# Properties

 Label 4032.2.p Level 4032 Weight 2 Character orbit p Rep. character $$\chi_{4032}(1567,\cdot)$$ Character field $$\Q$$ Dimension 80 Newform subspaces 11 Sturm bound 1536 Trace bound 25

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$4032 = 2^{6} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4032.p (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$56$$ Character field: $$\Q$$ Newform subspaces: $$11$$ Sturm bound: $$1536$$ Trace bound: $$25$$ Distinguishing $$T_p$$: $$5$$, $$11$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(4032, [\chi])$$.

Total New Old
Modular forms 816 80 736
Cusp forms 720 80 640
Eisenstein series 96 0 96

## Trace form

 $$80q + O(q^{10})$$ $$80q + 80q^{25} + 16q^{49} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(4032, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
4032.2.p.a $$4$$ $$32.196$$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{7}+(-\beta _{1}-2\beta _{2})q^{11}-4q^{13}+\cdots$$
4032.2.p.b $$4$$ $$32.196$$ $$\Q(\zeta_{12})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{12}^{2}q^{7}-2q^{13}+(-2\zeta_{12}+2\zeta_{12}^{2}+\cdots)q^{19}+\cdots$$
4032.2.p.c $$4$$ $$32.196$$ $$\Q(\zeta_{12})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{12}q^{7}+2q^{13}+(2\zeta_{12}-2\zeta_{12}^{2}+\cdots)q^{19}+\cdots$$
4032.2.p.d $$4$$ $$32.196$$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}+\beta _{2})q^{7}+(-\beta _{1}-2\beta _{2})q^{11}+\cdots$$
4032.2.p.e $$8$$ $$32.196$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{24}^{4}q^{5}+\zeta_{24}^{5}q^{7}+\zeta_{24}^{3}q^{11}+\cdots$$
4032.2.p.f $$8$$ $$32.196$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{24}^{6}q^{5}-\zeta_{24}^{4}q^{7}-\zeta_{24}^{3}q^{11}+\cdots$$
4032.2.p.g $$8$$ $$32.196$$ $$\Q(\zeta_{24})$$ $$\Q(\sqrt{-6})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{24}^{6}q^{5}+\zeta_{24}^{4}q^{7}-\zeta_{24}^{3}q^{11}+\cdots$$
4032.2.p.h $$8$$ $$32.196$$ 8.0.629407744.1 $$\Q(\sqrt{-14})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{5}-\beta _{4}q^{7}+(-\beta _{1}+\beta _{6})q^{13}+\cdots$$
4032.2.p.i $$8$$ $$32.196$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{24}^{4}q^{5}-\zeta_{24}^{5}q^{7}-\zeta_{24}^{3}q^{11}+\cdots$$
4032.2.p.j $$12$$ $$32.196$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{5}+\beta _{10}q^{7}+\beta _{11}q^{11}-2q^{13}+\cdots$$
4032.2.p.k $$12$$ $$32.196$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{5}-\beta _{10}q^{7}+\beta _{11}q^{11}+2q^{13}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(4032, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(4032, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(56, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(168, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(224, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(448, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(504, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(672, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1344, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(2016, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ ($$( 1 + 5 T^{2} )^{4}$$)($$( 1 + 5 T^{2} )^{4}$$)($$( 1 + 5 T^{2} )^{4}$$)($$( 1 + 5 T^{2} )^{4}$$)($$( 1 + 2 T^{2} + 25 T^{4} )^{4}$$)($$( 1 + 8 T^{2} + 25 T^{4} )^{4}$$)($$( 1 + 2 T^{2} + 25 T^{4} )^{4}$$)($$( 1 + 22 T^{4} + 625 T^{8} )^{2}$$)($$( 1 + 2 T^{2} + 25 T^{4} )^{4}$$)($$( 1 + 3 T^{2} - 12 T^{3} + 15 T^{4} + 125 T^{6} )^{4}$$)($$( 1 + 3 T^{2} + 12 T^{3} + 15 T^{4} + 125 T^{6} )^{4}$$)
$7$ ($$1 - 10 T^{2} + 49 T^{4}$$)($$1 + 2 T^{2} + 49 T^{4}$$)($$1 + 2 T^{2} + 49 T^{4}$$)($$1 - 10 T^{2} + 49 T^{4}$$)($$( 1 + 2 T^{2} + 49 T^{4} )^{2}$$)($$( 1 - 10 T^{2} + 49 T^{4} )^{2}$$)($$( 1 - 10 T^{2} + 49 T^{4} )^{2}$$)($$( 1 + 7 T^{2} )^{4}$$)($$( 1 + 2 T^{2} + 49 T^{4} )^{2}$$)($$1 + 6 T^{2} - 33 T^{4} - 556 T^{6} - 1617 T^{8} + 14406 T^{10} + 117649 T^{12}$$)($$1 + 6 T^{2} - 33 T^{4} - 556 T^{6} - 1617 T^{8} + 14406 T^{10} + 117649 T^{12}$$)
$11$ ($$( 1 - 2 T^{2} + 121 T^{4} )^{2}$$)($$( 1 + 11 T^{2} )^{4}$$)($$( 1 + 11 T^{2} )^{4}$$)($$( 1 - 2 T^{2} + 121 T^{4} )^{2}$$)($$( 1 - 2 T^{2} + 121 T^{4} )^{4}$$)($$( 1 + 10 T^{2} + 121 T^{4} )^{4}$$)($$( 1 + 10 T^{2} + 121 T^{4} )^{4}$$)($$( 1 + 11 T^{2} )^{8}$$)($$( 1 - 2 T^{2} + 121 T^{4} )^{4}$$)($$( 1 + 30 T^{2} + 327 T^{4} + 2548 T^{6} + 39567 T^{8} + 439230 T^{10} + 1771561 T^{12} )^{2}$$)($$( 1 + 30 T^{2} + 327 T^{4} + 2548 T^{6} + 39567 T^{8} + 439230 T^{10} + 1771561 T^{12} )^{2}$$)
$13$ ($$( 1 + 4 T + 13 T^{2} )^{4}$$)($$( 1 + 2 T + 13 T^{2} )^{4}$$)($$( 1 - 2 T + 13 T^{2} )^{4}$$)($$( 1 - 4 T + 13 T^{2} )^{4}$$)($$( 1 + 6 T + 13 T^{2} )^{8}$$)($$( 1 + 8 T^{2} + 169 T^{4} )^{4}$$)($$( 1 + 13 T^{2} )^{8}$$)($$( 1 + 310 T^{4} + 28561 T^{8} )^{2}$$)($$( 1 - 6 T + 13 T^{2} )^{8}$$)($$( 1 + 2 T + 13 T^{2} )^{12}$$)($$( 1 - 2 T + 13 T^{2} )^{12}$$)
$17$ ($$( 1 - 10 T^{2} + 289 T^{4} )^{2}$$)($$( 1 - 17 T^{2} )^{4}$$)($$( 1 - 17 T^{2} )^{4}$$)($$( 1 - 10 T^{2} + 289 T^{4} )^{2}$$)($$( 1 - 10 T^{2} + 289 T^{4} )^{4}$$)($$( 1 - 10 T^{2} + 289 T^{4} )^{4}$$)($$( 1 - 17 T^{2} )^{8}$$)($$( 1 - 17 T^{2} )^{8}$$)($$( 1 - 10 T^{2} + 289 T^{4} )^{4}$$)($$( 1 - 30 T^{2} + 831 T^{4} - 12628 T^{6} + 240159 T^{8} - 2505630 T^{10} + 24137569 T^{12} )^{2}$$)($$( 1 - 30 T^{2} + 831 T^{4} - 12628 T^{6} + 240159 T^{8} - 2505630 T^{10} + 24137569 T^{12} )^{2}$$)
$19$ ($$( 1 - 22 T^{2} + 361 T^{4} )^{2}$$)($$( 1 + 26 T^{2} + 361 T^{4} )^{2}$$)($$( 1 + 26 T^{2} + 361 T^{4} )^{2}$$)($$( 1 - 22 T^{2} + 361 T^{4} )^{2}$$)($$( 1 - 19 T^{2} )^{8}$$)($$( 1 - 20 T^{2} + 361 T^{4} )^{4}$$)($$( 1 - 19 T^{2} )^{8}$$)($$( 1 - 650 T^{4} + 130321 T^{8} )^{2}$$)($$( 1 - 19 T^{2} )^{8}$$)($$( 1 - 66 T^{2} + 2343 T^{4} - 54844 T^{6} + 845823 T^{8} - 8601186 T^{10} + 47045881 T^{12} )^{2}$$)($$( 1 - 66 T^{2} + 2343 T^{4} - 54844 T^{6} + 845823 T^{8} - 8601186 T^{10} + 47045881 T^{12} )^{2}$$)
$23$ ($$( 1 - 10 T^{2} + 529 T^{4} )^{2}$$)($$( 1 - 23 T^{2} )^{4}$$)($$( 1 - 23 T^{2} )^{4}$$)($$( 1 - 10 T^{2} + 529 T^{4} )^{2}$$)($$( 1 + 26 T^{2} + 529 T^{4} )^{4}$$)($$( 1 - 23 T^{2} )^{8}$$)($$( 1 - 23 T^{2} )^{8}$$)($$( 1 - 10 T^{2} + 529 T^{4} )^{4}$$)($$( 1 + 26 T^{2} + 529 T^{4} )^{4}$$)($$( 1 - 78 T^{2} + 3567 T^{4} - 98836 T^{6} + 1886943 T^{8} - 21827598 T^{10} + 148035889 T^{12} )^{2}$$)($$( 1 - 78 T^{2} + 3567 T^{4} - 98836 T^{6} + 1886943 T^{8} - 21827598 T^{10} + 148035889 T^{12} )^{2}$$)
$29$ ($$( 1 - 34 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 29 T^{2} )^{4}$$)($$( 1 - 29 T^{2} )^{4}$$)($$( 1 - 34 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 29 T^{2} )^{8}$$)($$( 1 - 29 T^{2} )^{8}$$)($$( 1 + 50 T^{2} + 841 T^{4} )^{4}$$)($$( 1 - 29 T^{2} )^{8}$$)($$( 1 - 29 T^{2} )^{8}$$)($$( 1 - 42 T^{2} + 1575 T^{4} - 60076 T^{6} + 1324575 T^{8} - 29705802 T^{10} + 594823321 T^{12} )^{2}$$)($$( 1 - 42 T^{2} + 1575 T^{4} - 60076 T^{6} + 1324575 T^{8} - 29705802 T^{10} + 594823321 T^{12} )^{2}$$)
$31$ ($$( 1 + 38 T^{2} + 961 T^{4} )^{2}$$)($$( 1 - 46 T^{2} + 961 T^{4} )^{2}$$)($$( 1 - 46 T^{2} + 961 T^{4} )^{2}$$)($$( 1 + 38 T^{2} + 961 T^{4} )^{2}$$)($$( 1 + 50 T^{2} + 961 T^{4} )^{4}$$)($$( 1 + 38 T^{2} + 961 T^{4} )^{4}$$)($$( 1 + 38 T^{2} + 961 T^{4} )^{4}$$)($$( 1 + 31 T^{2} )^{8}$$)($$( 1 + 50 T^{2} + 961 T^{4} )^{4}$$)($$( 1 + 54 T^{2} + 3471 T^{4} + 101684 T^{6} + 3335631 T^{8} + 49870134 T^{10} + 887503681 T^{12} )^{2}$$)($$( 1 + 54 T^{2} + 3471 T^{4} + 101684 T^{6} + 3335631 T^{8} + 49870134 T^{10} + 887503681 T^{12} )^{2}$$)
$37$ ($$( 1 + 22 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 - 10 T + 37 T^{2} )^{2}( 1 + 10 T + 37 T^{2} )^{2}$$)($$( 1 - 10 T + 37 T^{2} )^{2}( 1 + 10 T + 37 T^{2} )^{2}$$)($$( 1 + 22 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 - 10 T + 37 T^{2} )^{4}( 1 + 10 T + 37 T^{2} )^{4}$$)($$( 1 - 10 T + 37 T^{2} )^{4}( 1 + 10 T + 37 T^{2} )^{4}$$)($$( 1 - 37 T^{2} )^{8}$$)($$( 1 - 37 T^{2} )^{8}$$)($$( 1 - 10 T + 37 T^{2} )^{4}( 1 + 10 T + 37 T^{2} )^{4}$$)($$( 1 - 126 T^{2} + 7863 T^{4} - 337412 T^{6} + 10764447 T^{8} - 236144286 T^{10} + 2565726409 T^{12} )^{2}$$)($$( 1 - 126 T^{2} + 7863 T^{4} - 337412 T^{6} + 10764447 T^{8} - 236144286 T^{10} + 2565726409 T^{12} )^{2}$$)
$41$ ($$( 1 - 58 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 - 41 T^{2} )^{4}$$)($$( 1 - 41 T^{2} )^{4}$$)($$( 1 - 58 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 - 58 T^{2} + 1681 T^{4} )^{4}$$)($$( 1 + 14 T^{2} + 1681 T^{4} )^{4}$$)($$( 1 - 41 T^{2} )^{8}$$)($$( 1 - 41 T^{2} )^{8}$$)($$( 1 - 58 T^{2} + 1681 T^{4} )^{4}$$)($$( 1 - 126 T^{2} + 6639 T^{4} - 258580 T^{6} + 11160159 T^{8} - 356045886 T^{10} + 4750104241 T^{12} )^{2}$$)($$( 1 - 126 T^{2} + 6639 T^{4} - 258580 T^{6} + 11160159 T^{8} - 356045886 T^{10} + 4750104241 T^{12} )^{2}$$)
$43$ ($$( 1 - 10 T^{2} + 1849 T^{4} )^{2}$$)($$( 1 - 22 T^{2} + 1849 T^{4} )^{2}$$)($$( 1 - 22 T^{2} + 1849 T^{4} )^{2}$$)($$( 1 - 10 T^{2} + 1849 T^{4} )^{2}$$)($$( 1 + 74 T^{2} + 1849 T^{4} )^{4}$$)($$( 1 + 74 T^{2} + 1849 T^{4} )^{4}$$)($$( 1 + 43 T^{2} )^{8}$$)($$( 1 + 43 T^{2} )^{8}$$)($$( 1 + 74 T^{2} + 1849 T^{4} )^{4}$$)($$( 1 + 78 T^{2} + 4503 T^{4} + 248420 T^{6} + 8326047 T^{8} + 266666478 T^{10} + 6321363049 T^{12} )^{2}$$)($$( 1 + 78 T^{2} + 4503 T^{4} + 248420 T^{6} + 8326047 T^{8} + 266666478 T^{10} + 6321363049 T^{12} )^{2}$$)
$47$ ($$( 1 - 2 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 + 47 T^{2} )^{4}$$)($$( 1 + 47 T^{2} )^{4}$$)($$( 1 - 2 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 - 2 T^{2} + 2209 T^{4} )^{4}$$)($$( 1 + 70 T^{2} + 2209 T^{4} )^{4}$$)($$( 1 + 47 T^{2} )^{8}$$)($$( 1 + 47 T^{2} )^{8}$$)($$( 1 - 2 T^{2} + 2209 T^{4} )^{4}$$)($$( 1 + 90 T^{2} + 7791 T^{4} + 391852 T^{6} + 17210319 T^{8} + 439171290 T^{10} + 10779215329 T^{12} )^{2}$$)($$( 1 + 90 T^{2} + 7791 T^{4} + 391852 T^{6} + 17210319 T^{8} + 439171290 T^{10} + 10779215329 T^{12} )^{2}$$)
$53$ ($$( 1 - 82 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 53 T^{2} )^{4}$$)($$( 1 - 53 T^{2} )^{4}$$)($$( 1 - 82 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 10 T^{2} + 2809 T^{4} )^{4}$$)($$( 1 + 86 T^{2} + 2809 T^{4} )^{4}$$)($$( 1 - 94 T^{2} + 2809 T^{4} )^{4}$$)($$( 1 - 53 T^{2} )^{8}$$)($$( 1 - 10 T^{2} + 2809 T^{4} )^{4}$$)($$( 1 - 186 T^{2} + 19575 T^{4} - 1258444 T^{6} + 54986175 T^{8} - 1467629466 T^{10} + 22164361129 T^{12} )^{2}$$)($$( 1 - 186 T^{2} + 19575 T^{4} - 1258444 T^{6} + 54986175 T^{8} - 1467629466 T^{10} + 22164361129 T^{12} )^{2}$$)
$59$ ($$( 1 + 26 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 - 59 T^{2} )^{4}$$)($$( 1 - 59 T^{2} )^{4}$$)($$( 1 + 26 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 - 86 T^{2} + 3481 T^{4} )^{4}$$)($$( 1 - 68 T^{2} + 3481 T^{4} )^{4}$$)($$( 1 + 10 T^{2} + 3481 T^{4} )^{4}$$)($$( 1 - 1130 T^{4} + 12117361 T^{8} )^{2}$$)($$( 1 - 86 T^{2} + 3481 T^{4} )^{4}$$)($$( 1 - 258 T^{2} + 31863 T^{4} - 2365180 T^{6} + 110915103 T^{8} - 3126279138 T^{10} + 42180533641 T^{12} )^{2}$$)($$( 1 - 258 T^{2} + 31863 T^{4} - 2365180 T^{6} + 110915103 T^{8} - 3126279138 T^{10} + 42180533641 T^{12} )^{2}$$)
$61$ ($$( 1 - 4 T + 61 T^{2} )^{4}$$)($$( 1 - 14 T + 61 T^{2} )^{4}$$)($$( 1 + 14 T + 61 T^{2} )^{4}$$)($$( 1 + 4 T + 61 T^{2} )^{4}$$)($$( 1 + 6 T + 61 T^{2} )^{8}$$)($$( 1 + 104 T^{2} + 3721 T^{4} )^{4}$$)($$( 1 + 61 T^{2} )^{8}$$)($$( 1 - 7370 T^{4} + 13845841 T^{8} )^{2}$$)($$( 1 - 6 T + 61 T^{2} )^{8}$$)($$( 1 - 18 T + 195 T^{2} - 1484 T^{3} + 11895 T^{4} - 66978 T^{5} + 226981 T^{6} )^{4}$$)($$( 1 + 18 T + 195 T^{2} + 1484 T^{3} + 11895 T^{4} + 66978 T^{5} + 226981 T^{6} )^{4}$$)
$67$ ($$( 1 + 67 T^{2} )^{4}$$)($$( 1 + 122 T^{2} + 4489 T^{4} )^{2}$$)($$( 1 + 122 T^{2} + 4489 T^{4} )^{2}$$)($$( 1 + 67 T^{2} )^{4}$$)($$( 1 + 26 T^{2} + 4489 T^{4} )^{4}$$)($$( 1 + 26 T^{2} + 4489 T^{4} )^{4}$$)($$( 1 + 67 T^{2} )^{8}$$)($$( 1 + 67 T^{2} )^{8}$$)($$( 1 + 26 T^{2} + 4489 T^{4} )^{4}$$)($$( 1 + 78 T^{2} + 3399 T^{4} + 64676 T^{6} + 15258111 T^{8} + 1571787438 T^{10} + 90458382169 T^{12} )^{2}$$)($$( 1 + 78 T^{2} + 3399 T^{4} + 64676 T^{6} + 15258111 T^{8} + 1571787438 T^{10} + 90458382169 T^{12} )^{2}$$)
$71$ ($$( 1 - 106 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 71 T^{2} )^{4}$$)($$( 1 - 71 T^{2} )^{4}$$)($$( 1 - 106 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 70 T^{2} + 5041 T^{4} )^{4}$$)($$( 1 - 106 T^{2} + 5041 T^{4} )^{4}$$)($$( 1 - 71 T^{2} )^{8}$$)($$( 1 + 110 T^{2} + 5041 T^{4} )^{4}$$)($$( 1 - 70 T^{2} + 5041 T^{4} )^{4}$$)($$( 1 - 222 T^{2} + 23727 T^{4} - 1839796 T^{6} + 119607807 T^{8} - 5641393182 T^{10} + 128100283921 T^{12} )^{2}$$)($$( 1 - 222 T^{2} + 23727 T^{4} - 1839796 T^{6} + 119607807 T^{8} - 5641393182 T^{10} + 128100283921 T^{12} )^{2}$$)
$73$ ($$( 1 - 73 T^{2} )^{4}$$)($$( 1 - 10 T + 73 T^{2} )^{2}( 1 + 10 T + 73 T^{2} )^{2}$$)($$( 1 - 10 T + 73 T^{2} )^{2}( 1 + 10 T + 73 T^{2} )^{2}$$)($$( 1 - 73 T^{2} )^{4}$$)($$( 1 - 10 T + 73 T^{2} )^{4}( 1 + 10 T + 73 T^{2} )^{4}$$)($$( 1 - 122 T^{2} + 5329 T^{4} )^{4}$$)($$( 1 - 14 T + 73 T^{2} )^{4}( 1 + 14 T + 73 T^{2} )^{4}$$)($$( 1 - 73 T^{2} )^{8}$$)($$( 1 - 10 T + 73 T^{2} )^{4}( 1 + 10 T + 73 T^{2} )^{4}$$)($$( 1 - 150 T^{2} + 16575 T^{4} - 1350452 T^{6} + 88328175 T^{8} - 4259736150 T^{10} + 151334226289 T^{12} )^{2}$$)($$( 1 - 150 T^{2} + 16575 T^{4} - 1350452 T^{6} + 88328175 T^{8} - 4259736150 T^{10} + 151334226289 T^{12} )^{2}$$)
$79$ ($$( 1 - 154 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 - 142 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 - 142 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 - 154 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 - 142 T^{2} + 6241 T^{4} )^{4}$$)($$( 1 + 38 T^{2} + 6241 T^{4} )^{4}$$)($$( 1 - 58 T^{2} + 6241 T^{4} )^{4}$$)($$( 1 - 130 T^{2} + 6241 T^{4} )^{4}$$)($$( 1 - 142 T^{2} + 6241 T^{4} )^{4}$$)($$( 1 - 234 T^{2} + 22767 T^{4} - 1649932 T^{6} + 142088847 T^{8} - 9114318954 T^{10} + 243087455521 T^{12} )^{2}$$)($$( 1 - 234 T^{2} + 22767 T^{4} - 1649932 T^{6} + 142088847 T^{8} - 9114318954 T^{10} + 243087455521 T^{12} )^{2}$$)
$83$ ($$( 1 - 22 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 - 83 T^{2} )^{4}$$)($$( 1 - 83 T^{2} )^{4}$$)($$( 1 - 22 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 - 134 T^{2} + 6889 T^{4} )^{4}$$)($$( 1 - 68 T^{2} + 6889 T^{4} )^{4}$$)($$( 1 - 134 T^{2} + 6889 T^{4} )^{4}$$)($$( 1 - 13130 T^{4} + 47458321 T^{8} )^{2}$$)($$( 1 - 134 T^{2} + 6889 T^{4} )^{4}$$)($$( 1 - 258 T^{2} + 42087 T^{4} - 4123708 T^{6} + 289937343 T^{8} - 12244246818 T^{10} + 326940373369 T^{12} )^{2}$$)($$( 1 - 258 T^{2} + 42087 T^{4} - 4123708 T^{6} + 289937343 T^{8} - 12244246818 T^{10} + 326940373369 T^{12} )^{2}$$)
$89$ ($$( 1 - 154 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 - 89 T^{2} )^{4}$$)($$( 1 - 89 T^{2} )^{4}$$)($$( 1 - 154 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 + 38 T^{2} + 7921 T^{4} )^{4}$$)($$( 1 + 38 T^{2} + 7921 T^{4} )^{4}$$)($$( 1 - 89 T^{2} )^{8}$$)($$( 1 - 89 T^{2} )^{8}$$)($$( 1 + 38 T^{2} + 7921 T^{4} )^{4}$$)($$( 1 - 222 T^{2} + 32463 T^{4} - 3429460 T^{6} + 257139423 T^{8} - 13928777502 T^{10} + 496981290961 T^{12} )^{2}$$)($$( 1 - 222 T^{2} + 32463 T^{4} - 3429460 T^{6} + 257139423 T^{8} - 13928777502 T^{10} + 496981290961 T^{12} )^{2}$$)
$97$ ($$( 1 - 98 T^{2} + 9409 T^{4} )^{2}$$)($$( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2}$$)($$( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2}$$)($$( 1 - 98 T^{2} + 9409 T^{4} )^{2}$$)($$( 1 - 14 T + 97 T^{2} )^{4}( 1 + 14 T + 97 T^{2} )^{4}$$)($$( 1 - 170 T^{2} + 9409 T^{4} )^{4}$$)($$( 1 - 2 T + 97 T^{2} )^{4}( 1 + 2 T + 97 T^{2} )^{4}$$)($$( 1 - 97 T^{2} )^{8}$$)($$( 1 - 14 T + 97 T^{2} )^{4}( 1 + 14 T + 97 T^{2} )^{4}$$)($$( 1 - 390 T^{2} + 77391 T^{4} - 9349652 T^{6} + 728171919 T^{8} - 34526419590 T^{10} + 832972004929 T^{12} )^{2}$$)($$( 1 - 390 T^{2} + 77391 T^{4} - 9349652 T^{6} + 728171919 T^{8} - 34526419590 T^{10} + 832972004929 T^{12} )^{2}$$)