# Properties

 Label 1764.2.e Level $1764$ Weight $2$ Character orbit 1764.e Rep. character $\chi_{1764}(1079,\cdot)$ Character field $\Q$ Dimension $82$ Newform subspaces $10$ Sturm bound $672$ Trace bound $25$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1764.e (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$12$$ Character field: $$\Q$$ Newform subspaces: $$10$$ Sturm bound: $$672$$ Trace bound: $$25$$ Distinguishing $$T_p$$: $$5$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1764, [\chi])$$.

Total New Old
Modular forms 368 82 286
Cusp forms 304 82 222
Eisenstein series 64 0 64

## Trace form

 $$82q - 4q^{4} + O(q^{10})$$ $$82q - 4q^{4} + 4q^{10} + 8q^{13} + 16q^{16} - 16q^{22} - 90q^{25} + 36q^{34} - 12q^{37} + 32q^{46} - 60q^{58} - 36q^{61} + 8q^{64} + 32q^{73} - 72q^{76} - 60q^{82} + 52q^{85} - 8q^{88} + 24q^{94} - 16q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1764, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
1764.2.e.a $$2$$ $$14.086$$ $$\Q(\sqrt{-2})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta q^{2}-2q^{4}+3\beta q^{5}-2\beta q^{8}-6q^{10}+\cdots$$
1764.2.e.b $$2$$ $$14.086$$ $$\Q(\sqrt{-2})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta q^{2}-2q^{4}+\beta q^{5}-2\beta q^{8}-2q^{10}+\cdots$$
1764.2.e.c $$2$$ $$14.086$$ $$\Q(\sqrt{-2})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta q^{2}-2q^{4}+3\beta q^{5}+2\beta q^{8}+6q^{10}+\cdots$$
1764.2.e.d $$4$$ $$14.086$$ $$\Q(\zeta_{8})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{8}^{2}q^{2}-2q^{4}+2\zeta_{8}q^{5}+2\zeta_{8}^{2}q^{8}+\cdots$$
1764.2.e.e $$4$$ $$14.086$$ $$\Q(\zeta_{8})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{8}^{2}q^{2}-2q^{4}-\zeta_{8}q^{5}-2\zeta_{8}^{2}q^{8}+\cdots$$
1764.2.e.f $$8$$ $$14.086$$ 8.0.157351936.1 $$\Q(\sqrt{-7})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{6}q^{2}-\beta _{1}q^{4}-\beta _{5}q^{8}+(\beta _{5}+\beta _{6}+\cdots)q^{11}+\cdots$$
1764.2.e.g $$12$$ $$14.086$$ 12.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{6}q^{2}+(1+\beta _{10})q^{4}-\beta _{4}q^{5}+(\beta _{6}+\cdots)q^{8}+\cdots$$
1764.2.e.h $$16$$ $$14.086$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{2}+\beta _{2}q^{4}-\beta _{5}q^{5}+(\beta _{3}-\beta _{5}+\cdots)q^{8}+\cdots$$
1764.2.e.i $$16$$ $$14.086$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+\beta _{2}q^{4}-\beta _{5}q^{5}+(-\beta _{3}+\beta _{5}+\cdots)q^{8}+\cdots$$
1764.2.e.j $$16$$ $$14.086$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{3}q^{2}+(1-\beta _{8})q^{4}+\beta _{2}q^{5}+(-\beta _{3}+\cdots)q^{8}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1764, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1764, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(36, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(84, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(252, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(588, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + 2 T^{2}$$)($$1 + 2 T^{2}$$)($$1 + 2 T^{2}$$)($$( 1 + 2 T^{2} )^{2}$$)($$( 1 + 2 T^{2} )^{2}$$)($$1 + T^{4} + 16 T^{8}$$)($$1 - 4 T^{2} + 13 T^{4} - 28 T^{6} + 52 T^{8} - 64 T^{10} + 64 T^{12}$$)($$1 + T^{4} - 10 T^{6} + 4 T^{8} - 40 T^{10} + 16 T^{12} + 256 T^{16}$$)($$1 + T^{4} - 10 T^{6} + 4 T^{8} - 40 T^{10} + 16 T^{12} + 256 T^{16}$$)($$( 1 - 4 T^{2} + 9 T^{4} - 16 T^{6} + 16 T^{8} )^{2}$$)
$3$ 1
$5$ ($$1 + 8 T^{2} + 25 T^{4}$$)($$1 - 8 T^{2} + 25 T^{4}$$)($$1 + 8 T^{2} + 25 T^{4}$$)($$( 1 - 2 T + 5 T^{2} )^{2}( 1 + 2 T + 5 T^{2} )^{2}$$)($$( 1 - 4 T + 5 T^{2} )^{2}( 1 + 4 T + 5 T^{2} )^{2}$$)($$( 1 - 5 T^{2} )^{8}$$)($$( 1 - 12 T^{2} + 95 T^{4} - 568 T^{6} + 2375 T^{8} - 7500 T^{10} + 15625 T^{12} )^{2}$$)($$( 1 - 14 T^{2} + 117 T^{4} - 698 T^{6} + 3576 T^{8} - 17450 T^{10} + 73125 T^{12} - 218750 T^{14} + 390625 T^{16} )^{2}$$)($$( 1 - 14 T^{2} + 117 T^{4} - 698 T^{6} + 3576 T^{8} - 17450 T^{10} + 73125 T^{12} - 218750 T^{14} + 390625 T^{16} )^{2}$$)($$( 1 - 12 T^{2} + 74 T^{4} - 300 T^{6} + 625 T^{8} )^{4}$$)
$7$ 1
$11$ ($$( 1 + 11 T^{2} )^{2}$$)($$( 1 + 11 T^{2} )^{2}$$)($$( 1 + 11 T^{2} )^{2}$$)($$( 1 + 11 T^{2} )^{4}$$)($$( 1 + 11 T^{2} )^{4}$$)($$( 1 - 206 T^{4} + 14641 T^{8} )^{2}$$)($$( 1 + 38 T^{2} + 715 T^{4} + 9068 T^{6} + 86515 T^{8} + 556358 T^{10} + 1771561 T^{12} )^{2}$$)($$( 1 + 30 T^{2} + 757 T^{4} + 11666 T^{6} + 154360 T^{8} + 1411586 T^{10} + 11083237 T^{12} + 53146830 T^{14} + 214358881 T^{16} )^{2}$$)($$( 1 + 30 T^{2} + 757 T^{4} + 11666 T^{6} + 154360 T^{8} + 1411586 T^{10} + 11083237 T^{12} + 53146830 T^{14} + 214358881 T^{16} )^{2}$$)($$( 1 + 8 T^{2} + 66 T^{4} + 968 T^{6} + 14641 T^{8} )^{4}$$)
$13$ ($$( 1 + 6 T + 13 T^{2} )^{2}$$)($$( 1 - 4 T + 13 T^{2} )^{2}$$)($$( 1 - 6 T + 13 T^{2} )^{2}$$)($$( 1 + 24 T^{2} + 169 T^{4} )^{2}$$)($$( 1 - 24 T^{2} + 169 T^{4} )^{2}$$)($$( 1 + 13 T^{2} )^{8}$$)($$( 1 + 11 T^{2} + 16 T^{3} + 143 T^{4} + 2197 T^{6} )^{4}$$)($$( 1 + 2 T + 27 T^{2} + 38 T^{3} + 378 T^{4} + 494 T^{5} + 4563 T^{6} + 4394 T^{7} + 28561 T^{8} )^{4}$$)($$( 1 - 2 T + 27 T^{2} - 38 T^{3} + 378 T^{4} - 494 T^{5} + 4563 T^{6} - 4394 T^{7} + 28561 T^{8} )^{4}$$)($$( 1 + 24 T^{2} + 169 T^{4} )^{8}$$)
$17$ ($$1 - 16 T^{2} + 289 T^{4}$$)($$1 + 16 T^{2} + 289 T^{4}$$)($$1 - 16 T^{2} + 289 T^{4}$$)($$( 1 - 2 T + 17 T^{2} )^{2}( 1 + 2 T + 17 T^{2} )^{2}$$)($$( 1 - 8 T + 17 T^{2} )^{2}( 1 + 8 T + 17 T^{2} )^{2}$$)($$( 1 - 17 T^{2} )^{8}$$)($$( 1 - 68 T^{2} + 2167 T^{4} - 44168 T^{6} + 626263 T^{8} - 5679428 T^{10} + 24137569 T^{12} )^{2}$$)($$( 1 - 76 T^{2} + 3096 T^{4} - 83940 T^{6} + 1656750 T^{8} - 24258660 T^{10} + 258581016 T^{12} - 1834455244 T^{14} + 6975757441 T^{16} )^{2}$$)($$( 1 - 76 T^{2} + 3096 T^{4} - 83940 T^{6} + 1656750 T^{8} - 24258660 T^{10} + 258581016 T^{12} - 1834455244 T^{14} + 6975757441 T^{16} )^{2}$$)($$( 1 - 12 T^{2} + 506 T^{4} - 3468 T^{6} + 83521 T^{8} )^{4}$$)
$19$ ($$( 1 - 19 T^{2} )^{2}$$)($$( 1 - 19 T^{2} )^{2}$$)($$( 1 - 19 T^{2} )^{2}$$)($$( 1 - 19 T^{2} )^{4}$$)($$( 1 - 19 T^{2} )^{4}$$)($$( 1 - 19 T^{2} )^{8}$$)($$( 1 - 50 T^{2} + 1639 T^{4} - 35804 T^{6} + 591679 T^{8} - 6516050 T^{10} + 47045881 T^{12} )^{2}$$)($$( 1 - 34 T^{2} + 1041 T^{4} - 25110 T^{6} + 610032 T^{8} - 9064710 T^{10} + 135664161 T^{12} - 1599559954 T^{14} + 16983563041 T^{16} )^{2}$$)($$( 1 - 34 T^{2} + 1041 T^{4} - 25110 T^{6} + 610032 T^{8} - 9064710 T^{10} + 135664161 T^{12} - 1599559954 T^{14} + 16983563041 T^{16} )^{2}$$)($$( 1 - 28 T^{2} + 870 T^{4} - 10108 T^{6} + 130321 T^{8} )^{4}$$)
$23$ ($$( 1 + 23 T^{2} )^{2}$$)($$( 1 + 23 T^{2} )^{2}$$)($$( 1 + 23 T^{2} )^{2}$$)($$( 1 + 23 T^{2} )^{4}$$)($$( 1 + 23 T^{2} )^{4}$$)($$( 1 - 734 T^{4} + 279841 T^{8} )^{2}$$)($$( 1 - 2 T^{2} + 979 T^{4} + 3772 T^{6} + 517891 T^{8} - 559682 T^{10} + 148035889 T^{12} )^{2}$$)($$( 1 + 116 T^{2} + 6936 T^{4} + 269340 T^{6} + 7319790 T^{8} + 142480860 T^{10} + 1940977176 T^{12} + 17172163124 T^{14} + 78310985281 T^{16} )^{2}$$)($$( 1 + 116 T^{2} + 6936 T^{4} + 269340 T^{6} + 7319790 T^{8} + 142480860 T^{10} + 1940977176 T^{12} + 17172163124 T^{14} + 78310985281 T^{16} )^{2}$$)($$( 1 + 56 T^{2} + 1650 T^{4} + 29624 T^{6} + 279841 T^{8} )^{4}$$)
$29$ ($$1 + 40 T^{2} + 841 T^{4}$$)($$1 + 40 T^{2} + 841 T^{4}$$)($$1 + 40 T^{2} + 841 T^{4}$$)($$( 1 + 40 T^{2} + 841 T^{4} )^{2}$$)($$( 1 + 40 T^{2} + 841 T^{4} )^{2}$$)($$( 1 + 1234 T^{4} + 707281 T^{8} )^{2}$$)($$( 1 - 56 T^{2} + 841 T^{4} )^{6}$$)($$( 1 - 146 T^{2} + 10849 T^{4} - 522290 T^{6} + 17802004 T^{8} - 439245890 T^{10} + 7673291569 T^{12} - 86844204866 T^{14} + 500246412961 T^{16} )^{2}$$)($$( 1 - 146 T^{2} + 10849 T^{4} - 522290 T^{6} + 17802004 T^{8} - 439245890 T^{10} + 7673291569 T^{12} - 86844204866 T^{14} + 500246412961 T^{16} )^{2}$$)($$( 1 - 88 T^{2} + 3426 T^{4} - 74008 T^{6} + 707281 T^{8} )^{4}$$)
$31$ ($$( 1 - 31 T^{2} )^{2}$$)($$( 1 - 31 T^{2} )^{2}$$)($$( 1 - 31 T^{2} )^{2}$$)($$( 1 - 31 T^{2} )^{4}$$)($$( 1 - 31 T^{2} )^{4}$$)($$( 1 - 31 T^{2} )^{8}$$)($$( 1 - 58 T^{2} + 3727 T^{4} - 113644 T^{6} + 3581647 T^{8} - 53564218 T^{10} + 887503681 T^{12} )^{2}$$)($$( 1 - 132 T^{2} + 6430 T^{4} - 130264 T^{6} + 1736719 T^{8} - 125183704 T^{10} + 5938240030 T^{12} - 117150485892 T^{14} + 852891037441 T^{16} )^{2}$$)($$( 1 - 132 T^{2} + 6430 T^{4} - 130264 T^{6} + 1736719 T^{8} - 125183704 T^{10} + 5938240030 T^{12} - 117150485892 T^{14} + 852891037441 T^{16} )^{2}$$)($$( 1 + 20 T^{2} + 1590 T^{4} + 19220 T^{6} + 923521 T^{8} )^{4}$$)
$37$ ($$( 1 + 2 T + 37 T^{2} )^{2}$$)($$( 1 - 2 T + 37 T^{2} )^{2}$$)($$( 1 + 2 T + 37 T^{2} )^{2}$$)($$( 1 + 12 T + 37 T^{2} )^{4}$$)($$( 1 - 12 T + 37 T^{2} )^{4}$$)($$( 1 - 38 T^{2} + 1369 T^{4} )^{4}$$)($$( 1 - 2 T + 47 T^{2} - 276 T^{3} + 1739 T^{4} - 2738 T^{5} + 50653 T^{6} )^{4}$$)($$( 1 + 2 T + 115 T^{2} + 282 T^{3} + 5758 T^{4} + 10434 T^{5} + 157435 T^{6} + 101306 T^{7} + 1874161 T^{8} )^{4}$$)($$( 1 + 2 T + 115 T^{2} + 282 T^{3} + 5758 T^{4} + 10434 T^{5} + 157435 T^{6} + 101306 T^{7} + 1874161 T^{8} )^{4}$$)($$( 1 + 62 T^{2} + 1369 T^{4} )^{8}$$)
$41$ ($$1 + 80 T^{2} + 1681 T^{4}$$)($$1 - 80 T^{2} + 1681 T^{4}$$)($$1 + 80 T^{2} + 1681 T^{4}$$)($$( 1 - 8 T + 41 T^{2} )^{2}( 1 + 8 T + 41 T^{2} )^{2}$$)($$( 1 - 10 T + 41 T^{2} )^{2}( 1 + 10 T + 41 T^{2} )^{2}$$)($$( 1 - 41 T^{2} )^{8}$$)($$( 1 - 132 T^{2} + 10823 T^{4} - 527752 T^{6} + 18193463 T^{8} - 373000452 T^{10} + 4750104241 T^{12} )^{2}$$)($$( 1 - 188 T^{2} + 15912 T^{4} - 846644 T^{6} + 36194574 T^{8} - 1423208564 T^{10} + 44963509032 T^{12} - 893019597308 T^{14} + 7984925229121 T^{16} )^{2}$$)($$( 1 - 188 T^{2} + 15912 T^{4} - 846644 T^{6} + 36194574 T^{8} - 1423208564 T^{10} + 44963509032 T^{12} - 893019597308 T^{14} + 7984925229121 T^{16} )^{2}$$)($$( 1 - 60 T^{2} + 3674 T^{4} - 100860 T^{6} + 2825761 T^{8} )^{4}$$)
$43$ ($$( 1 - 43 T^{2} )^{2}$$)($$( 1 - 43 T^{2} )^{2}$$)($$( 1 - 43 T^{2} )^{2}$$)($$( 1 - 43 T^{2} )^{4}$$)($$( 1 - 43 T^{2} )^{4}$$)($$( 1 + 58 T^{2} + 1849 T^{4} )^{4}$$)($$( 1 - 98 T^{2} + 4567 T^{4} - 172988 T^{6} + 8444383 T^{8} - 335042498 T^{10} + 6321363049 T^{12} )^{2}$$)($$( 1 - 202 T^{2} + 17289 T^{4} - 881742 T^{6} + 37179072 T^{8} - 1630340958 T^{10} + 59107650489 T^{12} - 1276915335898 T^{14} + 11688200277601 T^{16} )^{2}$$)($$( 1 - 202 T^{2} + 17289 T^{4} - 881742 T^{6} + 37179072 T^{8} - 1630340958 T^{10} + 59107650489 T^{12} - 1276915335898 T^{14} + 11688200277601 T^{16} )^{2}$$)($$( 1 - 76 T^{2} + 4950 T^{4} - 140524 T^{6} + 3418801 T^{8} )^{4}$$)
$47$ ($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 47 T^{2} )^{4}$$)($$( 1 + 47 T^{2} )^{4}$$)($$( 1 + 47 T^{2} )^{8}$$)($$( 1 + 10 T^{2} + 4591 T^{4} + 70732 T^{6} + 10141519 T^{8} + 48796810 T^{10} + 10779215329 T^{12} )^{2}$$)($$( 1 + 200 T^{2} + 22212 T^{4} + 1666392 T^{6} + 91084806 T^{8} + 3681059928 T^{10} + 108387474372 T^{12} + 2155843065800 T^{14} + 23811286661761 T^{16} )^{2}$$)($$( 1 + 200 T^{2} + 22212 T^{4} + 1666392 T^{6} + 91084806 T^{8} + 3681059928 T^{10} + 108387474372 T^{12} + 2155843065800 T^{14} + 23811286661761 T^{16} )^{2}$$)($$( 1 + 68 T^{2} + 2502 T^{4} + 150212 T^{6} + 4879681 T^{8} )^{4}$$)
$53$ ($$1 - 56 T^{2} + 2809 T^{4}$$)($$1 - 56 T^{2} + 2809 T^{4}$$)($$1 - 56 T^{2} + 2809 T^{4}$$)($$( 1 + 56 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 + 56 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 5582 T^{4} + 7890481 T^{8} )^{2}$$)($$( 1 - 184 T^{2} + 16171 T^{4} - 975856 T^{6} + 45424339 T^{8} - 1451848504 T^{10} + 22164361129 T^{12} )^{2}$$)($$( 1 - 282 T^{2} + 34177 T^{4} - 2497906 T^{6} + 140957380 T^{8} - 7016617954 T^{10} + 269672969137 T^{12} - 6250349838378 T^{14} + 62259690411361 T^{16} )^{2}$$)($$( 1 - 282 T^{2} + 34177 T^{4} - 2497906 T^{6} + 140957380 T^{8} - 7016617954 T^{10} + 269672969137 T^{12} - 6250349838378 T^{14} + 62259690411361 T^{16} )^{2}$$)($$( 1 - 128 T^{2} + 7986 T^{4} - 359552 T^{6} + 7890481 T^{8} )^{4}$$)
$59$ ($$( 1 + 59 T^{2} )^{2}$$)($$( 1 + 59 T^{2} )^{2}$$)($$( 1 + 59 T^{2} )^{2}$$)($$( 1 + 59 T^{2} )^{4}$$)($$( 1 + 59 T^{2} )^{4}$$)($$( 1 + 59 T^{2} )^{8}$$)($$( 1 + 146 T^{2} + 7799 T^{4} + 306396 T^{6} + 27148319 T^{8} + 1769134706 T^{10} + 42180533641 T^{12} )^{2}$$)($$( 1 + 186 T^{2} + 18161 T^{4} + 1259962 T^{6} + 77631844 T^{8} + 4385927722 T^{10} + 220063393121 T^{12} + 7845579257226 T^{14} + 146830437604321 T^{16} )^{2}$$)($$( 1 + 186 T^{2} + 18161 T^{4} + 1259962 T^{6} + 77631844 T^{8} + 4385927722 T^{10} + 220063393121 T^{12} + 7845579257226 T^{14} + 146830437604321 T^{16} )^{2}$$)($$( 1 + 20 T^{2} + 150 T^{4} + 69620 T^{6} + 12117361 T^{8} )^{4}$$)
$61$ ($$( 1 + 12 T + 61 T^{2} )^{2}$$)($$( 1 - 10 T + 61 T^{2} )^{2}$$)($$( 1 - 12 T + 61 T^{2} )^{2}$$)($$( 1 + 120 T^{2} + 3721 T^{4} )^{2}$$)($$( 1 - 120 T^{2} + 3721 T^{4} )^{2}$$)($$( 1 + 61 T^{2} )^{8}$$)($$( 1 + 14 T + 187 T^{2} + 1700 T^{3} + 11407 T^{4} + 52094 T^{5} + 226981 T^{6} )^{4}$$)($$( 1 + 4 T + 162 T^{2} + 588 T^{3} + 12546 T^{4} + 35868 T^{5} + 602802 T^{6} + 907924 T^{7} + 13845841 T^{8} )^{4}$$)($$( 1 - 4 T + 162 T^{2} - 588 T^{3} + 12546 T^{4} - 35868 T^{5} + 602802 T^{6} - 907924 T^{7} + 13845841 T^{8} )^{4}$$)($$( 1 - 72 T^{2} + 3938 T^{4} - 267912 T^{6} + 13845841 T^{8} )^{4}$$)
$67$ ($$( 1 - 67 T^{2} )^{2}$$)($$( 1 - 67 T^{2} )^{2}$$)($$( 1 - 67 T^{2} )^{2}$$)($$( 1 - 67 T^{2} )^{4}$$)($$( 1 - 67 T^{2} )^{4}$$)($$( 1 - 4 T + 67 T^{2} )^{4}( 1 + 4 T + 67 T^{2} )^{4}$$)($$( 1 - 186 T^{2} + 13175 T^{4} - 680684 T^{6} + 59142575 T^{8} - 3748108506 T^{10} + 90458382169 T^{12} )^{2}$$)($$( 1 - 254 T^{2} + 32925 T^{4} - 3201102 T^{6} + 246095804 T^{8} - 14369746878 T^{10} + 663475658925 T^{12} - 22976429070926 T^{14} + 406067677556641 T^{16} )^{2}$$)($$( 1 - 254 T^{2} + 32925 T^{4} - 3201102 T^{6} + 246095804 T^{8} - 14369746878 T^{10} + 663475658925 T^{12} - 22976429070926 T^{14} + 406067677556641 T^{16} )^{2}$$)($$( 1 - 196 T^{2} + 17814 T^{4} - 879844 T^{6} + 20151121 T^{8} )^{4}$$)
$71$ ($$( 1 + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{4}$$)($$( 1 + 71 T^{2} )^{4}$$)($$( 1 + 2914 T^{4} + 25411681 T^{8} )^{2}$$)($$( 1 + 158 T^{2} + 5683 T^{4} - 107140 T^{6} + 28648003 T^{8} + 4015045598 T^{10} + 128100283921 T^{12} )^{2}$$)($$( 1 + 360 T^{2} + 62596 T^{4} + 7074488 T^{6} + 580384198 T^{8} + 35662494008 T^{10} + 1590669583876 T^{12} + 46116102211560 T^{14} + 645753531245761 T^{16} )^{2}$$)($$( 1 + 360 T^{2} + 62596 T^{4} + 7074488 T^{6} + 580384198 T^{8} + 35662494008 T^{10} + 1590669583876 T^{12} + 46116102211560 T^{14} + 645753531245761 T^{16} )^{2}$$)($$( 1 + 8 T^{2} + 7026 T^{4} + 40328 T^{6} + 25411681 T^{8} )^{4}$$)
$73$ ($$( 1 - 6 T + 73 T^{2} )^{2}$$)($$( 1 - 16 T + 73 T^{2} )^{2}$$)($$( 1 + 6 T + 73 T^{2} )^{2}$$)($$( 1 - 96 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 + 96 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 + 73 T^{2} )^{8}$$)($$( 1 + 47 T^{2} + 352 T^{3} + 3431 T^{4} + 389017 T^{6} )^{4}$$)($$( 1 + 2 T + 141 T^{2} - 58 T^{3} + 12336 T^{4} - 4234 T^{5} + 751389 T^{6} + 778034 T^{7} + 28398241 T^{8} )^{4}$$)($$( 1 - 2 T + 141 T^{2} + 58 T^{3} + 12336 T^{4} + 4234 T^{5} + 751389 T^{6} - 778034 T^{7} + 28398241 T^{8} )^{4}$$)($$( 1 + 120 T^{2} + 7346 T^{4} + 639480 T^{6} + 28398241 T^{8} )^{4}$$)
$79$ ($$( 1 - 79 T^{2} )^{2}$$)($$( 1 - 79 T^{2} )^{2}$$)($$( 1 - 79 T^{2} )^{2}$$)($$( 1 - 79 T^{2} )^{4}$$)($$( 1 - 79 T^{2} )^{4}$$)($$( 1 - 8 T + 79 T^{2} )^{4}( 1 + 8 T + 79 T^{2} )^{4}$$)($$( 1 - 162 T^{2} + 19359 T^{4} - 2006332 T^{6} + 120819519 T^{8} - 6309913122 T^{10} + 243087455521 T^{12} )^{2}$$)($$( 1 - 332 T^{2} + 56998 T^{4} - 6691240 T^{6} + 595616087 T^{8} - 41760028840 T^{10} + 2220076716838 T^{12} - 80705035232972 T^{14} + 1517108809906561 T^{16} )^{2}$$)($$( 1 - 332 T^{2} + 56998 T^{4} - 6691240 T^{6} + 595616087 T^{8} - 41760028840 T^{10} + 2220076716838 T^{12} - 80705035232972 T^{14} + 1517108809906561 T^{16} )^{2}$$)($$( 1 - 100 T^{2} + 8070 T^{4} - 624100 T^{6} + 38950081 T^{8} )^{4}$$)
$83$ ($$( 1 + 83 T^{2} )^{2}$$)($$( 1 + 83 T^{2} )^{2}$$)($$( 1 + 83 T^{2} )^{2}$$)($$( 1 + 83 T^{2} )^{4}$$)($$( 1 + 83 T^{2} )^{4}$$)($$( 1 + 83 T^{2} )^{8}$$)($$( 1 + 370 T^{2} + 65191 T^{4} + 6834652 T^{6} + 449100799 T^{8} + 17559578770 T^{10} + 326940373369 T^{12} )^{2}$$)($$( 1 + 110 T^{2} + 7989 T^{4} + 466194 T^{6} + 66686616 T^{8} + 3211610466 T^{10} + 379144526469 T^{12} + 35963441070590 T^{14} + 2252292232139041 T^{16} )^{2}$$)($$( 1 + 110 T^{2} + 7989 T^{4} + 466194 T^{6} + 66686616 T^{8} + 3211610466 T^{10} + 379144526469 T^{12} + 35963441070590 T^{14} + 2252292232139041 T^{16} )^{2}$$)($$( 1 + 44 T^{2} + 12534 T^{4} + 303116 T^{6} + 47458321 T^{8} )^{4}$$)
$89$ ($$1 - 160 T^{2} + 7921 T^{4}$$)($$1 + 160 T^{2} + 7921 T^{4}$$)($$1 - 160 T^{2} + 7921 T^{4}$$)($$( 1 - 16 T + 89 T^{2} )^{2}( 1 + 16 T + 89 T^{2} )^{2}$$)($$( 1 - 10 T + 89 T^{2} )^{2}( 1 + 10 T + 89 T^{2} )^{2}$$)($$( 1 - 89 T^{2} )^{8}$$)($$( 1 - 308 T^{2} + 49895 T^{4} - 5241384 T^{6} + 395218295 T^{8} - 19324610228 T^{10} + 496981290961 T^{12} )^{2}$$)($$( 1 - 352 T^{2} + 72316 T^{4} - 10162848 T^{6} + 1047015622 T^{8} - 80499919008 T^{10} + 4537267900156 T^{12} - 174937414418272 T^{14} + 3936588805702081 T^{16} )^{2}$$)($$( 1 - 352 T^{2} + 72316 T^{4} - 10162848 T^{6} + 1047015622 T^{8} - 80499919008 T^{10} + 4537267900156 T^{12} - 174937414418272 T^{14} + 3936588805702081 T^{16} )^{2}$$)($$( 1 - 252 T^{2} + 31130 T^{4} - 1996092 T^{6} + 62742241 T^{8} )^{4}$$)
$97$ ($$( 1 - 18 T + 97 T^{2} )^{2}$$)($$( 1 + 8 T + 97 T^{2} )^{2}$$)($$( 1 + 18 T + 97 T^{2} )^{2}$$)($$( 1 + 144 T^{2} + 9409 T^{4} )^{2}$$)($$( 1 - 144 T^{2} + 9409 T^{4} )^{2}$$)($$( 1 + 97 T^{2} )^{8}$$)($$( 1 + 135 T^{2} - 128 T^{3} + 13095 T^{4} + 912673 T^{6} )^{4}$$)($$( 1 - 22 T + 373 T^{2} - 3806 T^{3} + 41336 T^{4} - 369182 T^{5} + 3509557 T^{6} - 20078806 T^{7} + 88529281 T^{8} )^{4}$$)($$( 1 + 22 T + 373 T^{2} + 3806 T^{3} + 41336 T^{4} + 369182 T^{5} + 3509557 T^{6} + 20078806 T^{7} + 88529281 T^{8} )^{4}$$)($$( 1 + 72 T^{2} + 15314 T^{4} + 677448 T^{6} + 88529281 T^{8} )^{4}$$)