Properties

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

Related objects

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: 10.0.3899266318336.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} - 2x^{9} + 2x^{8} + 6x^{7} + 19x^{6} - 12x^{5} + 4x^{4} + 2x^{3} + 9x^{2} - 6x + 2$$ x^10 - 2*x^9 + 2*x^8 + 6*x^7 + 19*x^6 - 12*x^5 + 4*x^4 + 2*x^3 + 9*x^2 - 6*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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2x^{9} + 2x^{8} + 6x^{7} + 19x^{6} - 12x^{5} + 4x^{4} + 2x^{3} + 9x^{2} - 6x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 4381 \nu^{9} + 19149 \nu^{8} - 24628 \nu^{7} - 15014 \nu^{6} - 7864 \nu^{5} + 263961 \nu^{4} + \cdots - 8160 ) / 57533$$ (-4381*v^9 + 19149*v^8 - 24628*v^7 - 15014*v^6 - 7864*v^5 + 263961*v^4 - 26027*v^3 - 6465*v^2 + 8336*v - 8160) / 57533 $$\beta_{3}$$ $$=$$ $$( 4394 \nu^{9} - 9304 \nu^{8} + 11897 \nu^{7} + 22741 \nu^{6} + 82151 \nu^{5} - 49465 \nu^{4} + \cdots - 25960 ) / 57533$$ (4394*v^9 - 9304*v^8 + 11897*v^7 + 22741*v^6 + 82151*v^5 - 49465*v^4 + 85003*v^3 + 9045*v^2 + 38929*v - 25960) / 57533 $$\beta_{4}$$ $$=$$ $$( - 4908 \nu^{9} + 9502 \nu^{8} - 13053 \nu^{7} - 14036 \nu^{6} - 116141 \nu^{5} + 39251 \nu^{4} + \cdots + 20774 ) / 57533$$ (-4908*v^9 + 9502*v^8 - 13053*v^7 - 14036*v^6 - 116141*v^5 + 39251*v^4 - 26991*v^3 + 185462*v^2 - 71084*v + 20774) / 57533 $$\beta_{5}$$ $$=$$ $$( - 9291 \nu^{9} + 31562 \nu^{8} - 40148 \nu^{7} - 39090 \nu^{6} - 86752 \nu^{5} + 380853 \nu^{4} + \cdots + 124078 ) / 57533$$ (-9291*v^9 + 31562*v^8 - 40148*v^7 - 39090*v^6 - 86752*v^5 + 380853*v^4 - 110773*v^3 - 16127*v^2 + 27344*v + 124078) / 57533 $$\beta_{6}$$ $$=$$ $$( 12980 \nu^{9} - 21566 \nu^{8} + 16656 \nu^{7} + 89777 \nu^{6} + 269361 \nu^{5} - 73609 \nu^{4} + \cdots - 38951 ) / 57533$$ (12980*v^9 - 21566*v^8 + 16656*v^7 + 89777*v^6 + 269361*v^5 - 73609*v^4 + 2455*v^3 + 110963*v^2 + 125865*v - 38951) / 57533 $$\beta_{7}$$ $$=$$ $$( 35262 \nu^{9} - 61938 \nu^{8} + 58262 \nu^{7} + 216331 \nu^{6} + 737014 \nu^{5} - 235934 \nu^{4} + \cdots - 124636 ) / 57533$$ (35262*v^9 - 61938*v^8 + 58262*v^7 + 216331*v^6 + 737014*v^5 - 235934*v^4 + 116904*v^3 - 12024*v^2 + 419276*v - 124636) / 57533 $$\beta_{8}$$ $$=$$ $$( - 49059 \nu^{9} + 93221 \nu^{8} - 75860 \nu^{7} - 322605 \nu^{6} - 948470 \nu^{5} + 584107 \nu^{4} + \cdots + 303094 ) / 57533$$ (-49059*v^9 + 93221*v^8 - 75860*v^7 - 322605*v^6 - 948470*v^5 + 584107*v^4 + 135152*v^3 - 123888*v^2 - 448613*v + 303094) / 57533 $$\beta_{9}$$ $$=$$ $$( - 49338 \nu^{9} + 67808 \nu^{8} - 46042 \nu^{7} - 342393 \nu^{6} - 1131012 \nu^{5} - 62807 \nu^{4} + \cdots - 26786 ) / 57533$$ (-49338*v^9 + 67808*v^8 - 46042*v^7 - 342393*v^6 - 1131012*v^5 - 62807*v^4 - 10883*v^3 - 130577*v^2 - 422973*v - 26786) / 57533
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{7} + 2\beta_{6} - \beta_{4} + \beta_{3} + \beta_1$$ -b7 + 2*b6 - b4 + b3 + b1 $$\nu^{3}$$ $$=$$ $$-\beta_{8} - 2\beta_{7} + 2\beta_{5} - \beta_{4} + 7\beta_{3} - \beta_{2}$$ -b8 - 2*b7 + 2*b5 - b4 + 7*b3 - b2 $$\nu^{4}$$ $$=$$ $$\beta_{9} - \beta_{8} + 8\beta_{5} + 10\beta_{3} - 7\beta_{2} - 10\beta _1 - 8$$ b9 - b8 + 8*b5 + 10*b3 - 7*b2 - 10*b1 - 8 $$\nu^{5}$$ $$=$$ $$7\beta_{9} + 16\beta_{7} - 5\beta_{6} + 16\beta_{5} + 10\beta_{4} - 10\beta_{2} - 48\beta _1 - 5$$ 7*b9 + 16*b7 - 5*b6 + 16*b5 + 10*b4 - 10*b2 - 48*b1 - 5 $$\nu^{6}$$ $$=$$ $$10\beta_{9} + 10\beta_{8} + 61\beta_{7} - 44\beta_{6} + 48\beta_{4} - 82\beta_{3} - 82\beta_1$$ 10*b9 + 10*b8 + 61*b7 - 44*b6 + 48*b4 - 82*b3 - 82*b1 $$\nu^{7}$$ $$=$$ $$48\beta_{8} + 120\beta_{7} - 55\beta_{6} - 120\beta_{5} + 82\beta_{4} - 345\beta_{3} + 82\beta_{2} + 55$$ 48*b8 + 120*b7 - 55*b6 - 120*b5 + 82*b4 - 345*b3 + 82*b2 + 55 $$\nu^{8}$$ $$=$$ $$-82\beta_{9} + 82\beta_{8} - 461\beta_{5} - 636\beta_{3} + 345\beta_{2} + 636\beta _1 + 286$$ -82*b9 + 82*b8 - 461*b5 - 636*b3 + 345*b2 + 636*b1 + 286 $$\nu^{9}$$ $$=$$ $$-345\beta_{9} - 899\beta_{7} + 466\beta_{6} - 899\beta_{5} - 636\beta_{4} + 636\beta_{2} + 2545\beta _1 + 466$$ -345*b9 - 899*b7 + 466*b6 - 899*b5 - 636*b4 + 636*b2 + 2545*b1 + 466

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
 −1.20964 + 1.20964i 1.93833 − 1.93833i −0.604479 − 0.604479i 0.313948 − 0.313948i 0.561843 + 0.561843i 0.561843 − 0.561843i 0.313948 + 0.313948i −0.604479 + 0.604479i 1.93833 + 1.93833i −1.20964 − 1.20964i
2.51908i 0 −4.34577 1.27413 + 1.83755i 0 0.173311i 5.90919i 0 4.62893 3.20964i
784.2 2.15351i 0 −2.63760 0.0286357 2.23588i 0 1.51591i 1.37308i 0 −4.81500 0.0616673i
784.3 1.71457i 0 −0.939748 −1.51903 + 1.64090i 0 0.654317i 1.81788i 0 2.81344 + 2.60448i
784.4 0.754474i 0 1.43077 2.23474 + 0.0770824i 0 4.18524i 2.58843i 0 0.0581566 1.68605i
784.5 0.712495i 0 1.49235 −2.01848 0.962154i 0 2.77986i 2.48828i 0 −0.685530 + 1.43816i
784.6 0.712495i 0 1.49235 −2.01848 + 0.962154i 0 2.77986i 2.48828i 0 −0.685530 1.43816i
784.7 0.754474i 0 1.43077 2.23474 0.0770824i 0 4.18524i 2.58843i 0 0.0581566 + 1.68605i
784.8 1.71457i 0 −0.939748 −1.51903 1.64090i 0 0.654317i 1.81788i 0 2.81344 2.60448i
784.9 2.15351i 0 −2.63760 0.0286357 + 2.23588i 0 1.51591i 1.37308i 0 −4.81500 + 0.0616673i
784.10 2.51908i 0 −4.34577 1.27413 1.83755i 0 0.173311i 5.90919i 0 4.62893 + 3.20964i
 $$n$$: 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.j 10
3.b odd 2 1 435.2.c.e 10
5.b even 2 1 inner 1305.2.c.j 10
5.c odd 4 1 6525.2.a.bl 5
5.c odd 4 1 6525.2.a.bs 5
15.d odd 2 1 435.2.c.e 10
15.e even 4 1 2175.2.a.w 5
15.e even 4 1 2175.2.a.z 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.c.e 10 3.b odd 2 1
435.2.c.e 10 15.d odd 2 1
1305.2.c.j 10 1.a even 1 1 trivial
1305.2.c.j 10 5.b even 2 1 inner
2175.2.a.w 5 15.e even 4 1
2175.2.a.z 5 15.e even 4 1
6525.2.a.bl 5 5.c odd 4 1
6525.2.a.bs 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 $$S_{2}^{\mathrm{new}}(1305, [\chi])$$:

 $$T_{2}^{10} + 15T_{2}^{8} + 77T_{2}^{6} + 157T_{2}^{4} + 111T_{2}^{2} + 25$$ T2^10 + 15*T2^8 + 77*T2^6 + 157*T2^4 + 111*T2^2 + 25 $$T_{7}^{10} + 28T_{7}^{8} + 206T_{7}^{6} + 400T_{7}^{4} + 145T_{7}^{2} + 4$$ T7^10 + 28*T7^8 + 206*T7^6 + 400*T7^4 + 145*T7^2 + 4 $$T_{11}^{5} + 12T_{11}^{4} + 48T_{11}^{3} + 72T_{11}^{2} + 35T_{11} + 4$$ T11^5 + 12*T11^4 + 48*T11^3 + 72*T11^2 + 35*T11 + 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} + 15 T^{8} + \cdots + 25$$
$3$ $$T^{10}$$
$5$ $$T^{10} - T^{8} + \cdots + 3125$$
$7$ $$T^{10} + 28 T^{8} + \cdots + 4$$
$11$ $$(T^{5} + 12 T^{4} + 48 T^{3} + \cdots + 4)^{2}$$
$13$ $$T^{10} + 16 T^{8} + \cdots + 16$$
$17$ $$T^{10} + 56 T^{8} + \cdots + 88804$$
$19$ $$(T^{5} - 2 T^{4} + \cdots + 304)^{2}$$
$23$ $$T^{10} + 76 T^{8} + \cdots + 1600$$
$29$ $$(T - 1)^{10}$$
$31$ $$(T^{5} - 2 T^{4} + \cdots - 6304)^{2}$$
$37$ $$T^{10} + 180 T^{8} + \cdots + 341056$$
$41$ $$(T^{5} - 14 T^{4} + \cdots - 6176)^{2}$$
$43$ $$T^{10} + 292 T^{8} + \cdots + 46895104$$
$47$ $$T^{10} + 220 T^{8} + \cdots + 7246864$$
$53$ $$T^{10} + \cdots + 226683136$$
$59$ $$(T^{5} - 4 T^{4} + \cdots + 2000)^{2}$$
$61$ $$(T^{5} + 12 T^{4} + \cdots + 6872)^{2}$$
$67$ $$T^{10} + 228 T^{8} + \cdots + 1716100$$
$71$ $$(T^{5} + 30 T^{4} + \cdots - 6592)^{2}$$
$73$ $$T^{10} + 420 T^{8} + \cdots + 11343424$$
$79$ $$(T^{5} - 18 T^{4} + \cdots - 52048)^{2}$$
$83$ $$T^{10} + 216 T^{8} + \cdots + 64$$
$89$ $$(T^{5} + 22 T^{4} + \cdots + 40682)^{2}$$
$97$ $$T^{10} + 112 T^{8} + \cdots + 107584$$