# Properties

 Label 304.6.i.d Level $304$ Weight $6$ Character orbit 304.i Analytic conductor $48.757$ Analytic rank $0$ Dimension $18$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 304.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$48.7566812231$$ Analytic rank: $$0$$ Dimension: $$18$$ Relative dimension: $$9$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ Defining polynomial: $$x^{18} - 5 x^{17} - 3057 x^{16} + 14564 x^{15} + 3829838 x^{14} - 15907074 x^{13} - 2546775754 x^{12} + 7879525640 x^{11} + 976140188391 x^{10} - 1502012839499 x^{9} - 219575028272723 x^{8} - 104014033904304 x^{7} + 28115925509664606 x^{6} + 76771209356887914 x^{5} - 1810997176122162810 x^{4} - 9301294316540690604 x^{3} + 37701980111568799233 x^{2} + 354684213311311368351 x + 668276771042404734183$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{16}\cdot 3^{3}\cdot 5^{2}$$ Twist minimal: no (minimal twist has level 76) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{17}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{3} + ( -\beta_{1} + \beta_{5} - \beta_{7} ) q^{5} + ( -18 + 2 \beta_{2} + \beta_{6} ) q^{7} + ( -99 - 99 \beta_{1} + 3 \beta_{3} - \beta_{7} + \beta_{12} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{3} + ( -\beta_{1} + \beta_{5} - \beta_{7} ) q^{5} + ( -18 + 2 \beta_{2} + \beta_{6} ) q^{7} + ( -99 - 99 \beta_{1} + 3 \beta_{3} - \beta_{7} + \beta_{12} ) q^{9} + ( 18 + \beta_{2} + \beta_{8} + \beta_{15} ) q^{11} + ( 23 + 23 \beta_{1} - 5 \beta_{3} + 3 \beta_{7} + \beta_{11} - \beta_{12} ) q^{13} + ( 15 + 15 \beta_{1} + 11 \beta_{3} - 3 \beta_{7} + \beta_{12} + \beta_{14} ) q^{15} + ( 1 - 16 \beta_{1} - 15 \beta_{2} + 16 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} + 2 \beta_{10} - \beta_{12} + \beta_{13} - \beta_{15} + \beta_{17} ) q^{17} + ( 172 + 248 \beta_{1} - 7 \beta_{2} + 14 \beta_{3} - \beta_{4} + 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - \beta_{8} - 3 \beta_{9} - \beta_{11} + \beta_{14} - 2 \beta_{15} + \beta_{17} ) q^{19} + ( 1 + 622 \beta_{1} + 30 \beta_{2} - 29 \beta_{3} - 8 \beta_{4} - 3 \beta_{5} + 6 \beta_{6} + 3 \beta_{7} + 6 \beta_{9} + 2 \beta_{10} - 8 \beta_{12} + \beta_{13} - 5 \beta_{15} - 2 \beta_{16} + \beta_{17} ) q^{21} + ( 377 + 378 \beta_{1} - \beta_{2} - 16 \beta_{3} + \beta_{8} - 6 \beta_{9} - \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - \beta_{14} + \beta_{17} ) q^{23} + ( -775 - 775 \beta_{1} + 20 \beta_{3} - 10 \beta_{7} - 4 \beta_{8} - 10 \beta_{9} + \beta_{11} + 2 \beta_{12} - 2 \beta_{14} ) q^{25} + ( -790 - \beta_{1} + 80 \beta_{2} - \beta_{3} - 8 \beta_{4} - 18 \beta_{5} + 15 \beta_{6} + 4 \beta_{8} - \beta_{10} - \beta_{11} + 2 \beta_{13} + \beta_{14} + 4 \beta_{15} - \beta_{16} - 2 \beta_{17} ) q^{27} + ( -833 - 832 \beta_{1} - \beta_{2} - 59 \beta_{3} + 12 \beta_{7} + 3 \beta_{8} + 6 \beta_{9} - \beta_{10} + 4 \beta_{11} + 7 \beta_{12} - 2 \beta_{14} + \beta_{17} ) q^{29} + ( 561 + \beta_{1} + 18 \beta_{2} + \beta_{3} + 7 \beta_{4} - 13 \beta_{5} + 11 \beta_{6} + 9 \beta_{8} + \beta_{10} + \beta_{11} - 4 \beta_{13} + 2 \beta_{14} + 9 \beta_{15} - 2 \beta_{16} + 2 \beta_{17} ) q^{31} + ( 1 + 318 \beta_{1} + 46 \beta_{2} - 45 \beta_{3} - 12 \beta_{4} - 28 \beta_{5} - 24 \beta_{6} + 28 \beta_{7} - 24 \beta_{9} + 2 \beta_{10} + 6 \beta_{11} - 12 \beta_{12} + 7 \beta_{13} - \beta_{15} + \beta_{17} ) q^{33} + ( 1728 \beta_{1} + 41 \beta_{2} - 41 \beta_{3} - 27 \beta_{4} - 97 \beta_{5} - 15 \beta_{6} + 97 \beta_{7} - 15 \beta_{9} + 6 \beta_{11} - 27 \beta_{12} + 6 \beta_{13} + 5 \beta_{15} - \beta_{16} ) q^{35} + ( 1454 - 81 \beta_{2} + 9 \beta_{4} - 42 \beta_{5} - 20 \beta_{6} - 8 \beta_{8} - 3 \beta_{13} + 4 \beta_{14} - 8 \beta_{15} - 4 \beta_{16} ) q^{37} + ( 1634 + \beta_{1} - 193 \beta_{2} + \beta_{3} + 20 \beta_{4} - 6 \beta_{5} + 9 \beta_{6} - 15 \beta_{8} + \beta_{10} + \beta_{11} + 4 \beta_{13} - 3 \beta_{14} - 15 \beta_{15} + 3 \beta_{16} + 2 \beta_{17} ) q^{39} + ( 830 \beta_{1} - 47 \beta_{2} + 47 \beta_{3} - 15 \beta_{4} + 55 \beta_{5} - 55 \beta_{7} - 15 \beta_{12} + 16 \beta_{15} ) q^{41} + ( 1 - 881 \beta_{1} - 148 \beta_{2} + 149 \beta_{3} - 6 \beta_{4} + 16 \beta_{5} - 29 \beta_{6} - 16 \beta_{7} - 29 \beta_{9} + 2 \beta_{10} + 5 \beta_{11} - 6 \beta_{12} + 6 \beta_{13} - 11 \beta_{15} - 5 \beta_{16} + \beta_{17} ) q^{43} + ( -3339 + \beta_{1} + 272 \beta_{2} + \beta_{3} - 27 \beta_{4} - 103 \beta_{5} + 90 \beta_{6} - 5 \beta_{8} + \beta_{10} + \beta_{11} + 10 \beta_{13} + 2 \beta_{14} - 5 \beta_{15} - 2 \beta_{16} + 2 \beta_{17} ) q^{45} + ( -4173 - 4173 \beta_{1} - 53 \beta_{3} - 19 \beta_{7} - 32 \beta_{8} - 30 \beta_{9} + 12 \beta_{11} + 21 \beta_{12} + \beta_{14} ) q^{47} + ( 6809 - \beta_{1} - 473 \beta_{2} - \beta_{3} + 66 \beta_{4} - 44 \beta_{5} - 14 \beta_{6} + 9 \beta_{8} - \beta_{10} - \beta_{11} + 6 \beta_{13} - 2 \beta_{14} + 9 \beta_{15} + 2 \beta_{16} - 2 \beta_{17} ) q^{49} + ( -5229 - 5230 \beta_{1} + \beta_{2} - 71 \beta_{3} - 163 \beta_{7} - 16 \beta_{8} + 96 \beta_{9} + \beta_{10} + 2 \beta_{11} + 59 \beta_{12} - 2 \beta_{14} - \beta_{17} ) q^{51} + ( -2213 - 2212 \beta_{1} - \beta_{2} + 324 \beta_{3} + 228 \beta_{7} - 13 \beta_{8} - 66 \beta_{9} - \beta_{10} - 5 \beta_{11} + 22 \beta_{12} + 10 \beta_{14} + \beta_{17} ) q^{53} + ( 10 + 70 \beta_{1} + 172 \beta_{2} - 162 \beta_{3} - 14 \beta_{4} + 202 \beta_{5} + 71 \beta_{6} - 202 \beta_{7} + 71 \beta_{9} + 20 \beta_{10} + 10 \beta_{11} - 14 \beta_{12} + 20 \beta_{13} - 30 \beta_{15} - 4 \beta_{16} + 10 \beta_{17} ) q^{55} + ( -4138 - 1961 \beta_{1} - 108 \beta_{2} - 117 \beta_{3} - \beta_{4} - 175 \beta_{5} + 126 \beta_{6} - 92 \beta_{7} - 3 \beta_{8} + 78 \beta_{9} - \beta_{10} + 2 \beta_{11} + 54 \beta_{12} + 2 \beta_{13} + 8 \beta_{14} + 13 \beta_{15} - 2 \beta_{16} + 10 \beta_{17} ) q^{57} + ( 11 - 8219 \beta_{1} - 167 \beta_{2} + 178 \beta_{3} + 28 \beta_{4} - 102 \beta_{5} - 129 \beta_{6} + 102 \beta_{7} - 129 \beta_{9} + 22 \beta_{10} + 3 \beta_{11} + 28 \beta_{12} + 14 \beta_{13} - 10 \beta_{15} - 17 \beta_{16} + 11 \beta_{17} ) q^{59} + ( -859 - 849 \beta_{1} - 10 \beta_{2} + 42 \beta_{3} + 19 \beta_{7} - 2 \beta_{8} + 120 \beta_{9} - 10 \beta_{10} - 4 \beta_{11} + 30 \beta_{12} - 4 \beta_{14} + 10 \beta_{17} ) q^{61} + ( 5288 + 5288 \beta_{1} - 1660 \beta_{3} + 504 \beta_{7} + 240 \beta_{9} + 12 \beta_{11} - 28 \beta_{12} - 16 \beta_{14} ) q^{63} + ( 10477 - 10 \beta_{1} + 139 \beta_{2} - 10 \beta_{3} + 55 \beta_{4} - 29 \beta_{5} - 186 \beta_{6} + 10 \beta_{8} - 10 \beta_{10} - 10 \beta_{11} + 5 \beta_{13} + 10 \beta_{14} + 10 \beta_{15} - 10 \beta_{16} - 20 \beta_{17} ) q^{65} + ( 3019 + 3029 \beta_{1} - 10 \beta_{2} + 542 \beta_{3} - 103 \beta_{7} - 27 \beta_{8} - 29 \beta_{9} - 10 \beta_{10} + 34 \beta_{11} - 71 \beta_{12} - 19 \beta_{14} + 10 \beta_{17} ) q^{67} + ( 6101 + 10 \beta_{1} - 1004 \beta_{2} + 10 \beta_{3} - 2 \beta_{4} + 547 \beta_{5} - 108 \beta_{6} + 14 \beta_{8} + 10 \beta_{10} + 10 \beta_{11} - 8 \beta_{13} + 16 \beta_{14} + 14 \beta_{15} - 16 \beta_{16} + 20 \beta_{17} ) q^{69} + ( 11 - 5633 \beta_{1} - 1382 \beta_{2} + 1393 \beta_{3} + 10 \beta_{4} + 172 \beta_{5} + 201 \beta_{6} - 172 \beta_{7} + 201 \beta_{9} + 22 \beta_{10} + 33 \beta_{11} + 10 \beta_{12} + 44 \beta_{13} - 13 \beta_{15} - \beta_{16} + 11 \beta_{17} ) q^{71} + ( -1 + 1430 \beta_{1} + 1134 \beta_{2} - 1135 \beta_{3} - 2 \beta_{4} + 700 \beta_{5} + 154 \beta_{6} - 700 \beta_{7} + 154 \beta_{9} - 2 \beta_{10} + 27 \beta_{11} - 2 \beta_{12} + 26 \beta_{13} + 17 \beta_{15} - 6 \beta_{16} - \beta_{17} ) q^{73} + ( -6562 + 1181 \beta_{2} - 28 \beta_{4} + 620 \beta_{5} + 60 \beta_{6} - 31 \beta_{8} - 24 \beta_{13} + 32 \beta_{14} - 31 \beta_{15} - 32 \beta_{16} ) q^{75} + ( -1529 + 11 \beta_{1} + 1437 \beta_{2} + 11 \beta_{3} - 2 \beta_{4} + 325 \beta_{5} - 6 \beta_{6} - 39 \beta_{8} + 11 \beta_{10} + 11 \beta_{11} + 35 \beta_{13} - 26 \beta_{14} - 39 \beta_{15} + 26 \beta_{16} + 22 \beta_{17} ) q^{77} + ( 1 + 3095 \beta_{1} + 1476 \beta_{2} - 1475 \beta_{3} + 10 \beta_{4} - 540 \beta_{5} - 21 \beta_{6} + 540 \beta_{7} - 21 \beta_{9} + 2 \beta_{10} + 7 \beta_{11} + 10 \beta_{12} + 8 \beta_{13} + 57 \beta_{15} - 3 \beta_{16} + \beta_{17} ) q^{79} + ( 10 + 1696 \beta_{1} + 2235 \beta_{2} - 2225 \beta_{3} - 89 \beta_{4} - 457 \beta_{5} + 60 \beta_{6} + 457 \beta_{7} + 60 \beta_{9} + 20 \beta_{10} + 30 \beta_{11} - 89 \beta_{12} + 40 \beta_{13} - 34 \beta_{15} - 44 \beta_{16} + 10 \beta_{17} ) q^{81} + ( -4505 + 11 \beta_{1} + 280 \beta_{2} + 11 \beta_{3} - 134 \beta_{4} + 68 \beta_{5} - 117 \beta_{6} - 41 \beta_{8} + 11 \beta_{10} + 11 \beta_{11} + 74 \beta_{13} + 15 \beta_{14} - 41 \beta_{15} - 15 \beta_{16} + 22 \beta_{17} ) q^{83} + ( -5427 - 5427 \beta_{1} + 1275 \beta_{3} + 303 \beta_{7} - 144 \beta_{8} + 81 \beta_{11} + 15 \beta_{12} + 12 \beta_{14} ) q^{85} + ( 19212 - 11 \beta_{1} + 2351 \beta_{2} - 11 \beta_{3} + 124 \beta_{4} + 598 \beta_{5} - 24 \beta_{6} + 51 \beta_{8} - 11 \beta_{10} - 11 \beta_{11} + 40 \beta_{13} - 19 \beta_{14} + 51 \beta_{15} + 19 \beta_{16} - 22 \beta_{17} ) q^{87} + ( -17066 - 17077 \beta_{1} + 11 \beta_{2} + 883 \beta_{3} + 347 \beta_{7} - 81 \beta_{8} - 24 \beta_{9} + 11 \beta_{10} - 23 \beta_{11} + 151 \beta_{12} - 16 \beta_{14} - 11 \beta_{17} ) q^{89} + ( 596 + 606 \beta_{1} - 10 \beta_{2} - 94 \beta_{3} - 1170 \beta_{7} - 86 \beta_{8} + 50 \beta_{9} - 10 \beta_{10} - 28 \beta_{11} + 138 \beta_{12} + 40 \beta_{14} + 10 \beta_{17} ) q^{91} + ( 34 + 5084 \beta_{1} - 765 \beta_{2} + 799 \beta_{3} - 111 \beta_{4} - 886 \beta_{5} + 72 \beta_{6} + 886 \beta_{7} + 72 \beta_{9} + 68 \beta_{10} + 81 \beta_{11} - 111 \beta_{12} + 115 \beta_{13} - 154 \beta_{15} - 36 \beta_{16} + 34 \beta_{17} ) q^{93} + ( -10194 - 15960 \beta_{1} + 2632 \beta_{2} + 506 \beta_{3} - 53 \beta_{4} + 541 \beta_{5} + 36 \beta_{6} + 334 \beta_{7} + \beta_{8} - 87 \beta_{9} - 11 \beta_{10} + 42 \beta_{11} + 76 \beta_{12} + 8 \beta_{13} + 20 \beta_{14} + 98 \beta_{15} - 21 \beta_{16} + 35 \beta_{17} ) q^{95} + ( 45 - 11983 \beta_{1} - 1061 \beta_{2} + 1106 \beta_{3} - \beta_{4} + 721 \beta_{5} + 134 \beta_{6} - 721 \beta_{7} + 134 \beta_{9} + 90 \beta_{10} + 19 \beta_{11} - \beta_{12} + 64 \beta_{13} - 5 \beta_{15} - 50 \beta_{16} + 45 \beta_{17} ) q^{97} + ( 21036 + 21070 \beta_{1} - 34 \beta_{2} - 2191 \beta_{3} - 99 \beta_{7} - 3 \beta_{8} - 27 \beta_{9} - 34 \beta_{10} + 34 \beta_{11} + 21 \beta_{12} + 17 \beta_{14} + 34 \beta_{17} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18q + 11q^{3} + 11q^{5} - 336q^{7} - 902q^{9} + O(q^{10})$$ $$18q + 11q^{3} + 11q^{5} - 336q^{7} - 902q^{9} + 320q^{11} + 227q^{13} + 101q^{15} + 179q^{17} + 868q^{19} - 5700q^{21} + 3425q^{23} - 7054q^{25} - 14722q^{27} - 7349q^{29} + 9960q^{31} - 2998q^{33} - 15888q^{35} + 26444q^{37} + 30246q^{39} - 7311q^{41} + 8283q^{43} - 62164q^{45} - 37603q^{47} + 124738q^{49} - 47227q^{51} - 20337q^{53} - 716q^{55} - 57555q^{57} + 74455q^{59} - 7569q^{61} + 52544q^{63} + 188998q^{65} + 26177q^{67} + 116282q^{69} + 53463q^{71} - 14103q^{73} - 120912q^{75} - 31960q^{77} - 31825q^{79} - 21137q^{81} - 82600q^{83} - 50787q^{85} + 339766q^{87} - 155197q^{89} + 2800q^{91} - 46460q^{93} - 49315q^{95} + 111241q^{97} + 193544q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} - 5 x^{17} - 3057 x^{16} + 14564 x^{15} + 3829838 x^{14} - 15907074 x^{13} - 2546775754 x^{12} + 7879525640 x^{11} + 976140188391 x^{10} - 1502012839499 x^{9} - 219575028272723 x^{8} - 104014033904304 x^{7} + 28115925509664606 x^{6} + 76771209356887914 x^{5} - 1810997176122162810 x^{4} - 9301294316540690604 x^{3} + 37701980111568799233 x^{2} + 354684213311311368351 x + 668276771042404734183$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-$$$$96\!\cdots\!86$$$$\nu^{17} +$$$$10\!\cdots\!81$$$$\nu^{16} +$$$$28\!\cdots\!56$$$$\nu^{15} -$$$$30\!\cdots\!00$$$$\nu^{14} -$$$$34\!\cdots\!68$$$$\nu^{13} +$$$$34\!\cdots\!52$$$$\nu^{12} +$$$$22\!\cdots\!12$$$$\nu^{11} -$$$$20\!\cdots\!32$$$$\nu^{10} -$$$$79\!\cdots\!14$$$$\nu^{9} +$$$$59\!\cdots\!38$$$$\nu^{8} +$$$$16\!\cdots\!20$$$$\nu^{7} -$$$$87\!\cdots\!76$$$$\nu^{6} -$$$$20\!\cdots\!00$$$$\nu^{5} +$$$$47\!\cdots\!96$$$$\nu^{4} +$$$$13\!\cdots\!24$$$$\nu^{3} +$$$$67\!\cdots\!60$$$$\nu^{2} -$$$$36\!\cdots\!98$$$$\nu -$$$$10\!\cdots\!18$$$$)/$$$$11\!\cdots\!75$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$96\!\cdots\!86$$$$\nu^{17} +$$$$10\!\cdots\!81$$$$\nu^{16} +$$$$28\!\cdots\!56$$$$\nu^{15} -$$$$30\!\cdots\!00$$$$\nu^{14} -$$$$34\!\cdots\!68$$$$\nu^{13} +$$$$34\!\cdots\!52$$$$\nu^{12} +$$$$22\!\cdots\!12$$$$\nu^{11} -$$$$20\!\cdots\!32$$$$\nu^{10} -$$$$79\!\cdots\!14$$$$\nu^{9} +$$$$59\!\cdots\!38$$$$\nu^{8} +$$$$16\!\cdots\!20$$$$\nu^{7} -$$$$87\!\cdots\!76$$$$\nu^{6} -$$$$20\!\cdots\!00$$$$\nu^{5} +$$$$47\!\cdots\!96$$$$\nu^{4} +$$$$13\!\cdots\!24$$$$\nu^{3} +$$$$67\!\cdots\!60$$$$\nu^{2} -$$$$35\!\cdots\!23$$$$\nu -$$$$10\!\cdots\!18$$$$)/$$$$11\!\cdots\!75$$ $$\beta_{3}$$ $$=$$ $$($$$$18\!\cdots\!17$$$$\nu^{17} -$$$$24\!\cdots\!82$$$$\nu^{16} -$$$$54\!\cdots\!32$$$$\nu^{15} +$$$$71\!\cdots\!00$$$$\nu^{14} +$$$$65\!\cdots\!96$$$$\nu^{13} -$$$$82\!\cdots\!44$$$$\nu^{12} -$$$$41\!\cdots\!64$$$$\nu^{11} +$$$$47\!\cdots\!04$$$$\nu^{10} +$$$$15\!\cdots\!08$$$$\nu^{9} -$$$$14\!\cdots\!86$$$$\nu^{8} -$$$$32\!\cdots\!40$$$$\nu^{7} +$$$$21\!\cdots\!72$$$$\nu^{6} +$$$$40\!\cdots\!00$$$$\nu^{5} -$$$$13\!\cdots\!12$$$$\nu^{4} -$$$$27\!\cdots\!28$$$$\nu^{3} -$$$$12\!\cdots\!20$$$$\nu^{2} +$$$$79\!\cdots\!81$$$$\nu +$$$$21\!\cdots\!21$$$$)/$$$$39\!\cdots\!25$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$35\!\cdots\!04$$$$\nu^{17} +$$$$18\!\cdots\!09$$$$\nu^{16} +$$$$10\!\cdots\!84$$$$\nu^{15} -$$$$56\!\cdots\!00$$$$\nu^{14} -$$$$13\!\cdots\!52$$$$\nu^{13} +$$$$68\!\cdots\!78$$$$\nu^{12} +$$$$89\!\cdots\!68$$$$\nu^{11} -$$$$40\!\cdots\!48$$$$\nu^{10} -$$$$34\!\cdots\!96$$$$\nu^{9} +$$$$12\!\cdots\!07$$$$\nu^{8} +$$$$76\!\cdots\!80$$$$\nu^{7} -$$$$17\!\cdots\!64$$$$\nu^{6} -$$$$97\!\cdots\!00$$$$\nu^{5} +$$$$62\!\cdots\!94$$$$\nu^{4} +$$$$63\!\cdots\!36$$$$\nu^{3} +$$$$79\!\cdots\!40$$$$\nu^{2} -$$$$16\!\cdots\!72$$$$\nu -$$$$67\!\cdots\!27$$$$)/$$$$53\!\cdots\!00$$ $$\beta_{5}$$ $$=$$ $$($$$$33\!\cdots\!28$$$$\nu^{17} -$$$$33\!\cdots\!01$$$$\nu^{16} -$$$$10\!\cdots\!28$$$$\nu^{15} +$$$$10\!\cdots\!52$$$$\nu^{14} +$$$$12\!\cdots\!96$$$$\nu^{13} -$$$$11\!\cdots\!46$$$$\nu^{12} -$$$$81\!\cdots\!36$$$$\nu^{11} +$$$$67\!\cdots\!28$$$$\nu^{10} +$$$$30\!\cdots\!84$$$$\nu^{9} -$$$$20\!\cdots\!19$$$$\nu^{8} -$$$$66\!\cdots\!44$$$$\nu^{7} +$$$$30\!\cdots\!64$$$$\nu^{6} +$$$$83\!\cdots\!32$$$$\nu^{5} -$$$$16\!\cdots\!62$$$$\nu^{4} -$$$$55\!\cdots\!72$$$$\nu^{3} -$$$$30\!\cdots\!76$$$$\nu^{2} +$$$$15\!\cdots\!64$$$$\nu +$$$$43\!\cdots\!19$$$$)/$$$$21\!\cdots\!52$$ $$\beta_{6}$$ $$=$$ $$($$$$19\!\cdots\!68$$$$\nu^{17} -$$$$18\!\cdots\!53$$$$\nu^{16} -$$$$59\!\cdots\!28$$$$\nu^{15} +$$$$54\!\cdots\!00$$$$\nu^{14} +$$$$73\!\cdots\!84$$$$\nu^{13} -$$$$63\!\cdots\!26$$$$\nu^{12} -$$$$48\!\cdots\!56$$$$\nu^{11} +$$$$36\!\cdots\!16$$$$\nu^{10} +$$$$18\!\cdots\!32$$$$\nu^{9} -$$$$11\!\cdots\!19$$$$\nu^{8} -$$$$39\!\cdots\!60$$$$\nu^{7} +$$$$16\!\cdots\!88$$$$\nu^{6} +$$$$49\!\cdots\!00$$$$\nu^{5} -$$$$87\!\cdots\!98$$$$\nu^{4} -$$$$32\!\cdots\!12$$$$\nu^{3} -$$$$22\!\cdots\!80$$$$\nu^{2} +$$$$88\!\cdots\!24$$$$\nu +$$$$26\!\cdots\!59$$$$)/$$$$12\!\cdots\!00$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$28\!\cdots\!79$$$$\nu^{17} +$$$$32\!\cdots\!29$$$$\nu^{16} +$$$$84\!\cdots\!04$$$$\nu^{15} -$$$$96\!\cdots\!20$$$$\nu^{14} -$$$$10\!\cdots\!82$$$$\nu^{13} +$$$$11\!\cdots\!58$$$$\nu^{12} +$$$$64\!\cdots\!28$$$$\nu^{11} -$$$$65\!\cdots\!88$$$$\nu^{10} -$$$$23\!\cdots\!61$$$$\nu^{9} +$$$$20\!\cdots\!07$$$$\nu^{8} +$$$$49\!\cdots\!00$$$$\nu^{7} -$$$$30\!\cdots\!44$$$$\nu^{6} -$$$$59\!\cdots\!30$$$$\nu^{5} +$$$$19\!\cdots\!34$$$$\nu^{4} +$$$$39\!\cdots\!16$$$$\nu^{3} -$$$$16\!\cdots\!80$$$$\nu^{2} -$$$$11\!\cdots\!67$$$$\nu -$$$$28\!\cdots\!47$$$$)/$$$$10\!\cdots\!60$$ $$\beta_{8}$$ $$=$$ $$($$$$85\!\cdots\!71$$$$\nu^{17} +$$$$41\!\cdots\!59$$$$\nu^{16} -$$$$27\!\cdots\!16$$$$\nu^{15} -$$$$11\!\cdots\!00$$$$\nu^{14} +$$$$36\!\cdots\!98$$$$\nu^{13} +$$$$12\!\cdots\!78$$$$\nu^{12} -$$$$26\!\cdots\!32$$$$\nu^{11} -$$$$70\!\cdots\!48$$$$\nu^{10} +$$$$10\!\cdots\!29$$$$\nu^{9} +$$$$25\!\cdots\!57$$$$\nu^{8} -$$$$26\!\cdots\!20$$$$\nu^{7} -$$$$57\!\cdots\!64$$$$\nu^{6} +$$$$36\!\cdots\!50$$$$\nu^{5} +$$$$87\!\cdots\!94$$$$\nu^{4} -$$$$25\!\cdots\!64$$$$\nu^{3} -$$$$76\!\cdots\!60$$$$\nu^{2} +$$$$72\!\cdots\!03$$$$\nu +$$$$28\!\cdots\!23$$$$)/$$$$17\!\cdots\!00$$ $$\beta_{9}$$ $$=$$ $$($$$$72\!\cdots\!31$$$$\nu^{17} -$$$$87\!\cdots\!01$$$$\nu^{16} -$$$$21\!\cdots\!76$$$$\nu^{15} +$$$$25\!\cdots\!00$$$$\nu^{14} +$$$$26\!\cdots\!78$$$$\nu^{13} -$$$$29\!\cdots\!42$$$$\nu^{12} -$$$$16\!\cdots\!52$$$$\nu^{11} +$$$$17\!\cdots\!72$$$$\nu^{10} +$$$$61\!\cdots\!69$$$$\nu^{9} -$$$$52\!\cdots\!23$$$$\nu^{8} -$$$$13\!\cdots\!20$$$$\nu^{7} +$$$$79\!\cdots\!96$$$$\nu^{6} +$$$$16\!\cdots\!50$$$$\nu^{5} -$$$$47\!\cdots\!66$$$$\nu^{4} -$$$$10\!\cdots\!04$$$$\nu^{3} -$$$$17\!\cdots\!60$$$$\nu^{2} +$$$$30\!\cdots\!83$$$$\nu +$$$$82\!\cdots\!03$$$$)/$$$$12\!\cdots\!00$$ $$\beta_{10}$$ $$=$$ $$($$$$-$$$$32\!\cdots\!57$$$$\nu^{17} -$$$$43\!\cdots\!78$$$$\nu^{16} +$$$$11\!\cdots\!72$$$$\nu^{15} +$$$$12\!\cdots\!00$$$$\nu^{14} -$$$$16\!\cdots\!66$$$$\nu^{13} -$$$$14\!\cdots\!76$$$$\nu^{12} +$$$$12\!\cdots\!44$$$$\nu^{11} +$$$$89\!\cdots\!16$$$$\nu^{10} -$$$$58\!\cdots\!43$$$$\nu^{9} -$$$$31\!\cdots\!94$$$$\nu^{8} +$$$$15\!\cdots\!40$$$$\nu^{7} +$$$$69\!\cdots\!88$$$$\nu^{6} -$$$$23\!\cdots\!50$$$$\nu^{5} -$$$$93\!\cdots\!48$$$$\nu^{4} +$$$$18\!\cdots\!88$$$$\nu^{3} +$$$$71\!\cdots\!20$$$$\nu^{2} -$$$$53\!\cdots\!01$$$$\nu -$$$$24\!\cdots\!66$$$$)/$$$$48\!\cdots\!00$$ $$\beta_{11}$$ $$=$$ $$($$$$81\!\cdots\!41$$$$\nu^{17} -$$$$14\!\cdots\!71$$$$\nu^{16} -$$$$23\!\cdots\!56$$$$\nu^{15} +$$$$43\!\cdots\!40$$$$\nu^{14} +$$$$26\!\cdots\!18$$$$\nu^{13} -$$$$50\!\cdots\!42$$$$\nu^{12} -$$$$15\!\cdots\!92$$$$\nu^{11} +$$$$29\!\cdots\!72$$$$\nu^{10} +$$$$48\!\cdots\!39$$$$\nu^{9} -$$$$92\!\cdots\!73$$$$\nu^{8} -$$$$87\!\cdots\!60$$$$\nu^{7} +$$$$15\!\cdots\!76$$$$\nu^{6} +$$$$92\!\cdots\!70$$$$\nu^{5} -$$$$12\!\cdots\!46$$$$\nu^{4} -$$$$61\!\cdots\!44$$$$\nu^{3} +$$$$35\!\cdots\!80$$$$\nu^{2} +$$$$21\!\cdots\!73$$$$\nu +$$$$18\!\cdots\!93$$$$)/$$$$96\!\cdots\!40$$ $$\beta_{12}$$ $$=$$ $$($$$$-$$$$14\!\cdots\!51$$$$\nu^{17} +$$$$16\!\cdots\!21$$$$\nu^{16} +$$$$43\!\cdots\!96$$$$\nu^{15} -$$$$47\!\cdots\!00$$$$\nu^{14} -$$$$52\!\cdots\!38$$$$\nu^{13} +$$$$55\!\cdots\!82$$$$\nu^{12} +$$$$33\!\cdots\!92$$$$\nu^{11} -$$$$31\!\cdots\!12$$$$\nu^{10} -$$$$12\!\cdots\!49$$$$\nu^{9} +$$$$95\!\cdots\!83$$$$\nu^{8} +$$$$25\!\cdots\!20$$$$\nu^{7} -$$$$13\!\cdots\!16$$$$\nu^{6} -$$$$31\!\cdots\!50$$$$\nu^{5} +$$$$79\!\cdots\!86$$$$\nu^{4} +$$$$21\!\cdots\!84$$$$\nu^{3} +$$$$88\!\cdots\!60$$$$\nu^{2} -$$$$57\!\cdots\!43$$$$\nu -$$$$16\!\cdots\!63$$$$)/$$$$53\!\cdots\!00$$ $$\beta_{13}$$ $$=$$ $$($$$$15\!\cdots\!04$$$$\nu^{17} -$$$$16\!\cdots\!59$$$$\nu^{16} -$$$$47\!\cdots\!84$$$$\nu^{15} +$$$$50\!\cdots\!00$$$$\nu^{14} +$$$$58\!\cdots\!52$$$$\nu^{13} -$$$$58\!\cdots\!78$$$$\nu^{12} -$$$$37\!\cdots\!68$$$$\nu^{11} +$$$$33\!\cdots\!48$$$$\nu^{10} +$$$$13\!\cdots\!96$$$$\nu^{9} -$$$$10\!\cdots\!57$$$$\nu^{8} -$$$$29\!\cdots\!80$$$$\nu^{7} +$$$$15\!\cdots\!64$$$$\nu^{6} +$$$$36\!\cdots\!00$$$$\nu^{5} -$$$$86\!\cdots\!94$$$$\nu^{4} -$$$$24\!\cdots\!36$$$$\nu^{3} -$$$$10\!\cdots\!40$$$$\nu^{2} +$$$$66\!\cdots\!72$$$$\nu +$$$$18\!\cdots\!77$$$$)/$$$$48\!\cdots\!00$$ $$\beta_{14}$$ $$=$$ $$($$$$18\!\cdots\!99$$$$\nu^{17} -$$$$20\!\cdots\!29$$$$\nu^{16} -$$$$55\!\cdots\!04$$$$\nu^{15} +$$$$59\!\cdots\!00$$$$\nu^{14} +$$$$67\!\cdots\!62$$$$\nu^{13} -$$$$68\!\cdots\!18$$$$\nu^{12} -$$$$42\!\cdots\!08$$$$\nu^{11} +$$$$39\!\cdots\!88$$$$\nu^{10} +$$$$15\!\cdots\!01$$$$\nu^{9} -$$$$11\!\cdots\!67$$$$\nu^{8} -$$$$33\!\cdots\!80$$$$\nu^{7} +$$$$17\!\cdots\!84$$$$\nu^{6} +$$$$40\!\cdots\!50$$$$\nu^{5} -$$$$99\!\cdots\!14$$$$\nu^{4} -$$$$26\!\cdots\!16$$$$\nu^{3} -$$$$10\!\cdots\!40$$$$\nu^{2} +$$$$70\!\cdots\!07$$$$\nu +$$$$19\!\cdots\!87$$$$)/$$$$53\!\cdots\!00$$ $$\beta_{15}$$ $$=$$ $$($$$$62\!\cdots\!09$$$$\nu^{17} -$$$$79\!\cdots\!14$$$$\nu^{16} -$$$$18\!\cdots\!64$$$$\nu^{15} +$$$$23\!\cdots\!00$$$$\nu^{14} +$$$$22\!\cdots\!42$$$$\nu^{13} -$$$$27\!\cdots\!88$$$$\nu^{12} -$$$$13\!\cdots\!28$$$$\nu^{11} +$$$$15\!\cdots\!08$$$$\nu^{10} +$$$$49\!\cdots\!91$$$$\nu^{9} -$$$$47\!\cdots\!22$$$$\nu^{8} -$$$$10\!\cdots\!80$$$$\nu^{7} +$$$$72\!\cdots\!44$$$$\nu^{6} +$$$$12\!\cdots\!50$$$$\nu^{5} -$$$$47\!\cdots\!24$$$$\nu^{4} -$$$$81\!\cdots\!56$$$$\nu^{3} +$$$$34\!\cdots\!60$$$$\nu^{2} +$$$$22\!\cdots\!37$$$$\nu +$$$$52\!\cdots\!42$$$$)/$$$$17\!\cdots\!00$$ $$\beta_{16}$$ $$=$$ $$($$$$23\!\cdots\!43$$$$\nu^{17} -$$$$23\!\cdots\!78$$$$\nu^{16} -$$$$69\!\cdots\!28$$$$\nu^{15} +$$$$70\!\cdots\!00$$$$\nu^{14} +$$$$84\!\cdots\!34$$$$\nu^{13} -$$$$82\!\cdots\!76$$$$\nu^{12} -$$$$54\!\cdots\!56$$$$\nu^{11} +$$$$47\!\cdots\!16$$$$\nu^{10} +$$$$19\!\cdots\!57$$$$\nu^{9} -$$$$14\!\cdots\!94$$$$\nu^{8} -$$$$42\!\cdots\!60$$$$\nu^{7} +$$$$21\!\cdots\!88$$$$\nu^{6} +$$$$51\!\cdots\!50$$$$\nu^{5} -$$$$11\!\cdots\!48$$$$\nu^{4} -$$$$33\!\cdots\!12$$$$\nu^{3} -$$$$16\!\cdots\!80$$$$\nu^{2} +$$$$91\!\cdots\!99$$$$\nu +$$$$25\!\cdots\!34$$$$)/$$$$53\!\cdots\!00$$ $$\beta_{17}$$ $$=$$ $$($$$$44\!\cdots\!48$$$$\nu^{17} -$$$$41\!\cdots\!33$$$$\nu^{16} -$$$$13\!\cdots\!08$$$$\nu^{15} +$$$$12\!\cdots\!00$$$$\nu^{14} +$$$$16\!\cdots\!24$$$$\nu^{13} -$$$$14\!\cdots\!86$$$$\nu^{12} -$$$$10\!\cdots\!16$$$$\nu^{11} +$$$$82\!\cdots\!76$$$$\nu^{10} +$$$$40\!\cdots\!52$$$$\nu^{9} -$$$$24\!\cdots\!59$$$$\nu^{8} -$$$$89\!\cdots\!60$$$$\nu^{7} +$$$$34\!\cdots\!68$$$$\nu^{6} +$$$$11\!\cdots\!00$$$$\nu^{5} -$$$$16\!\cdots\!78$$$$\nu^{4} -$$$$76\!\cdots\!32$$$$\nu^{3} -$$$$72\!\cdots\!80$$$$\nu^{2} +$$$$21\!\cdots\!64$$$$\nu +$$$$63\!\cdots\!99$$$$)/$$$$48\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{2} - \beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_{1} + 340$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{17} - \beta_{16} + 4 \beta_{15} + \beta_{14} + 2 \beta_{13} + 3 \beta_{12} - \beta_{11} - \beta_{10} + 4 \beta_{8} - 3 \beta_{7} + 15 \beta_{6} - 12 \beta_{5} - 2 \beta_{4} - \beta_{3} + 557 \beta_{2} - 1024 \beta_{1} - 253$$ $$\nu^{4}$$ $$=$$ $$-32 \beta_{17} + 36 \beta_{16} + 66 \beta_{15} - 40 \beta_{14} - 24 \beta_{13} - 14 \beta_{12} - 2 \beta_{11} - 22 \beta_{10} + 60 \beta_{9} + 50 \beta_{8} + 54 \beta_{7} + 60 \beta_{6} + 1060 \beta_{5} + 772 \beta_{4} - 2248 \beta_{3} + 70 \beta_{2} - 1055 \beta_{1} + 186850$$ $$\nu^{5}$$ $$=$$ $$-1528 \beta_{17} - 882 \beta_{16} + 3510 \beta_{15} + 1072 \beta_{14} + 1380 \beta_{13} + 3910 \beta_{12} - 784 \beta_{11} - 884 \beta_{10} + 150 \beta_{9} + 3220 \beta_{8} - 5450 \beta_{7} + 16446 \beta_{6} - 17144 \beta_{5} - 1172 \beta_{4} + 4491 \beta_{3} + 352487 \beta_{2} - 942783 \beta_{1} - 666057$$ $$\nu^{6}$$ $$=$$ $$-34714 \beta_{17} + 44788 \beta_{16} + 80540 \beta_{15} - 50680 \beta_{14} - 27176 \beta_{13} - 18867 \beta_{12} - 1676 \beta_{11} - 24278 \beta_{10} + 98676 \beta_{9} + 60230 \beta_{8} + 119619 \beta_{7} + 32820 \beta_{6} + 800317 \beta_{5} + 558981 \beta_{4} - 2162861 \beta_{3} - 335779 \beta_{2} + 1142479 \beta_{1} + 118830231$$ $$\nu^{7}$$ $$=$$ $$-994635 \beta_{17} - 653413 \beta_{16} + 2517178 \beta_{15} + 989546 \beta_{14} + 948458 \beta_{13} + 4019988 \beta_{12} - 536171 \beta_{11} - 680826 \beta_{10} - 114051 \beta_{9} + 2021438 \beta_{8} - 6078142 \beta_{7} + 14387379 \beta_{6} - 16798627 \beta_{5} - 922665 \beta_{4} + 9454103 \beta_{3} + 234800478 \beta_{2} - 838236427 \beta_{1} - 785864070$$ $$\nu^{8}$$ $$=$$ $$-29172564 \beta_{17} + 41601552 \beta_{16} + 70749708 \beta_{15} - 48255960 \beta_{14} - 22892928 \beta_{13} - 23725704 \beta_{12} + 78468 \beta_{11} - 20651112 \beta_{10} + 116937960 \beta_{9} + 52310568 \beta_{8} + 160377384 \beta_{7} + 8131296 \beta_{6} + 568060732 \beta_{5} + 401043340 \beta_{4} - 1961624516 \beta_{3} - 466530068 \beta_{2} + 2934862403 \beta_{1} + 80637457063$$ $$\nu^{9}$$ $$=$$ $$-618658592 \beta_{17} - 470162752 \beta_{16} + 1716048064 \beta_{15} + 880579132 \beta_{14} + 688894880 \beta_{13} + 3788663472 \beta_{12} - 357850600 \beta_{11} - 509551792 \beta_{10} - 455086404 \beta_{9} + 1153342540 \beta_{8} - 5943857544 \beta_{7} + 11480105328 \beta_{6} - 14483664972 \beta_{5} - 825292028 \beta_{4} + 12868784111 \beta_{3} + 160666526300 \beta_{2} - 727981821340 \beta_{1} - 771187047865$$ $$\nu^{10}$$ $$=$$ $$-22418713520 \beta_{17} + 34494760608 \beta_{16} + 55338418284 \beta_{15} - 41398565944 \beta_{14} - 17503373088 \beta_{13} - 27731256491 \beta_{12} + 1894197148 \beta_{11} - 15986191708 \beta_{10} + 119713814088 \beta_{9} + 40575302924 \beta_{8} + 178171971867 \beta_{7} - 4926610704 \beta_{6} + 399121993168 \beta_{5} + 286816393504 \beta_{4} - 1727163481375 \beta_{3} - 471454445774 \beta_{2} + 4126186427404 \beta_{1} + 56891017155247$$ $$\nu^{11}$$ $$=$$ $$-377519170879 \beta_{17} - 336992032644 \beta_{16} + 1155968918484 \beta_{15} + 765725112727 \beta_{14} + 510819377064 \beta_{13} + 3412089626143 \beta_{12} - 243477445444 \beta_{11} - 379238293601 \beta_{10} - 725144817837 \beta_{9} + 612919885540 \beta_{8} - 5534347208621 \beta_{7} + 8734319102748 \beta_{6} - 11671294239251 \beta_{5} - 734139997835 \beta_{4} + 14833750743480 \beta_{3} + 111552747521255 \beta_{2} - 623416405473291 \beta_{1} - 701448549462327$$ $$\nu^{12}$$ $$=$$ $$-16530979031806 \beta_{17} + 26943465802252 \beta_{16} + 41073808391000 \beta_{15} - 33788585935348 \beta_{14} - 12874778881880 \beta_{13} - 30200883687834 \beta_{12} + 3167799283336 \beta_{11} - 11821702353446 \beta_{10} + 113130677985696 \beta_{9} + 29970542462162 \beta_{8} + 179165176361730 \beta_{7} - 10063263374868 \beta_{6} + 280509609768850 \beta_{5} + 204495130956474 \beta_{4} - 1493960735185730 \beta_{3} - 420757325673664 \beta_{2} + 4804448540503186 \beta_{1} + 41191036147953147$$ $$\nu^{13}$$ $$=$$ $$-227161506858060 \beta_{17} - 241313300363986 \beta_{16} + 778913741676130 \beta_{15} + 652880632558082 \beta_{14} + 377972692878356 \beta_{13} + 2988631058363106 \beta_{12} - 173121274374728 \beta_{11} - 282229619185908 \beta_{10} - 894550220002920 \beta_{9} + 295825300631102 \beta_{8} - 5028319795938634 \beta_{7} + 6457608380650470 \beta_{6} - 9006261598381798 \beta_{5} - 625354183144170 \beta_{4} + 15632609479844522 \beta_{3} + 77975512615080351 \beta_{2} - 528854141910888019 \beta_{1} - 612290192855082186$$ $$\nu^{14}$$ $$=$$ $$-11924163338759814 \beta_{17} + 20276580000352716 \beta_{16} + 29654398036272990 \beta_{15} - 26849476603786740 \beta_{14} - 9292292781634344 \beta_{13} - 31057889832385896 \beta_{12} + 3836235635149662 \beta_{11} - 8510531423413668 \beta_{10} + 101851991325759120 \beta_{9} + 21621557015087508 \beta_{8} + 169491732095943696 \beta_{7} - 10895858744565660 \beta_{6} + 197205979815936631 \beta_{5} + 145166859593831815 \beta_{4} - 1276346277595749116 \beta_{3} - 349201457496077873 \beta_{2} + 5083167734480699117 \beta_{1} + 30356422411291333960$$ $$\nu^{15}$$ $$=$$ $$-134337047286067190 \beta_{17} - 172029755462749261 \beta_{16} + 526844498334935788 \beta_{15} + 547259357495628343 \beta_{14} + 276446695688436938 \beta_{13} + 2568749709002749179 \beta_{12} - 129928525542001801 \beta_{11} - 210248075098192945 \beta_{10} - 974590115678527446 \beta_{9} + 116688494581587490 \beta_{8} - 4495550022695203263 \beta_{7} + 4680405122276445963 \beta_{6} - 6736217119392736242 \beta_{5} - 507091344898090088 \beta_{4} + 15557206326608905715 \beta_{3} + 54567004074539318027 \beta_{2} - 445325307133025125654 \beta_{1} - 520411189563070171501$$ $$\nu^{16}$$ $$=$$ $$-8489994057881508368 \beta_{17} + 14861123373140241168 \beta_{16} + 21051391869633265560 \beta_{15} - 21013583281499809120 \beta_{14} - 6618398613336247008 \beta_{13} - 30543565850567124632 \beta_{12} + 4031120679293532904 \beta_{11} - 6011915076740283352 \beta_{10} + 88856712076336154736 \beta_{9} + 15415018437573782312 \beta_{8} + 154012125023812787832 \beta_{7} - 9672845245668833232 \beta_{6} + 138205994832001447504 \beta_{5} + 102420655543218528400 \beta_{4} - 1079962492312305728176 \beta_{3} - 274634401341843077960 \beta_{2} + 5066583875308257898897 \beta_{1} + 22643208026464637135128$$ $$\nu^{17}$$ $$=$$ $$-77397562232635515808 \beta_{17} - 121317764430180614232 \beta_{16} + 357888472509806807448 \beta_{15} + 452065681067487758464 \beta_{14} + 199096557335779185072 \beta_{13} + 2177607713457669429064 \beta_{12} - 102602961013845426736 \beta_{11} - 156797558799851069312 \beta_{10} - 986637057711787709688 \beta_{9} + 19684093886188525792 \beta_{8} - 3967937020309701174584 \beta_{7} + 3338672139826052013480 \beta_{6} - 4913692378588862945648 \beta_{5} - 391268558621268315008 \beta_{4} + 14868569080219820797545 \beta_{3} + 38062113876296900728799 \beta_{2} - 372582371742800441835951 \beta_{1} - 433840404146188304522631$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/304\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$191$$ $$229$$ $$\chi(n)$$ $$-1 - \beta_{1}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 26.3623 − 0.866025i 22.4078 − 0.866025i 17.4003 − 0.866025i 9.90852 − 0.866025i −5.10643 − 0.866025i −6.49256 − 0.866025i −12.0901 − 0.866025i −21.8370 − 0.866025i −28.0529 − 0.866025i 26.3623 + 0.866025i 22.4078 + 0.866025i 17.4003 + 0.866025i 9.90852 + 0.866025i −5.10643 + 0.866025i −6.49256 + 0.866025i −12.0901 + 0.866025i −21.8370 + 0.866025i −28.0529 + 0.866025i
0 −12.4311 + 21.5314i 0 −18.0963 + 31.3437i 0 208.885 0 −187.566 324.875i 0
49.2 0 −10.4539 + 18.1067i 0 50.5928 87.6293i 0 −95.5451 0 −97.0688 168.128i 0
49.3 0 −7.95017 + 13.7701i 0 −15.4645 + 26.7852i 0 −132.225 0 −4.91049 8.50521i 0
49.4 0 −4.20426 + 7.28199i 0 −18.6530 + 32.3080i 0 −15.8772 0 86.1484 + 149.213i 0
49.5 0 3.30322 5.72134i 0 12.6095 21.8402i 0 40.7176 0 99.6775 + 172.647i 0
49.6 0 3.99628 6.92176i 0 31.6056 54.7425i 0 80.2775 0 89.5595 + 155.122i 0
49.7 0 6.79505 11.7694i 0 −47.4871 + 82.2500i 0 −189.860 0 29.1546 + 50.4972i 0
49.8 0 11.6685 20.2105i 0 −25.2470 + 43.7291i 0 187.942 0 −150.808 261.208i 0
49.9 0 14.7764 25.5935i 0 35.6401 61.7304i 0 −252.315 0 −315.186 545.918i 0
273.1 0 −12.4311 21.5314i 0 −18.0963 31.3437i 0 208.885 0 −187.566 + 324.875i 0
273.2 0 −10.4539 18.1067i 0 50.5928 + 87.6293i 0 −95.5451 0 −97.0688 + 168.128i 0
273.3 0 −7.95017 13.7701i 0 −15.4645 26.7852i 0 −132.225 0 −4.91049 + 8.50521i 0
273.4 0 −4.20426 7.28199i 0 −18.6530 32.3080i 0 −15.8772 0 86.1484 149.213i 0
273.5 0 3.30322 + 5.72134i 0 12.6095 + 21.8402i 0 40.7176 0 99.6775 172.647i 0
273.6 0 3.99628 + 6.92176i 0 31.6056 + 54.7425i 0 80.2775 0 89.5595 155.122i 0
273.7 0 6.79505 + 11.7694i 0 −47.4871 82.2500i 0 −189.860 0 29.1546 50.4972i 0
273.8 0 11.6685 + 20.2105i 0 −25.2470 43.7291i 0 187.942 0 −150.808 + 261.208i 0
273.9 0 14.7764 + 25.5935i 0 35.6401 + 61.7304i 0 −252.315 0 −315.186 + 545.918i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 273.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.6.i.d 18
4.b odd 2 1 76.6.e.a 18
12.b even 2 1 684.6.k.f 18
19.c even 3 1 inner 304.6.i.d 18
76.g odd 6 1 76.6.e.a 18
228.m even 6 1 684.6.k.f 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.6.e.a 18 4.b odd 2 1
76.6.e.a 18 76.g odd 6 1
304.6.i.d 18 1.a even 1 1 trivial
304.6.i.d 18 19.c even 3 1 inner
684.6.k.f 18 12.b even 2 1
684.6.k.f 18 228.m even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$15\!\cdots\!01$$$$T_{3}^{9} +$$$$82\!\cdots\!75$$$$T_{3}^{8} -$$$$39\!\cdots\!36$$$$T_{3}^{7} +$$$$13\!\cdots\!18$$$$T_{3}^{6} -$$$$64\!\cdots\!38$$$$T_{3}^{5} +$$$$10\!\cdots\!86$$$$T_{3}^{4} -$$$$41\!\cdots\!16$$$$T_{3}^{3} +$$$$55\!\cdots\!61$$$$T_{3}^{2} -$$$$18\!\cdots\!43$$$$T_{3} +$$$$11\!\cdots\!89$$">$$T_{3}^{18} - \cdots$$ acting on $$S_{6}^{\mathrm{new}}(304, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{18}$$
$3$ $$11\!\cdots\!89$$$$-$$$$18\!\cdots\!43$$$$T + 55353296119122128061 T^{2} - 4135023977940829116 T^{3} + 1064365845873830586 T^{4} - 64277359976697138 T^{5} + 13339239476598918 T^{6} - 391884723575136 T^{7} + 82262584011775 T^{8} - 1520828506301 T^{9} + 365497827591 T^{10} - 3216311392 T^{11} + 919778846 T^{12} - 7740906 T^{13} + 1629002 T^{14} - 8740 T^{15} + 1605 T^{16} - 11 T^{17} + T^{18}$$
$5$ $$53\!\cdots\!96$$$$+$$$$19\!\cdots\!64$$$$T +$$$$17\!\cdots\!60$$$$T^{2} +$$$$51\!\cdots\!56$$$$T^{3} +$$$$32\!\cdots\!64$$$$T^{4} +$$$$85\!\cdots\!40$$$$T^{5} +$$$$33\!\cdots\!96$$$$T^{6} +$$$$56\!\cdots\!24$$$$T^{7} + 16796878410512090940 T^{8} + 182235863003254716 T^{9} + 5792633935619680 T^{10} + 38635011538024 T^{11} + 1306014959701 T^{12} + 3086592365 T^{13} + 211569494 T^{14} + 174581 T^{15} + 17650 T^{16} - 11 T^{17} + T^{18}$$
$7$ $$( 1233055145631744000 + 53167521425264640 T - 1928158954033152 T^{2} - 19652044114368 T^{3} + 334655586976 T^{4} + 2542736868 T^{5} - 13902400 T^{6} - 92704 T^{7} + 168 T^{8} + T^{9} )^{2}$$
$11$ $$( -$$$$15\!\cdots\!96$$$$+ 30973025188546606224 T + 2007063245270803884 T^{2} - 8001493002002373 T^{3} - 31276588367368 T^{4} + 134453704519 T^{5} + 146426924 T^{6} - 687587 T^{7} - 160 T^{8} + T^{9} )^{2}$$
$13$ $$60\!\cdots\!64$$$$+$$$$71\!\cdots\!76$$$$T +$$$$81\!\cdots\!20$$$$T^{2} +$$$$18\!\cdots\!84$$$$T^{3} +$$$$98\!\cdots\!00$$$$T^{4} +$$$$65\!\cdots\!04$$$$T^{5} +$$$$74\!\cdots\!24$$$$T^{6} +$$$$17\!\cdots\!76$$$$T^{7} +$$$$30\!\cdots\!40$$$$T^{8} +$$$$19\!\cdots\!40$$$$T^{9} +$$$$91\!\cdots\!68$$$$T^{10} + 15606742310908401088 T^{11} + 1766913125495305309 T^{12} - 49775665263507 T^{13} + 2463225262058 T^{14} - 100940479 T^{15} + 1961430 T^{16} - 227 T^{17} + T^{18}$$
$17$ $$16\!\cdots\!84$$$$-$$$$28\!\cdots\!92$$$$T +$$$$83\!\cdots\!04$$$$T^{2} -$$$$63\!\cdots\!00$$$$T^{3} +$$$$15\!\cdots\!04$$$$T^{4} -$$$$10\!\cdots\!76$$$$T^{5} +$$$$19\!\cdots\!16$$$$T^{6} -$$$$84\!\cdots\!84$$$$T^{7} +$$$$13\!\cdots\!40$$$$T^{8} -$$$$46\!\cdots\!68$$$$T^{9} +$$$$71\!\cdots\!40$$$$T^{10} -$$$$16\!\cdots\!60$$$$T^{11} +$$$$24\!\cdots\!29$$$$T^{12} - 37790308727158707 T^{13} + 63161431950210 T^{14} - 5114803983 T^{15} + 9862822 T^{16} - 179 T^{17} + T^{18}$$
$19$ $$34\!\cdots\!99$$$$-$$$$12\!\cdots\!68$$$$T +$$$$41\!\cdots\!72$$$$T^{2} -$$$$24\!\cdots\!24$$$$T^{3} +$$$$30\!\cdots\!48$$$$T^{4} -$$$$20\!\cdots\!76$$$$T^{5} +$$$$14\!\cdots\!03$$$$T^{6} -$$$$12\!\cdots\!80$$$$T^{7} +$$$$60\!\cdots\!84$$$$T^{8} -$$$$54\!\cdots\!88$$$$T^{9} +$$$$24\!\cdots\!16$$$$T^{10} -$$$$20\!\cdots\!80$$$$T^{11} + 96465687567916357097 T^{12} - 54428798269344676 T^{13} + 32274638969252 T^{14} - 10562267824 T^{15} + 7248128 T^{16} - 868 T^{17} + T^{18}$$
$23$ $$75\!\cdots\!00$$$$-$$$$45\!\cdots\!00$$$$T +$$$$32\!\cdots\!00$$$$T^{2} +$$$$24\!\cdots\!00$$$$T^{3} +$$$$50\!\cdots\!96$$$$T^{4} +$$$$59\!\cdots\!96$$$$T^{5} +$$$$22\!\cdots\!32$$$$T^{6} -$$$$75\!\cdots\!08$$$$T^{7} +$$$$75\!\cdots\!28$$$$T^{8} -$$$$10\!\cdots\!04$$$$T^{9} +$$$$13\!\cdots\!72$$$$T^{10} -$$$$16\!\cdots\!72$$$$T^{11} +$$$$15\!\cdots\!81$$$$T^{12} - 1959012574736536137 T^{13} + 1113768359604122 T^{14} - 89398709041 T^{15} + 44751186 T^{16} - 3425 T^{17} + T^{18}$$
$29$ $$96\!\cdots\!24$$$$+$$$$14\!\cdots\!80$$$$T +$$$$18\!\cdots\!96$$$$T^{2} +$$$$10\!\cdots\!80$$$$T^{3} +$$$$69\!\cdots\!64$$$$T^{4} +$$$$30\!\cdots\!76$$$$T^{5} +$$$$14\!\cdots\!88$$$$T^{6} +$$$$52\!\cdots\!68$$$$T^{7} +$$$$18\!\cdots\!24$$$$T^{8} +$$$$43\!\cdots\!64$$$$T^{9} +$$$$11\!\cdots\!88$$$$T^{10} +$$$$20\!\cdots\!64$$$$T^{11} +$$$$44\!\cdots\!49$$$$T^{12} + 61471755887734205509 T^{13} + 10053122334757166 T^{14} + 861197973469 T^{15} + 125209066 T^{16} + 7349 T^{17} + T^{18}$$
$31$ $$($$$$91\!\cdots\!60$$$$-$$$$78\!\cdots\!92$$$$T +$$$$49\!\cdots\!88$$$$T^{2} -$$$$44\!\cdots\!00$$$$T^{3} - 49426332686599495056 T^{4} + 6591076477990692 T^{5} + 989114300688 T^{6} - 167296276 T^{7} - 4980 T^{8} + T^{9} )^{2}$$
$37$ $$( -$$$$70\!\cdots\!00$$$$-$$$$14\!\cdots\!80$$$$T +$$$$38\!\cdots\!28$$$$T^{2} +$$$$31\!\cdots\!88$$$$T^{3} - 91986213539390854248 T^{4} - 4630799241237276 T^{5} + 3092826215800 T^{6} - 174702136 T^{7} - 13222 T^{8} + T^{9} )^{2}$$
$41$ $$33\!\cdots\!25$$$$-$$$$74\!\cdots\!25$$$$T +$$$$16\!\cdots\!25$$$$T^{2} -$$$$26\!\cdots\!00$$$$T^{3} +$$$$31\!\cdots\!14$$$$T^{4} -$$$$72\!\cdots\!30$$$$T^{5} +$$$$45\!\cdots\!06$$$$T^{6} -$$$$69\!\cdots\!44$$$$T^{7} +$$$$22\!\cdots\!23$$$$T^{8} -$$$$12\!\cdots\!27$$$$T^{9} +$$$$55\!\cdots\!99$$$$T^{10} -$$$$64\!\cdots\!52$$$$T^{11} +$$$$11\!\cdots\!54$$$$T^{12} +$$$$45\!\cdots\!54$$$$T^{13} + 102037756916691790 T^{14} + 1992308609992 T^{15} + 386322313 T^{16} + 7311 T^{17} + T^{18}$$
$43$ $$51\!\cdots\!00$$$$-$$$$14\!\cdots\!00$$$$T +$$$$38\!\cdots\!00$$$$T^{2} -$$$$62\!\cdots\!00$$$$T^{3} +$$$$90\!\cdots\!00$$$$T^{4} -$$$$41\!\cdots\!40$$$$T^{5} +$$$$18\!\cdots\!76$$$$T^{6} -$$$$47\!\cdots\!96$$$$T^{7} +$$$$10\!\cdots\!80$$$$T^{8} -$$$$14\!\cdots\!84$$$$T^{9} +$$$$18\!\cdots\!84$$$$T^{10} -$$$$17\!\cdots\!12$$$$T^{11} +$$$$17\!\cdots\!89$$$$T^{12} -$$$$12\!\cdots\!91$$$$T^{13} + 112961728986806062 T^{14} - 4760975065791 T^{15} + 379698838 T^{16} - 8283 T^{17} + T^{18}$$
$47$ $$16\!\cdots\!16$$$$-$$$$23\!\cdots\!44$$$$T +$$$$96\!\cdots\!24$$$$T^{2} -$$$$78\!\cdots\!88$$$$T^{3} +$$$$21\!\cdots\!36$$$$T^{4} +$$$$67\!\cdots\!76$$$$T^{5} +$$$$40\!\cdots\!32$$$$T^{6} +$$$$26\!\cdots\!32$$$$T^{7} +$$$$39\!\cdots\!12$$$$T^{8} +$$$$20\!\cdots\!40$$$$T^{9} +$$$$20\!\cdots\!08$$$$T^{10} +$$$$95\!\cdots\!48$$$$T^{11} +$$$$70\!\cdots\!29$$$$T^{12} +$$$$25\!\cdots\!99$$$$T^{13} + 1438596564851851442 T^{14} + 41903660395535 T^{15} + 1912969374 T^{16} + 37603 T^{17} + T^{18}$$
$53$ $$50\!\cdots\!64$$$$+$$$$39\!\cdots\!44$$$$T +$$$$17\!\cdots\!60$$$$T^{2} +$$$$65\!\cdots\!96$$$$T^{3} +$$$$33\!\cdots\!16$$$$T^{4} +$$$$11\!\cdots\!92$$$$T^{5} +$$$$37\!\cdots\!08$$$$T^{6} +$$$$73\!\cdots\!92$$$$T^{7} +$$$$27\!\cdots\!96$$$$T^{8} +$$$$69\!\cdots\!00$$$$T^{9} +$$$$10\!\cdots\!88$$$$T^{10} +$$$$24\!\cdots\!56$$$$T^{11} +$$$$25\!\cdots\!69$$$$T^{12} +$$$$59\!\cdots\!49$$$$T^{13} + 3460270021546904626 T^{14} + 43357881060629 T^{15} + 2267852878 T^{16} + 20337 T^{17} + T^{18}$$
$59$ $$21\!\cdots\!89$$$$-$$$$40\!\cdots\!55$$$$T +$$$$71\!\cdots\!53$$$$T^{2} -$$$$47\!\cdots\!00$$$$T^{3} +$$$$44\!\cdots\!82$$$$T^{4} -$$$$25\!\cdots\!38$$$$T^{5} +$$$$17\!\cdots\!26$$$$T^{6} -$$$$79\!\cdots\!68$$$$T^{7} +$$$$39\!\cdots\!87$$$$T^{8} -$$$$13\!\cdots\!77$$$$T^{9} +$$$$53\!\cdots\!07$$$$T^{10} -$$$$15\!\cdots\!52$$$$T^{11} +$$$$47\!\cdots\!62$$$$T^{12} -$$$$10\!\cdots\!18$$$$T^{13} + 23327287366044483930 T^{14} - 357651739695276 T^{15} + 7129477477 T^{16} - 74455 T^{17} + T^{18}$$
$61$ $$21\!\cdots\!56$$$$-$$$$20\!\cdots\!04$$$$T +$$$$11\!\cdots\!76$$$$T^{2} -$$$$16\!\cdots\!40$$$$T^{3} +$$$$57\!\cdots\!80$$$$T^{4} -$$$$63\!\cdots\!04$$$$T^{5} +$$$$81\!\cdots\!92$$$$T^{6} -$$$$33\!\cdots\!76$$$$T^{7} +$$$$26\!\cdots\!64$$$$T^{8} -$$$$57\!\cdots\!12$$$$T^{9} +$$$$57\!\cdots\!12$$$$T^{10} -$$$$69\!\cdots\!56$$$$T^{11} +$$$$74\!\cdots\!81$$$$T^{12} -$$$$16\!\cdots\!47$$$$T^{13} + 6930515824571977902 T^{14} - 1910399499327 T^{15} + 3182325786 T^{16} + 7569 T^{17} + T^{18}$$
$67$ $$92\!\cdots\!81$$$$-$$$$49\!\cdots\!29$$$$T +$$$$58\!\cdots\!85$$$$T^{2} -$$$$13\!\cdots\!60$$$$T^{3} +$$$$15\!\cdots\!62$$$$T^{4} -$$$$27\!\cdots\!90$$$$T^{5} +$$$$25\!\cdots\!86$$$$T^{6} -$$$$31\!\cdots\!92$$$$T^{7} +$$$$27\!\cdots\!35$$$$T^{8} -$$$$30\!\cdots\!31$$$$T^{9} +$$$$21\!\cdots\!79$$$$T^{10} -$$$$17\!\cdots\!60$$$$T^{11} +$$$$10\!\cdots\!10$$$$T^{12} -$$$$82\!\cdots\!10$$$$T^{13} + 38408940894285316426 T^{14} - 181133911740404 T^{15} + 7768368661 T^{16} - 26177 T^{17} + T^{18}$$
$71$ $$17\!\cdots\!00$$$$+$$$$39\!\cdots\!80$$$$T +$$$$13\!\cdots\!96$$$$T^{2} +$$$$43\!\cdots\!60$$$$T^{3} +$$$$28\!\cdots\!72$$$$T^{4} +$$$$21\!\cdots\!92$$$$T^{5} +$$$$27\!\cdots\!00$$$$T^{6} +$$$$71\!\cdots\!28$$$$T^{7} +$$$$42\!\cdots\!72$$$$T^{8} +$$$$65\!\cdots\!32$$$$T^{9} +$$$$41\!\cdots\!64$$$$T^{10} +$$$$44\!\cdots\!68$$$$T^{11} +$$$$19\!\cdots\!33$$$$T^{12} +$$$$39\!\cdots\!97$$$$T^{13} + 53973233604256978870 T^{14} - 53391804438599 T^{15} + 10847901010 T^{16} - 53463 T^{17} + T^{18}$$
$73$ $$43\!\cdots\!25$$$$+$$$$74\!\cdots\!75$$$$T +$$$$29\!\cdots\!25$$$$T^{2} +$$$$13\!\cdots\!20$$$$T^{3} +$$$$17\!\cdots\!06$$$$T^{4} +$$$$62\!\cdots\!86$$$$T^{5} +$$$$33\!\cdots\!42$$$$T^{6} +$$$$67\!\cdots\!72$$$$T^{7} +$$$$29\!\cdots\!47$$$$T^{8} +$$$$45\!\cdots\!81$$$$T^{9} +$$$$17\!\cdots\!35$$$$T^{10} +$$$$15\!\cdots\!60$$$$T^{11} +$$$$50\!\cdots\!46$$$$T^{12} +$$$$28\!\cdots\!26$$$$T^{13} +$$$$11\!\cdots\!74$$$$T^{14} + 307313940776492 T^{15} + 12536846917 T^{16} + 14103 T^{17} + T^{18}$$
$79$ $$12\!\cdots\!44$$$$-$$$$90\!\cdots\!32$$$$T +$$$$33\!\cdots\!84$$$$T^{2} +$$$$34\!\cdots\!16$$$$T^{3} +$$$$49\!\cdots\!56$$$$T^{4} +$$$$22\!\cdots\!72$$$$T^{5} +$$$$11\!\cdots\!08$$$$T^{6} +$$$$26\!\cdots\!68$$$$T^{7} +$$$$12\!\cdots\!16$$$$T^{8} +$$$$20\!\cdots\!36$$$$T^{9} +$$$$86\!\cdots\!68$$$$T^{10} +$$$$76\!\cdots\!32$$$$T^{11} +$$$$31\!\cdots\!41$$$$T^{12} +$$$$18\!\cdots\!25$$$$T^{13} + 83272908481639449190 T^{14} + 246647107958493 T^{15} + 11209186574 T^{16} + 31825 T^{17} + T^{18}$$
$83$ $$( -$$$$24\!\cdots\!00$$$$+$$$$20\!\cdots\!20$$$$T -$$$$98\!\cdots\!28$$$$T^{2} -$$$$39\!\cdots\!37$$$$T^{3} +$$$$15\!\cdots\!20$$$$T^{4} + 38020731823258649847 T^{5} - 511917348889560 T^{6} - 11706552339 T^{7} + 41300 T^{8} + T^{9} )^{2}$$
$89$ $$86\!\cdots\!44$$$$+$$$$57\!\cdots\!40$$$$T +$$$$64\!\cdots\!92$$$$T^{2} +$$$$27\!\cdots\!72$$$$T^{3} +$$$$26\!\cdots\!88$$$$T^{4} +$$$$11\!\cdots\!40$$$$T^{5} +$$$$49\!\cdots\!48$$$$T^{6} +$$$$13\!\cdots\!36$$$$T^{7} +$$$$35\!\cdots\!68$$$$T^{8} +$$$$69\!\cdots\!20$$$$T^{9} +$$$$14\!\cdots\!24$$$$T^{10} +$$$$22\!\cdots\!88$$$$T^{11} +$$$$35\!\cdots\!61$$$$T^{12} +$$$$38\!\cdots\!45$$$$T^{13} +$$$$41\!\cdots\!70$$$$T^{14} + 3121899045395049 T^{15} + 29541644542 T^{16} + 155197 T^{17} + T^{18}$$
$97$ $$93\!\cdots\!25$$$$+$$$$95\!\cdots\!75$$$$T +$$$$10\!\cdots\!25$$$$T^{2} +$$$$26\!\cdots\!00$$$$T^{3} +$$$$17\!\cdots\!14$$$$T^{4} +$$$$38\!\cdots\!90$$$$T^{5} +$$$$22\!\cdots\!50$$$$T^{6} +$$$$28\!\cdots\!48$$$$T^{7} +$$$$71\!\cdots\!71$$$$T^{8} +$$$$31\!\cdots\!17$$$$T^{9} +$$$$10\!\cdots\!83$$$$T^{10} +$$$$19\!\cdots\!24$$$$T^{11} +$$$$12\!\cdots\!86$$$$T^{12} -$$$$12\!\cdots\!34$$$$T^{13} +$$$$85\!\cdots\!22$$$$T^{14} - 1646469577269280 T^{15} + 42695596097 T^{16} - 111241 T^{17} + T^{18}$$