# Properties

 Label 1849.4.a.e Level 1849 Weight 4 Character orbit 1849.a Self dual yes Analytic conductor 109.095 Analytic rank 0 Dimension 10 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1849 = 43^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1849.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$109.094531601$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 62 x^{8} + 1289 x^{6} - 11252 x^{4} + 39376 x^{2} - 35688$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{5}$$ 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{4} q^{3} + ( 5 + \beta_{7} + \beta_{9} ) q^{4} + ( \beta_{1} + \beta_{5} ) q^{5} + ( 3 - \beta_{2} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{6} + ( \beta_{1} + \beta_{3} ) q^{7} + ( 4 \beta_{1} + \beta_{4} + \beta_{6} ) q^{8} + ( 6 - \beta_{2} - 3 \beta_{7} + \beta_{8} - \beta_{9} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{4} q^{3} + ( 5 + \beta_{7} + \beta_{9} ) q^{4} + ( \beta_{1} + \beta_{5} ) q^{5} + ( 3 - \beta_{2} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{6} + ( \beta_{1} + \beta_{3} ) q^{7} + ( 4 \beta_{1} + \beta_{4} + \beta_{6} ) q^{8} + ( 6 - \beta_{2} - 3 \beta_{7} + \beta_{8} - \beta_{9} ) q^{9} + ( 11 - \beta_{2} - 3 \beta_{7} + 3 \beta_{9} ) q^{10} + ( -1 - 4 \beta_{2} - \beta_{7} - 4 \beta_{8} + 2 \beta_{9} ) q^{11} + ( 6 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{12} + ( 16 - 4 \beta_{2} - \beta_{7} - 3 \beta_{8} - 3 \beta_{9} ) q^{13} + ( 11 - 12 \beta_{2} + 2 \beta_{7} + 3 \beta_{9} ) q^{14} + ( 7 + 4 \beta_{2} - 5 \beta_{7} + 11 \beta_{8} + 3 \beta_{9} ) q^{15} + ( 20 - 6 \beta_{2} + 5 \beta_{7} + 8 \beta_{8} + 14 \beta_{9} ) q^{16} + ( 33 + 5 \beta_{2} - 11 \beta_{7} - 2 \beta_{8} ) q^{17} + ( -6 \beta_{1} + \beta_{3} + 2 \beta_{5} - \beta_{6} ) q^{18} + ( -8 \beta_{1} + \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} ) q^{19} + ( 17 \beta_{1} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} ) q^{20} + ( 9 + 10 \beta_{2} + 16 \beta_{7} + 7 \beta_{8} + 3 \beta_{9} ) q^{21} + ( 14 \beta_{1} + 4 \beta_{3} - 5 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} ) q^{22} + ( 46 - 9 \beta_{2} + 14 \beta_{7} + 7 \beta_{8} + 7 \beta_{9} ) q^{23} + ( 56 - 7 \beta_{2} - 12 \beta_{8} + 14 \beta_{9} ) q^{24} + ( 44 - 9 \beta_{2} - 2 \beta_{7} + 15 \beta_{8} - 3 \beta_{9} ) q^{25} + ( -\beta_{1} + 4 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} ) q^{26} + ( -5 \beta_{1} - 4 \beta_{3} - 5 \beta_{4} + 7 \beta_{5} - 2 \beta_{6} ) q^{27} + ( 11 \beta_{1} + 4 \beta_{3} + 14 \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{28} + ( -14 \beta_{1} + 6 \beta_{3} - 7 \beta_{4} - 8 \beta_{5} - 2 \beta_{6} ) q^{29} + ( 2 \beta_{1} - 4 \beta_{3} + 13 \beta_{4} + 8 \beta_{5} + 3 \beta_{6} ) q^{30} + ( 8 - 7 \beta_{2} - 25 \beta_{7} - 5 \beta_{8} - \beta_{9} ) q^{31} + ( 55 \beta_{1} + 6 \beta_{3} + 19 \beta_{4} + 9 \beta_{5} + 6 \beta_{6} ) q^{32} + ( -9 \beta_{1} - 5 \beta_{3} - 4 \beta_{4} - 12 \beta_{5} + 6 \beta_{6} ) q^{33} + ( 31 \beta_{1} - 5 \beta_{3} - 20 \beta_{4} + 11 \beta_{5} ) q^{34} + ( 33 - 4 \beta_{2} - 20 \beta_{7} - 21 \beta_{8} + 7 \beta_{9} ) q^{35} + ( -137 - \beta_{2} + 4 \beta_{7} - 14 \beta_{8} - 9 \beta_{9} ) q^{36} + ( 3 \beta_{1} - 3 \beta_{3} + 21 \beta_{4} - 6 \beta_{5} - 4 \beta_{6} ) q^{37} + ( -128 - 3 \beta_{2} - 37 \beta_{7} - 16 \beta_{8} - 36 \beta_{9} ) q^{38} + ( -19 \beta_{1} - 5 \beta_{3} + 30 \beta_{4} - 14 \beta_{5} ) q^{39} + ( 144 - 15 \beta_{2} + 67 \beta_{7} + 14 \beta_{8} + 40 \beta_{9} ) q^{40} + ( 82 - 19 \beta_{2} + 22 \beta_{7} + 27 \beta_{8} + 5 \beta_{9} ) q^{41} + ( 39 \beta_{1} - 10 \beta_{3} + 20 \beta_{4} - 13 \beta_{5} + 3 \beta_{6} ) q^{42} + ( 171 - 24 \beta_{2} + 18 \beta_{7} + 34 \beta_{8} + 41 \beta_{9} ) q^{44} + ( 14 \beta_{1} - \beta_{3} + 27 \beta_{4} + 23 \beta_{5} - 8 \beta_{6} ) q^{45} + ( 79 \beta_{1} + 9 \beta_{3} + 37 \beta_{4} - 7 \beta_{5} + 7 \beta_{6} ) q^{46} + ( -129 - 11 \beta_{2} + 2 \beta_{7} + 25 \beta_{8} - 17 \beta_{9} ) q^{47} + ( 109 \beta_{1} - \beta_{3} - 9 \beta_{4} + 22 \beta_{5} + 6 \beta_{6} ) q^{48} + ( 140 + 40 \beta_{2} - 22 \beta_{7} - 51 \beta_{8} - 15 \beta_{9} ) q^{49} + ( -15 \beta_{1} + 9 \beta_{3} + 37 \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{50} + ( -53 \beta_{1} - 6 \beta_{3} + 52 \beta_{4} + 21 \beta_{5} + 2 \beta_{6} ) q^{51} + ( -169 + 4 \beta_{2} - 8 \beta_{7} - 27 \beta_{9} ) q^{52} + ( 129 + 12 \beta_{2} + 9 \beta_{7} - 44 \beta_{8} - 16 \beta_{9} ) q^{53} + ( -96 + 56 \beta_{2} - 61 \beta_{7} - 22 \beta_{8} - 38 \beta_{9} ) q^{54} + ( 77 \beta_{1} + 13 \beta_{3} - 56 \beta_{4} - 20 \beta_{5} + 4 \beta_{6} ) q^{55} + ( 102 + 18 \beta_{2} + 44 \beta_{7} + 46 \beta_{8} + 62 \beta_{9} ) q^{56} + ( -94 + 40 \beta_{2} - 7 \beta_{7} + 56 \beta_{8} - 20 \beta_{9} ) q^{57} + ( -209 - 47 \beta_{2} - 4 \beta_{7} - 26 \beta_{8} - 59 \beta_{9} ) q^{58} + ( 71 + 6 \beta_{2} + 74 \beta_{7} - 11 \beta_{8} + \beta_{9} ) q^{59} + ( 16 - 20 \beta_{2} + 53 \beta_{7} - 44 \beta_{8} + 50 \beta_{9} ) q^{60} + ( -83 \beta_{1} + 9 \beta_{3} + 12 \beta_{4} + 16 \beta_{5} + 2 \beta_{6} ) q^{61} + ( -20 \beta_{1} + 7 \beta_{3} - 28 \beta_{4} + 24 \beta_{5} - \beta_{6} ) q^{62} + ( 45 \beta_{1} - \beta_{3} - 54 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} ) q^{63} + ( 612 - 82 \beta_{2} + 65 \beta_{7} + 10 \beta_{8} + 94 \beta_{9} ) q^{64} + ( -19 \beta_{1} + \beta_{3} - 72 \beta_{4} + 32 \beta_{5} - 2 \beta_{6} ) q^{65} + ( -65 + 46 \beta_{2} + 68 \beta_{7} + 28 \beta_{8} + 55 \beta_{9} ) q^{66} + ( 321 + 58 \beta_{2} + 51 \beta_{7} + 44 \beta_{8} + 6 \beta_{9} ) q^{67} + ( 67 + 29 \beta_{2} + 30 \beta_{7} - 24 \beta_{8} + 23 \beta_{9} ) q^{68} + ( 116 \beta_{1} + 5 \beta_{3} + 15 \beta_{4} - 9 \beta_{5} ) q^{69} + ( 93 \beta_{1} + 4 \beta_{3} - 58 \beta_{4} + 27 \beta_{5} + 7 \beta_{6} ) q^{70} + ( -76 \beta_{1} + 2 \beta_{3} + 106 \beta_{4} + 6 \beta_{5} - 4 \beta_{6} ) q^{71} + ( -112 \beta_{1} - 7 \beta_{3} - 23 \beta_{4} - 29 \beta_{5} - \beta_{6} ) q^{72} + ( -69 \beta_{1} + 9 \beta_{3} + 46 \beta_{4} + 10 \beta_{5} + 2 \beta_{6} ) q^{73} + ( 100 + 41 \beta_{2} + 38 \beta_{7} + 18 \beta_{8} - 62 \beta_{9} ) q^{74} + ( 78 \beta_{1} - 11 \beta_{3} + 107 \beta_{4} + 37 \beta_{5} - 18 \beta_{6} ) q^{75} + ( -288 \beta_{1} - 5 \beta_{3} - 50 \beta_{4} - 23 \beta_{5} - 20 \beta_{6} ) q^{76} + ( 195 \beta_{1} - 3 \beta_{3} - 94 \beta_{4} + 34 \beta_{5} - 6 \beta_{6} ) q^{77} + ( -119 + 44 \beta_{2} + 92 \beta_{7} + 60 \beta_{8} - 27 \beta_{9} ) q^{78} + ( -143 + 45 \beta_{2} - 67 \beta_{7} + 29 \beta_{8} - 25 \beta_{9} ) q^{79} + ( 272 \beta_{1} + 7 \beta_{3} + 126 \beta_{4} - 11 \beta_{5} + 16 \beta_{6} ) q^{80} + ( -360 + 32 \beta_{2} - 9 \beta_{7} + 43 \beta_{8} + 11 \beta_{9} ) q^{81} + ( 61 \beta_{1} + 19 \beta_{3} + 95 \beta_{4} - 17 \beta_{5} + 5 \beta_{6} ) q^{82} + ( 608 + 14 \beta_{2} - 27 \beta_{7} + 47 \beta_{8} + 33 \beta_{9} ) q^{83} + ( 556 + 18 \beta_{2} + 14 \beta_{7} + 2 \beta_{8} + 40 \beta_{9} ) q^{84} + ( 134 \beta_{1} + 15 \beta_{3} + 85 \beta_{4} + 19 \beta_{5} - 20 \beta_{6} ) q^{85} + ( -291 + 51 \beta_{2} + 143 \beta_{7} - 35 \beta_{8} - 33 \beta_{9} ) q^{87} + ( 231 \beta_{1} - 8 \beta_{3} + 150 \beta_{4} - \beta_{5} + 25 \beta_{6} ) q^{88} + ( 111 \beta_{1} - 19 \beta_{3} + 68 \beta_{4} - 18 \beta_{5} - 22 \beta_{6} ) q^{89} + ( 179 + 2 \beta_{2} - 81 \beta_{7} + 6 \beta_{8} - 51 \beta_{9} ) q^{90} + ( 159 \beta_{1} + 15 \beta_{3} - 90 \beta_{4} + 4 \beta_{5} - 20 \beta_{6} ) q^{91} + ( 801 - 101 \beta_{2} + 127 \beta_{7} + 60 \beta_{8} + 183 \beta_{9} ) q^{92} + ( -89 \beta_{1} - 32 \beta_{3} + 90 \beta_{4} + 27 \beta_{5} + 4 \beta_{6} ) q^{93} + ( -290 \beta_{1} + 11 \beta_{3} + 63 \beta_{4} - 19 \beta_{5} - 17 \beta_{6} ) q^{94} + ( 208 + 3 \beta_{2} - 148 \beta_{7} - 4 \beta_{8} - 94 \beta_{9} ) q^{95} + ( 930 + 25 \beta_{2} + 44 \beta_{7} + 114 \beta_{8} + 132 \beta_{9} ) q^{96} + ( -13 - 125 \beta_{2} + 96 \beta_{7} + 32 \beta_{8} - 10 \beta_{9} ) q^{97} + ( 170 \beta_{1} - 40 \beta_{3} - 164 \beta_{4} + 7 \beta_{5} - 15 \beta_{6} ) q^{98} + ( -144 - 6 \beta_{2} + 5 \beta_{7} - 173 \beta_{8} - 27 \beta_{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q + 44q^{4} + 30q^{6} + 80q^{9} + O(q^{10})$$ $$10q + 44q^{4} + 30q^{6} + 80q^{9} + 118q^{10} - 18q^{11} + 166q^{13} + 120q^{14} + 120q^{15} + 196q^{16} + 356q^{17} + 28q^{21} + 436q^{23} + 498q^{24} + 532q^{25} + 176q^{31} + 320q^{35} - 1422q^{36} - 1118q^{38} + 1178q^{40} + 868q^{41} + 1740q^{44} - 1142q^{47} + 1234q^{49} - 1612q^{52} + 1086q^{53} - 840q^{54} + 868q^{56} - 728q^{57} - 1966q^{58} + 356q^{59} - 288q^{60} + 5876q^{64} - 1012q^{66} + 3054q^{67} + 350q^{68} + 962q^{74} - 1352q^{78} - 1086q^{79} - 3478q^{81} + 6282q^{83} + 5396q^{84} - 3658q^{87} + 2236q^{90} + 7578q^{92} + 2838q^{95} + 9266q^{96} - 116q^{97} - 2086q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 62 x^{8} + 1289 x^{6} - 11252 x^{4} + 39376 x^{2} - 35688$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-71 \nu^{8} + 2915 \nu^{6} - 22388 \nu^{4} - 19240 \nu^{2} + 6504$$$$)/63744$$ $$\beta_{3}$$ $$=$$ $$($$$$-35 \nu^{9} + 2447 \nu^{7} - 54692 \nu^{5} + 407096 \nu^{3} - 633336 \nu$$$$)/31872$$ $$\beta_{4}$$ $$=$$ $$($$$$47 \nu^{9} - 2603 \nu^{7} + 43924 \nu^{5} - 264984 \nu^{3} + 441304 \nu$$$$)/21248$$ $$\beta_{5}$$ $$=$$ $$($$$$-157 \nu^{9} + 8017 \nu^{7} - 110332 \nu^{5} + 336328 \nu^{3} + 752376 \nu$$$$)/63744$$ $$\beta_{6}$$ $$=$$ $$($$$$-47 \nu^{9} + 2603 \nu^{7} - 43924 \nu^{5} + 286232 \nu^{3} - 866264 \nu$$$$)/21248$$ $$\beta_{7}$$ $$=$$ $$($$$$149 \nu^{8} - 7913 \nu^{6} + 121052 \nu^{4} - 565640 \nu^{2} + 317640$$$$)/31872$$ $$\beta_{8}$$ $$=$$ $$($$$$47 \nu^{8} - 2603 \nu^{6} + 43924 \nu^{4} - 264984 \nu^{2} + 420056$$$$)/10624$$ $$\beta_{9}$$ $$=$$ $$($$$$-149 \nu^{8} + 7913 \nu^{6} - 121052 \nu^{4} + 597512 \nu^{2} - 731976$$$$)/31872$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{9} + \beta_{7} + 13$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{4} + 20 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$38 \beta_{9} + 8 \beta_{8} + 29 \beta_{7} - 6 \beta_{2} + 268$$ $$\nu^{5}$$ $$=$$ $$38 \beta_{6} + 9 \beta_{5} + 51 \beta_{4} + 6 \beta_{3} + 503 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$1230 \beta_{9} + 330 \beta_{8} + 841 \beta_{7} - 322 \beta_{2} + 6852$$ $$\nu^{7}$$ $$=$$ $$1230 \beta_{6} + 389 \beta_{5} + 1823 \beta_{4} + 322 \beta_{3} + 14091 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$38246 \beta_{9} + 11026 \beta_{8} + 25113 \beta_{7} - 12226 \beta_{2} + 193380$$ $$\nu^{9}$$ $$=$$ $$38246 \beta_{6} + 13133 \beta_{5} + 59391 \beta_{4} + 12226 \beta_{3} + 413691 \beta_{1}$$

## 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
 −5.51834 −3.79990 −3.05512 −2.53388 −1.16377 1.16377 2.53388 3.05512 3.79990 5.51834
−5.51834 −3.84074 22.4521 −10.9281 21.1945 −14.0538 −79.7518 −12.2488 60.3051
1.2 −3.79990 3.62572 6.43924 1.22749 −13.7774 33.4949 5.93074 −13.8541 −4.66434
1.3 −3.05512 −4.99167 1.33374 15.7813 15.2501 −15.1156 20.3662 −2.08325 −48.2136
1.4 −2.53388 6.96727 −1.57947 −11.2250 −17.6542 −27.5005 24.2732 21.5429 28.4428
1.5 −1.16377 −8.58157 −6.64564 −19.8751 9.98696 5.27515 17.0441 46.6433 23.1301
1.6 1.16377 8.58157 −6.64564 19.8751 9.98696 −5.27515 −17.0441 46.6433 23.1301
1.7 2.53388 −6.96727 −1.57947 11.2250 −17.6542 27.5005 −24.2732 21.5429 28.4428
1.8 3.05512 4.99167 1.33374 −15.7813 15.2501 15.1156 −20.3662 −2.08325 −48.2136
1.9 3.79990 −3.62572 6.43924 −1.22749 −13.7774 −33.4949 −5.93074 −13.8541 −4.66434
1.10 5.51834 3.84074 22.4521 10.9281 21.1945 14.0538 79.7518 −12.2488 60.3051
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$43$$ $$1$$

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1849.4.a.e 10
43.b odd 2 1 inner 1849.4.a.e 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1849.4.a.e 10 1.a even 1 1 trivial
1849.4.a.e 10 43.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} - 62 T_{2}^{8} + 1289 T_{2}^{6} - 11252 T_{2}^{4} + 39376 T_{2}^{2} - 35688$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1849))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 18 T^{2} + 201 T^{4} + 956 T^{6} - 752 T^{8} - 46184 T^{10} - 48128 T^{12} + 3915776 T^{14} + 52690944 T^{16} + 301989888 T^{18} + 1073741824 T^{20}$$
$3$ $$1 + 95 T^{2} + 5922 T^{4} + 256154 T^{6} + 9211033 T^{8} + 267499734 T^{10} + 6714843057 T^{12} + 136130737914 T^{14} + 2294304135858 T^{16} + 26830805965695 T^{18} + 205891132094649 T^{20}$$
$5$ $$1 + 359 T^{2} + 84962 T^{4} + 14944950 T^{6} + 2171265625 T^{8} + 291181218750 T^{10} + 33926025390625 T^{12} + 3648669433593750 T^{14} + 324104309082031250 T^{16} + 21398067474365234375 T^{18} +$$$$93\!\cdots\!25$$$$T^{20}$$
$7$ $$1 + 1098 T^{2} + 653001 T^{4} + 284656576 T^{6} + 112730018758 T^{8} + 41089741820396 T^{10} + 13262573976859942 T^{12} + 3940013422069283776 T^{14} +$$$$10\!\cdots\!49$$$$T^{16} +$$$$21\!\cdots\!98$$$$T^{18} +$$$$22\!\cdots\!49$$$$T^{20}$$
$11$ $$( 1 + 9 T + 2384 T^{2} - 13893 T^{3} + 4770573 T^{4} + 8230948 T^{5} + 6349632663 T^{6} - 24612296973 T^{7} + 5621347295344 T^{8} + 28245855390489 T^{9} + 4177248169415651 T^{10} )^{2}$$
$13$ $$( 1 - 83 T + 9454 T^{2} - 596069 T^{3} + 40607719 T^{4} - 1819510392 T^{5} + 89215158643 T^{6} - 2877111213821 T^{7} + 100254937072342 T^{8} - 1933741065165923 T^{9} + 51185893014090757 T^{10} )^{2}$$
$17$ $$( 1 - 178 T + 25996 T^{2} - 2205671 T^{3} + 187666299 T^{4} - 12083652557 T^{5} + 922004526987 T^{6} - 53239535953799 T^{7} + 3082810437416012 T^{8} - 103706758226897458 T^{9} + 2862423051509815793 T^{10} )^{2}$$
$19$ $$1 + 31843 T^{2} + 535896854 T^{4} + 6338154832806 T^{6} + 58161246612493177 T^{8} +$$$$43\!\cdots\!98$$$$T^{10} +$$$$27\!\cdots\!37$$$$T^{12} +$$$$14\!\cdots\!66$$$$T^{14} +$$$$55\!\cdots\!14$$$$T^{16} +$$$$15\!\cdots\!03$$$$T^{18} +$$$$23\!\cdots\!01$$$$T^{20}$$
$23$ $$( 1 - 218 T + 56774 T^{2} - 8869709 T^{3} + 1374111229 T^{4} - 151692574827 T^{5} + 16718811323243 T^{6} - 1313035256986301 T^{7} + 102258641201900362 T^{8} - 4777388126180429978 T^{9} +$$$$26\!\cdots\!07$$$$T^{10} )^{2}$$
$29$ $$1 + 58895 T^{2} + 4143404282 T^{4} + 149358075060438 T^{6} + 5650373294419812385 T^{8} +$$$$13\!\cdots\!58$$$$T^{10} +$$$$33\!\cdots\!85$$$$T^{12} +$$$$52\!\cdots\!58$$$$T^{14} +$$$$87\!\cdots\!02$$$$T^{16} +$$$$73\!\cdots\!95$$$$T^{18} +$$$$74\!\cdots\!01$$$$T^{20}$$
$31$ $$( 1 - 88 T + 98878 T^{2} - 8424585 T^{3} + 4938863141 T^{4} - 338339755141 T^{5} + 147133671833531 T^{6} - 7476850198397385 T^{7} + 2614296960002827138 T^{8} - 69314324973392378968 T^{9} +$$$$23\!\cdots\!51$$$$T^{10} )^{2}$$
$37$ $$1 + 268175 T^{2} + 38484069102 T^{4} + 3697995118261426 T^{6} +$$$$26\!\cdots\!13$$$$T^{8} +$$$$15\!\cdots\!06$$$$T^{10} +$$$$68\!\cdots\!17$$$$T^{12} +$$$$24\!\cdots\!06$$$$T^{14} +$$$$64\!\cdots\!58$$$$T^{16} +$$$$11\!\cdots\!75$$$$T^{18} +$$$$11\!\cdots\!49$$$$T^{20}$$
$41$ $$( 1 - 434 T + 301696 T^{2} - 89839919 T^{3} + 38480688127 T^{4} - 8694561007673 T^{5} + 2652127506400967 T^{6} - 426748980252996479 T^{7} + 98769820078920457856 T^{8} -$$$$97\!\cdots\!54$$$$T^{9} +$$$$15\!\cdots\!01$$$$T^{10} )^{2}$$
$43$ 1
$47$ $$( 1 + 571 T + 457832 T^{2} + 175639812 T^{3} + 88718533663 T^{4} + 25744123054834 T^{5} + 9211024320493649 T^{6} + 1893259353893078148 T^{7} +$$$$51\!\cdots\!44$$$$T^{8} +$$$$66\!\cdots\!11$$$$T^{9} +$$$$12\!\cdots\!43$$$$T^{10} )^{2}$$
$53$ $$( 1 - 543 T + 571878 T^{2} - 201247789 T^{3} + 138842080523 T^{4} - 38480412395312 T^{5} + 20670392422022671 T^{6} - 4460528671808793781 T^{7} +$$$$18\!\cdots\!74$$$$T^{8} -$$$$26\!\cdots\!63$$$$T^{9} +$$$$73\!\cdots\!57$$$$T^{10} )^{2}$$
$59$ $$( 1 - 178 T + 522743 T^{2} - 65549680 T^{3} + 143047005586 T^{4} - 16551907015452 T^{5} + 29378850960247094 T^{6} - 2764920482396784880 T^{7} +$$$$45\!\cdots\!77$$$$T^{8} -$$$$31\!\cdots\!18$$$$T^{9} +$$$$36\!\cdots\!99$$$$T^{10} )^{2}$$
$61$ $$1 + 1345434 T^{2} + 828365815089 T^{4} + 318175733454512224 T^{6} +$$$$90\!\cdots\!94$$$$T^{8} +$$$$21\!\cdots\!72$$$$T^{10} +$$$$46\!\cdots\!34$$$$T^{12} +$$$$84\!\cdots\!04$$$$T^{14} +$$$$11\!\cdots\!09$$$$T^{16} +$$$$94\!\cdots\!94$$$$T^{18} +$$$$36\!\cdots\!01$$$$T^{20}$$
$67$ $$( 1 - 1527 T + 1792198 T^{2} - 1327080289 T^{3} + 882751774077 T^{4} - 480430713542888 T^{5} + 265499071826720751 T^{6} -$$$$12\!\cdots\!41$$$$T^{7} +$$$$48\!\cdots\!06$$$$T^{8} -$$$$12\!\cdots\!47$$$$T^{9} +$$$$24\!\cdots\!43$$$$T^{10} )^{2}$$
$71$ $$1 + 1315422 T^{2} + 988176145549 T^{4} + 536051614941630344 T^{6} +$$$$23\!\cdots\!62$$$$T^{8} +$$$$87\!\cdots\!52$$$$T^{10} +$$$$29\!\cdots\!02$$$$T^{12} +$$$$87\!\cdots\!04$$$$T^{14} +$$$$20\!\cdots\!89$$$$T^{16} +$$$$35\!\cdots\!82$$$$T^{18} +$$$$34\!\cdots\!01$$$$T^{20}$$
$73$ $$1 + 2944426 T^{2} + 4040518324729 T^{4} + 3453590446571069184 T^{6} +$$$$20\!\cdots\!18$$$$T^{8} +$$$$92\!\cdots\!44$$$$T^{10} +$$$$31\!\cdots\!02$$$$T^{12} +$$$$79\!\cdots\!64$$$$T^{14} +$$$$14\!\cdots\!01$$$$T^{16} +$$$$15\!\cdots\!66$$$$T^{18} +$$$$79\!\cdots\!49$$$$T^{20}$$
$79$ $$( 1 + 543 T + 1834146 T^{2} + 680102676 T^{3} + 1501938388313 T^{4} + 421382183769066 T^{5} + 740514201035453207 T^{6} +$$$$16\!\cdots\!96$$$$T^{7} +$$$$21\!\cdots\!74$$$$T^{8} +$$$$32\!\cdots\!63$$$$T^{9} +$$$$29\!\cdots\!99$$$$T^{10} )^{2}$$
$83$ $$( 1 - 3141 T + 6192162 T^{2} - 8512740683 T^{3} + 9123301378337 T^{4} - 7671831348357484 T^{5} + 5216585125215178219 T^{6} -$$$$27\!\cdots\!27$$$$T^{7} +$$$$11\!\cdots\!86$$$$T^{8} -$$$$33\!\cdots\!01$$$$T^{9} +$$$$61\!\cdots\!07$$$$T^{10} )^{2}$$
$89$ $$1 + 863470 T^{2} + 526173734377 T^{4} + 475356130178542528 T^{6} +$$$$26\!\cdots\!10$$$$T^{8} +$$$$87\!\cdots\!08$$$$T^{10} +$$$$13\!\cdots\!10$$$$T^{12} +$$$$11\!\cdots\!88$$$$T^{14} +$$$$64\!\cdots\!37$$$$T^{16} +$$$$52\!\cdots\!70$$$$T^{18} +$$$$30\!\cdots\!01$$$$T^{20}$$
$97$ $$( 1 + 58 T + 2219536 T^{2} - 133485569 T^{3} + 1934414030539 T^{4} - 340582610040763 T^{5} + 1765487456494120747 T^{6} -$$$$11\!\cdots\!01$$$$T^{7} +$$$$16\!\cdots\!12$$$$T^{8} +$$$$40\!\cdots\!78$$$$T^{9} +$$$$63\!\cdots\!93$$$$T^{10} )^{2}$$