# Properties

 Label 74.4.b.a Level $74$ Weight $4$ Character orbit 74.b Analytic conductor $4.366$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$74 = 2 \cdot 37$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 74.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.36614134042$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Defining polynomial: $$x^{10} + 212 x^{8} + 15052 x^{6} + 392769 x^{4} + 2690496 x^{2} + 2985984$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{7}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{2} + ( 1 - \beta_{2} ) q^{3} -4 q^{4} + ( 2 \beta_{5} + \beta_{7} ) q^{5} + ( \beta_{1} - \beta_{5} ) q^{6} + ( \beta_{2} + \beta_{3} ) q^{7} + 4 \beta_{5} q^{8} + ( 16 - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} ) q^{9} +O(q^{10})$$ $$q -\beta_{5} q^{2} + ( 1 - \beta_{2} ) q^{3} -4 q^{4} + ( 2 \beta_{5} + \beta_{7} ) q^{5} + ( \beta_{1} - \beta_{5} ) q^{6} + ( \beta_{2} + \beta_{3} ) q^{7} + 4 \beta_{5} q^{8} + ( 16 - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} ) q^{9} + ( 8 + \beta_{2} + \beta_{6} ) q^{10} + ( -5 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{11} + ( -4 + 4 \beta_{2} ) q^{12} + ( 12 \beta_{5} + \beta_{8} + \beta_{9} ) q^{13} + ( -\beta_{1} - \beta_{5} + 2 \beta_{8} ) q^{14} + ( -\beta_{1} - 6 \beta_{5} + 3 \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{15} + 16 q^{16} + ( 5 \beta_{1} - 12 \beta_{5} + \beta_{9} ) q^{17} + ( \beta_{1} - 18 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{18} + ( 2 \beta_{1} + 10 \beta_{5} - 4 \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{19} + ( -8 \beta_{5} - 4 \beta_{7} ) q^{20} + ( -32 - 6 \beta_{2} + \beta_{3} + 4 \beta_{4} + 5 \beta_{6} ) q^{21} + ( -\beta_{1} + 5 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{22} + ( -4 \beta_{1} + 6 \beta_{7} + \beta_{8} ) q^{23} + ( -4 \beta_{1} + 4 \beta_{5} ) q^{24} + ( -66 + 10 \beta_{2} + 3 \beta_{3} - 4 \beta_{6} ) q^{25} + ( 48 - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{6} ) q^{26} + ( 76 - 21 \beta_{2} - 2 \beta_{3} - \beta_{4} - 4 \beta_{6} ) q^{27} + ( -4 \beta_{2} - 4 \beta_{3} ) q^{28} + ( -\beta_{1} + 30 \beta_{5} - 3 \beta_{8} + 6 \beta_{9} ) q^{29} + ( -22 + \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 5 \beta_{6} ) q^{30} + ( 2 \beta_{1} - 44 \beta_{5} - \beta_{7} + 4 \beta_{8} - \beta_{9} ) q^{31} -16 \beta_{5} q^{32} + ( -52 - 2 \beta_{2} - 6 \beta_{3} - \beta_{4} - 5 \beta_{6} ) q^{33} + ( -50 + 19 \beta_{2} - 2 \beta_{4} - \beta_{6} ) q^{34} + ( -7 \beta_{1} + 50 \beta_{5} - 14 \beta_{7} - 14 \beta_{8} + \beta_{9} ) q^{35} + ( -64 + 8 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} + 4 \beta_{6} ) q^{36} + ( 15 + 16 \beta_{2} - 3 \beta_{3} + \beta_{4} + 28 \beta_{5} + 4 \beta_{6} + 5 \beta_{7} - 5 \beta_{8} + 4 \beta_{9} ) q^{37} + ( 34 + 3 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 5 \beta_{6} ) q^{38} + ( -14 \beta_{1} + 16 \beta_{5} - 13 \beta_{7} + 6 \beta_{8} + 4 \beta_{9} ) q^{39} + ( -32 - 4 \beta_{2} - 4 \beta_{6} ) q^{40} + ( -125 - 18 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - 3 \beta_{6} ) q^{41} + ( 11 \beta_{1} + 35 \beta_{5} - 12 \beta_{7} + 2 \beta_{8} + 8 \beta_{9} ) q^{42} + ( -16 \beta_{1} + 24 \beta_{5} + 12 \beta_{7} + 8 \beta_{8} - 9 \beta_{9} ) q^{43} + ( 20 - 4 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} ) q^{44} + ( 18 \beta_{1} - 52 \beta_{5} + \beta_{7} - 15 \beta_{8} - 9 \beta_{9} ) q^{45} + ( 2 - 10 \beta_{2} - 2 \beta_{3} + 6 \beta_{6} ) q^{46} + ( 58 + 27 \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{6} ) q^{47} + ( 16 - 16 \beta_{2} ) q^{48} + ( 231 - 14 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} + 9 \beta_{6} ) q^{49} + ( -14 \beta_{1} + 63 \beta_{5} + 16 \beta_{7} + 6 \beta_{8} ) q^{50} + ( 5 \beta_{1} - 224 \beta_{5} + 2 \beta_{7} + 14 \beta_{8} - 9 \beta_{9} ) q^{51} + ( -48 \beta_{5} - 4 \beta_{8} - 4 \beta_{9} ) q^{52} + ( -74 - 8 \beta_{2} - 9 \beta_{3} - 4 \beta_{4} + 5 \beta_{6} ) q^{53} + ( 17 \beta_{1} - 75 \beta_{5} + 14 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} ) q^{54} + ( 22 \beta_{1} + 10 \beta_{5} + 8 \beta_{7} + 23 \beta_{8} - 3 \beta_{9} ) q^{55} + ( 4 \beta_{1} + 4 \beta_{5} - 8 \beta_{8} ) q^{56} + ( -31 \beta_{1} - 30 \beta_{5} - 6 \beta_{7} + 8 \beta_{8} - \beta_{9} ) q^{57} + ( 102 - 10 \beta_{2} + 6 \beta_{3} - 12 \beta_{4} - 6 \beta_{6} ) q^{58} + ( 25 \beta_{1} + 70 \beta_{5} - 4 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} ) q^{59} + ( 4 \beta_{1} + 24 \beta_{5} - 12 \beta_{7} + 4 \beta_{8} + 8 \beta_{9} ) q^{60} + ( 11 \beta_{1} + 46 \beta_{5} + 5 \beta_{7} + 10 \beta_{8} + 9 \beta_{9} ) q^{61} + ( -166 + 8 \beta_{2} - 8 \beta_{3} + 2 \beta_{4} ) q^{62} + ( 270 + 92 \beta_{2} + 8 \beta_{3} + 8 \beta_{4} + 26 \beta_{6} ) q^{63} -64 q^{64} + ( 28 - 76 \beta_{2} + 19 \beta_{3} - 5 \beta_{4} - 20 \beta_{6} ) q^{65} + ( -3 \beta_{1} + 57 \beta_{5} + 18 \beta_{7} - 12 \beta_{8} - 2 \beta_{9} ) q^{66} + ( 123 + 44 \beta_{2} - 6 \beta_{3} - 15 \beta_{4} + 10 \beta_{6} ) q^{67} + ( -20 \beta_{1} + 48 \beta_{5} - 4 \beta_{9} ) q^{68} + ( 11 \beta_{1} + 124 \beta_{5} + 5 \beta_{7} - 12 \beta_{8} - \beta_{9} ) q^{69} + ( 170 - 43 \beta_{2} + 28 \beta_{3} - 2 \beta_{4} - 15 \beta_{6} ) q^{70} + ( -136 + 17 \beta_{2} - \beta_{3} + 8 \beta_{4} + 2 \beta_{6} ) q^{71} + ( -4 \beta_{1} + 72 \beta_{5} - 8 \beta_{7} - 8 \beta_{8} + 8 \beta_{9} ) q^{72} + ( 215 + 32 \beta_{2} + 23 \beta_{3} - 15 \beta_{4} - 9 \beta_{6} ) q^{73} + ( 94 - 12 \beta_{1} + \beta_{2} + 10 \beta_{3} - 8 \beta_{4} - 11 \beta_{5} + \beta_{6} - 14 \beta_{7} - 6 \beta_{8} + 2 \beta_{9} ) q^{74} + ( -504 + 23 \beta_{2} - 16 \beta_{3} + 7 \beta_{4} + 6 \beta_{6} ) q^{75} + ( -8 \beta_{1} - 40 \beta_{5} + 16 \beta_{7} + 8 \beta_{8} - 4 \beta_{9} ) q^{76} + ( -580 + 76 \beta_{2} + \beta_{3} - 2 \beta_{4} - 27 \beta_{6} ) q^{77} + ( 68 - 73 \beta_{2} - 12 \beta_{3} - 8 \beta_{4} - 17 \beta_{6} ) q^{78} + ( 37 \beta_{1} + 60 \beta_{5} + 18 \beta_{7} - \beta_{8} - 5 \beta_{9} ) q^{79} + ( 32 \beta_{5} + 16 \beta_{7} ) q^{80} + ( 463 - 76 \beta_{2} - 25 \beta_{3} - 11 \beta_{4} - 20 \beta_{6} ) q^{81} + ( 15 \beta_{1} + 127 \beta_{5} + 20 \beta_{7} + 4 \beta_{8} + 8 \beta_{9} ) q^{82} + ( -144 - 33 \beta_{2} - 3 \beta_{3} + 14 \beta_{4} + 36 \beta_{6} ) q^{83} + ( 128 + 24 \beta_{2} - 4 \beta_{3} - 16 \beta_{4} - 20 \beta_{6} ) q^{84} + ( 4 - 30 \beta_{2} + 14 \beta_{3} + 14 \beta_{4} + 30 \beta_{6} ) q^{85} + ( 130 - 43 \beta_{2} - 16 \beta_{3} + 18 \beta_{4} + 21 \beta_{6} ) q^{86} + ( -87 \beta_{1} + 134 \beta_{5} - 35 \beta_{7} + 16 \beta_{8} - \beta_{9} ) q^{87} + ( 4 \beta_{1} - 20 \beta_{5} + 8 \beta_{7} + 8 \beta_{8} + 8 \beta_{9} ) q^{88} + ( -51 \beta_{1} - 250 \beta_{5} + 22 \beta_{7} - 8 \beta_{8} + 7 \beta_{9} ) q^{89} + ( -220 + 82 \beta_{2} + 30 \beta_{3} + 18 \beta_{4} + 10 \beta_{6} ) q^{90} + ( 69 \beta_{1} - 340 \beta_{5} - 32 \beta_{7} - 20 \beta_{8} - 3 \beta_{9} ) q^{91} + ( 16 \beta_{1} - 24 \beta_{7} - 4 \beta_{8} ) q^{92} + ( 70 \beta_{1} - 148 \beta_{5} - 11 \beta_{7} + 9 \beta_{8} + 9 \beta_{9} ) q^{93} + ( -29 \beta_{1} - 57 \beta_{5} + 12 \beta_{7} + 2 \beta_{8} + 4 \beta_{9} ) q^{94} + ( 486 - 56 \beta_{2} - 34 \beta_{3} + 16 \beta_{6} ) q^{95} + ( 16 \beta_{1} - 16 \beta_{5} ) q^{96} + ( -14 \beta_{1} + 70 \beta_{5} + 26 \beta_{7} - 8 \beta_{8} - 16 \beta_{9} ) q^{97} + ( 23 \beta_{1} - 238 \beta_{5} - 44 \beta_{7} + 6 \beta_{8} - 8 \beta_{9} ) q^{98} + ( 52 + 10 \beta_{2} - \beta_{3} - 7 \beta_{4} - 48 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q + 14q^{3} - 40q^{4} - 4q^{7} + 172q^{9} + O(q^{10})$$ $$10q + 14q^{3} - 40q^{4} - 4q^{7} + 172q^{9} + 76q^{10} - 50q^{11} - 56q^{12} + 160q^{16} - 312q^{21} - 700q^{25} + 492q^{26} + 848q^{27} + 16q^{28} - 240q^{30} - 508q^{33} - 568q^{34} - 688q^{36} + 82q^{37} + 336q^{38} - 304q^{40} - 1194q^{41} + 200q^{44} + 60q^{46} + 464q^{47} + 224q^{48} + 2382q^{49} - 692q^{53} + 1108q^{58} - 1700q^{62} + 2300q^{63} - 640q^{64} + 604q^{65} + 1114q^{67} + 1880q^{70} - 1460q^{71} + 2082q^{73} + 968q^{74} - 5160q^{75} - 6096q^{77} + 1004q^{78} + 4978q^{81} - 1364q^{83} + 1248q^{84} + 104q^{85} + 1400q^{86} - 2600q^{90} + 5084q^{95} + 508q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 212 x^{8} + 15052 x^{6} + 392769 x^{4} + 2690496 x^{2} + 2985984$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$49 \nu^{8} + 5060 \nu^{6} + 73924 \nu^{4} - 1042479 \nu^{2} + 4950720$$$$)/5557896$$ $$\beta_{3}$$ $$=$$ $$($$$$247 \nu^{8} + 31808 \nu^{6} + 744424 \nu^{4} - 20315457 \nu^{2} - 277495956$$$$)/8336844$$ $$\beta_{4}$$ $$=$$ $$($$$$-203 \nu^{8} - 43018 \nu^{6} - 2996984 \nu^{4} - 69411501 \nu^{2} - 170771814$$$$)/4168422$$ $$\beta_{5}$$ $$=$$ $$($$$$955 \nu^{9} + 174236 \nu^{7} + 11460100 \nu^{5} + 332514171 \nu^{3} + 3169891584 \nu$$$$)/ 1600674048$$ $$\beta_{6}$$ $$=$$ $$($$$$653 \nu^{8} + 117844 \nu^{6} + 6738392 \nu^{4} + 126844389 \nu^{2} + 414195120$$$$)/8336844$$ $$\beta_{7}$$ $$=$$ $$($$$$4877 \nu^{9} + 1100164 \nu^{7} + 81458780 \nu^{5} + 2086850637 \nu^{3} + 9887785344 \nu$$$$)/ 533558016$$ $$\beta_{8}$$ $$=$$ $$($$$$6325 \nu^{9} + 1283300 \nu^{7} + 85626172 \nu^{5} + 2049826869 \nu^{3} + 13178567808 \nu$$$$)/ 400168512$$ $$\beta_{9}$$ $$=$$ $$($$$$11011 \nu^{9} + 2019260 \nu^{7} + 118682980 \nu^{5} + 2281437891 \nu^{3} + 4155048576 \nu$$$$)/ 533558016$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{4} - \beta_{3} - 42$$ $$\nu^{3}$$ $$=$$ $$($$$$-4 \beta_{9} + 10 \beta_{8} - 8 \beta_{7} - 4 \beta_{5} - 71 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-91 \beta_{6} - 94 \beta_{4} + 70 \beta_{3} + 54 \beta_{2} + 2952$$ $$\nu^{5}$$ $$=$$ $$($$$$358 \beta_{9} - 1078 \beta_{8} + 866 \beta_{7} + 2908 \beta_{5} + 5453 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$7759 \beta_{6} + 7936 \beta_{4} - 5197 \beta_{3} - 7632 \beta_{2} - 226551$$ $$\nu^{7}$$ $$=$$ $$($$$$-31636 \beta_{9} + 99040 \beta_{8} - 74552 \beta_{7} - 387340 \beta_{5} - 433313 \beta_{1}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$-642673 \beta_{6} - 656425 \beta_{4} + 409789 \beta_{3} + 820080 \beta_{2} + 17946726$$ $$\nu^{9}$$ $$=$$ $$($$$$2868556 \beta_{9} - 8615170 \beta_{8} + 5995088 \beta_{7} + 40517284 \beta_{5} + 35021375 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/74\mathbb{Z}\right)^\times$$.

 $$n$$ $$39$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
73.1
 9.15606i 2.87017i 1.17028i − 6.35485i − 8.84166i − 9.15606i − 2.87017i − 1.17028i 6.35485i 8.84166i
2.00000i −8.15606 −4.00000 6.18414i 16.3121i 23.1710 8.00000i 39.5214 12.3683
73.2 2.00000i −1.87017 −4.00000 8.73820i 3.74034i −6.27788 8.00000i −23.5025 −17.4764
73.3 2.00000i −0.170276 −4.00000 19.7476i 0.340553i −28.6201 8.00000i −26.9710 39.4952
73.4 2.00000i 7.35485 −4.00000 16.2134i 14.7097i 31.9123 8.00000i 27.0938 32.4267
73.5 2.00000i 9.84166 −4.00000 14.4069i 19.6833i −22.1853 8.00000i 69.8583 −28.8138
73.6 2.00000i −8.15606 −4.00000 6.18414i 16.3121i 23.1710 8.00000i 39.5214 12.3683
73.7 2.00000i −1.87017 −4.00000 8.73820i 3.74034i −6.27788 8.00000i −23.5025 −17.4764
73.8 2.00000i −0.170276 −4.00000 19.7476i 0.340553i −28.6201 8.00000i −26.9710 39.4952
73.9 2.00000i 7.35485 −4.00000 16.2134i 14.7097i 31.9123 8.00000i 27.0938 32.4267
73.10 2.00000i 9.84166 −4.00000 14.4069i 19.6833i −22.1853 8.00000i 69.8583 −28.8138
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 73.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.4.b.a 10
3.b odd 2 1 666.4.c.d 10
4.b odd 2 1 592.4.g.d 10
37.b even 2 1 inner 74.4.b.a 10
111.d odd 2 1 666.4.c.d 10
148.b odd 2 1 592.4.g.d 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.4.b.a 10 1.a even 1 1 trivial
74.4.b.a 10 37.b even 2 1 inner
592.4.g.d 10 4.b odd 2 1
592.4.g.d 10 148.b odd 2 1
666.4.c.d 10 3.b odd 2 1
666.4.c.d 10 111.d odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(74, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 + T^{2} )^{5}$$
$3$ $$( 188 + 1183 T + 449 T^{2} - 86 T^{3} - 7 T^{4} + T^{5} )^{2}$$
$5$ $$62132541696 + 3133410192 T^{2} + 51066280 T^{4} + 339537 T^{6} + 975 T^{8} + T^{10}$$
$7$ $$( 2947488 + 485780 T - 6348 T^{2} - 1451 T^{3} + 2 T^{4} + T^{5} )^{2}$$
$11$ $$( -33993108 + 3624921 T - 35739 T^{2} - 3808 T^{3} + 25 T^{4} + T^{5} )^{2}$$
$13$ $$1191021821081856 + 26550815907600 T^{2} + 57084976264 T^{4} + 42132721 T^{6} + 11551 T^{8} + T^{10}$$
$17$ $$60664777665675264 + 733465712279808 T^{2} + 783747736128 T^{4} + 234489856 T^{6} + 26532 T^{8} + T^{10}$$
$19$ $$718725483611136 + 172413887187456 T^{2} + 464181651520 T^{4} + 178718608 T^{6} + 23368 T^{8} + T^{10}$$
$23$ $$2779415743614944256 + 5763800563784064 T^{2} + 2516251502500 T^{4} + 445409221 T^{6} + 34939 T^{8} + T^{10}$$
$29$ $$12\!\cdots\!56$$$$+ 423461029721582592 T^{2} + 168338326783824 T^{4} + 8938553353 T^{6} + 164807 T^{8} + T^{10}$$
$31$ $$1113969264828235776 + 6104520149045760 T^{2} + 8770620909028 T^{4} + 1665729477 T^{6} + 87711 T^{8} + T^{10}$$
$37$ $$33\!\cdots\!93$$$$-$$$$53\!\cdots\!42$$$$T + 4636644990668962129 T^{2} - 2527466296788992 T^{3} - 170328868139530 T^{4} + 84384858820 T^{5} - 3362661010 T^{6} - 985088 T^{7} + 35677 T^{8} - 82 T^{9} + T^{10}$$
$41$ $$( -671986566 + 199536597 T - 13756121 T^{2} + 45156 T^{3} + 597 T^{4} + T^{5} )^{2}$$
$43$ $$41\!\cdots\!04$$$$+$$$$41\!\cdots\!68$$$$T^{2} + 11331067177848448 T^{4} + 126532754336 T^{6} + 600404 T^{8} + T^{10}$$
$47$ $$( 69952577472 + 2243029428 T + 9489802 T^{2} - 79063 T^{3} - 232 T^{4} + T^{5} )^{2}$$
$53$ $$( 494344579464 - 3292490232 T - 64098356 T^{2} - 148071 T^{3} + 346 T^{4} + T^{5} )^{2}$$
$59$ $$42\!\cdots\!04$$$$+ 13854061732344901632 T^{2} + 1548974330284608 T^{4} + 51099493504 T^{6} + 566688 T^{8} + T^{10}$$
$61$ $$31\!\cdots\!04$$$$+$$$$17\!\cdots\!72$$$$T^{2} + 32612777977305904 T^{4} + 251783214737 T^{6} + 839615 T^{8} + T^{10}$$
$67$ $$( 16621395102016 + 175465058416 T + 298622764 T^{2} - 900827 T^{3} - 557 T^{4} + T^{5} )^{2}$$
$71$ $$( -731949946272 - 11796373476 T - 51053096 T^{2} + 53941 T^{3} + 730 T^{4} + T^{5} )^{2}$$
$73$ $$( -21354004243006 + 69006200669 T + 325928401 T^{2} - 518880 T^{3} - 1041 T^{4} + T^{5} )^{2}$$
$79$ $$39\!\cdots\!76$$$$+$$$$11\!\cdots\!52$$$$T^{2} + 218336938850924188 T^{4} + 1164152826069 T^{6} + 2006811 T^{8} + T^{10}$$
$83$ $$( 465096834945216 + 803768665824 T - 1305429296 T^{2} - 1966055 T^{3} + 682 T^{4} + T^{5} )^{2}$$
$89$ $$50\!\cdots\!84$$$$+$$$$25\!\cdots\!92$$$$T^{2} + 1906051113225084672 T^{4} + 4796325838048 T^{6} + 3973872 T^{8} + T^{10}$$
$97$ $$41\!\cdots\!76$$$$+$$$$18\!\cdots\!64$$$$T^{2} + 289105564546096128 T^{4} + 1678196800656 T^{6} + 2573068 T^{8} + T^{10}$$