# Properties

 Label 1840.2.i.b Level $1840$ Weight $2$ Character orbit 1840.i Analytic conductor $14.692$ Analytic rank $0$ Dimension $16$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(1471,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1840.1471");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1840.i (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: $$14.6924739719$$ Analytic rank: $$0$$ 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: $$x^{16} - 8x^{14} + 33x^{12} - 98x^{10} + 272x^{8} - 882x^{6} + 2673x^{4} - 5832x^{2} + 6561$$ x^16 - 8*x^14 + 33*x^12 - 98*x^10 + 272*x^8 - 882*x^6 + 2673*x^4 - 5832*x^2 + 6561 Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: yes 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{3} + \beta_{2} q^{5} + \beta_{13} q^{7} + ( - \beta_{4} - 1) q^{9}+O(q^{10})$$ q + b7 * q^3 + b2 * q^5 + b13 * q^7 + (-b4 - 1) * q^9 $$q + \beta_{7} q^{3} + \beta_{2} q^{5} + \beta_{13} q^{7} + ( - \beta_{4} - 1) q^{9} + ( - \beta_{6} - \beta_1) q^{11} + (\beta_{3} - 1) q^{13} + \beta_1 q^{15} + (\beta_{12} - \beta_{5} + \beta_{2}) q^{17} + ( - \beta_{6} + \beta_1) q^{19} + ( - \beta_{8} + \beta_{5} - 2 \beta_{2}) q^{21} + ( - \beta_{15} - \beta_{10} - \beta_{9}) q^{23} - q^{25} + (\beta_{14} - \beta_{10} - \beta_{7}) q^{27} + ( - \beta_{11} + \beta_{4} + \beta_{3} + 2) q^{29} + (\beta_{15} - \beta_{7}) q^{31} + (\beta_{8} - \beta_{5} + 4 \beta_{2}) q^{33} + \beta_{15} q^{35} + (\beta_{12} + \beta_{8} + \beta_{2}) q^{37} + ( - 2 \beta_{15} + \beta_{14} + \cdots - 2 \beta_{7}) q^{39}+ \cdots + (2 \beta_{13} - 2 \beta_{6} + 2 \beta_1) q^{99}+O(q^{100})$$ q + b7 * q^3 + b2 * q^5 + b13 * q^7 + (-b4 - 1) * q^9 + (-b6 - b1) * q^11 + (b3 - 1) * q^13 + b1 * q^15 + (b12 - b5 + b2) * q^17 + (-b6 + b1) * q^19 + (-b8 + b5 - 2*b2) * q^21 + (-b15 - b10 - b9) * q^23 - q^25 + (b14 - b10 - b7) * q^27 + (-b11 + b4 + b3 + 2) * q^29 + (b15 - b7) * q^31 + (b8 - b5 + 4*b2) * q^33 + b15 * q^35 + (b12 + b8 + b2) * q^37 + (-2*b15 + b14 - b10 + b9 - 2*b7) * q^39 + (-b11 - b4 - b3 - 2) * q^41 + (-b14 + 2*b13 - b10 - b6 - b1) * q^43 + (-b8 - b2) * q^45 + (-2*b9 - b7) * q^47 + (-b11 + b3 + 1) * q^49 + (-2*b14 + 2*b13 - 2*b10 - 2*b1) * q^51 + (-b12 - b5 - b2) * q^53 + (b9 + b7) * q^55 + (-b8 - b5 - 4*b2) * q^57 + (b15 + b14 - b10 + b9 + 3*b7) * q^59 + (b8 + b5 + 2*b2) * q^61 + (b13 - b6 - 3*b1) * q^63 - b5 * q^65 + (b13 - 2*b6 + 2*b1) * q^67 + (-b12 + b11 + b8 + b5 - b3 - 1) * q^69 + (b15 + 2*b9 + b7) * q^71 + (-2*b11 + b3 - 3) * q^73 - b7 * q^75 + (2*b11 - 2*b4 - 2*b3 - 4) * q^77 + (-b6 + b1) * q^79 + (2*b11 - 2*b3 + 3) * q^81 + (b13 - b6 + 3*b1) * q^83 + (b11 - b3) * q^85 + (-2*b15 + b14 - b10 + 3*b7) * q^87 + (2*b12 + 3*b8 - b5 + 2*b2) * q^89 + (-2*b14 + 2*b13 - 2*b10 - b6 + 3*b1) * q^91 + (2*b4 + b3 + 5) * q^93 + (b9 - b7) * q^95 + (-3*b8 + b5 - 2*b2) * q^97 + (2*b13 - 2*b6 + 2*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 16 q^{9}+O(q^{10})$$ 16 * q - 16 * q^9 $$16 q - 16 q^{9} - 8 q^{13} - 16 q^{25} + 36 q^{29} - 44 q^{41} + 20 q^{49} - 20 q^{69} - 48 q^{73} - 72 q^{77} + 40 q^{81} - 4 q^{85} + 88 q^{93}+O(q^{100})$$ 16 * q - 16 * q^9 - 8 * q^13 - 16 * q^25 + 36 * q^29 - 44 * q^41 + 20 * q^49 - 20 * q^69 - 48 * q^73 - 72 * q^77 + 40 * q^81 - 4 * q^85 + 88 * q^93

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 8x^{14} + 33x^{12} - 98x^{10} + 272x^{8} - 882x^{6} + 2673x^{4} - 5832x^{2} + 6561$$ :

 $$\beta_{1}$$ $$=$$ $$( -5\nu^{15} - 41\nu^{13} + 78\nu^{11} + 328\nu^{9} + 1232\nu^{7} - 8550\nu^{5} + 23733\nu^{3} - 28431\nu ) / 69984$$ (-5*v^15 - 41*v^13 + 78*v^11 + 328*v^9 + 1232*v^7 - 8550*v^5 + 23733*v^3 - 28431*v) / 69984 $$\beta_{2}$$ $$=$$ $$( -5\nu^{14} + 7\nu^{12} - 90\nu^{10} + 184\nu^{8} - 232\nu^{6} + 834\nu^{4} - 2619\nu^{2} + 8505 ) / 15552$$ (-5*v^14 + 7*v^12 - 90*v^10 + 184*v^8 - 232*v^6 + 834*v^4 - 2619*v^2 + 8505) / 15552 $$\beta_{3}$$ $$=$$ $$( -4\nu^{14} + 5\nu^{12} + 3\nu^{10} + 149\nu^{8} - 386\nu^{6} + 477\nu^{4} + 567\nu^{2} + 729 ) / 2916$$ (-4*v^14 + 5*v^12 + 3*v^10 + 149*v^8 - 386*v^6 + 477*v^4 + 567*v^2 + 729) / 2916 $$\beta_{4}$$ $$=$$ $$( \nu^{14} - 8\nu^{12} + 33\nu^{10} - 98\nu^{8} + 272\nu^{6} - 882\nu^{4} + 1944\nu^{2} - 4374 ) / 729$$ (v^14 - 8*v^12 + 33*v^10 - 98*v^8 + 272*v^6 - 882*v^4 + 1944*v^2 - 4374) / 729 $$\beta_{5}$$ $$=$$ $$( \nu^{14} - 8\nu^{12} + 33\nu^{10} - 98\nu^{8} + 272\nu^{6} - 882\nu^{4} + 3402\nu^{2} - 5832 ) / 729$$ (v^14 - 8*v^12 + 33*v^10 - 98*v^8 + 272*v^6 - 882*v^4 + 3402*v^2 - 5832) / 729 $$\beta_{6}$$ $$=$$ $$( -\nu^{15} + 8\nu^{13} - 33\nu^{11} + 98\nu^{9} - 272\nu^{7} + 882\nu^{5} - 2673\nu^{3} + 8019\nu ) / 2187$$ (-v^15 + 8*v^13 - 33*v^11 + 98*v^9 - 272*v^7 + 882*v^5 - 2673*v^3 + 8019*v) / 2187 $$\beta_{7}$$ $$=$$ $$( \nu^{15} - 8\nu^{13} + 33\nu^{11} - 98\nu^{9} + 272\nu^{7} - 882\nu^{5} + 2673\nu^{3} - 3645\nu ) / 2187$$ (v^15 - 8*v^13 + 33*v^11 - 98*v^9 + 272*v^7 - 882*v^5 + 2673*v^3 - 3645*v) / 2187 $$\beta_{8}$$ $$=$$ $$( 13\nu^{14} - 41\nu^{12} + 168\nu^{10} - 410\nu^{8} + 1736\nu^{6} - 6480\nu^{4} + 14661\nu^{2} - 22599 ) / 5832$$ (13*v^14 - 41*v^12 + 168*v^10 - 410*v^8 + 1736*v^6 - 6480*v^4 + 14661*v^2 - 22599) / 5832 $$\beta_{9}$$ $$=$$ $$( 5\nu^{15} - 13\nu^{13} + 111\nu^{11} - 166\nu^{9} + 415\nu^{7} - 2007\nu^{5} + 5913\nu^{3} - 13122\nu ) / 8748$$ (5*v^15 - 13*v^13 + 111*v^11 - 166*v^9 + 415*v^7 - 2007*v^5 + 5913*v^3 - 13122*v) / 8748 $$\beta_{10}$$ $$=$$ $$( - 145 \nu^{15} + 1403 \nu^{13} - 4866 \nu^{11} + 9512 \nu^{9} - 25832 \nu^{7} + 85770 \nu^{5} + \cdots + 499365 \nu ) / 139968$$ (-145*v^15 + 1403*v^13 - 4866*v^11 + 9512*v^9 - 25832*v^7 + 85770*v^5 - 322623*v^3 + 499365*v) / 139968 $$\beta_{11}$$ $$=$$ $$( -14\nu^{14} + 67\nu^{12} - 183\nu^{10} + 535\nu^{8} - 1342\nu^{6} + 5859\nu^{4} - 14661\nu^{2} + 19683 ) / 2916$$ (-14*v^14 + 67*v^12 - 183*v^10 + 535*v^8 - 1342*v^6 + 5859*v^4 - 14661*v^2 + 19683) / 2916 $$\beta_{12}$$ $$=$$ $$( 149 \nu^{14} - 895 \nu^{12} + 2946 \nu^{10} - 7312 \nu^{8} + 22600 \nu^{6} - 71370 \nu^{4} + \cdots - 333153 ) / 23328$$ (149*v^14 - 895*v^12 + 2946*v^10 - 7312*v^8 + 22600*v^6 - 71370*v^4 + 211491*v^2 - 333153) / 23328 $$\beta_{13}$$ $$=$$ $$( 229 \nu^{15} - 1175 \nu^{13} + 3354 \nu^{11} - 8456 \nu^{9} + 24632 \nu^{7} - 113346 \nu^{5} + \cdots - 257337 \nu ) / 139968$$ (229*v^15 - 1175*v^13 + 3354*v^11 - 8456*v^9 + 24632*v^7 - 113346*v^5 + 243243*v^3 - 257337*v) / 139968 $$\beta_{14}$$ $$=$$ $$( 239 \nu^{15} - 1093 \nu^{13} + 3198 \nu^{11} - 9112 \nu^{9} + 22168 \nu^{7} - 96246 \nu^{5} + \cdots - 480411 \nu ) / 139968$$ (239*v^15 - 1093*v^13 + 3198*v^11 - 9112*v^9 + 22168*v^7 - 96246*v^5 + 335745*v^3 - 480411*v) / 139968 $$\beta_{15}$$ $$=$$ $$( 2\nu^{15} - 10\nu^{13} + 27\nu^{11} - 70\nu^{9} + 253\nu^{7} - 771\nu^{5} + 2016\nu^{3} - 2835\nu ) / 972$$ (2*v^15 - 10*v^13 + 27*v^11 - 70*v^9 + 253*v^7 - 771*v^5 + 2016*v^3 - 2835*v) / 972
 $$\nu$$ $$=$$ $$( \beta_{7} + \beta_{6} ) / 2$$ (b7 + b6) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{5} - \beta_{4} + 2 ) / 2$$ (b5 - b4 + 2) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{14} - \beta_{13} + \beta_{7} + \beta_{6} + \beta_1$$ b14 - b13 + b7 + b6 + b1 $$\nu^{4}$$ $$=$$ $$( 2\beta_{12} + 2\beta_{11} - 2\beta_{8} + \beta_{5} - 3\beta_{4} - 2\beta_{3} - 4\beta_{2} ) / 2$$ (2*b12 + 2*b11 - 2*b8 + b5 - 3*b4 - 2*b3 - 4*b2) / 2 $$\nu^{5}$$ $$=$$ $$( 4\beta_{15} + 2\beta_{14} - 8\beta_{13} - 2\beta_{10} + \beta_{7} + 3\beta_{6} - 4\beta_1 ) / 2$$ (4*b15 + 2*b14 - 8*b13 - 2*b10 + b7 + 3*b6 - 4*b1) / 2 $$\nu^{6}$$ $$=$$ $$3\beta_{12} + 5\beta_{11} + 3\beta_{8} - \beta_{5} - \beta_{4} - 3\beta_{3} + 10\beta_{2} + 2$$ 3*b12 + 5*b11 + 3*b8 - b5 - b4 - 3*b3 + 10*b2 + 2 $$\nu^{7}$$ $$=$$ $$( 24\beta_{15} - 10\beta_{14} - 20\beta_{13} + 6\beta_{10} + 4\beta_{9} + 17\beta_{7} + 5\beta_{6} + 16\beta_1 ) / 2$$ (24*b15 - 10*b14 - 20*b13 + 6*b10 + 4*b9 + 17*b7 + 5*b6 + 16*b1) / 2 $$\nu^{8}$$ $$=$$ $$( 14\beta_{12} + 14\beta_{11} + 26\beta_{8} - 17\beta_{5} - 15\beta_{4} + 18\beta_{3} + 36\beta_{2} - 44 ) / 2$$ (14*b12 + 14*b11 + 26*b8 - 17*b5 - 15*b4 + 18*b3 + 36*b2 - 44) / 2 $$\nu^{9}$$ $$=$$ $$18\beta_{15} - 8\beta_{14} - 13\beta_{13} - 3\beta_{10} + 22\beta_{9} - 21\beta_{7} + 9\beta_{6} + 57\beta_1$$ 18*b15 - 8*b14 - 13*b13 - 3*b10 + 22*b9 - 21*b7 + 9*b6 + 57*b1 $$\nu^{10}$$ $$=$$ $$( 20\beta_{12} + 32\beta_{11} + 12\beta_{8} - 37\beta_{5} + 23\beta_{4} + 56\beta_{3} - 296\beta_{2} + 106 ) / 2$$ (20*b12 + 32*b11 + 12*b8 - 37*b5 + 23*b4 + 56*b3 - 296*b2 + 106) / 2 $$\nu^{11}$$ $$=$$ $$( -4\beta_{15} - 92\beta_{14} - 4\beta_{13} - 64\beta_{10} + 220\beta_{9} + 25\beta_{7} + 71\beta_{6} - 12\beta_1 ) / 2$$ (-4*b15 - 92*b14 - 4*b13 - 64*b10 + 220*b9 + 25*b7 + 71*b6 - 12*b1) / 2 $$\nu^{12}$$ $$=$$ $$-48\beta_{12} + 34\beta_{11} + 104\beta_{8} + 56\beta_{5} + 6\beta_{4} - 14\beta_{3} - 416\beta_{2} + 203$$ -48*b12 + 34*b11 + 104*b8 + 56*b5 + 6*b4 - 14*b3 - 416*b2 + 203 $$\nu^{13}$$ $$=$$ $$( 44 \beta_{15} + 32 \beta_{14} - 236 \beta_{13} + 332 \beta_{10} + 700 \beta_{9} + 613 \beta_{7} + \cdots + 76 \beta_1 ) / 2$$ (44*b15 + 32*b14 - 236*b13 + 332*b10 + 700*b9 + 613*b7 + 197*b6 + 76*b1) / 2 $$\nu^{14}$$ $$=$$ $$( 76\beta_{12} - 96\beta_{11} + 420\beta_{8} - 67\beta_{5} - 833\beta_{4} - 440\beta_{3} - 2328\beta_{2} - 790 ) / 2$$ (76*b12 - 96*b11 + 420*b8 - 67*b5 - 833*b4 - 440*b3 - 2328*b2 - 790) / 2 $$\nu^{15}$$ $$=$$ $$506 \beta_{15} + 431 \beta_{14} - 287 \beta_{13} + 392 \beta_{10} + 782 \beta_{9} - 553 \beta_{7} + \cdots - 525 \beta_1$$ 506*b15 + 431*b14 - 287*b13 + 392*b10 + 782*b9 - 553*b7 + 291*b6 - 525*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$$.

 $$n$$ $$737$$ $$1151$$ $$1201$$ $$1381$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1471.1
 0.739948 − 1.56604i −0.739948 − 1.56604i −1.38594 − 1.03883i 1.38594 − 1.03883i 1.59950 − 0.664536i −1.59950 − 0.664536i −1.72432 − 0.163515i 1.72432 − 0.163515i 1.72432 + 0.163515i −1.72432 + 0.163515i −1.59950 + 0.664536i 1.59950 + 0.664536i 1.38594 + 1.03883i −1.38594 + 1.03883i −0.739948 + 1.56604i 0.739948 + 1.56604i
0 3.13208i 0 1.00000i 0 −1.01363 0 −6.80991 0
1471.2 0 3.13208i 0 1.00000i 0 1.01363 0 −6.80991 0
1471.3 0 2.07767i 0 1.00000i 0 −3.88693 0 −1.31671 0
1471.4 0 2.07767i 0 1.00000i 0 3.88693 0 −1.31671 0
1471.5 0 1.32907i 0 1.00000i 0 3.37473 0 1.23357 0
1471.6 0 1.32907i 0 1.00000i 0 −3.37473 0 1.23357 0
1471.7 0 0.327030i 0 1.00000i 0 2.33999 0 2.89305 0
1471.8 0 0.327030i 0 1.00000i 0 −2.33999 0 2.89305 0
1471.9 0 0.327030i 0 1.00000i 0 −2.33999 0 2.89305 0
1471.10 0 0.327030i 0 1.00000i 0 2.33999 0 2.89305 0
1471.11 0 1.32907i 0 1.00000i 0 −3.37473 0 1.23357 0
1471.12 0 1.32907i 0 1.00000i 0 3.37473 0 1.23357 0
1471.13 0 2.07767i 0 1.00000i 0 3.88693 0 −1.31671 0
1471.14 0 2.07767i 0 1.00000i 0 −3.88693 0 −1.31671 0
1471.15 0 3.13208i 0 1.00000i 0 1.01363 0 −6.80991 0
1471.16 0 3.13208i 0 1.00000i 0 −1.01363 0 −6.80991 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1471.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
23.b odd 2 1 inner
92.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1840.2.i.b 16
4.b odd 2 1 inner 1840.2.i.b 16
23.b odd 2 1 inner 1840.2.i.b 16
92.b even 2 1 inner 1840.2.i.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1840.2.i.b 16 1.a even 1 1 trivial
1840.2.i.b 16 4.b odd 2 1 inner
1840.2.i.b 16 23.b odd 2 1 inner
1840.2.i.b 16 92.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 16T_{3}^{6} + 69T_{3}^{4} + 82T_{3}^{2} + 8$$ acting on $$S_{2}^{\mathrm{new}}(1840, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$(T^{8} + 16 T^{6} + 69 T^{4} + \cdots + 8)^{2}$$
$5$ $$(T^{2} + 1)^{8}$$
$7$ $$(T^{8} - 33 T^{6} + \cdots + 968)^{2}$$
$11$ $$(T^{8} - 44 T^{6} + \cdots + 3200)^{2}$$
$13$ $$(T^{4} + 2 T^{3} + \cdots + 128)^{4}$$
$17$ $$(T^{8} + 117 T^{6} + \cdots + 179776)^{2}$$
$19$ $$(T^{8} - 52 T^{6} + \cdots + 2048)^{2}$$
$23$ $$T^{16} + \cdots + 78310985281$$
$29$ $$(T^{4} - 9 T^{3} + \cdots - 110)^{4}$$
$31$ $$(T^{8} + 61 T^{6} + \cdots + 10368)^{2}$$
$37$ $$(T^{8} + 101 T^{6} + \cdots + 55696)^{2}$$
$41$ $$(T^{4} + 11 T^{3} + \cdots - 3244)^{4}$$
$43$ $$(T^{8} - 186 T^{6} + \cdots + 850208)^{2}$$
$47$ $$(T^{8} + 136 T^{6} + \cdots + 2312)^{2}$$
$53$ $$(T^{8} + 189 T^{6} + \cdots + 331776)^{2}$$
$59$ $$(T^{8} + 323 T^{6} + \cdots + 4147200)^{2}$$
$61$ $$(T^{8} + 192 T^{6} + \cdots + 270400)^{2}$$
$67$ $$(T^{8} - 217 T^{6} + \cdots + 1555848)^{2}$$
$71$ $$(T^{8} + 109 T^{6} + \cdots + 2592)^{2}$$
$73$ $$(T^{4} + 12 T^{3} + \cdots + 4244)^{4}$$
$79$ $$(T^{8} - 52 T^{6} + \cdots + 2048)^{2}$$
$83$ $$(T^{8} - 233 T^{6} + \cdots + 2562848)^{2}$$
$89$ $$(T^{8} + 492 T^{6} + \cdots + 34857216)^{2}$$
$97$ $$(T^{8} + 448 T^{6} + \cdots + 97298496)^{2}$$