# Properties

 Label 3626.2.a.bi Level $3626$ Weight $2$ Character orbit 3626.a Self dual yes Analytic conductor $28.954$ Analytic rank $0$ Dimension $14$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3626,2,Mod(1,3626)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3626, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3626.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3626 = 2 \cdot 7^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3626.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: $$28.9537557729$$ Analytic rank: $$0$$ Dimension: $$14$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{14} - 32x^{12} + 379x^{10} - 2028x^{8} + 4673x^{6} - 3288x^{4} + 352x^{2} - 8$$ x^14 - 32*x^12 + 379*x^10 - 2028*x^8 + 4673*x^6 - 3288*x^4 + 352*x^2 - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes 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 $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + \beta_1 q^{3} + q^{4} + \beta_{10} q^{5} - \beta_1 q^{6} - q^{8} + (\beta_{7} + \beta_{6} + 2) q^{9}+O(q^{10})$$ q - q^2 + b1 * q^3 + q^4 + b10 * q^5 - b1 * q^6 - q^8 + (b7 + b6 + 2) * q^9 $$q - q^{2} + \beta_1 q^{3} + q^{4} + \beta_{10} q^{5} - \beta_1 q^{6} - q^{8} + (\beta_{7} + \beta_{6} + 2) q^{9} - \beta_{10} q^{10} + (\beta_{8} - \beta_{2} + 1) q^{11} + \beta_1 q^{12} + ( - \beta_{12} + \beta_{9} + \beta_1) q^{13} + (\beta_{8} + \beta_{7} - \beta_{6} + \cdots + 1) q^{15}+ \cdots + ( - 2 \beta_{8} + 2 \beta_{7} + \cdots + 1) q^{99}+O(q^{100})$$ q - q^2 + b1 * q^3 + q^4 + b10 * q^5 - b1 * q^6 - q^8 + (b7 + b6 + 2) * q^9 - b10 * q^10 + (b8 - b2 + 1) * q^11 + b1 * q^12 + (-b12 + b9 + b1) * q^13 + (b8 + b7 - b6 + b5 + 1) * q^15 + q^16 + (b11 + b9 + b1) * q^17 + (-b7 - b6 - 2) * q^18 + (b13 - b10 + 2*b9 - b4) * q^19 + b10 * q^20 + (-b8 + b2 - 1) * q^22 + (b8 - b5) * q^23 - b1 * q^24 + (b7 - 2*b2 + 3) * q^25 + (b12 - b9 - b1) * q^26 + (b11 + b10 - b9 + 2*b4 + 2*b1) * q^27 + (b8 + b5 + b2) * q^29 + (-b8 - b7 + b6 - b5 - 1) * q^30 + (-b12 - b11 + b9) * q^31 - q^32 + (-b13 + b10 - b9 - 3*b4) * q^33 + (-b11 - b9 - b1) * q^34 + (b7 + b6 + 2) * q^36 + q^37 + (-b13 + b10 - 2*b9 + b4) * q^38 + (-2*b8 + b6 + b5 - b3 + 3) * q^39 - b10 * q^40 + (b13 - b11 - b10 + b9 + b4 + 2*b1) * q^41 + (-b8 + b6 - b5 + b3 + b2 + 2) * q^43 + (b8 - b2 + 1) * q^44 + (-b13 + 2*b12 - b11 + 2*b10 - 3*b9 + 4*b4) * q^45 + (-b8 + b5) * q^46 + (2*b10 - 2*b9 - 2*b4) * q^47 + b1 * q^48 + (-b7 + 2*b2 - 3) * q^50 + (-b8 + 3*b6 - b5 + b3 + b2 + 4) * q^51 + (-b12 + b9 + b1) * q^52 + (b8 + b6 - b5 + b3 - b2 + 2) * q^53 + (-b11 - b10 + b9 - 2*b4 - 2*b1) * q^54 + (-3*b13 + b12 + 3*b10 - b9 - b1) * q^55 + (-2*b8 - 3*b7 - b6 - b3 + 2*b2 - 6) * q^57 + (-b8 - b5 - b2) * q^58 + (b13 + b10 + 3*b4) * q^59 + (b8 + b7 - b6 + b5 + 1) * q^60 + (b13 - b12 - b10 - 5*b4 - b1) * q^61 + (b12 + b11 - b9) * q^62 + q^64 + (b8 - 3*b6 - b3 - b2 - 1) * q^65 + (b13 - b10 + b9 + 3*b4) * q^66 + (-b8 + b7 - b6 - b3 - b2 + 3) * q^67 + (b11 + b9 + b1) * q^68 + (b13 + 2*b12 - 2*b9 + 4*b4) * q^69 + (-b8 + b7 + b5 - 2*b3 - b2 - 2) * q^71 + (-b7 - b6 - 2) * q^72 + (2*b13 - b12 - 2*b10 + 2*b4 + 3*b1) * q^73 - q^74 + (-2*b13 - b12 + 2*b10 - 2*b9 - 7*b4 + 2*b1) * q^75 + (b13 - b10 + 2*b9 - b4) * q^76 + (2*b8 - b6 - b5 + b3 - 3) * q^78 + (-b8 - b7 + b6 + b5 + b3 + b2 + 4) * q^79 + b10 * q^80 + (2*b8 + b7 + 2*b6 - 2*b5 + 3*b3 + b2 + 8) * q^81 + (-b13 + b11 + b10 - b9 - b4 - 2*b1) * q^82 + (-b13 + b11 + b10 + b9 - 2*b4 + b1) * q^83 + (2*b8 - 2*b7 - 2*b6 - 2*b5 + 2*b2 - 2) * q^85 + (b8 - b6 + b5 - b3 - b2 - 2) * q^86 + (b12 + 2*b10 + 5*b4 + b1) * q^87 + (-b8 + b2 - 1) * q^88 + (b11 + 2*b10 + b9 - 2*b4 - b1) * q^89 + (b13 - 2*b12 + b11 - 2*b10 + 3*b9 - 4*b4) * q^90 + (b8 - b5) * q^92 + (-2*b8 - b7 - 3*b6 + 3*b5 - 3*b3 - 3) * q^93 + (-2*b10 + 2*b9 + 2*b4) * q^94 + (-2*b8 - 4*b7 - 2*b5 + 2*b3 + 2*b2 - 2) * q^95 - b1 * q^96 + (2*b13 - b11 - 2*b10 + b9 + 2*b4 + b1) * q^97 + (-2*b8 + 2*b7 + b5 - 2*b2 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14 q - 14 q^{2} + 14 q^{4} - 14 q^{8} + 22 q^{9}+O(q^{10})$$ 14 * q - 14 * q^2 + 14 * q^4 - 14 * q^8 + 22 * q^9 $$14 q - 14 q^{2} + 14 q^{4} - 14 q^{8} + 22 q^{9} + 16 q^{11} + 8 q^{15} + 14 q^{16} - 22 q^{18} - 16 q^{22} + 46 q^{25} - 8 q^{29} - 8 q^{30} - 14 q^{32} + 22 q^{36} + 14 q^{37} + 48 q^{39} + 20 q^{43} + 16 q^{44} - 46 q^{50} + 44 q^{51} + 24 q^{53} - 68 q^{57} + 8 q^{58} + 8 q^{60} + 14 q^{64} + 52 q^{67} - 16 q^{71} - 22 q^{72} - 14 q^{74} - 48 q^{78} + 48 q^{79} + 82 q^{81} - 24 q^{85} - 20 q^{86} - 16 q^{88} - 16 q^{93} - 24 q^{95} + 16 q^{99}+O(q^{100})$$ 14 * q - 14 * q^2 + 14 * q^4 - 14 * q^8 + 22 * q^9 + 16 * q^11 + 8 * q^15 + 14 * q^16 - 22 * q^18 - 16 * q^22 + 46 * q^25 - 8 * q^29 - 8 * q^30 - 14 * q^32 + 22 * q^36 + 14 * q^37 + 48 * q^39 + 20 * q^43 + 16 * q^44 - 46 * q^50 + 44 * q^51 + 24 * q^53 - 68 * q^57 + 8 * q^58 + 8 * q^60 + 14 * q^64 + 52 * q^67 - 16 * q^71 - 22 * q^72 - 14 * q^74 - 48 * q^78 + 48 * q^79 + 82 * q^81 - 24 * q^85 - 20 * q^86 - 16 * q^88 - 16 * q^93 - 24 * q^95 + 16 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} - 32x^{12} + 379x^{10} - 2028x^{8} + 4673x^{6} - 3288x^{4} + 352x^{2} - 8$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -89\nu^{12} + 2242\nu^{10} - 16515\nu^{8} + 16360\nu^{6} + 168433\nu^{4} - 275752\nu^{2} - 13280 ) / 86786$$ (-89*v^12 + 2242*v^10 - 16515*v^8 + 16360*v^6 + 168433*v^4 - 275752*v^2 - 13280) / 86786 $$\beta_{3}$$ $$=$$ $$( 1660\nu^{12} - 53031\nu^{10} + 626898\nu^{8} - 3349965\nu^{6} + 7740820\nu^{4} - 5583120\nu^{2} + 469535 ) / 43393$$ (1660*v^12 - 53031*v^10 + 626898*v^8 - 3349965*v^6 + 7740820*v^4 - 5583120*v^2 + 469535) / 43393 $$\beta_{4}$$ $$=$$ $$( - 1660 \nu^{13} + 53031 \nu^{11} - 626898 \nu^{9} + 3349965 \nu^{7} - 7740820 \nu^{5} + \cdots - 860072 \nu ) / 86786$$ (-1660*v^13 + 53031*v^11 - 626898*v^9 + 3349965*v^7 - 7740820*v^5 + 5626513*v^3 - 860072*v) / 86786 $$\beta_{5}$$ $$=$$ $$( -2073\nu^{12} + 66012\nu^{10} - 774928\nu^{8} + 4072529\nu^{6} - 8980925\nu^{4} + 5364093\nu^{2} - 239041 ) / 43393$$ (-2073*v^12 + 66012*v^10 - 774928*v^8 + 4072529*v^6 - 8980925*v^4 + 5364093*v^2 - 239041) / 43393 $$\beta_{6}$$ $$=$$ $$( - 5357 \nu^{12} + 170192 \nu^{10} - 1996203 \nu^{8} + 10529374 \nu^{6} - 23629681 \nu^{4} + \cdots - 1000630 ) / 86786$$ (-5357*v^12 + 170192*v^10 - 1996203*v^8 + 10529374*v^6 - 23629681*v^4 + 15303516*v^2 - 1000630) / 86786 $$\beta_{7}$$ $$=$$ $$( 5357 \nu^{12} - 170192 \nu^{10} + 1996203 \nu^{8} - 10529374 \nu^{6} + 23629681 \nu^{4} + \cdots + 566700 ) / 86786$$ (5357*v^12 - 170192*v^10 + 1996203*v^8 - 10529374*v^6 + 23629681*v^4 - 15216730*v^2 + 566700) / 86786 $$\beta_{8}$$ $$=$$ $$( - 6403 \nu^{12} + 204900 \nu^{10} - 2424191 \nu^{8} + 12922086 \nu^{6} - 29410293 \nu^{4} + \cdots - 1119374 ) / 86786$$ (-6403*v^12 + 204900*v^10 - 2424191*v^8 + 12922086*v^6 - 29410293*v^4 + 19529734*v^2 - 1119374) / 86786 $$\beta_{9}$$ $$=$$ $$( - 12675 \nu^{13} + 403296 \nu^{11} - 4732969 \nu^{9} + 24912948 \nu^{7} - 55351751 \nu^{5} + \cdots - 457292 \nu ) / 173572$$ (-12675*v^13 + 403296*v^11 - 4732969*v^9 + 24912948*v^7 - 55351751*v^5 + 33945872*v^3 - 457292*v) / 173572 $$\beta_{10}$$ $$=$$ $$( 14165 \nu^{13} - 452950 \nu^{11} + 5361615 \nu^{9} - 28689902 \nu^{7} + 66209837 \nu^{5} + \cdots + 4461428 \nu ) / 173572$$ (14165*v^13 - 452950*v^11 + 5361615*v^9 - 28689902*v^7 + 66209837*v^5 - 46800082*v^3 + 4461428*v) / 173572 $$\beta_{11}$$ $$=$$ $$( - 10100 \nu^{13} + 322061 \nu^{11} - 3793496 \nu^{9} + 20101495 \nu^{7} - 45299154 \nu^{5} + \cdots - 1433504 \nu ) / 86786$$ (-10100*v^13 + 322061*v^11 - 3793496*v^9 + 20101495*v^7 - 45299154*v^5 + 29206737*v^3 - 1433504*v) / 86786 $$\beta_{12}$$ $$=$$ $$( - 16503 \nu^{13} + 526961 \nu^{11} - 6217687 \nu^{9} + 33023581 \nu^{7} - 74709447 \nu^{5} + \cdots - 1858590 \nu ) / 86786$$ (-16503*v^13 + 526961*v^11 - 6217687*v^9 + 33023581*v^7 - 74709447*v^5 + 48649685*v^3 - 1858590*v) / 86786 $$\beta_{13}$$ $$=$$ $$( 12357 \nu^{13} - 394937 \nu^{11} + 4667831 \nu^{9} - 24878523 \nu^{7} + 56790990 \nu^{5} + \cdots + 3029442 \nu ) / 43393$$ (12357*v^13 - 394937*v^11 + 4667831*v^9 - 24878523*v^7 + 56790990*v^5 - 38529001*v^3 + 3029442*v) / 43393
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{7} + \beta_{6} + 5$$ b7 + b6 + 5 $$\nu^{3}$$ $$=$$ $$\beta_{11} + \beta_{10} - \beta_{9} + 2\beta_{4} + 8\beta_1$$ b11 + b10 - b9 + 2*b4 + 8*b1 $$\nu^{4}$$ $$=$$ $$2\beta_{8} + 10\beta_{7} + 11\beta_{6} - 2\beta_{5} + 3\beta_{3} + \beta_{2} + 44$$ 2*b8 + 10*b7 + 11*b6 - 2*b5 + 3*b3 + b2 + 44 $$\nu^{5}$$ $$=$$ $$3\beta_{13} + 4\beta_{12} + 14\beta_{11} + 12\beta_{10} - 16\beta_{9} + 32\beta_{4} + 74\beta_1$$ 3*b13 + 4*b12 + 14*b11 + 12*b10 - 16*b9 + 32*b4 + 74*b1 $$\nu^{6}$$ $$=$$ $$35\beta_{8} + 102\beta_{7} + 116\beta_{6} - 35\beta_{5} + 47\beta_{3} + 23\beta_{2} + 425$$ 35*b8 + 102*b7 + 116*b6 - 35*b5 + 47*b3 + 23*b2 + 425 $$\nu^{7}$$ $$=$$ $$58\beta_{13} + 79\beta_{12} + 163\beta_{11} + 135\beta_{10} - 205\beta_{9} + 445\beta_{4} + 735\beta_1$$ 58*b13 + 79*b12 + 163*b11 + 135*b10 - 205*b9 + 445*b4 + 735*b1 $$\nu^{8}$$ $$=$$ $$477\beta_{8} + 1075\beta_{7} + 1215\beta_{6} - 454\beta_{5} + 589\beta_{3} + 377\beta_{2} + 4325$$ 477*b8 + 1075*b7 + 1215*b6 - 454*b5 + 589*b3 + 377*b2 + 4325 $$\nu^{9}$$ $$=$$ $$831\beta_{13} + 1168\beta_{12} + 1804\beta_{11} + 1547\beta_{10} - 2455\beta_{9} + 5757\beta_{4} + 7618\beta_1$$ 831*b13 + 1168*b12 + 1804*b11 + 1547*b10 - 2455*b9 + 5757*b4 + 7618*b1 $$\nu^{10}$$ $$=$$ $$6001\beta_{8} + 11620\beta_{7} + 12770\beta_{6} - 5347\beta_{5} + 6894\beta_{3} + 5301\beta_{2} + 45520$$ 6001*b8 + 11620*b7 + 12770*b6 - 5347*b5 + 6894*b3 + 5301*b2 + 45520 $$\nu^{11}$$ $$=$$ $$10648 \beta_{13} + 15499 \beta_{12} + 19664 \beta_{11} + 18018 \beta_{10} - 28712 \beta_{9} + \cdots + 81087 \beta_1$$ 10648*b13 + 15499*b12 + 19664*b11 + 18018*b10 - 28712*b9 + 71291*b4 + 81087*b1 $$\nu^{12}$$ $$=$$ $$72877\beta_{8} + 127817\beta_{7} + 135274\beta_{6} - 60670\beta_{5} + 78688\beta_{3} + 68726\beta_{2} + 489893$$ 72877*b8 + 127817*b7 + 135274*b6 - 60670*b5 + 78688*b3 + 68726*b2 + 489893 $$\nu^{13}$$ $$=$$ $$129396 \beta_{13} + 194816 \beta_{12} + 213962 \beta_{11} + 211255 \beta_{10} - 332595 \beta_{9} + \cdots + 878296 \beta_1$$ 129396*b13 + 194816*b12 + 213962*b11 + 211255*b10 - 332595*b9 + 858901*b4 + 878296*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
 −3.37101 −2.94589 −2.51188 −2.13577 −0.981300 −0.303902 −0.178025 0.178025 0.303902 0.981300 2.13577 2.51188 2.94589 3.37101
−1.00000 −3.37101 1.00000 −2.15229 3.37101 0 −1.00000 8.36369 2.15229
1.2 −1.00000 −2.94589 1.00000 3.60911 2.94589 0 −1.00000 5.67825 −3.60911
1.3 −1.00000 −2.51188 1.00000 −4.25891 2.51188 0 −1.00000 3.30954 4.25891
1.4 −1.00000 −2.13577 1.00000 0.474614 2.13577 0 −1.00000 1.56152 −0.474614
1.5 −1.00000 −0.981300 1.00000 3.28753 0.981300 0 −1.00000 −2.03705 −3.28753
1.6 −1.00000 −0.303902 1.00000 −1.19352 0.303902 0 −1.00000 −2.90764 1.19352
1.7 −1.00000 −0.178025 1.00000 −3.12188 0.178025 0 −1.00000 −2.96831 3.12188
1.8 −1.00000 0.178025 1.00000 3.12188 −0.178025 0 −1.00000 −2.96831 −3.12188
1.9 −1.00000 0.303902 1.00000 1.19352 −0.303902 0 −1.00000 −2.90764 −1.19352
1.10 −1.00000 0.981300 1.00000 −3.28753 −0.981300 0 −1.00000 −2.03705 3.28753
1.11 −1.00000 2.13577 1.00000 −0.474614 −2.13577 0 −1.00000 1.56152 0.474614
1.12 −1.00000 2.51188 1.00000 4.25891 −2.51188 0 −1.00000 3.30954 −4.25891
1.13 −1.00000 2.94589 1.00000 −3.60911 −2.94589 0 −1.00000 5.67825 3.60911
1.14 −1.00000 3.37101 1.00000 2.15229 −3.37101 0 −1.00000 8.36369 −2.15229
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$7$$ $$1$$
$$37$$ $$-1$$

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3626.2.a.bi 14
7.b odd 2 1 inner 3626.2.a.bi 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3626.2.a.bi 14 1.a even 1 1 trivial
3626.2.a.bi 14 7.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3626))$$:

 $$T_{3}^{14} - 32T_{3}^{12} + 379T_{3}^{10} - 2028T_{3}^{8} + 4673T_{3}^{6} - 3288T_{3}^{4} + 352T_{3}^{2} - 8$$ T3^14 - 32*T3^12 + 379*T3^10 - 2028*T3^8 + 4673*T3^6 - 3288*T3^4 + 352*T3^2 - 8 $$T_{5}^{14} - 58T_{5}^{12} + 1315T_{5}^{10} - 14722T_{5}^{8} + 83913T_{5}^{6} - 222610T_{5}^{4} + 210272T_{5}^{2} - 36992$$ T5^14 - 58*T5^12 + 1315*T5^10 - 14722*T5^8 + 83913*T5^6 - 222610*T5^4 + 210272*T5^2 - 36992 $$T_{11}^{7} - 8T_{11}^{6} - 25T_{11}^{5} + 268T_{11}^{4} - 23T_{11}^{3} - 1918T_{11}^{2} + 1508T_{11} + 1680$$ T11^7 - 8*T11^6 - 25*T11^5 + 268*T11^4 - 23*T11^3 - 1918*T11^2 + 1508*T11 + 1680

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{14}$$
$3$ $$T^{14} - 32 T^{12} + \cdots - 8$$
$5$ $$T^{14} - 58 T^{12} + \cdots - 36992$$
$7$ $$T^{14}$$
$11$ $$(T^{7} - 8 T^{6} + \cdots + 1680)^{2}$$
$13$ $$T^{14} - 130 T^{12} + \cdots - 24893568$$
$17$ $$T^{14} - 190 T^{12} + \cdots - 14450688$$
$19$ $$T^{14} + \cdots - 177020928$$
$23$ $$(T^{7} - 93 T^{5} + \cdots - 11424)^{2}$$
$29$ $$(T^{7} + 4 T^{6} + \cdots + 109760)^{2}$$
$31$ $$T^{14} + \cdots - 4302579848$$
$37$ $$(T - 1)^{14}$$
$41$ $$T^{14} - 254 T^{12} + \cdots - 71760200$$
$43$ $$(T^{7} - 10 T^{6} + \cdots - 3584)^{2}$$
$47$ $$T^{14} + \cdots - 3699376128$$
$53$ $$(T^{7} - 12 T^{6} + \cdots + 13824)^{2}$$
$59$ $$T^{14} + \cdots - 1363673088$$
$61$ $$T^{14} + \cdots - 593004060800$$
$67$ $$(T^{7} - 26 T^{6} + \cdots + 17408)^{2}$$
$71$ $$(T^{7} + 8 T^{6} + \cdots + 150528)^{2}$$
$73$ $$T^{14} + \cdots - 14236085154312$$
$79$ $$(T^{7} - 24 T^{6} + \cdots + 648760)^{2}$$
$83$ $$T^{14} + \cdots - 460621952$$
$89$ $$T^{14} + \cdots - 5602746368$$
$97$ $$T^{14} + \cdots - 384950476800$$