# Properties

 Label 1143.2.a.k Level $1143$ Weight $2$ Character orbit 1143.a Self dual yes Analytic conductor $9.127$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1143 = 3^{2} \cdot 127$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1143.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.12690095103$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 26 x^{14} + 269 x^{12} - 1408 x^{10} + 3924 x^{8} - 5655 x^{6} + 3886 x^{4} - 1107 x^{2} + 108$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} -\beta_{14} q^{5} + ( 1 - \beta_{6} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} -\beta_{14} q^{5} + ( 1 - \beta_{6} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} + ( 1 - \beta_{5} ) q^{10} + ( \beta_{13} - \beta_{15} ) q^{11} + ( 1 - \beta_{10} ) q^{13} + ( \beta_{1} + \beta_{7} + \beta_{8} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{14} + ( 1 + \beta_{2} + \beta_{4} + \beta_{6} + \beta_{10} ) q^{16} + ( \beta_{1} - \beta_{3} - \beta_{7} - \beta_{8} + \beta_{12} + \beta_{14} - \beta_{15} ) q^{17} + ( 1 + \beta_{5} - \beta_{6} - \beta_{11} ) q^{19} + ( \beta_{1} - \beta_{8} + \beta_{12} + \beta_{15} ) q^{20} + ( 2 + \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{9} + \beta_{11} ) q^{22} + ( -\beta_{1} + \beta_{8} - \beta_{12} - \beta_{14} + \beta_{15} ) q^{23} + ( 3 - \beta_{2} - \beta_{4} + \beta_{6} + \beta_{9} + \beta_{11} ) q^{25} + ( 2 \beta_{1} - \beta_{7} + \beta_{14} ) q^{26} + ( 1 + 2 \beta_{4} + \beta_{6} + 2 \beta_{9} + 2 \beta_{10} ) q^{28} + ( -\beta_{7} - \beta_{13} ) q^{29} + ( 2 + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{9} ) q^{31} + ( \beta_{1} + \beta_{7} - \beta_{8} ) q^{32} + ( 1 - \beta_{4} - \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{34} + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{7} + 2 \beta_{12} + \beta_{13} + \beta_{14} ) q^{35} + ( 1 - \beta_{2} + \beta_{4} + \beta_{6} - \beta_{9} ) q^{37} + ( \beta_{1} - \beta_{3} + 2 \beta_{8} - \beta_{12} - 2 \beta_{13} + \beta_{14} ) q^{38} + ( -1 + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{40} + ( -\beta_{1} + \beta_{7} - \beta_{8} - \beta_{12} - \beta_{14} ) q^{41} + ( 1 + 2 \beta_{4} - \beta_{5} - 2 \beta_{11} ) q^{43} + ( \beta_{3} - \beta_{7} - \beta_{8} + 3 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{44} + ( -2 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{10} ) q^{46} + ( -2 \beta_{1} + 2 \beta_{3} + \beta_{7} - 2 \beta_{12} - \beta_{14} + \beta_{15} ) q^{47} + ( 2 - 2 \beta_{2} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{49} + ( 2 \beta_{1} - 2 \beta_{3} - \beta_{7} + \beta_{8} + \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{50} + ( 4 + 2 \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{52} + ( \beta_{7} - \beta_{13} ) q^{53} + ( 2 - 2 \beta_{2} - \beta_{5} + \beta_{6} + 2 \beta_{10} + 2 \beta_{11} ) q^{55} + ( 3 \beta_{1} + \beta_{7} + \beta_{8} + \beta_{12} - 3 \beta_{13} + \beta_{14} + \beta_{15} ) q^{56} + ( -1 - \beta_{6} - 2 \beta_{10} - 2 \beta_{11} ) q^{58} + ( \beta_{1} + \beta_{8} + \beta_{12} + \beta_{14} - \beta_{15} ) q^{59} + ( 3 - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{9} - 2 \beta_{10} ) q^{61} + ( \beta_{1} + \beta_{3} - \beta_{12} + 2 \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{62} + ( -\beta_{2} + \beta_{4} - \beta_{5} + \beta_{10} + \beta_{11} ) q^{64} + ( -2 \beta_{1} + 2 \beta_{3} - \beta_{7} - \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{65} + ( 3 - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{67} + ( -2 \beta_{1} + \beta_{3} - 2 \beta_{7} + 2 \beta_{13} ) q^{68} + ( 2 - 2 \beta_{2} - \beta_{5} + \beta_{6} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{70} + ( -2 \beta_{1} + \beta_{3} - 2 \beta_{12} - \beta_{13} + 2 \beta_{15} ) q^{71} + ( 5 - 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{10} ) q^{73} + ( -2 \beta_{1} + \beta_{3} - 3 \beta_{8} + 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{74} + ( -2 - \beta_{2} + 2 \beta_{5} - 2 \beta_{6} + 3 \beta_{9} ) q^{76} + ( -\beta_{1} - 2 \beta_{3} + \beta_{8} + \beta_{12} + 2 \beta_{13} - \beta_{15} ) q^{77} + ( 1 - 3 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{79} + ( -\beta_{1} + 2 \beta_{3} - \beta_{8} - \beta_{12} - \beta_{15} ) q^{80} + ( -1 - \beta_{2} + 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{82} + ( -5 \beta_{1} + 2 \beta_{3} + \beta_{8} - \beta_{12} + 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{83} + ( 2 \beta_{2} - 2 \beta_{6} - 2 \beta_{9} - 2 \beta_{10} - 4 \beta_{11} ) q^{85} + ( 3 \beta_{1} - \beta_{8} + \beta_{12} - 4 \beta_{13} - 2 \beta_{14} + 3 \beta_{15} ) q^{86} + ( 1 + 2 \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} - 3 \beta_{9} - 3 \beta_{10} ) q^{88} + ( 2 \beta_{1} + 2 \beta_{12} + \beta_{14} + 2 \beta_{15} ) q^{89} + ( \beta_{5} - 3 \beta_{6} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{91} + ( -2 \beta_{1} - 4 \beta_{3} - 2 \beta_{7} + 2 \beta_{12} + 2 \beta_{14} - 2 \beta_{15} ) q^{92} + ( -2 + \beta_{2} + 2 \beta_{4} + \beta_{6} - \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{94} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{8} + 2 \beta_{12} - 2 \beta_{13} ) q^{95} + ( 6 + 2 \beta_{2} + 2 \beta_{6} + 2 \beta_{10} ) q^{97} + ( 2 \beta_{1} + 4 \beta_{7} + \beta_{8} - \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + 3 \beta_{15} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 20q^{4} + 10q^{7} + O(q^{10})$$ $$16q + 20q^{4} + 10q^{7} + 14q^{10} + 20q^{13} + 28q^{16} + 12q^{19} + 18q^{22} + 52q^{25} + 42q^{28} + 18q^{31} + 10q^{34} + 16q^{37} + 6q^{40} + 26q^{43} - 24q^{46} + 54q^{49} + 52q^{52} + 20q^{55} - 14q^{58} + 36q^{61} - 4q^{64} + 26q^{67} + 36q^{70} + 60q^{73} - 20q^{76} + 12q^{79} - 20q^{82} - 12q^{85} + 8q^{88} - 24q^{91} - 26q^{94} + 108q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 26 x^{14} + 269 x^{12} - 1408 x^{10} + 3924 x^{8} - 5655 x^{6} + 3886 x^{4} - 1107 x^{2} + 108$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5 \nu$$ $$\beta_{4}$$ $$=$$ $$($$$$5 \nu^{14} - 127 \nu^{12} + 1288 \nu^{10} - 6632 \nu^{8} + 18204 \nu^{6} - 25503 \nu^{4} + 15677 \nu^{2} - 2676$$$$)/48$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{14} - 25 \nu^{12} + 248 \nu^{10} - 1240 \nu^{8} + 3276 \nu^{6} - 4363 \nu^{4} + 2499 \nu^{2} - 388$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$7 \nu^{14} - 185 \nu^{12} + 1928 \nu^{10} - 10024 \nu^{8} + 27060 \nu^{6} - 35781 \nu^{4} + 19675 \nu^{2} - 2892$$$$)/48$$ $$\beta_{7}$$ $$=$$ $$($$$$5 \nu^{15} - 127 \nu^{13} + 1276 \nu^{11} - 6416 \nu^{9} + 16812 \nu^{7} - 21711 \nu^{5} + 11813 \nu^{3} - 1728 \nu$$$$)/24$$ $$\beta_{8}$$ $$=$$ $$($$$$5 \nu^{15} - 127 \nu^{13} + 1276 \nu^{11} - 6416 \nu^{9} + 16812 \nu^{7} - 21735 \nu^{5} + 12005 \nu^{3} - 1992 \nu$$$$)/24$$ $$\beta_{9}$$ $$=$$ $$($$$$\nu^{14} - 25 \nu^{12} + 248 \nu^{10} - 1236 \nu^{8} + 3224 \nu^{6} - 4163 \nu^{4} + 2287 \nu^{2} - 360$$$$)/4$$ $$\beta_{10}$$ $$=$$ $$($$$$-\nu^{14} + 26 \nu^{12} - 268 \nu^{10} + 1388 \nu^{8} - 3772 \nu^{6} + 5111 \nu^{4} - 2974 \nu^{2} + 488$$$$)/4$$ $$\beta_{11}$$ $$=$$ $$($$$$13 \nu^{14} - 335 \nu^{12} + 3416 \nu^{10} - 17464 \nu^{8} + 46764 \nu^{6} - 62487 \nu^{4} + 36205 \nu^{2} - 6036$$$$)/48$$ $$\beta_{12}$$ $$=$$ $$($$$$25 \nu^{15} - 641 \nu^{13} + 6500 \nu^{11} - 32968 \nu^{9} + 87012 \nu^{7} - 112827 \nu^{5} + 61555 \nu^{3} - 9504 \nu$$$$)/72$$ $$\beta_{13}$$ $$=$$ $$($$$$59 \nu^{15} - 1525 \nu^{13} + 15592 \nu^{11} - 79832 \nu^{9} + 213444 \nu^{7} - 282849 \nu^{5} + 160703 \nu^{3} - 26028 \nu$$$$)/144$$ $$\beta_{14}$$ $$=$$ $$($$$$11 \nu^{15} - 283 \nu^{13} + 2884 \nu^{11} - 14744 \nu^{9} + 39444 \nu^{7} - 52377 \nu^{5} + 29657 \nu^{3} - 4680 \nu$$$$)/24$$ $$\beta_{15}$$ $$=$$ $$($$$$47 \nu^{15} - 1213 \nu^{13} + 12400 \nu^{11} - 63584 \nu^{9} + 170604 \nu^{7} - 227373 \nu^{5} + 129911 \nu^{3} - 21060 \nu$$$$)/72$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{10} + \beta_{6} + \beta_{4} + 7 \beta_{2} + 15$$ $$\nu^{5}$$ $$=$$ $$-\beta_{8} + \beta_{7} + 8 \beta_{3} + 29 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$\beta_{11} + 11 \beta_{10} + 10 \beta_{6} - \beta_{5} + 11 \beta_{4} + 45 \beta_{2} + 86$$ $$\nu^{7}$$ $$=$$ $$2 \beta_{15} - 3 \beta_{14} - \beta_{12} - 11 \beta_{8} + 13 \beta_{7} + 57 \beta_{3} + 176 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$13 \beta_{11} + 93 \beta_{10} + \beta_{9} + 80 \beta_{6} - 15 \beta_{5} + 93 \beta_{4} + 288 \beta_{2} + 520$$ $$\nu^{9}$$ $$=$$ $$29 \beta_{15} - 43 \beta_{14} - 2 \beta_{13} - 11 \beta_{12} - 93 \beta_{8} + 119 \beta_{7} + 393 \beta_{3} + 1098 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$117 \beta_{11} + 712 \beta_{10} + 17 \beta_{9} + 594 \beta_{6} - 154 \beta_{5} + 717 \beta_{4} + 1855 \beta_{2} + 3236$$ $$\nu^{11}$$ $$=$$ $$294 \beta_{15} - 426 \beta_{14} - 40 \beta_{13} - 86 \beta_{12} - 714 \beta_{8} + 952 \beta_{7} + 2672 \beta_{3} + 6985 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$912 \beta_{11} + 5188 \beta_{10} + 192 \beta_{9} + 4252 \beta_{6} - 1348 \beta_{5} + 5284 \beta_{4} + 12035 \beta_{2} + 20521$$ $$\nu^{13}$$ $$=$$ $$2572 \beta_{15} - 3632 \beta_{14} - 504 \beta_{13} - 596 \beta_{12} - 5216 \beta_{8} + 7132 \beta_{7} + 18039 \beta_{3} + 45071 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$6628 \beta_{11} + 36771 \beta_{10} + 1824 \beta_{9} + 29791 \beta_{6} - 10824 \beta_{5} + 37931 \beta_{4} + 78577 \beta_{2} + 131897$$ $$\nu^{15}$$ $$=$$ $$20788 \beta_{15} - 28628 \beta_{14} - 5160 \beta_{13} - 3944 \beta_{12} - 36967 \beta_{8} + 51539 \beta_{7} + 121312 \beta_{3} + 293859 \beta_{1}$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.59858 −2.57927 −2.28010 −2.01982 −1.36017 −0.979723 −0.518678 −0.487100 0.487100 0.518678 0.979723 1.36017 2.01982 2.28010 2.57927 2.59858
−2.59858 0 4.75262 −0.560562 0 4.70134 −7.15292 0 1.45667
1.2 −2.57927 0 4.65263 −1.53253 0 −2.06948 −6.84184 0 3.95280
1.3 −2.28010 0 3.19885 3.42195 0 0.843402 −2.73351 0 −7.80238
1.4 −2.01982 0 2.07968 −3.33517 0 −0.226097 −0.160938 0 6.73645
1.5 −1.36017 0 −0.149936 −3.81085 0 4.20131 2.92428 0 5.18341
1.6 −0.979723 0 −1.04014 1.37010 0 3.49660 2.97850 0 −1.34231
1.7 −0.518678 0 −1.73097 4.37543 0 −5.00960 1.93517 0 −2.26944
1.8 −0.487100 0 −1.76273 −2.22708 0 −0.937481 1.83283 0 1.08481
1.9 0.487100 0 −1.76273 2.22708 0 −0.937481 −1.83283 0 1.08481
1.10 0.518678 0 −1.73097 −4.37543 0 −5.00960 −1.93517 0 −2.26944
1.11 0.979723 0 −1.04014 −1.37010 0 3.49660 −2.97850 0 −1.34231
1.12 1.36017 0 −0.149936 3.81085 0 4.20131 −2.92428 0 5.18341
1.13 2.01982 0 2.07968 3.33517 0 −0.226097 0.160938 0 6.73645
1.14 2.28010 0 3.19885 −3.42195 0 0.843402 2.73351 0 −7.80238
1.15 2.57927 0 4.65263 1.53253 0 −2.06948 6.84184 0 3.95280
1.16 2.59858 0 4.75262 0.560562 0 4.70134 7.15292 0 1.45667
 $$n$$: 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
$$3$$ $$1$$
$$127$$ $$-1$$

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1143.2.a.k 16
3.b odd 2 1 inner 1143.2.a.k 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1143.2.a.k 16 1.a even 1 1 trivial
1143.2.a.k 16 3.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1143))$$:

 $$T_{2}^{16} - \cdots$$ $$T_{5}^{16} - \cdots$$ $$T_{7}^{8} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$108 - 1107 T^{2} + 3886 T^{4} - 5655 T^{6} + 3924 T^{8} - 1408 T^{10} + 269 T^{12} - 26 T^{14} + T^{16}$$
$3$ $$T^{16}$$
$5$ $$248832 - 1154304 T^{2} + 1351552 T^{4} - 682800 T^{6} + 173128 T^{8} - 23541 T^{10} + 1742 T^{12} - 66 T^{14} + T^{16}$$
$7$ $$( 128 + 544 T - 288 T^{2} - 824 T^{3} + 112 T^{4} + 158 T^{5} - 29 T^{6} - 5 T^{7} + T^{8} )^{2}$$
$11$ $$101059248 - 122364816 T^{2} + 55175557 T^{4} - 12414755 T^{6} + 1552162 T^{8} - 112855 T^{10} + 4746 T^{12} - 107 T^{14} + T^{16}$$
$13$ $$( -6902 + 457 T + 5944 T^{2} - 1781 T^{3} - 719 T^{4} + 319 T^{5} - 5 T^{6} - 10 T^{7} + T^{8} )^{2}$$
$17$ $$212487168 - 433528128 T^{2} + 290537248 T^{4} - 77114408 T^{6} + 8465409 T^{8} - 457766 T^{10} + 12927 T^{12} - 182 T^{14} + T^{16}$$
$19$ $$( -2624 + 15520 T - 3404 T^{2} - 6634 T^{3} + 1337 T^{4} + 538 T^{5} - 91 T^{6} - 6 T^{7} + T^{8} )^{2}$$
$23$ $$28390195200 - 34180890624 T^{2} + 7411290112 T^{4} - 721507584 T^{6} + 38723520 T^{8} - 1226272 T^{10} + 22880 T^{12} - 233 T^{14} + T^{16}$$
$29$ $$1108992 - 94894848 T^{2} + 119604928 T^{4} - 49477968 T^{6} + 7777844 T^{8} - 530381 T^{10} + 16230 T^{12} - 214 T^{14} + T^{16}$$
$31$ $$( -74656 + 26020 T + 30386 T^{2} - 15897 T^{3} + 435 T^{4} + 739 T^{5} - 69 T^{6} - 9 T^{7} + T^{8} )^{2}$$
$37$ $$( 4799818 + 725129 T - 410874 T^{2} - 51715 T^{3} + 12887 T^{4} + 1145 T^{5} - 183 T^{6} - 8 T^{7} + T^{8} )^{2}$$
$41$ $$256633651200 - 177261086976 T^{2} + 42610370704 T^{4} - 4370811012 T^{6} + 208919265 T^{8} - 5045746 T^{10} + 63727 T^{12} - 402 T^{14} + T^{16}$$
$43$ $$( 538624 + 679712 T - 87712 T^{2} - 157112 T^{3} + 12224 T^{4} + 2810 T^{5} - 215 T^{6} - 13 T^{7} + T^{8} )^{2}$$
$47$ $$4140380534832 - 1197296430480 T^{2} + 140001608797 T^{4} - 8600164887 T^{6} + 302113442 T^{8} - 6195907 T^{10} + 72090 T^{12} - 431 T^{14} + T^{16}$$
$53$ $$40368000000 - 27543360000 T^{2} + 7093403200 T^{4} - 889950688 T^{6} + 59759420 T^{8} - 2183525 T^{10} + 41294 T^{12} - 350 T^{14} + T^{16}$$
$59$ $$50331648 - 3230466048 T^{2} + 5123657728 T^{4} - 1226862848 T^{6} + 103417088 T^{8} - 3712864 T^{10} + 60188 T^{12} - 421 T^{14} + T^{16}$$
$61$ $$( 532930 - 1349477 T + 632752 T^{2} - 52907 T^{3} - 21519 T^{4} + 4305 T^{5} - 133 T^{6} - 18 T^{7} + T^{8} )^{2}$$
$67$ $$( 304768 + 72736 T - 147360 T^{2} - 45208 T^{3} + 8000 T^{4} + 1622 T^{5} - 147 T^{6} - 13 T^{7} + T^{8} )^{2}$$
$71$ $$220283977728 - 189843224940 T^{2} + 46274433385 T^{4} - 4980719683 T^{6} + 265718230 T^{8} - 7109911 T^{10} + 90238 T^{12} - 511 T^{14} + T^{16}$$
$73$ $$( 55534 + 67655 T - 76616 T^{2} - 78319 T^{3} - 8279 T^{4} + 2457 T^{5} + 123 T^{6} - 30 T^{7} + T^{8} )^{2}$$
$79$ $$( -798208 + 1535392 T - 173684 T^{2} - 120446 T^{3} + 17961 T^{4} + 1602 T^{5} - 263 T^{6} - 6 T^{7} + T^{8} )^{2}$$
$83$ $$6491311177728 - 6856221904896 T^{2} + 2142042686464 T^{4} - 200033593088 T^{6} + 5239407232 T^{8} - 59503728 T^{10} + 335204 T^{12} - 925 T^{14} + T^{16}$$
$89$ $$2174768483328 - 3318295530240 T^{2} + 1342420507840 T^{4} - 99030253136 T^{6} + 2986727828 T^{8} - 42645005 T^{10} + 288158 T^{12} - 886 T^{14} + T^{16}$$
$97$ $$( -12139520 + 13035264 T - 5290944 T^{2} + 966624 T^{3} - 59104 T^{4} - 5248 T^{5} + 1000 T^{6} - 54 T^{7} + T^{8} )^{2}$$