 10 1.a even 1 1 trivial
1305.2.c.i 10 5.b even 2 1 inner
2175.2.a.x 5 15.e even 4 1
2175.2.a.y 5 15.e even 4 1
6525.2.a.bn 5 5.c odd 4 1
6525.2.a.br 5 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1305, [\chi])$$:

 $$T_{2}^{10} + 18T_{2}^{8} + 111T_{2}^{6} + 266T_{2}^{4} + 193T_{2}^{2} + 4$$ T2^10 + 18*T2^8 + 111*T2^6 + 266*T2^4 + 193*T2^2 + 4 $$T_{7}^{10} + 44T_{7}^{8} + 734T_{7}^{6} + 5824T_{7}^{4} + 22017T_{7}^{2} + 31684$$ T7^10 + 44*T7^8 + 734*T7^6 + 5824*T7^4 + 22017*T7^2 + 31684 $$T_{11}^{5} - 5T_{11}^{4} - 16T_{11}^{3} + 88T_{11}^{2} - 5T_{11} - 191$$ T11^5 - 5*T11^4 - 16*T11^3 + 88*T11^2 - 5*T11 - 191

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} + 18 T^{8} + \cdots + 4$$
$3$ $$T^{10}$$
$5$ $$T^{10} + 2 T^{9} + \cdots + 3125$$
$7$ $$T^{10} + 44 T^{8} + \cdots + 31684$$
$11$ $$(T^{5} - 5 T^{4} + \cdots - 191)^{2}$$
$13$ $$T^{10} + 48 T^{8} + \cdots + 256$$
$17$ $$T^{10} + 60 T^{8} + \cdots + 16$$
$19$ $$(T^{5} + 2 T^{4} + \cdots - 352)^{2}$$
$23$ $$T^{10} + 105 T^{8} + \cdots + 126736$$
$29$ $$(T + 1)^{10}$$
$31$ $$(T^{5} - 2 T^{4} + \cdots + 2672)^{2}$$
$37$ $$T^{10} + 201 T^{8} + \cdots + 12475024$$
$41$ $$(T^{5} - 19 T^{4} + \cdots - 656)^{2}$$
$43$ $$T^{10} + \cdots + 248755984$$
$47$ $$T^{10} + 212 T^{8} + \cdots + 8549776$$
$53$ $$T^{10} + 149 T^{8} + \cdots + 8690704$$
$59$ $$(T^{5} + 22 T^{4} + \cdots - 8744)^{2}$$
$61$ $$(T^{5} + 8 T^{4} + \cdots + 200)^{2}$$
$67$ $$T^{10} + 400 T^{8} + \cdots + 4129024$$
$71$ $$(T^{5} - 18 T^{4} + \cdots - 176)^{2}$$
$73$ $$T^{10} + 393 T^{8} + \cdots + 1281424$$
$79$ $$(T^{5} + 16 T^{4} + \cdots - 14488)^{2}$$
$83$ $$T^{10} + \cdots + 1746905616$$
$89$ $$(T^{5} - 2 T^{4} + \cdots - 165482)^{2}$$
$97$ $$T^{10} + \cdots + 147962896$$
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