# Properties

 Label 252.9.z.e Level $252$ Weight $9$ Character orbit 252.z Analytic conductor $102.659$ Analytic rank $0$ Dimension $20$ CM no Inner twists $4$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 252.z (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$102.659409735$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$10$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Defining polynomial: $$x^{20} - 10 x^{19} - 1751639 x^{18} + 15765036 x^{17} + 1288400424989 x^{16} - 10307560742020 x^{15} - 516815190850700874 x^{14} + 3617886719518590784 x^{13} + 122992744079100130844521 x^{12} - 738003497470953185965030 x^{11} - 17716220051138913180459215577 x^{10} + 88587865373982244785457265884 x^{9} + 1507689855516745584812805692686751 x^{8} - 6031290957377412523625955897396396 x^{7} - 69448792452425943361067315312590506770 x^{6} + 208367487247705803791441930629487279968 x^{5} + 1331037342615679684155036722672299751825731 x^{4} - 2662421971413455508046552715438433455340210 x^{3} - 47007135303470257351659397734299414285784783 x^{2} + 48338381018096768126520835100261119832719644 x + 515596507492955532983572496163412311048918663$$ Coefficient ring: $$\Z[a_1, \ldots, a_{31}]$$ Coefficient ring index: $$2^{36}\cdot 3^{26}\cdot 7^{16}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{4} ) q^{5} + ( -577 - 430 \beta_{3} + \beta_{7} ) q^{7} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{4} ) q^{5} + ( -577 - 430 \beta_{3} + \beta_{7} ) q^{7} + ( -2 \beta_{1} - \beta_{4} - \beta_{11} ) q^{11} + ( -875 - 1750 \beta_{3} - 4 \beta_{5} - 3 \beta_{6} + \beta_{7} + 4 \beta_{8} + 4 \beta_{9} - 4 \beta_{12} - \beta_{14} - \beta_{15} ) q^{13} + ( 9 \beta_{4} + 2 \beta_{10} + \beta_{11} - \beta_{17} ) q^{17} + ( -3565 + 3577 \beta_{3} + 15 \beta_{5} + 19 \beta_{6} - 19 \beta_{7} + 2 \beta_{9} + 4 \beta_{12} - 10 \beta_{14} ) q^{19} + ( -41 \beta_{1} + 41 \beta_{4} - \beta_{10} - \beta_{13} ) q^{23} + ( 23 - 134842 \beta_{3} - 37 \beta_{5} - 37 \beta_{6} + 6 \beta_{7} - \beta_{8} - 2 \beta_{9} - 6 \beta_{12} - 48 \beta_{14} - 24 \beta_{15} ) q^{25} + ( -29 \beta_{1} + \beta_{2} - 58 \beta_{4} - 9 \beta_{10} - 9 \beta_{11} + 2 \beta_{17} + \beta_{18} - \beta_{19} ) q^{29} + ( -315576 - 157781 \beta_{3} + \beta_{5} - \beta_{6} - 138 \beta_{7} - 9 \beta_{8} - 138 \beta_{12} - 23 \beta_{15} ) q^{31} + ( 154 \beta_{1} + \beta_{2} + 356 \beta_{4} + 9 \beta_{10} + 20 \beta_{11} - \beta_{13} + 3 \beta_{16} - 8 \beta_{17} + \beta_{18} ) q^{35} + ( 680173 + 680168 \beta_{3} + 126 \beta_{5} + 189 \beta_{6} + 189 \beta_{7} + 110 \beta_{8} + 55 \beta_{9} + 315 \beta_{12} + 60 \beta_{14} + 120 \beta_{15} ) q^{37} + ( -733 \beta_{1} + 10 \beta_{2} + 30 \beta_{10} - 30 \beta_{11} - 8 \beta_{13} - 4 \beta_{16} - \beta_{18} - \beta_{19} ) q^{41} + ( -29966 - 226 \beta_{5} - 354 \beta_{6} + 580 \beta_{7} - 97 \beta_{8} + 97 \beta_{9} + 226 \beta_{12} + 95 \beta_{14} - 95 \beta_{15} ) q^{43} + ( 14 \beta_{1} - 47 \beta_{2} + 14 \beta_{4} - 11 \beta_{10} - 22 \beta_{11} - 3 \beta_{13} - 6 \beta_{16} - 48 \beta_{17} - \beta_{18} + 2 \beta_{19} ) q^{47} + ( -727895 + 469448 \beta_{3} - 91 \beta_{5} + 98 \beta_{6} - 392 \beta_{7} - 343 \beta_{9} - 392 \beta_{12} + 686 \beta_{14} + 343 \beta_{15} ) q^{49} + ( 312 \beta_{1} + 69 \beta_{2} + 156 \beta_{4} - 144 \beta_{11} + 20 \beta_{13} + 20 \beta_{16} + 34 \beta_{17} + \beta_{19} ) q^{53} + ( -403043 - 806086 \beta_{3} + 1421 \beta_{5} + 648 \beta_{6} - 773 \beta_{7} - 920 \beta_{8} - 920 \beta_{9} + 1421 \beta_{12} - 156 \beta_{14} - 156 \beta_{15} ) q^{55} + ( -\beta_{2} - 3137 \beta_{4} + 368 \beta_{10} + 184 \beta_{11} + 17 \beta_{13} - 17 \beta_{16} + 44 \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{59} + ( 1085615 - 1085135 \beta_{3} - 2327 \beta_{5} - 1389 \beta_{6} + 1389 \beta_{7} + 187 \beta_{9} + 938 \beta_{12} - 293 \beta_{14} ) q^{61} + ( 1630 \beta_{1} - 126 \beta_{2} - 1630 \beta_{4} - 1209 \beta_{10} + 8 \beta_{13} + 125 \beta_{17} - \beta_{18} ) q^{65} + ( 1548 + 2601397 \beta_{3} + 2490 \beta_{5} + 2490 \beta_{6} - 3025 \beta_{7} + 864 \beta_{8} + 1728 \beta_{9} + 3025 \beta_{12} - 1368 \beta_{14} - 684 \beta_{15} ) q^{67} + ( 1826 \beta_{1} + 76 \beta_{2} + 3652 \beta_{4} + 343 \beta_{10} + 343 \beta_{11} - 13 \beta_{16} + 152 \beta_{17} - 15 \beta_{18} + 15 \beta_{19} ) q^{71} + ( -3526803 - 1763581 \beta_{3} - 1413 \beta_{5} + 1413 \beta_{6} + 585 \beta_{7} + 1088 \beta_{8} + 585 \beta_{12} + 1447 \beta_{15} ) q^{73} + ( 9976 \beta_{1} + 42 \beta_{2} + 4797 \beta_{4} + 807 \beta_{10} + 1896 \beta_{11} + 12 \beta_{13} - 215 \beta_{17} - 16 \beta_{18} + \beta_{19} ) q^{77} + ( -12993035 - 12988829 \beta_{3} + 898 \beta_{5} - 2141 \beta_{6} - 2141 \beta_{7} + 4810 \beta_{8} + 2405 \beta_{9} - 1243 \beta_{12} - 1801 \beta_{14} - 3602 \beta_{15} ) q^{79} + ( -598 \beta_{1} + 56 \beta_{2} + 660 \beta_{10} - 660 \beta_{11} + 58 \beta_{13} + 29 \beta_{16} + 16 \beta_{18} + 16 \beta_{19} ) q^{83} + ( -4189291 - 11125 \beta_{5} - 5009 \beta_{6} + 16134 \beta_{7} - 4155 \beta_{8} + 4155 \beta_{9} + 11125 \beta_{12} - 779 \beta_{14} + 779 \beta_{15} ) q^{85} + ( 4396 \beta_{1} - 318 \beta_{2} + 4396 \beta_{4} - 2198 \beta_{10} - 4396 \beta_{11} + 20 \beta_{13} + 40 \beta_{16} - 302 \beta_{17} + 16 \beta_{18} - 32 \beta_{19} ) q^{89} + ( -1686010 - 7592602 \beta_{3} + 2046 \beta_{5} + 6370 \beta_{6} - 3788 \beta_{7} + 980 \beta_{8} - 7007 \beta_{9} + 16660 \beta_{12} - 1813 \beta_{14} - 1176 \beta_{15} ) q^{91} + ( -61828 \beta_{1} + 255 \beta_{2} - 30914 \beta_{4} - 1657 \beta_{11} - 187 \beta_{13} - 187 \beta_{16} + 136 \beta_{17} - 17 \beta_{19} ) q^{95} + ( 5988500 + 11977000 \beta_{3} + 32318 \beta_{5} + 14311 \beta_{6} - 18007 \beta_{7} - 10506 \beta_{8} - 10506 \beta_{9} + 32318 \beta_{12} + 6653 \beta_{14} + 6653 \beta_{15} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20 q - 7238 q^{7} + O(q^{10})$$ $$20 q - 7238 q^{7} - 107256 q^{19} + 1348832 q^{25} - 4734426 q^{31} + 6802748 q^{37} - 596128 q^{43} - 19249258 q^{49} + 32573772 q^{61} - 25999304 q^{67} - 52899612 q^{73} - 129945530 q^{79} - 83755728 q^{85} + 42306852 q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 10 x^{19} - 1751639 x^{18} + 15765036 x^{17} + 1288400424989 x^{16} - 10307560742020 x^{15} - 516815190850700874 x^{14} + 3617886719518590784 x^{13} + 122992744079100130844521 x^{12} - 738003497470953185965030 x^{11} - 17716220051138913180459215577 x^{10} + 88587865373982244785457265884 x^{9} + 1507689855516745584812805692686751 x^{8} - 6031290957377412523625955897396396 x^{7} - 69448792452425943361067315312590506770 x^{6} + 208367487247705803791441930629487279968 x^{5} + 1331037342615679684155036722672299751825731 x^{4} - 2662421971413455508046552715438433455340210 x^{3} - 47007135303470257351659397734299414285784783 x^{2} + 48338381018096768126520835100261119832719644 x + 515596507492955532983572496163412311048918663$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-$$$$12\!\cdots\!75$$$$\nu^{18} +$$$$10\!\cdots\!75$$$$\nu^{17} +$$$$19\!\cdots\!47$$$$\nu^{16} -$$$$15\!\cdots\!76$$$$\nu^{15} -$$$$12\!\cdots\!54$$$$\nu^{14} +$$$$90\!\cdots\!58$$$$\nu^{13} +$$$$45\!\cdots\!02$$$$\nu^{12} -$$$$27\!\cdots\!24$$$$\nu^{11} -$$$$91\!\cdots\!73$$$$\nu^{10} +$$$$45\!\cdots\!49$$$$\nu^{9} +$$$$10\!\cdots\!57$$$$\nu^{8} -$$$$41\!\cdots\!72$$$$\nu^{7} -$$$$62\!\cdots\!42$$$$\nu^{6} +$$$$18\!\cdots\!46$$$$\nu^{5} +$$$$15\!\cdots\!86$$$$\nu^{4} -$$$$30\!\cdots\!72$$$$\nu^{3} -$$$$28\!\cdots\!83$$$$\nu^{2} +$$$$28\!\cdots\!51$$$$\nu +$$$$51\!\cdots\!39$$$$)/$$$$27\!\cdots\!72$$ $$\beta_{2}$$ $$=$$ $$($$$$14\!\cdots\!49$$$$\nu^{18} -$$$$13\!\cdots\!41$$$$\nu^{17} -$$$$26\!\cdots\!01$$$$\nu^{16} +$$$$20\!\cdots\!04$$$$\nu^{15} +$$$$19\!\cdots\!66$$$$\nu^{14} -$$$$13\!\cdots\!18$$$$\nu^{13} -$$$$76\!\cdots\!10$$$$\nu^{12} +$$$$46\!\cdots\!24$$$$\nu^{11} +$$$$18\!\cdots\!63$$$$\nu^{10} -$$$$91\!\cdots\!31$$$$\nu^{9} -$$$$26\!\cdots\!67$$$$\nu^{8} +$$$$10\!\cdots\!80$$$$\nu^{7} +$$$$22\!\cdots\!46$$$$\nu^{6} -$$$$66\!\cdots\!38$$$$\nu^{5} -$$$$10\!\cdots\!14$$$$\nu^{4} +$$$$20\!\cdots\!56$$$$\nu^{3} +$$$$18\!\cdots\!89$$$$\nu^{2} -$$$$18\!\cdots\!57$$$$\nu -$$$$34\!\cdots\!13$$$$)/$$$$10\!\cdots\!00$$ $$\beta_{3}$$ $$=$$ $$($$$$41\!\cdots\!46$$$$\nu^{19} -$$$$39\!\cdots\!37$$$$\nu^{18} -$$$$72\!\cdots\!75$$$$\nu^{17} +$$$$62\!\cdots\!31$$$$\nu^{16} +$$$$53\!\cdots\!12$$$$\nu^{15} -$$$$40\!\cdots\!34$$$$\nu^{14} -$$$$21\!\cdots\!26$$$$\nu^{13} +$$$$13\!\cdots\!86$$$$\nu^{12} +$$$$51\!\cdots\!74$$$$\nu^{11} -$$$$28\!\cdots\!23$$$$\nu^{10} -$$$$73\!\cdots\!01$$$$\nu^{9} +$$$$33\!\cdots\!81$$$$\nu^{8} +$$$$62\!\cdots\!64$$$$\nu^{7} -$$$$21\!\cdots\!74$$$$\nu^{6} -$$$$28\!\cdots\!22$$$$\nu^{5} +$$$$71\!\cdots\!62$$$$\nu^{4} +$$$$54\!\cdots\!02$$$$\nu^{3} -$$$$82\!\cdots\!49$$$$\nu^{2} -$$$$88\!\cdots\!15$$$$\nu -$$$$63\!\cdots\!61$$$$)/$$$$21\!\cdots\!20$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$20\!\cdots\!23$$$$\nu^{19} +$$$$16\!\cdots\!31$$$$\nu^{18} +$$$$36\!\cdots\!75$$$$\nu^{17} -$$$$25\!\cdots\!48$$$$\nu^{16} -$$$$26\!\cdots\!46$$$$\nu^{15} +$$$$17\!\cdots\!82$$$$\nu^{14} +$$$$10\!\cdots\!58$$$$\nu^{13} -$$$$60\!\cdots\!88$$$$\nu^{12} -$$$$25\!\cdots\!97$$$$\nu^{11} +$$$$12\!\cdots\!29$$$$\nu^{10} +$$$$36\!\cdots\!73$$$$\nu^{9} -$$$$13\!\cdots\!48$$$$\nu^{8} -$$$$30\!\cdots\!62$$$$\nu^{7} +$$$$82\!\cdots\!82$$$$\nu^{6} +$$$$14\!\cdots\!26$$$$\nu^{5} -$$$$20\!\cdots\!16$$$$\nu^{4} -$$$$27\!\cdots\!31$$$$\nu^{3} +$$$$41\!\cdots\!67$$$$\nu^{2} +$$$$11\!\cdots\!95$$$$\nu -$$$$74\!\cdots\!32$$$$)/$$$$72\!\cdots\!40$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$43\!\cdots\!25$$$$\nu^{19} +$$$$41\!\cdots\!08$$$$\nu^{18} +$$$$76\!\cdots\!38$$$$\nu^{17} -$$$$65\!\cdots\!59$$$$\nu^{16} -$$$$56\!\cdots\!66$$$$\nu^{15} +$$$$42\!\cdots\!28$$$$\nu^{14} +$$$$22\!\cdots\!72$$$$\nu^{13} -$$$$14\!\cdots\!26$$$$\nu^{12} -$$$$53\!\cdots\!67$$$$\nu^{11} +$$$$29\!\cdots\!92$$$$\nu^{10} +$$$$77\!\cdots\!02$$$$\nu^{9} -$$$$34\!\cdots\!65$$$$\nu^{8} -$$$$65\!\cdots\!22$$$$\nu^{7} +$$$$23\!\cdots\!72$$$$\nu^{6} +$$$$30\!\cdots\!76$$$$\nu^{5} -$$$$75\!\cdots\!74$$$$\nu^{4} -$$$$58\!\cdots\!97$$$$\nu^{3} +$$$$87\!\cdots\!72$$$$\nu^{2} +$$$$94\!\cdots\!46$$$$\nu -$$$$74\!\cdots\!27$$$$)/$$$$31\!\cdots\!00$$ $$\beta_{6}$$ $$=$$ $$($$$$18\!\cdots\!80$$$$\nu^{19} -$$$$17\!\cdots\!89$$$$\nu^{18} -$$$$31\!\cdots\!47$$$$\nu^{17} +$$$$27\!\cdots\!85$$$$\nu^{16} +$$$$23\!\cdots\!60$$$$\nu^{15} -$$$$17\!\cdots\!34$$$$\nu^{14} -$$$$94\!\cdots\!26$$$$\nu^{13} +$$$$61\!\cdots\!30$$$$\nu^{12} +$$$$22\!\cdots\!80$$$$\nu^{11} -$$$$12\!\cdots\!47$$$$\nu^{10} -$$$$32\!\cdots\!53$$$$\nu^{9} +$$$$14\!\cdots\!19$$$$\nu^{8} +$$$$27\!\cdots\!80$$$$\nu^{7} -$$$$96\!\cdots\!18$$$$\nu^{6} -$$$$12\!\cdots\!66$$$$\nu^{5} +$$$$31\!\cdots\!58$$$$\nu^{4} +$$$$24\!\cdots\!60$$$$\nu^{3} -$$$$36\!\cdots\!89$$$$\nu^{2} -$$$$39\!\cdots\!67$$$$\nu +$$$$10\!\cdots\!73$$$$)/$$$$51\!\cdots\!00$$ $$\beta_{7}$$ $$=$$ $$($$$$18\!\cdots\!80$$$$\nu^{19} -$$$$17\!\cdots\!31$$$$\nu^{18} -$$$$31\!\cdots\!69$$$$\nu^{17} +$$$$26\!\cdots\!11$$$$\nu^{16} +$$$$23\!\cdots\!84$$$$\nu^{15} -$$$$17\!\cdots\!06$$$$\nu^{14} -$$$$94\!\cdots\!54$$$$\nu^{13} +$$$$60\!\cdots\!94$$$$\nu^{12} +$$$$22\!\cdots\!68$$$$\nu^{11} -$$$$12\!\cdots\!25$$$$\nu^{10} -$$$$32\!\cdots\!91$$$$\nu^{9} +$$$$14\!\cdots\!09$$$$\nu^{8} +$$$$27\!\cdots\!88$$$$\nu^{7} -$$$$96\!\cdots\!86$$$$\nu^{6} -$$$$12\!\cdots\!50$$$$\nu^{5} +$$$$31\!\cdots\!54$$$$\nu^{4} +$$$$24\!\cdots\!68$$$$\nu^{3} -$$$$36\!\cdots\!27$$$$\nu^{2} -$$$$39\!\cdots\!01$$$$\nu -$$$$66\!\cdots\!89$$$$)/$$$$51\!\cdots\!00$$ $$\beta_{8}$$ $$=$$ $$($$$$-$$$$19\!\cdots\!15$$$$\nu^{19} +$$$$19\!\cdots\!61$$$$\nu^{18} +$$$$34\!\cdots\!85$$$$\nu^{17} -$$$$30\!\cdots\!02$$$$\nu^{16} -$$$$25\!\cdots\!58$$$$\nu^{15} +$$$$19\!\cdots\!06$$$$\nu^{14} +$$$$10\!\cdots\!74$$$$\nu^{13} -$$$$69\!\cdots\!48$$$$\nu^{12} -$$$$24\!\cdots\!01$$$$\nu^{11} +$$$$13\!\cdots\!67$$$$\nu^{10} +$$$$34\!\cdots\!35$$$$\nu^{9} -$$$$16\!\cdots\!82$$$$\nu^{8} -$$$$29\!\cdots\!66$$$$\nu^{7} +$$$$10\!\cdots\!90$$$$\nu^{6} +$$$$13\!\cdots\!26$$$$\nu^{5} -$$$$35\!\cdots\!76$$$$\nu^{4} -$$$$26\!\cdots\!31$$$$\nu^{3} +$$$$39\!\cdots\!73$$$$\nu^{2} +$$$$42\!\cdots\!09$$$$\nu -$$$$17\!\cdots\!70$$$$)/$$$$51\!\cdots\!00$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$19\!\cdots\!15$$$$\nu^{19} +$$$$18\!\cdots\!24$$$$\nu^{18} +$$$$34\!\cdots\!18$$$$\nu^{17} -$$$$28\!\cdots\!41$$$$\nu^{16} -$$$$25\!\cdots\!94$$$$\nu^{15} +$$$$18\!\cdots\!64$$$$\nu^{14} +$$$$10\!\cdots\!16$$$$\nu^{13} -$$$$63\!\cdots\!94$$$$\nu^{12} -$$$$24\!\cdots\!33$$$$\nu^{11} +$$$$12\!\cdots\!84$$$$\nu^{10} +$$$$34\!\cdots\!42$$$$\nu^{9} -$$$$15\!\cdots\!67$$$$\nu^{8} -$$$$29\!\cdots\!78$$$$\nu^{7} +$$$$10\!\cdots\!92$$$$\nu^{6} +$$$$13\!\cdots\!52$$$$\nu^{5} -$$$$33\!\cdots\!70$$$$\nu^{4} -$$$$26\!\cdots\!43$$$$\nu^{3} +$$$$39\!\cdots\!80$$$$\nu^{2} +$$$$42\!\cdots\!10$$$$\nu +$$$$13\!\cdots\!23$$$$)/$$$$51\!\cdots\!00$$ $$\beta_{10}$$ $$=$$ $$($$$$13\!\cdots\!09$$$$\nu^{19} -$$$$82\!\cdots\!74$$$$\nu^{18} -$$$$24\!\cdots\!04$$$$\nu^{17} +$$$$14\!\cdots\!21$$$$\nu^{16} +$$$$17\!\cdots\!78$$$$\nu^{15} -$$$$10\!\cdots\!00$$$$\nu^{14} -$$$$71\!\cdots\!84$$$$\nu^{13} +$$$$39\!\cdots\!54$$$$\nu^{12} +$$$$16\!\cdots\!15$$$$\nu^{11} -$$$$89\!\cdots\!62$$$$\nu^{10} -$$$$24\!\cdots\!80$$$$\nu^{9} +$$$$12\!\cdots\!99$$$$\nu^{8} +$$$$20\!\cdots\!26$$$$\nu^{7} -$$$$98\!\cdots\!80$$$$\nu^{6} -$$$$95\!\cdots\!08$$$$\nu^{5} +$$$$42\!\cdots\!94$$$$\nu^{4} +$$$$18\!\cdots\!57$$$$\nu^{3} -$$$$75\!\cdots\!10$$$$\nu^{2} -$$$$29\!\cdots\!84$$$$\nu +$$$$13\!\cdots\!41$$$$)/$$$$28\!\cdots\!00$$ $$\beta_{11}$$ $$=$$ $$($$$$13\!\cdots\!09$$$$\nu^{19} +$$$$56\!\cdots\!03$$$$\nu^{18} -$$$$24\!\cdots\!97$$$$\nu^{17} -$$$$10\!\cdots\!52$$$$\nu^{16} +$$$$17\!\cdots\!70$$$$\nu^{15} +$$$$74\!\cdots\!18$$$$\nu^{14} -$$$$71\!\cdots\!98$$$$\nu^{13} -$$$$29\!\cdots\!76$$$$\nu^{12} +$$$$16\!\cdots\!67$$$$\nu^{11} +$$$$70\!\cdots\!37$$$$\nu^{10} -$$$$24\!\cdots\!43$$$$\nu^{9} -$$$$10\!\cdots\!92$$$$\nu^{8} +$$$$20\!\cdots\!66$$$$\nu^{7} +$$$$83\!\cdots\!78$$$$\nu^{6} -$$$$96\!\cdots\!82$$$$\nu^{5} -$$$$37\!\cdots\!28$$$$\nu^{4} +$$$$18\!\cdots\!45$$$$\nu^{3} +$$$$70\!\cdots\!87$$$$\nu^{2} -$$$$17\!\cdots\!45$$$$\nu -$$$$12\!\cdots\!08$$$$)/$$$$28\!\cdots\!00$$ $$\beta_{12}$$ $$=$$ $$($$$$-$$$$15\!\cdots\!05$$$$\nu^{19} +$$$$14\!\cdots\!01$$$$\nu^{18} +$$$$26\!\cdots\!89$$$$\nu^{17} -$$$$22\!\cdots\!96$$$$\nu^{16} -$$$$19\!\cdots\!74$$$$\nu^{15} +$$$$14\!\cdots\!26$$$$\nu^{14} +$$$$79\!\cdots\!34$$$$\nu^{13} -$$$$51\!\cdots\!84$$$$\nu^{12} -$$$$18\!\cdots\!23$$$$\nu^{11} +$$$$10\!\cdots\!55$$$$\nu^{10} +$$$$27\!\cdots\!71$$$$\nu^{9} -$$$$12\!\cdots\!84$$$$\nu^{8} -$$$$23\!\cdots\!18$$$$\nu^{7} +$$$$81\!\cdots\!66$$$$\nu^{6} +$$$$10\!\cdots\!90$$$$\nu^{5} -$$$$26\!\cdots\!64$$$$\nu^{4} -$$$$20\!\cdots\!73$$$$\nu^{3} +$$$$30\!\cdots\!57$$$$\nu^{2} +$$$$33\!\cdots\!41$$$$\nu +$$$$45\!\cdots\!84$$$$)/$$$$31\!\cdots\!00$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$19\!\cdots\!79$$$$\nu^{19} +$$$$21\!\cdots\!44$$$$\nu^{18} +$$$$33\!\cdots\!74$$$$\nu^{17} -$$$$37\!\cdots\!01$$$$\nu^{16} -$$$$24\!\cdots\!18$$$$\nu^{15} +$$$$26\!\cdots\!00$$$$\nu^{14} +$$$$98\!\cdots\!04$$$$\nu^{13} -$$$$10\!\cdots\!74$$$$\nu^{12} -$$$$23\!\cdots\!65$$$$\nu^{11} +$$$$24\!\cdots\!72$$$$\nu^{10} +$$$$33\!\cdots\!30$$$$\nu^{9} -$$$$32\!\cdots\!19$$$$\nu^{8} -$$$$28\!\cdots\!06$$$$\nu^{7} +$$$$23\!\cdots\!80$$$$\nu^{6} +$$$$13\!\cdots\!48$$$$\nu^{5} -$$$$86\!\cdots\!14$$$$\nu^{4} -$$$$25\!\cdots\!67$$$$\nu^{3} +$$$$10\!\cdots\!60$$$$\nu^{2} +$$$$40\!\cdots\!54$$$$\nu -$$$$19\!\cdots\!21$$$$)/$$$$18\!\cdots\!00$$ $$\beta_{14}$$ $$=$$ $$($$$$-$$$$69\!\cdots\!55$$$$\nu^{19} +$$$$65\!\cdots\!83$$$$\nu^{18} +$$$$12\!\cdots\!31$$$$\nu^{17} -$$$$10\!\cdots\!72$$$$\nu^{16} -$$$$89\!\cdots\!98$$$$\nu^{15} +$$$$67\!\cdots\!38$$$$\nu^{14} +$$$$35\!\cdots\!22$$$$\nu^{13} -$$$$23\!\cdots\!48$$$$\nu^{12} -$$$$85\!\cdots\!61$$$$\nu^{11} +$$$$46\!\cdots\!53$$$$\nu^{10} +$$$$12\!\cdots\!89$$$$\nu^{9} -$$$$55\!\cdots\!64$$$$\nu^{8} -$$$$10\!\cdots\!26$$$$\nu^{7} +$$$$36\!\cdots\!14$$$$\nu^{6} +$$$$47\!\cdots\!34$$$$\nu^{5} -$$$$12\!\cdots\!40$$$$\nu^{4} -$$$$91\!\cdots\!31$$$$\nu^{3} +$$$$13\!\cdots\!35$$$$\nu^{2} +$$$$14\!\cdots\!95$$$$\nu +$$$$47\!\cdots\!16$$$$)/$$$$49\!\cdots\!00$$ $$\beta_{15}$$ $$=$$ $$($$$$-$$$$69\!\cdots\!55$$$$\nu^{19} +$$$$65\!\cdots\!62$$$$\nu^{18} +$$$$12\!\cdots\!20$$$$\nu^{17} -$$$$10\!\cdots\!59$$$$\nu^{16} -$$$$89\!\cdots\!86$$$$\nu^{15} +$$$$66\!\cdots\!52$$$$\nu^{14} +$$$$35\!\cdots\!08$$$$\nu^{13} -$$$$23\!\cdots\!66$$$$\nu^{12} -$$$$85\!\cdots\!17$$$$\nu^{11} +$$$$46\!\cdots\!14$$$$\nu^{10} +$$$$12\!\cdots\!20$$$$\nu^{9} -$$$$54\!\cdots\!69$$$$\nu^{8} -$$$$10\!\cdots\!22$$$$\nu^{7} +$$$$36\!\cdots\!80$$$$\nu^{6} +$$$$47\!\cdots\!92$$$$\nu^{5} -$$$$11\!\cdots\!42$$$$\nu^{4} -$$$$91\!\cdots\!27$$$$\nu^{3} +$$$$13\!\cdots\!66$$$$\nu^{2} +$$$$14\!\cdots\!28$$$$\nu -$$$$62\!\cdots\!15$$$$)/$$$$49\!\cdots\!00$$ $$\beta_{16}$$ $$=$$ $$($$$$38\!\cdots\!58$$$$\nu^{19} -$$$$36\!\cdots\!01$$$$\nu^{18} -$$$$66\!\cdots\!31$$$$\nu^{17} +$$$$56\!\cdots\!39$$$$\nu^{16} +$$$$49\!\cdots\!88$$$$\nu^{15} -$$$$36\!\cdots\!42$$$$\nu^{14} -$$$$19\!\cdots\!42$$$$\nu^{13} +$$$$12\!\cdots\!18$$$$\nu^{12} +$$$$46\!\cdots\!42$$$$\nu^{11} -$$$$25\!\cdots\!75$$$$\nu^{10} -$$$$67\!\cdots\!13$$$$\nu^{9} +$$$$30\!\cdots\!17$$$$\nu^{8} +$$$$57\!\cdots\!52$$$$\nu^{7} -$$$$20\!\cdots\!62$$$$\nu^{6} -$$$$26\!\cdots\!90$$$$\nu^{5} +$$$$66\!\cdots\!46$$$$\nu^{4} +$$$$51\!\cdots\!62$$$$\nu^{3} -$$$$76\!\cdots\!13$$$$\nu^{2} -$$$$28\!\cdots\!99$$$$\nu +$$$$14\!\cdots\!23$$$$)/$$$$18\!\cdots\!00$$ $$\beta_{17}$$ $$=$$ $$($$$$27\!\cdots\!01$$$$\nu^{19} -$$$$65\!\cdots\!85$$$$\nu^{18} -$$$$48\!\cdots\!65$$$$\nu^{17} +$$$$10\!\cdots\!20$$$$\nu^{16} +$$$$35\!\cdots\!38$$$$\nu^{15} -$$$$76\!\cdots\!66$$$$\nu^{14} -$$$$14\!\cdots\!58$$$$\nu^{13} +$$$$29\!\cdots\!16$$$$\nu^{12} +$$$$34\!\cdots\!11$$$$\nu^{11} -$$$$66\!\cdots\!31$$$$\nu^{10} -$$$$49\!\cdots\!39$$$$\nu^{9} +$$$$90\!\cdots\!28$$$$\nu^{8} +$$$$41\!\cdots\!34$$$$\nu^{7} -$$$$72\!\cdots\!66$$$$\nu^{6} -$$$$19\!\cdots\!74$$$$\nu^{5} +$$$$31\!\cdots\!80$$$$\nu^{4} +$$$$37\!\cdots\!17$$$$\nu^{3} -$$$$54\!\cdots\!29$$$$\nu^{2} -$$$$15\!\cdots\!69$$$$\nu +$$$$10\!\cdots\!12$$$$)/$$$$56\!\cdots\!00$$ $$\beta_{18}$$ $$=$$ $$($$$$33\!\cdots\!09$$$$\nu^{19} -$$$$73\!\cdots\!41$$$$\nu^{18} -$$$$59\!\cdots\!01$$$$\nu^{17} +$$$$13\!\cdots\!04$$$$\nu^{16} +$$$$43\!\cdots\!46$$$$\nu^{15} -$$$$10\!\cdots\!78$$$$\nu^{14} -$$$$17\!\cdots\!90$$$$\nu^{13} +$$$$44\!\cdots\!84$$$$\nu^{12} +$$$$41\!\cdots\!23$$$$\nu^{11} -$$$$11\!\cdots\!91$$$$\nu^{10} -$$$$59\!\cdots\!07$$$$\nu^{9} +$$$$19\!\cdots\!60$$$$\nu^{8} +$$$$50\!\cdots\!86$$$$\nu^{7} -$$$$18\!\cdots\!98$$$$\nu^{6} -$$$$23\!\cdots\!54$$$$\nu^{5} +$$$$99\!\cdots\!56$$$$\nu^{4} +$$$$44\!\cdots\!09$$$$\nu^{3} -$$$$21\!\cdots\!97$$$$\nu^{2} -$$$$39\!\cdots\!53$$$$\nu +$$$$39\!\cdots\!20$$$$)/$$$$18\!\cdots\!00$$ $$\beta_{19}$$ $$=$$ $$($$$$-$$$$33\!\cdots\!09$$$$\nu^{19} -$$$$98\!\cdots\!70$$$$\nu^{18} +$$$$59\!\cdots\!00$$$$\nu^{17} +$$$$33\!\cdots\!35$$$$\nu^{16} -$$$$43\!\cdots\!02$$$$\nu^{15} -$$$$38\!\cdots\!96$$$$\nu^{14} +$$$$17\!\cdots\!92$$$$\nu^{13} +$$$$22\!\cdots\!06$$$$\nu^{12} -$$$$41\!\cdots\!59$$$$\nu^{11} -$$$$72\!\cdots\!66$$$$\nu^{10} +$$$$59\!\cdots\!16$$$$\nu^{9} +$$$$13\!\cdots\!53$$$$\nu^{8} -$$$$50\!\cdots\!06$$$$\nu^{7} -$$$$15\!\cdots\!96$$$$\nu^{6} +$$$$23\!\cdots\!36$$$$\nu^{5} +$$$$87\!\cdots\!90$$$$\nu^{4} -$$$$45\!\cdots\!93$$$$\nu^{3} -$$$$20\!\cdots\!74$$$$\nu^{2} +$$$$46\!\cdots\!76$$$$\nu +$$$$37\!\cdots\!87$$$$)/$$$$18\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{4} + 3 \beta_{3} + \beta_{1} + 3$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$24 \beta_{15} - 24 \beta_{14} + 31 \beta_{12} - \beta_{9} + \beta_{8} + 37 \beta_{7} - 6 \beta_{6} - 31 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} - 2 \beta_{1} + 525513$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$312 \beta_{19} - 312 \beta_{18} - 82 \beta_{17} + 72 \beta_{16} + 432 \beta_{15} + 216 \beta_{14} + 333 \beta_{12} + 12207 \beta_{11} + 12207 \beta_{10} + 9 \beta_{9} + 18 \beta_{8} + 279 \beta_{7} + 279 \beta_{6} + 54 \beta_{5} + 1688879 \beta_{4} + 4729203 \beta_{3} - 41 \beta_{2} + 844426 \beta_{1} + 4729401$$$$)/9$$ $$\nu^{4}$$ $$=$$ $$($$$$-1248 \beta_{18} - 164 \beta_{17} + 25204541 \beta_{15} - 25203245 \beta_{14} - 288 \beta_{13} + 61012892 \beta_{12} + 48828 \beta_{10} - 3420598 \beta_{9} + 3420652 \beta_{8} + 40115653 \beta_{7} + 20897689 \beta_{6} - 61012118 \beta_{5} + 3377752 \beta_{4} + 9458397 \beta_{3} - 1084 \beta_{2} - 3377788 \beta_{1} + 442327961807$$$$)/9$$ $$\nu^{5}$$ $$=$$ $$($$$$140042625 \beta_{19} - 140045745 \beta_{18} + 1780888 \beta_{17} - 11773292 \beta_{16} + 252040370 \beta_{15} + 126021265 \beta_{14} - 720 \beta_{13} + 200577875 \beta_{12} + 5309934535 \beta_{11} + 5310056605 \beta_{10} + 17103200 \beta_{9} + 34206310 \beta_{8} + 305062810 \beta_{7} + 305064100 \beta_{6} - 104484185 \beta_{5} + 531463073003 \beta_{4} + 2211421976391 \beta_{3} + 887939 \beta_{2} + 265718869864 \beta_{1} + 2211515128385$$$$)/9$$ $$\nu^{6}$$ $$=$$ $$($$$$-6240 \beta_{19} - 840268230 \beta_{18} + 5343074 \beta_{17} - 1440 \beta_{16} + 8522005327209 \beta_{15} - 8520871145544 \beta_{14} + 70635432 \beta_{13} + 25164989114637 \beta_{12} - 244140 \beta_{11} + 31860095490 \beta_{10} - 1345383708042 \beta_{9} + 1345537636437 \beta_{8} + 16158490285845 \beta_{7} + 9007727474772 \beta_{6} - 25164700835502 \beta_{5} + 1594380774632 \beta_{4} + 6634242283185 \beta_{3} - 845615084 \beta_{2} - 1594431441218 \beta_{1} + 139028330770172811$$$$)/9$$ $$\nu^{7}$$ $$=$$ $$($$$$52072729741221 \beta_{19} - 52075670690946 \beta_{18} + 28432828051446 \beta_{17} - 16679167623216 \beta_{16} + 119301899602949 \beta_{15} + 59652273014929 \beta_{14} + 247226532 \beta_{13} + 113105102375447 \beta_{12} + 1852958477116866 \beta_{11} + 1853069987878326 \beta_{10} + 9418524011183 \beta_{9} + 18836688855985 \beta_{8} + 176155713950272 \beta_{7} + 176156386603393 \beta_{6} - 63047744733155 \beta_{5} + 174442122316256613 \beta_{4} + 973097851837729389 \beta_{3} + 14213445031317 \beta_{2} + 87212690526061653 \beta_{1} + 973132603378742228$$$$)/9$$ $$\nu^{8}$$ $$=$$ $$($$$$-7842544248 \beta_{19} - 416597523053208 \beta_{18} + 113731287271056 \beta_{17} + 659268064 \beta_{16} + 2877769307876710303 \beta_{15} - 2877053496479083537 \beta_{14} + 133435318781856 \beta_{13} + 9454857725975145904 \beta_{12} - 297362486288 \beta_{11} + 14824262543274688 \beta_{10} - 417128560187269028 \beta_{9} + 417241580320405316 \beta_{8} + 6409228057973845097 \beta_{7} + 3046586490260333489 \beta_{6} - 9454657497889877560 \beta_{5} + 697761048829292920 \beta_{4} + 3892360447575665613 \beta_{3} - 530336802435552 \beta_{2} - 697805692294892800 \beta_{1} + 45624936241950913519630$$$$)/9$$ $$\nu^{9}$$ $$=$$ $$($$$$18503031228866217159 \beta_{19} - 18504905935365720465 \beta_{18} + 21793504891175213672 \beta_{17} - 9373129931013290876 \beta_{16} + 51794836885285963704 \beta_{15} + 25898492162915139735 \beta_{14} + 600457451156136 \beta_{13} + 57678668881733224863 \beta_{12} + 605409228217469709649 \beta_{11} + 605475938067981577975 \beta_{10} + 3754948184628707412 \beta_{9} + 7509557307996006192 \beta_{8} + 85094632065248735904 \beta_{7} + 85095232751118914928 \beta_{6} - 27413092709857989207 \beta_{5} + 58172699664182175123885 \beta_{4} + 410580142326494583319710 \beta_{3} + 10894110052394242891 \beta_{2} + 29081639794339633835343 \beta_{1} + 410590606943598970644558$$$$)/9$$ $$\nu^{10}$$ $$=$$ $$($$$$-6249045192696180 \beta_{19} - 185042810449630394790 \beta_{18} + 108966671471258937778 \beta_{17} + 2001526814979216 \beta_{16} + 974802994838814022258677 \beta_{15} - 974414533562150559677052 \beta_{14} + 93737303896511274536 \beta_{13} + 3450711810563854148978748 \beta_{12} - 222367060462404060 \beta_{11} + 6054537018971881644490 \beta_{10} - 114156299340710813287923 \beta_{9} + 114212621020524015832278 \beta_{8} + 2481094561247469001818006 \beta_{7} + 970179798119899977065082 \beta_{6} - 3450560484184703394362973 \beta_{5} + 290858265124205304456198 \beta_{4} + 2052871518975555747796776 \beta_{3} - 294020848877823113464 \beta_{2} - 290889665509812485032764 \beta_{1} + 15218323881856267442431339410$$$$)/9$$ $$\nu^{11}$$ $$=$$ $$($$$$6488045496180850901924637 \beta_{19} - 6489063248823295200557322 \beta_{18} + 11823804462157139785711024 \beta_{17} - 4298878934382940546864072 \beta_{16} + 21442342413291405573533841 \beta_{15} + 10721883389589910932514461 \beta_{14} + 515549667240229245140 \beta_{13} + 27288972306514588401499350 \beta_{12} + 193010659734526106944348427 \beta_{11} + 193043960299643548428189087 \beta_{10} + 1256201158909304760572970 \beta_{9} + 2512195803749398412320587 \beta_{8} + 37958280304711483827649209 \beta_{7} + 37958835170304226997606238 \beta_{6} - 10667245308725566463264637 \beta_{5} + 19530565736083753065605890586 \beta_{4} + 167382633622627462010261231856 \beta_{3} + 5909985482285100850577822 \beta_{2} + 9762883157827734977727104125 \beta_{1} + 167384570848620440891365298949$$$$)/9$$ $$\nu^{12}$$ $$=$$ $$($$$$-$$$$40\!\cdots\!08$$$$\beta_{19} -$$$$77\!\cdots\!04$$$$\beta_{18} +$$$$70\!\cdots\!76$$$$\beta_{17} +$$$$20\!\cdots\!24$$$$\beta_{16} +$$$$33\!\cdots\!58$$$$\beta_{15} -$$$$33\!\cdots\!28$$$$\beta_{14} +$$$$51\!\cdots\!04$$$$\beta_{13} +$$$$12\!\cdots\!74$$$$\beta_{12} -$$$$13\!\cdots\!48$$$$\beta_{11} +$$$$23\!\cdots\!64$$$$\beta_{10} -$$$$27\!\cdots\!44$$$$\beta_{9} +$$$$27\!\cdots\!34$$$$\beta_{8} +$$$$94\!\cdots\!30$$$$\beta_{7} +$$$$30\!\cdots\!04$$$$\beta_{6} -$$$$12\!\cdots\!44$$$$\beta_{5} +$$$$11\!\cdots\!92$$$$\beta_{4} +$$$$10\!\cdots\!50$$$$\beta_{3} -$$$$14\!\cdots\!08$$$$\beta_{2} -$$$$11\!\cdots\!84$$$$\beta_{1} +$$$$51\!\cdots\!73$$$$)/9$$ $$\nu^{13}$$ $$=$$ $$($$$$22\!\cdots\!10$$$$\beta_{19} -$$$$22\!\cdots\!24$$$$\beta_{18} +$$$$55\!\cdots\!44$$$$\beta_{17} -$$$$18\!\cdots\!52$$$$\beta_{16} +$$$$86\!\cdots\!34$$$$\beta_{15} +$$$$43\!\cdots\!06$$$$\beta_{14} +$$$$33\!\cdots\!40$$$$\beta_{13} +$$$$12\!\cdots\!62$$$$\beta_{12} +$$$$60\!\cdots\!20$$$$\beta_{11} +$$$$60\!\cdots\!44$$$$\beta_{10} +$$$$36\!\cdots\!90$$$$\beta_{9} +$$$$72\!\cdots\!94$$$$\beta_{8} +$$$$16\!\cdots\!16$$$$\beta_{7} +$$$$16\!\cdots\!38$$$$\beta_{6} -$$$$39\!\cdots\!42$$$$\beta_{5} +$$$$65\!\cdots\!12$$$$\beta_{4} +$$$$66\!\cdots\!77$$$$\beta_{3} +$$$$27\!\cdots\!94$$$$\beta_{2} +$$$$32\!\cdots\!27$$$$\beta_{1} +$$$$66\!\cdots\!95$$$$)/9$$ $$\nu^{14}$$ $$=$$ $$($$$$-$$$$23\!\cdots\!36$$$$\beta_{19} -$$$$31\!\cdots\!24$$$$\beta_{18} +$$$$38\!\cdots\!88$$$$\beta_{17} +$$$$15\!\cdots\!20$$$$\beta_{16} +$$$$11\!\cdots\!30$$$$\beta_{15} -$$$$11\!\cdots\!50$$$$\beta_{14} +$$$$25\!\cdots\!88$$$$\beta_{13} +$$$$44\!\cdots\!87$$$$\beta_{12} -$$$$70\!\cdots\!76$$$$\beta_{11} +$$$$85\!\cdots\!24$$$$\beta_{10} -$$$$56\!\cdots\!85$$$$\beta_{9} +$$$$56\!\cdots\!89$$$$\beta_{8} +$$$$35\!\cdots\!87$$$$\beta_{7} +$$$$93\!\cdots\!72$$$$\beta_{6} -$$$$44\!\cdots\!87$$$$\beta_{5} +$$$$46\!\cdots\!90$$$$\beta_{4} +$$$$46\!\cdots\!97$$$$\beta_{3} -$$$$70\!\cdots\!80$$$$\beta_{2} -$$$$46\!\cdots\!42$$$$\beta_{1} +$$$$17\!\cdots\!37$$$$)/9$$ $$\nu^{15}$$ $$=$$ $$($$$$78\!\cdots\!86$$$$\beta_{19} -$$$$79\!\cdots\!06$$$$\beta_{18} +$$$$23\!\cdots\!58$$$$\beta_{17} -$$$$72\!\cdots\!28$$$$\beta_{16} +$$$$33\!\cdots\!84$$$$\beta_{15} +$$$$16\!\cdots\!78$$$$\beta_{14} +$$$$19\!\cdots\!04$$$$\beta_{13} +$$$$52\!\cdots\!01$$$$\beta_{12} +$$$$19\!\cdots\!91$$$$\beta_{11} +$$$$19\!\cdots\!31$$$$\beta_{10} +$$$$84\!\cdots\!27$$$$\beta_{9} +$$$$16\!\cdots\!26$$$$\beta_{8} +$$$$66\!\cdots\!29$$$$\beta_{7} +$$$$66\!\cdots\!17$$$$\beta_{6} -$$$$14\!\cdots\!28$$$$\beta_{5} +$$$$22\!\cdots\!83$$$$\beta_{4} +$$$$25\!\cdots\!97$$$$\beta_{3} +$$$$11\!\cdots\!07$$$$\beta_{2} +$$$$11\!\cdots\!52$$$$\beta_{1} +$$$$25\!\cdots\!43$$$$)/9$$ $$\nu^{16}$$ $$=$$ $$($$$$-$$$$12\!\cdots\!64$$$$\beta_{19} -$$$$12\!\cdots\!76$$$$\beta_{18} +$$$$19\!\cdots\!24$$$$\beta_{17} +$$$$10\!\cdots\!60$$$$\beta_{16} +$$$$38\!\cdots\!77$$$$\beta_{15} -$$$$38\!\cdots\!23$$$$\beta_{14} +$$$$11\!\cdots\!68$$$$\beta_{13} +$$$$15\!\cdots\!20$$$$\beta_{12} -$$$$34\!\cdots\!24$$$$\beta_{11} +$$$$30\!\cdots\!56$$$$\beta_{10} -$$$$61\!\cdots\!76$$$$\beta_{9} +$$$$61\!\cdots\!88$$$$\beta_{8} +$$$$13\!\cdots\!75$$$$\beta_{7} +$$$$28\!\cdots\!23$$$$\beta_{6} -$$$$15\!\cdots\!84$$$$\beta_{5} +$$$$17\!\cdots\!08$$$$\beta_{4} +$$$$20\!\cdots\!47$$$$\beta_{3} -$$$$31\!\cdots\!12$$$$\beta_{2} -$$$$17\!\cdots\!20$$$$\beta_{1} +$$$$58\!\cdots\!57$$$$)/9$$ $$\nu^{17}$$ $$=$$ $$($$$$27\!\cdots\!33$$$$\beta_{19} -$$$$27\!\cdots\!31$$$$\beta_{18} +$$$$97\!\cdots\!36$$$$\beta_{17} -$$$$28\!\cdots\!08$$$$\beta_{16} +$$$$13\!\cdots\!04$$$$\beta_{15} +$$$$65\!\cdots\!81$$$$\beta_{14} +$$$$99\!\cdots\!04$$$$\beta_{13} +$$$$22\!\cdots\!57$$$$\beta_{12} +$$$$59\!\cdots\!43$$$$\beta_{11} +$$$$59\!\cdots\!81$$$$\beta_{10} +$$$$10\!\cdots\!16$$$$\beta_{9} +$$$$21\!\cdots\!20$$$$\beta_{8} +$$$$27\!\cdots\!64$$$$\beta_{7} +$$$$27\!\cdots\!36$$$$\beta_{6} -$$$$48\!\cdots\!97$$$$\beta_{5} +$$$$75\!\cdots\!25$$$$\beta_{4} +$$$$98\!\cdots\!53$$$$\beta_{3} +$$$$48\!\cdots\!05$$$$\beta_{2} +$$$$37\!\cdots\!34$$$$\beta_{1} +$$$$98\!\cdots\!93$$$$)/9$$ $$\nu^{18}$$ $$=$$ $$($$$$-$$$$64\!\cdots\!24$$$$\beta_{19} -$$$$49\!\cdots\!86$$$$\beta_{18} +$$$$87\!\cdots\!38$$$$\beta_{17} +$$$$59\!\cdots\!28$$$$\beta_{16} +$$$$13\!\cdots\!43$$$$\beta_{15} -$$$$13\!\cdots\!88$$$$\beta_{14} +$$$$51\!\cdots\!16$$$$\beta_{13} +$$$$56\!\cdots\!35$$$$\beta_{12} -$$$$15\!\cdots\!24$$$$\beta_{11} +$$$$10\!\cdots\!66$$$$\beta_{10} +$$$$21\!\cdots\!90$$$$\beta_{9} -$$$$21\!\cdots\!65$$$$\beta_{8} +$$$$48\!\cdots\!03$$$$\beta_{7} +$$$$87\!\cdots\!72$$$$\beta_{6} -$$$$56\!\cdots\!10$$$$\beta_{5} +$$$$67\!\cdots\!52$$$$\beta_{4} +$$$$89\!\cdots\!55$$$$\beta_{3} -$$$$13\!\cdots\!04$$$$\beta_{2} -$$$$67\!\cdots\!86$$$$\beta_{1} +$$$$19\!\cdots\!67$$$$)/9$$ $$\nu^{19}$$ $$=$$ $$($$$$95\!\cdots\!99$$$$\beta_{19} -$$$$95\!\cdots\!50$$$$\beta_{18} +$$$$38\!\cdots\!70$$$$\beta_{17} -$$$$10\!\cdots\!52$$$$\beta_{16} +$$$$49\!\cdots\!87$$$$\beta_{15} +$$$$24\!\cdots\!55$$$$\beta_{14} +$$$$48\!\cdots\!68$$$$\beta_{13} +$$$$91\!\cdots\!49$$$$\beta_{12} +$$$$18\!\cdots\!94$$$$\beta_{11} +$$$$18\!\cdots\!50$$$$\beta_{10} -$$$$40\!\cdots\!27$$$$\beta_{9} -$$$$81\!\cdots\!77$$$$\beta_{8} +$$$$10\!\cdots\!28$$$$\beta_{7} +$$$$10\!\cdots\!75$$$$\beta_{6} -$$$$16\!\cdots\!61$$$$\beta_{5} +$$$$25\!\cdots\!19$$$$\beta_{4} +$$$$37\!\cdots\!17$$$$\beta_{3} +$$$$19\!\cdots\!19$$$$\beta_{2} +$$$$12\!\cdots\!57$$$$\beta_{1} +$$$$37\!\cdots\!74$$$$)/9$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$73$$ $$127$$ $$\chi(n)$$ $$1$$ $$1 + \beta_{3}$$ $$1$$

## 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
 590.770 + 0.866025i 564.961 + 0.866025i 357.003 + 0.866025i 286.330 + 0.866025i 4.87946 + 0.866025i −3.87946 + 0.866025i −285.330 + 0.866025i −356.003 + 0.866025i −563.961 + 0.866025i −589.770 + 0.866025i 590.770 − 0.866025i 564.961 − 0.866025i 357.003 − 0.866025i 286.330 − 0.866025i 4.87946 − 0.866025i −3.87946 − 0.866025i −285.330 − 0.866025i −356.003 − 0.866025i −563.961 − 0.866025i −589.770 − 0.866025i
0 0 0 −885.405 + 511.189i 0 765.093 + 2275.84i 0 0 0
73.2 0 0 0 −846.691 + 488.837i 0 −1709.42 1686.02i 0 0 0
73.3 0 0 0 −534.755 + 308.741i 0 −750.736 2280.61i 0 0 0
73.4 0 0 0 −428.745 + 247.536i 0 1933.79 1423.12i 0 0 0
73.5 0 0 0 −6.56919 + 3.79272i 0 −2048.22 + 1252.83i 0 0 0
73.6 0 0 0 6.56919 3.79272i 0 −2048.22 + 1252.83i 0 0 0
73.7 0 0 0 428.745 247.536i 0 1933.79 1423.12i 0 0 0
73.8 0 0 0 534.755 308.741i 0 −750.736 2280.61i 0 0 0
73.9 0 0 0 846.691 488.837i 0 −1709.42 1686.02i 0 0 0
73.10 0 0 0 885.405 511.189i 0 765.093 + 2275.84i 0 0 0
145.1 0 0 0 −885.405 511.189i 0 765.093 2275.84i 0 0 0
145.2 0 0 0 −846.691 488.837i 0 −1709.42 + 1686.02i 0 0 0
145.3 0 0 0 −534.755 308.741i 0 −750.736 + 2280.61i 0 0 0
145.4 0 0 0 −428.745 247.536i 0 1933.79 + 1423.12i 0 0 0
145.5 0 0 0 −6.56919 3.79272i 0 −2048.22 1252.83i 0 0 0
145.6 0 0 0 6.56919 + 3.79272i 0 −2048.22 1252.83i 0 0 0
145.7 0 0 0 428.745 + 247.536i 0 1933.79 + 1423.12i 0 0 0
145.8 0 0 0 534.755 + 308.741i 0 −750.736 + 2280.61i 0 0 0
145.9 0 0 0 846.691 + 488.837i 0 −1709.42 + 1686.02i 0 0 0
145.10 0 0 0 885.405 + 511.189i 0 765.093 2275.84i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 145.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.9.z.e 20
3.b odd 2 1 inner 252.9.z.e 20
7.d odd 6 1 inner 252.9.z.e 20
21.g even 6 1 inner 252.9.z.e 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.9.z.e 20 1.a even 1 1 trivial
252.9.z.e 20 3.b odd 2 1 inner
252.9.z.e 20 7.d odd 6 1 inner
252.9.z.e 20 21.g even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$45\!\cdots\!38$$$$T_{5}^{16} -$$$$45\!\cdots\!33$$$$T_{5}^{14} +$$$$32\!\cdots\!34$$$$T_{5}^{12} -$$$$14\!\cdots\!45$$$$T_{5}^{10} +$$$$44\!\cdots\!25$$$$T_{5}^{8} -$$$$75\!\cdots\!00$$$$T_{5}^{6} +$$$$87\!\cdots\!00$$$$T_{5}^{4} -$$$$50\!\cdots\!00$$$$T_{5}^{2} +$$$$28\!\cdots\!00$$">$$T_{5}^{20} - \cdots$$ acting on $$S_{9}^{\mathrm{new}}(252, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20}$$
$3$ $$T^{20}$$
$5$ $$28\!\cdots\!00$$$$-$$$$50\!\cdots\!00$$$$T^{2} +$$$$87\!\cdots\!00$$$$T^{4} -$$$$75\!\cdots\!00$$$$T^{6} +$$$$44\!\cdots\!25$$$$T^{8} -$$$$14\!\cdots\!45$$$$T^{10} +$$$$32\!\cdots\!34$$$$T^{12} - 4538708430919915833 T^{14} + 4557812756238 T^{16} - 2627541 T^{18} + T^{20}$$
$7$ $$($$$$63\!\cdots\!01$$$$+$$$$39\!\cdots\!19$$$$T +$$$$21\!\cdots\!95$$$$T^{2} +$$$$70\!\cdots\!86$$$$T^{3} +$$$$38\!\cdots\!29$$$$T^{4} + 129916718471073765 T^{5} + 66473263967229 T^{6} + 21337173186 T^{7} + 11360895 T^{8} + 3619 T^{9} + T^{10} )^{2}$$
$11$ $$76\!\cdots\!00$$$$+$$$$24\!\cdots\!00$$$$T^{2} +$$$$75\!\cdots\!00$$$$T^{4} +$$$$17\!\cdots\!60$$$$T^{6} +$$$$37\!\cdots\!01$$$$T^{8} +$$$$29\!\cdots\!07$$$$T^{10} +$$$$15\!\cdots\!06$$$$T^{12} +$$$$45\!\cdots\!23$$$$T^{14} + 984480158423293686 T^{16} + 1209688767 T^{18} + T^{20}$$
$13$ $$($$$$19\!\cdots\!00$$$$+$$$$72\!\cdots\!04$$$$T^{2} +$$$$46\!\cdots\!24$$$$T^{4} + 9686523009231857025 T^{6} + 5740976634 T^{8} + T^{10} )^{2}$$
$17$ $$52\!\cdots\!00$$$$-$$$$40\!\cdots\!00$$$$T^{2} +$$$$20\!\cdots\!00$$$$T^{4} -$$$$60\!\cdots\!00$$$$T^{6} +$$$$13\!\cdots\!24$$$$T^{8} -$$$$20\!\cdots\!96$$$$T^{10} +$$$$23\!\cdots\!56$$$$T^{12} -$$$$19\!\cdots\!76$$$$T^{14} +$$$$11\!\cdots\!44$$$$T^{16} - 43075543896 T^{18} + T^{20}$$
$19$ $$($$$$42\!\cdots\!00$$$$-$$$$15\!\cdots\!00$$$$T +$$$$19\!\cdots\!04$$$$T^{2} -$$$$16\!\cdots\!60$$$$T^{3} -$$$$37\!\cdots\!88$$$$T^{4} +$$$$38\!\cdots\!20$$$$T^{5} +$$$$79\!\cdots\!29$$$$T^{6} - 1653071447654532 T^{7} - 29866128591 T^{8} + 53628 T^{9} + T^{10} )^{2}$$
$23$ $$87\!\cdots\!00$$$$+$$$$15\!\cdots\!00$$$$T^{2} +$$$$27\!\cdots\!00$$$$T^{4} +$$$$12\!\cdots\!00$$$$T^{6} +$$$$39\!\cdots\!00$$$$T^{8} +$$$$58\!\cdots\!40$$$$T^{10} +$$$$60\!\cdots\!56$$$$T^{12} +$$$$39\!\cdots\!28$$$$T^{14} +$$$$19\!\cdots\!88$$$$T^{16} + 540169907508 T^{18} + T^{20}$$
$29$ $$( -$$$$14\!\cdots\!00$$$$+$$$$26\!\cdots\!60$$$$T^{2} -$$$$81\!\cdots\!29$$$$T^{4} +$$$$98\!\cdots\!19$$$$T^{6} - 5171113544751 T^{8} + T^{10} )^{2}$$
$31$ $$($$$$27\!\cdots\!07$$$$-$$$$25\!\cdots\!95$$$$T -$$$$10\!\cdots\!61$$$$T^{2} +$$$$95\!\cdots\!10$$$$T^{3} +$$$$31\!\cdots\!99$$$$T^{4} -$$$$50\!\cdots\!73$$$$T^{5} -$$$$25\!\cdots\!23$$$$T^{6} + 286842812692529286 T^{7} + 1989072345345 T^{8} + 2367213 T^{9} + T^{10} )^{2}$$
$37$ $$($$$$15\!\cdots\!96$$$$-$$$$40\!\cdots\!24$$$$T +$$$$18\!\cdots\!96$$$$T^{2} -$$$$35\!\cdots\!00$$$$T^{3} +$$$$16\!\cdots\!56$$$$T^{4} -$$$$36\!\cdots\!04$$$$T^{5} +$$$$55\!\cdots\!01$$$$T^{6} - 16797007861750404510 T^{7} + 14862322970331 T^{8} - 3401374 T^{9} + T^{10} )^{2}$$
$41$ $$($$$$85\!\cdots\!00$$$$+$$$$17\!\cdots\!40$$$$T^{2} +$$$$12\!\cdots\!48$$$$T^{4} +$$$$61\!\cdots\!72$$$$T^{6} + 47701796739156 T^{8} + T^{10} )^{2}$$
$43$ $$($$$$18\!\cdots\!00$$$$-$$$$59\!\cdots\!40$$$$T + 61130048163516606068 T^{2} - 22420995393101 T^{3} + 149032 T^{4} + T^{5} )^{4}$$
$47$ $$38\!\cdots\!00$$$$-$$$$80\!\cdots\!00$$$$T^{2} +$$$$16\!\cdots\!00$$$$T^{4} -$$$$53\!\cdots\!80$$$$T^{6} +$$$$12\!\cdots\!24$$$$T^{8} -$$$$13\!\cdots\!88$$$$T^{10} +$$$$11\!\cdots\!32$$$$T^{12} -$$$$41\!\cdots\!52$$$$T^{14} +$$$$10\!\cdots\!28$$$$T^{16} - 120593971151388 T^{18} + T^{20}$$
$53$ $$76\!\cdots\!00$$$$+$$$$86\!\cdots\!00$$$$T^{2} +$$$$72\!\cdots\!00$$$$T^{4} +$$$$25\!\cdots\!00$$$$T^{6} +$$$$67\!\cdots\!25$$$$T^{8} +$$$$37\!\cdots\!75$$$$T^{10} +$$$$15\!\cdots\!50$$$$T^{12} +$$$$15\!\cdots\!35$$$$T^{14} +$$$$11\!\cdots\!86$$$$T^{16} + 396373925510199 T^{18} + T^{20}$$
$59$ $$34\!\cdots\!00$$$$-$$$$18\!\cdots\!00$$$$T^{2} +$$$$60\!\cdots\!00$$$$T^{4} -$$$$12\!\cdots\!80$$$$T^{6} +$$$$17\!\cdots\!89$$$$T^{8} -$$$$18\!\cdots\!49$$$$T^{10} +$$$$14\!\cdots\!46$$$$T^{12} -$$$$78\!\cdots\!09$$$$T^{14} +$$$$30\!\cdots\!94$$$$T^{16} - 689407494920109 T^{18} + T^{20}$$
$61$ $$($$$$25\!\cdots\!00$$$$-$$$$24\!\cdots\!40$$$$T +$$$$80\!\cdots\!64$$$$T^{2} -$$$$16\!\cdots\!04$$$$T^{3} +$$$$32\!\cdots\!04$$$$T^{4} +$$$$15\!\cdots\!04$$$$T^{5} +$$$$83\!\cdots\!56$$$$T^{6} +$$$$14\!\cdots\!32$$$$T^{7} + 2330597999520 T^{8} - 16286886 T^{9} + T^{10} )^{2}$$
$67$ $$($$$$54\!\cdots\!00$$$$-$$$$53\!\cdots\!00$$$$T +$$$$68\!\cdots\!00$$$$T^{2} -$$$$17\!\cdots\!00$$$$T^{3} +$$$$21\!\cdots\!00$$$$T^{4} -$$$$56\!\cdots\!80$$$$T^{5} +$$$$49\!\cdots\!09$$$$T^{6} -$$$$10\!\cdots\!16$$$$T^{7} + 881210578909677 T^{8} + 12999652 T^{9} + T^{10} )^{2}$$
$71$ $$( -$$$$15\!\cdots\!00$$$$+$$$$40\!\cdots\!00$$$$T^{2} -$$$$37\!\cdots\!40$$$$T^{4} +$$$$14\!\cdots\!16$$$$T^{6} - 2210194026495252 T^{8} + T^{10} )^{2}$$
$73$ $$($$$$49\!\cdots\!52$$$$+$$$$45\!\cdots\!56$$$$T -$$$$20\!\cdots\!24$$$$T^{2} -$$$$32\!\cdots\!40$$$$T^{3} +$$$$10\!\cdots\!88$$$$T^{4} +$$$$20\!\cdots\!84$$$$T^{5} +$$$$45\!\cdots\!29$$$$T^{6} -$$$$23\!\cdots\!30$$$$T^{7} - 670940250687993 T^{8} + 26449806 T^{9} + T^{10} )^{2}$$
$79$ $$($$$$24\!\cdots\!25$$$$-$$$$16\!\cdots\!15$$$$T +$$$$92\!\cdots\!51$$$$T^{2} -$$$$11\!\cdots\!10$$$$T^{3} +$$$$13\!\cdots\!77$$$$T^{4} +$$$$74\!\cdots\!55$$$$T^{5} +$$$$98\!\cdots\!53$$$$T^{6} +$$$$81\!\cdots\!70$$$$T^{7} + 6223661480453127 T^{8} + 64972765 T^{9} + T^{10} )^{2}$$
$83$ $$($$$$12\!\cdots\!00$$$$+$$$$25\!\cdots\!60$$$$T^{2} +$$$$99\!\cdots\!43$$$$T^{4} +$$$$13\!\cdots\!27$$$$T^{6} + 6856065632807181 T^{8} + T^{10} )^{2}$$
$89$ $$34\!\cdots\!00$$$$-$$$$21\!\cdots\!00$$$$T^{2} +$$$$11\!\cdots\!00$$$$T^{4} -$$$$90\!\cdots\!20$$$$T^{6} +$$$$47\!\cdots\!84$$$$T^{8} -$$$$14\!\cdots\!84$$$$T^{10} +$$$$33\!\cdots\!16$$$$T^{12} -$$$$46\!\cdots\!64$$$$T^{14} +$$$$46\!\cdots\!44$$$$T^{16} - 26478777167636724 T^{18} + T^{20}$$
$97$ $$($$$$45\!\cdots\!00$$$$+$$$$65\!\cdots\!24$$$$T^{2} +$$$$16\!\cdots\!79$$$$T^{4} +$$$$15\!\cdots\!15$$$$T^{6} + 65277242250659649 T^{8} + T^{10} )^{2}$$
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