# Properties

 Label 400.6.n.g Level $400$ Weight $6$ Character orbit 400.n Analytic conductor $64.154$ Analytic rank $0$ Dimension $20$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 400.n (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$64.1535279252$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$10$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ Defining polynomial: $$x^{20} + 271 x^{18} + 109637 x^{16} + 25993614 x^{14} + 5522961902 x^{12} + 881545050522 x^{10} + 133816049059481 x^{8} + 14779507781220031 x^{6} + 824105698447750789 x^{4} + 12044868290803250652 x^{2} + 579398322543528055824$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{67}\cdot 3^{4}\cdot 5^{4}$$ Twist minimal: no (minimal twist has level 80) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{6} q^{3} + ( -\beta_{5} + \beta_{15} ) q^{7} + ( 103 \beta_{4} + \beta_{18} ) q^{9} +O(q^{10})$$ $$q -\beta_{6} q^{3} + ( -\beta_{5} + \beta_{15} ) q^{7} + ( 103 \beta_{4} + \beta_{18} ) q^{9} + ( -\beta_{2} + \beta_{5} + \beta_{6} + \beta_{15} - \beta_{16} ) q^{11} + ( -40 - \beta_{1} + 40 \beta_{4} - 2 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{18} ) q^{13} + ( 112 - \beta_{1} + 112 \beta_{4} + 4 \beta_{7} + 5 \beta_{9} - 2 \beta_{10} + \beta_{12} + 2 \beta_{18} ) q^{17} + ( 5 \beta_{5} - 5 \beta_{6} - 2 \beta_{13} - 2 \beta_{14} - 3 \beta_{15} - 3 \beta_{16} - \beta_{17} - \beta_{19} ) q^{19} + ( -224 - 13 \beta_{1} - 12 \beta_{8} - 10 \beta_{9} + 2 \beta_{10} - 5 \beta_{11} + 5 \beta_{12} ) q^{21} + ( 3 \beta_{2} + \beta_{3} - 59 \beta_{6} + \beta_{14} - 5 \beta_{16} - 3 \beta_{17} + \beta_{19} ) q^{23} + ( 7 \beta_{2} - \beta_{3} - 142 \beta_{5} + 6 \beta_{13} + 7 \beta_{17} + \beta_{19} ) q^{27} + ( 19 \beta_{1} + 960 \beta_{4} - 16 \beta_{7} + 19 \beta_{8} - 19 \beta_{9} + 10 \beta_{11} + 10 \beta_{12} - 2 \beta_{18} ) q^{29} + ( -8 \beta_{2} - \beta_{3} - 96 \beta_{5} - 96 \beta_{6} + 4 \beta_{13} - 4 \beta_{14} - 23 \beta_{15} + 23 \beta_{16} ) q^{31} + ( 559 - 12 \beta_{1} - 559 \beta_{4} + 20 \beta_{7} - 31 \beta_{8} - 8 \beta_{9} - 8 \beta_{10} - 21 \beta_{11} - 8 \beta_{18} ) q^{33} + ( -2209 - 18 \beta_{1} - 2209 \beta_{4} - 25 \beta_{7} - 7 \beta_{9} + 14 \beta_{10} + 20 \beta_{12} - 14 \beta_{18} ) q^{37} + ( -225 \beta_{5} + 225 \beta_{6} - 11 \beta_{13} - 11 \beta_{14} + 5 \beta_{15} + 5 \beta_{16} ) q^{39} + ( -324 - 26 \beta_{1} - 9 \beta_{8} - 20 \beta_{9} - 11 \beta_{10} - 35 \beta_{11} + 35 \beta_{12} ) q^{41} + ( -8 \beta_{2} - \beta_{3} - 323 \beta_{6} - 36 \beta_{14} - 10 \beta_{16} + 8 \beta_{17} - \beta_{19} ) q^{43} + ( -19 \beta_{2} + 2 \beta_{3} - 97 \beta_{5} + 23 \beta_{13} + 11 \beta_{15} - 19 \beta_{17} - 2 \beta_{19} ) q^{47} + ( 35 \beta_{1} - 89 \beta_{4} - 120 \beta_{7} + 35 \beta_{8} - 35 \beta_{9} + 25 \beta_{11} + 25 \beta_{12} - 23 \beta_{18} ) q^{49} + ( 30 \beta_{2} + 3 \beta_{3} - 568 \beta_{5} - 568 \beta_{6} + 16 \beta_{13} - 16 \beta_{14} + 68 \beta_{15} - 68 \beta_{16} ) q^{51} + ( -9114 - 42 \beta_{1} + 9114 \beta_{4} + 17 \beta_{7} + 50 \beta_{8} + 25 \beta_{9} + 25 \beta_{10} - 43 \beta_{11} + 25 \beta_{18} ) q^{53} + ( 1719 + 110 \beta_{1} + 1719 \beta_{4} - 46 \beta_{7} - 156 \beta_{9} - 27 \beta_{10} + 23 \beta_{12} + 27 \beta_{18} ) q^{57} + ( -59 \beta_{5} + 59 \beta_{6} - 42 \beta_{13} - 42 \beta_{14} + 59 \beta_{15} + 59 \beta_{16} + 15 \beta_{17} + 11 \beta_{19} ) q^{59} + ( -2044 + 72 \beta_{1} - 55 \beta_{8} - 47 \beta_{9} + 8 \beta_{10} - 25 \beta_{11} + 25 \beta_{12} ) q^{61} + ( -5 \beta_{2} - 10 \beta_{3} + 163 \beta_{6} - 185 \beta_{14} + 179 \beta_{16} + 5 \beta_{17} - 10 \beta_{19} ) q^{63} + ( -26 \beta_{2} + 8 \beta_{3} + 1099 \beta_{5} + 92 \beta_{13} - 14 \beta_{15} - 26 \beta_{17} - 8 \beta_{19} ) q^{67} + ( -209 \beta_{1} + 20062 \beta_{4} + 321 \beta_{7} - 209 \beta_{8} + 209 \beta_{9} - 10 \beta_{11} - 10 \beta_{12} + 90 \beta_{18} ) q^{69} + ( 40 \beta_{2} + 5 \beta_{3} + 530 \beta_{5} + 530 \beta_{6} + 67 \beta_{13} - 67 \beta_{14} + 5 \beta_{15} - 5 \beta_{16} ) q^{71} + ( -13230 - 143 \beta_{1} + 13230 \beta_{4} + 182 \beta_{7} - 229 \beta_{8} - 39 \beta_{9} - 39 \beta_{10} + 57 \beta_{11} - 39 \beta_{18} ) q^{73} + ( -19981 - 197 \beta_{1} - 19981 \beta_{4} - 41 \beta_{7} + 156 \beta_{9} - 19 \beta_{10} - 79 \beta_{12} + 19 \beta_{18} ) q^{77} + ( -163 \beta_{5} + 163 \beta_{6} - 166 \beta_{13} - 166 \beta_{14} - 224 \beta_{15} - 224 \beta_{16} - 24 \beta_{17} + 3 \beta_{19} ) q^{79} + ( -26065 - 66 \beta_{1} + 259 \beta_{8} + 292 \beta_{9} + 33 \beta_{10} + 135 \beta_{11} - 135 \beta_{12} ) q^{81} + ( 9 \beta_{2} + 8 \beta_{3} + 163 \beta_{6} - 372 \beta_{14} - 416 \beta_{16} - 9 \beta_{17} + 8 \beta_{19} ) q^{83} + ( 74 \beta_{2} - 17 \beta_{3} - 1750 \beta_{5} + 342 \beta_{13} - 354 \beta_{15} + 74 \beta_{17} + 17 \beta_{19} ) q^{87} + ( 128 \beta_{1} + 28740 \beta_{4} - 182 \beta_{7} + 128 \beta_{8} - 128 \beta_{9} - 130 \beta_{11} - 130 \beta_{12} - 46 \beta_{18} ) q^{89} + ( -112 \beta_{2} - 23 \beta_{3} + 570 \beta_{5} + 570 \beta_{6} + 198 \beta_{13} - 198 \beta_{14} - 46 \beta_{15} + 46 \beta_{16} ) q^{91} + ( -35718 + 683 \beta_{1} + 35718 \beta_{4} - 703 \beta_{7} + 722 \beta_{8} + 20 \beta_{9} + 20 \beta_{10} + 216 \beta_{11} + 20 \beta_{18} ) q^{93} + ( -18762 + 218 \beta_{1} - 18762 \beta_{4} + 634 \beta_{7} + 416 \beta_{9} + 99 \beta_{10} - 117 \beta_{12} - 99 \beta_{18} ) q^{97} + ( 3562 \beta_{5} - 3562 \beta_{6} - 414 \beta_{13} - 414 \beta_{14} + 171 \beta_{15} + 171 \beta_{16} + \beta_{17} - 47 \beta_{19} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q + O(q^{10})$$ $$20q - 804q^{13} + 2236q^{17} - 4520q^{21} + 11096q^{33} - 44260q^{37} - 6760q^{41} - 182452q^{53} + 34288q^{57} - 41080q^{61} - 264372q^{73} - 399304q^{77} - 520220q^{81} - 713496q^{93} - 374772q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} + 271 x^{18} + 109637 x^{16} + 25993614 x^{14} + 5522961902 x^{12} + 881545050522 x^{10} + 133816049059481 x^{8} + 14779507781220031 x^{6} + 824105698447750789 x^{4} + 12044868290803250652 x^{2} + 579398322543528055824$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$21\!\cdots\!17$$$$\nu^{18} -$$$$27\!\cdots\!66$$$$\nu^{16} -$$$$92\!\cdots\!48$$$$\nu^{14} -$$$$14\!\cdots\!30$$$$\nu^{12} -$$$$22\!\cdots\!52$$$$\nu^{10} -$$$$31\!\cdots\!96$$$$\nu^{8} -$$$$40\!\cdots\!67$$$$\nu^{6} -$$$$23\!\cdots\!06$$$$\nu^{4} -$$$$33\!\cdots\!40$$$$\nu^{2} +$$$$21\!\cdots\!84$$$$)/$$$$11\!\cdots\!08$$ $$\beta_{2}$$ $$=$$ $$($$$$12\!\cdots\!45$$$$\nu^{18} -$$$$82\!\cdots\!16$$$$\nu^{16} +$$$$12\!\cdots\!04$$$$\nu^{14} -$$$$77\!\cdots\!26$$$$\nu^{12} -$$$$92\!\cdots\!08$$$$\nu^{10} -$$$$32\!\cdots\!56$$$$\nu^{8} -$$$$33\!\cdots\!39$$$$\nu^{6} -$$$$73\!\cdots\!00$$$$\nu^{4} -$$$$46\!\cdots\!32$$$$\nu^{2} -$$$$59\!\cdots\!24$$$$)/$$$$11\!\cdots\!60$$ $$\beta_{3}$$ $$=$$ $$($$$$24\!\cdots\!39$$$$\nu^{18} -$$$$75\!\cdots\!68$$$$\nu^{16} +$$$$28\!\cdots\!84$$$$\nu^{14} -$$$$57\!\cdots\!62$$$$\nu^{12} -$$$$91\!\cdots\!96$$$$\nu^{10} -$$$$26\!\cdots\!32$$$$\nu^{8} -$$$$27\!\cdots\!09$$$$\nu^{6} -$$$$74\!\cdots\!52$$$$\nu^{4} -$$$$52\!\cdots\!88$$$$\nu^{2} -$$$$60\!\cdots\!64$$$$)/$$$$22\!\cdots\!20$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$10\!\cdots\!57$$$$\nu^{19} +$$$$48\!\cdots\!81$$$$\nu^{17} +$$$$74\!\cdots\!38$$$$\nu^{15} +$$$$49\!\cdots\!50$$$$\nu^{13} +$$$$11\!\cdots\!18$$$$\nu^{11} +$$$$26\!\cdots\!44$$$$\nu^{9} +$$$$39\!\cdots\!11$$$$\nu^{7} +$$$$65\!\cdots\!81$$$$\nu^{5} +$$$$77\!\cdots\!98$$$$\nu^{3} +$$$$27\!\cdots\!16$$$$\nu$$$$)/$$$$16\!\cdots\!20$$ $$\beta_{5}$$ $$=$$ $$($$$$42\!\cdots\!31$$$$\nu^{19} +$$$$18\!\cdots\!83$$$$\nu^{18} +$$$$11\!\cdots\!49$$$$\nu^{17} +$$$$53\!\cdots\!64$$$$\nu^{16} +$$$$45\!\cdots\!44$$$$\nu^{15} +$$$$21\!\cdots\!68$$$$\nu^{14} +$$$$10\!\cdots\!86$$$$\nu^{13} +$$$$52\!\cdots\!46$$$$\nu^{12} +$$$$22\!\cdots\!30$$$$\nu^{11} +$$$$11\!\cdots\!08$$$$\nu^{10} +$$$$35\!\cdots\!92$$$$\nu^{9} +$$$$18\!\cdots\!56$$$$\nu^{8} +$$$$52\!\cdots\!07$$$$\nu^{7} +$$$$28\!\cdots\!07$$$$\nu^{6} +$$$$55\!\cdots\!57$$$$\nu^{5} +$$$$31\!\cdots\!36$$$$\nu^{4} +$$$$26\!\cdots\!00$$$$\nu^{3} +$$$$18\!\cdots\!44$$$$\nu^{2} -$$$$17\!\cdots\!08$$$$\nu +$$$$25\!\cdots\!12$$$$)/$$$$26\!\cdots\!80$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$42\!\cdots\!31$$$$\nu^{19} +$$$$18\!\cdots\!83$$$$\nu^{18} -$$$$11\!\cdots\!49$$$$\nu^{17} +$$$$53\!\cdots\!64$$$$\nu^{16} -$$$$45\!\cdots\!44$$$$\nu^{15} +$$$$21\!\cdots\!68$$$$\nu^{14} -$$$$10\!\cdots\!86$$$$\nu^{13} +$$$$52\!\cdots\!46$$$$\nu^{12} -$$$$22\!\cdots\!30$$$$\nu^{11} +$$$$11\!\cdots\!08$$$$\nu^{10} -$$$$35\!\cdots\!92$$$$\nu^{9} +$$$$18\!\cdots\!56$$$$\nu^{8} -$$$$52\!\cdots\!07$$$$\nu^{7} +$$$$28\!\cdots\!07$$$$\nu^{6} -$$$$55\!\cdots\!57$$$$\nu^{5} +$$$$31\!\cdots\!36$$$$\nu^{4} -$$$$26\!\cdots\!00$$$$\nu^{3} +$$$$18\!\cdots\!44$$$$\nu^{2} +$$$$17\!\cdots\!08$$$$\nu +$$$$25\!\cdots\!12$$$$)/$$$$26\!\cdots\!80$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$46\!\cdots\!17$$$$\nu^{19} +$$$$16\!\cdots\!91$$$$\nu^{17} +$$$$14\!\cdots\!48$$$$\nu^{15} +$$$$16\!\cdots\!70$$$$\nu^{13} +$$$$36\!\cdots\!58$$$$\nu^{11} +$$$$87\!\cdots\!44$$$$\nu^{9} +$$$$13\!\cdots\!91$$$$\nu^{7} +$$$$21\!\cdots\!51$$$$\nu^{5} +$$$$25\!\cdots\!08$$$$\nu^{3} +$$$$88\!\cdots\!96$$$$\nu$$$$)/$$$$28\!\cdots\!20$$ $$\beta_{8}$$ $$=$$ $$($$$$-$$$$65\!\cdots\!21$$$$\nu^{19} -$$$$68\!\cdots\!45$$$$\nu^{18} -$$$$82\!\cdots\!47$$$$\nu^{17} -$$$$10\!\cdots\!40$$$$\nu^{16} -$$$$54\!\cdots\!76$$$$\nu^{15} -$$$$63\!\cdots\!20$$$$\nu^{14} -$$$$85\!\cdots\!70$$$$\nu^{13} -$$$$11\!\cdots\!50$$$$\nu^{12} -$$$$19\!\cdots\!06$$$$\nu^{11} -$$$$24\!\cdots\!80$$$$\nu^{10} -$$$$24\!\cdots\!28$$$$\nu^{9} -$$$$36\!\cdots\!40$$$$\nu^{8} -$$$$37\!\cdots\!77$$$$\nu^{7} -$$$$50\!\cdots\!05$$$$\nu^{6} -$$$$27\!\cdots\!47$$$$\nu^{5} -$$$$46\!\cdots\!40$$$$\nu^{4} +$$$$14\!\cdots\!04$$$$\nu^{3} -$$$$43\!\cdots\!00$$$$\nu^{2} +$$$$30\!\cdots\!88$$$$\nu -$$$$73\!\cdots\!20$$$$)/$$$$21\!\cdots\!40$$ $$\beta_{9}$$ $$=$$ $$($$$$65\!\cdots\!21$$$$\nu^{19} -$$$$64\!\cdots\!10$$$$\nu^{18} +$$$$82\!\cdots\!47$$$$\nu^{17} -$$$$16\!\cdots\!70$$$$\nu^{16} +$$$$54\!\cdots\!76$$$$\nu^{15} -$$$$65\!\cdots\!60$$$$\nu^{14} +$$$$85\!\cdots\!70$$$$\nu^{13} -$$$$14\!\cdots\!00$$$$\nu^{12} +$$$$19\!\cdots\!06$$$$\nu^{11} -$$$$29\!\cdots\!40$$$$\nu^{10} +$$$$24\!\cdots\!28$$$$\nu^{9} -$$$$42\!\cdots\!20$$$$\nu^{8} +$$$$37\!\cdots\!77$$$$\nu^{7} -$$$$58\!\cdots\!90$$$$\nu^{6} +$$$$27\!\cdots\!47$$$$\nu^{5} -$$$$50\!\cdots\!70$$$$\nu^{4} -$$$$14\!\cdots\!04$$$$\nu^{3} -$$$$49\!\cdots\!00$$$$\nu^{2} -$$$$30\!\cdots\!88$$$$\nu +$$$$32\!\cdots\!00$$$$)/$$$$21\!\cdots\!40$$ $$\beta_{10}$$ $$=$$ $$($$$$-$$$$65\!\cdots\!21$$$$\nu^{19} +$$$$40\!\cdots\!66$$$$\nu^{18} -$$$$82\!\cdots\!47$$$$\nu^{17} +$$$$80\!\cdots\!94$$$$\nu^{16} -$$$$54\!\cdots\!76$$$$\nu^{15} +$$$$37\!\cdots\!04$$$$\nu^{14} -$$$$85\!\cdots\!70$$$$\nu^{13} +$$$$75\!\cdots\!76$$$$\nu^{12} -$$$$19\!\cdots\!06$$$$\nu^{11} +$$$$15\!\cdots\!40$$$$\nu^{10} -$$$$24\!\cdots\!28$$$$\nu^{9} +$$$$22\!\cdots\!72$$$$\nu^{8} -$$$$37\!\cdots\!77$$$$\nu^{7} +$$$$33\!\cdots\!42$$$$\nu^{6} -$$$$27\!\cdots\!47$$$$\nu^{5} +$$$$31\!\cdots\!42$$$$\nu^{4} +$$$$14\!\cdots\!04$$$$\nu^{3} +$$$$29\!\cdots\!40$$$$\nu^{2} +$$$$30\!\cdots\!88$$$$\nu -$$$$27\!\cdots\!88$$$$)/$$$$21\!\cdots\!40$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$10\!\cdots\!61$$$$\nu^{19} -$$$$10\!\cdots\!06$$$$\nu^{18} -$$$$28\!\cdots\!01$$$$\nu^{17} -$$$$23\!\cdots\!74$$$$\nu^{16} -$$$$11\!\cdots\!10$$$$\nu^{15} -$$$$10\!\cdots\!84$$$$\nu^{14} -$$$$27\!\cdots\!94$$$$\nu^{13} -$$$$23\!\cdots\!96$$$$\nu^{12} -$$$$59\!\cdots\!18$$$$\nu^{11} -$$$$46\!\cdots\!20$$$$\nu^{10} -$$$$98\!\cdots\!32$$$$\nu^{9} -$$$$67\!\cdots\!32$$$$\nu^{8} -$$$$15\!\cdots\!53$$$$\nu^{7} -$$$$10\!\cdots\!42$$$$\nu^{6} -$$$$17\!\cdots\!17$$$$\nu^{5} -$$$$93\!\cdots\!22$$$$\nu^{4} -$$$$11\!\cdots\!94$$$$\nu^{3} -$$$$87\!\cdots\!40$$$$\nu^{2} -$$$$43\!\cdots\!28$$$$\nu +$$$$14\!\cdots\!48$$$$)/$$$$28\!\cdots\!20$$ $$\beta_{12}$$ $$=$$ $$($$$$-$$$$10\!\cdots\!61$$$$\nu^{19} +$$$$10\!\cdots\!06$$$$\nu^{18} -$$$$28\!\cdots\!01$$$$\nu^{17} +$$$$23\!\cdots\!74$$$$\nu^{16} -$$$$11\!\cdots\!10$$$$\nu^{15} +$$$$10\!\cdots\!84$$$$\nu^{14} -$$$$27\!\cdots\!94$$$$\nu^{13} +$$$$23\!\cdots\!96$$$$\nu^{12} -$$$$59\!\cdots\!18$$$$\nu^{11} +$$$$46\!\cdots\!20$$$$\nu^{10} -$$$$98\!\cdots\!32$$$$\nu^{9} +$$$$67\!\cdots\!32$$$$\nu^{8} -$$$$15\!\cdots\!53$$$$\nu^{7} +$$$$10\!\cdots\!42$$$$\nu^{6} -$$$$17\!\cdots\!17$$$$\nu^{5} +$$$$93\!\cdots\!22$$$$\nu^{4} -$$$$11\!\cdots\!94$$$$\nu^{3} +$$$$87\!\cdots\!40$$$$\nu^{2} -$$$$43\!\cdots\!28$$$$\nu -$$$$14\!\cdots\!48$$$$)/$$$$28\!\cdots\!20$$ $$\beta_{13}$$ $$=$$ $$($$$$22\!\cdots\!60$$$$\nu^{19} +$$$$72\!\cdots\!68$$$$\nu^{18} +$$$$54\!\cdots\!60$$$$\nu^{17} +$$$$27\!\cdots\!16$$$$\nu^{16} +$$$$23\!\cdots\!40$$$$\nu^{15} +$$$$89\!\cdots\!92$$$$\nu^{14} +$$$$51\!\cdots\!20$$$$\nu^{13} +$$$$25\!\cdots\!36$$$$\nu^{12} +$$$$11\!\cdots\!80$$$$\nu^{11} +$$$$48\!\cdots\!40$$$$\nu^{10} +$$$$16\!\cdots\!20$$$$\nu^{9} +$$$$86\!\cdots\!92$$$$\nu^{8} +$$$$26\!\cdots\!20$$$$\nu^{7} +$$$$12\!\cdots\!44$$$$\nu^{6} +$$$$26\!\cdots\!00$$$$\nu^{5} +$$$$15\!\cdots\!16$$$$\nu^{4} +$$$$12\!\cdots\!40$$$$\nu^{3} +$$$$90\!\cdots\!28$$$$\nu^{2} -$$$$88\!\cdots\!40$$$$\nu +$$$$12\!\cdots\!64$$$$)/$$$$20\!\cdots\!05$$ $$\beta_{14}$$ $$=$$ $$($$$$22\!\cdots\!60$$$$\nu^{19} -$$$$72\!\cdots\!68$$$$\nu^{18} +$$$$54\!\cdots\!60$$$$\nu^{17} -$$$$27\!\cdots\!16$$$$\nu^{16} +$$$$23\!\cdots\!40$$$$\nu^{15} -$$$$89\!\cdots\!92$$$$\nu^{14} +$$$$51\!\cdots\!20$$$$\nu^{13} -$$$$25\!\cdots\!36$$$$\nu^{12} +$$$$11\!\cdots\!80$$$$\nu^{11} -$$$$48\!\cdots\!40$$$$\nu^{10} +$$$$16\!\cdots\!20$$$$\nu^{9} -$$$$86\!\cdots\!92$$$$\nu^{8} +$$$$26\!\cdots\!20$$$$\nu^{7} -$$$$12\!\cdots\!44$$$$\nu^{6} +$$$$26\!\cdots\!00$$$$\nu^{5} -$$$$15\!\cdots\!16$$$$\nu^{4} +$$$$12\!\cdots\!40$$$$\nu^{3} -$$$$90\!\cdots\!28$$$$\nu^{2} -$$$$88\!\cdots\!40$$$$\nu -$$$$12\!\cdots\!64$$$$)/$$$$20\!\cdots\!05$$ $$\beta_{15}$$ $$=$$ $$($$$$-$$$$25\!\cdots\!11$$$$\nu^{19} -$$$$16\!\cdots\!12$$$$\nu^{18} -$$$$71\!\cdots\!04$$$$\nu^{17} -$$$$41\!\cdots\!63$$$$\nu^{16} -$$$$27\!\cdots\!84$$$$\nu^{15} -$$$$17\!\cdots\!68$$$$\nu^{14} -$$$$67\!\cdots\!46$$$$\nu^{13} -$$$$39\!\cdots\!72$$$$\nu^{12} -$$$$13\!\cdots\!00$$$$\nu^{11} -$$$$86\!\cdots\!50$$$$\nu^{10} -$$$$22\!\cdots\!12$$$$\nu^{9} -$$$$13\!\cdots\!84$$$$\nu^{8} -$$$$32\!\cdots\!27$$$$\nu^{7} -$$$$20\!\cdots\!24$$$$\nu^{6} -$$$$36\!\cdots\!52$$$$\nu^{5} -$$$$21\!\cdots\!39$$$$\nu^{4} -$$$$17\!\cdots\!00$$$$\nu^{3} -$$$$12\!\cdots\!80$$$$\nu^{2} +$$$$11\!\cdots\!68$$$$\nu -$$$$17\!\cdots\!64$$$$)/$$$$22\!\cdots\!40$$ $$\beta_{16}$$ $$=$$ $$($$$$-$$$$25\!\cdots\!11$$$$\nu^{19} +$$$$16\!\cdots\!12$$$$\nu^{18} -$$$$71\!\cdots\!04$$$$\nu^{17} +$$$$41\!\cdots\!63$$$$\nu^{16} -$$$$27\!\cdots\!84$$$$\nu^{15} +$$$$17\!\cdots\!68$$$$\nu^{14} -$$$$67\!\cdots\!46$$$$\nu^{13} +$$$$39\!\cdots\!72$$$$\nu^{12} -$$$$13\!\cdots\!00$$$$\nu^{11} +$$$$86\!\cdots\!50$$$$\nu^{10} -$$$$22\!\cdots\!12$$$$\nu^{9} +$$$$13\!\cdots\!84$$$$\nu^{8} -$$$$32\!\cdots\!27$$$$\nu^{7} +$$$$20\!\cdots\!24$$$$\nu^{6} -$$$$36\!\cdots\!52$$$$\nu^{5} +$$$$21\!\cdots\!39$$$$\nu^{4} -$$$$17\!\cdots\!00$$$$\nu^{3} +$$$$12\!\cdots\!80$$$$\nu^{2} +$$$$11\!\cdots\!68$$$$\nu +$$$$17\!\cdots\!64$$$$)/$$$$22\!\cdots\!40$$ $$\beta_{17}$$ $$=$$ $$($$$$-$$$$39\!\cdots\!47$$$$\nu^{19} -$$$$80\!\cdots\!13$$$$\nu^{17} -$$$$40\!\cdots\!28$$$$\nu^{15} -$$$$80\!\cdots\!22$$$$\nu^{13} -$$$$19\!\cdots\!10$$$$\nu^{11} -$$$$26\!\cdots\!24$$$$\nu^{9} -$$$$43\!\cdots\!59$$$$\nu^{7} -$$$$39\!\cdots\!69$$$$\nu^{5} -$$$$19\!\cdots\!00$$$$\nu^{3} +$$$$14\!\cdots\!56$$$$\nu$$$$)/$$$$30\!\cdots\!40$$ $$\beta_{18}$$ $$=$$ $$($$$$41\!\cdots\!27$$$$\nu^{19} +$$$$11\!\cdots\!63$$$$\nu^{17} +$$$$46\!\cdots\!88$$$$\nu^{15} +$$$$10\!\cdots\!42$$$$\nu^{13} +$$$$23\!\cdots\!30$$$$\nu^{11} +$$$$37\!\cdots\!44$$$$\nu^{9} +$$$$56\!\cdots\!59$$$$\nu^{7} +$$$$63\!\cdots\!99$$$$\nu^{5} +$$$$37\!\cdots\!40$$$$\nu^{3} +$$$$14\!\cdots\!84$$$$\nu$$$$)/$$$$28\!\cdots\!20$$ $$\beta_{19}$$ $$=$$ $$($$$$15\!\cdots\!99$$$$\nu^{19} +$$$$30\!\cdots\!01$$$$\nu^{17} +$$$$14\!\cdots\!56$$$$\nu^{15} +$$$$30\!\cdots\!14$$$$\nu^{13} +$$$$66\!\cdots\!10$$$$\nu^{11} +$$$$95\!\cdots\!48$$$$\nu^{9} +$$$$15\!\cdots\!43$$$$\nu^{7} +$$$$13\!\cdots\!73$$$$\nu^{5} +$$$$67\!\cdots\!80$$$$\nu^{3} -$$$$44\!\cdots\!52$$$$\nu$$$$)/$$$$18\!\cdots\!40$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-12 \beta_{19} + 80 \beta_{18} + 16 \beta_{17} + 96 \beta_{16} + 96 \beta_{15} + 19 \beta_{14} + 19 \beta_{13} - 60 \beta_{12} - 60 \beta_{11} - 52 \beta_{9} + 52 \beta_{8} - 88 \beta_{7} - 492 \beta_{6} + 492 \beta_{5} - 408 \beta_{4} + 52 \beta_{1}$$$$)/3840$$ $$\nu^{2}$$ $$=$$ $$($$$$1968 \beta_{16} - 1968 \beta_{15} + 77 \beta_{14} - 77 \beta_{13} - 240 \beta_{12} + 240 \beta_{11} - 720 \beta_{10} - 1200 \beta_{9} - 480 \beta_{8} - 13416 \beta_{6} - 13416 \beta_{5} + 24 \beta_{3} - 608 \beta_{2} + 480 \beta_{1} - 104160$$$$)/3840$$ $$\nu^{3}$$ $$=$$ $$($$$$339 \beta_{19} + 120 \beta_{18} - 472 \beta_{17} - 1032 \beta_{16} - 1032 \beta_{15} - 1588 \beta_{14} - 1588 \beta_{13} + 90 \beta_{12} + 90 \beta_{11} - 2262 \beta_{9} + 2262 \beta_{8} - 3828 \beta_{7} + 4179 \beta_{6} - 4179 \beta_{5} + 662052 \beta_{4} + 2262 \beta_{1}$$$$)/480$$ $$\nu^{4}$$ $$=$$ $$($$$$54912 \beta_{16} - 54912 \beta_{15} + 36633 \beta_{14} - 36633 \beta_{13} + 79020 \beta_{12} - 79020 \beta_{11} + 77840 \beta_{10} + 240564 \beta_{9} + 162724 \beta_{8} - 744924 \beta_{6} - 744924 \beta_{5} - 38844 \beta_{3} + 52368 \beta_{2} + 249412 \beta_{1} - 55968264$$$$)/3840$$ $$\nu^{5}$$ $$=$$ $$($$$$-50448 \beta_{19} + 648560 \beta_{18} - 1483456 \beta_{17} - 4880496 \beta_{16} - 4880496 \beta_{15} - 362479 \beta_{14} - 362479 \beta_{13} + 1941240 \beta_{12} + 1941240 \beta_{11} + 2169368 \beta_{9} - 2169368 \beta_{8} + 2057792 \beta_{7} + 39809712 \beta_{6} - 39809712 \beta_{5} - 179150928 \beta_{4} - 2169368 \beta_{1}$$$$)/3840$$ $$\nu^{6}$$ $$=$$ $$($$$$-6352440 \beta_{16} + 6352440 \beta_{15} - 5280055 \beta_{14} + 5280055 \beta_{13} + 139395 \beta_{12} - 139395 \beta_{11} + 2214600 \beta_{10} + 9725613 \beta_{9} + 7511013 \beta_{8} + 40920630 \beta_{6} + 40920630 \beta_{5} + 726150 \beta_{3} + 3580120 \beta_{2} - 2893011 \beta_{1} - 419870898$$$$)/480$$ $$\nu^{7}$$ $$=$$ $$($$$$16934100 \beta_{19} - 97402960 \beta_{18} + 357716240 \beta_{17} + 322186560 \beta_{16} + 322186560 \beta_{15} + 532797245 \beta_{14} + 532797245 \beta_{13} + 34112340 \beta_{12} + 34112340 \beta_{11} + 308735708 \beta_{9} - 308735708 \beta_{8} + 1422146792 \beta_{7} + 34175220 \beta_{6} - 34175220 \beta_{5} - 221957110008 \beta_{4} - 308735708 \beta_{1}$$$$)/3840$$ $$\nu^{8}$$ $$=$$ $$($$$$-7550452272 \beta_{16} + 7550452272 \beta_{15} - 575458893 \beta_{14} + 575458893 \beta_{13} - 8852128560 \beta_{12} + 8852128560 \beta_{11} + 657386000 \beta_{10} - 9965868912 \beta_{9} - 10623254912 \beta_{8} + 65813060904 \beta_{6} + 65813060904 \beta_{5} + 728816424 \beta_{3} - 1110229728 \beta_{2} - 5439480896 \beta_{1} + 1268248071072$$$$)/3840$$ $$\nu^{9}$$ $$=$$ $$($$$$-425670693 \beta_{19} - 7813909080 \beta_{18} + 12704120624 \beta_{17} + 21259349184 \beta_{16} + 21259349184 \beta_{15} + 7781297711 \beta_{14} + 7781297711 \beta_{13} - 8239378860 \beta_{12} - 8239378860 \beta_{11} + 9096824328 \beta_{9} - 9096824328 \beta_{8} - 1525798488 \beta_{7} - 167008810773 \beta_{6} + 167008810773 \beta_{5} + 1288055722632 \beta_{4} - 9096824328 \beta_{1}$$$$)/480$$ $$\nu^{10}$$ $$=$$ $$($$$$2132499788352 \beta_{16} - 2132499788352 \beta_{15} + 3016806827423 \beta_{14} - 3016806827423 \beta_{13} + 305046798660 \beta_{12} - 305046798660 \beta_{11} - 510876137040 \beta_{10} - 1672259724900 \beta_{9} - 1161383587860 \beta_{8} - 5686955744244 \beta_{6} - 5686955744244 \beta_{5} - 172680569364 \beta_{3} - 1549752741392 \beta_{2} - 948932108340 \beta_{1} + 361597278178920$$$$)/3840$$ $$\nu^{11}$$ $$=$$ $$($$$$-2307104256480 \beta_{19} - 1703697317680 \beta_{18} - 21570378063680 \beta_{17} - 5096896541520 \beta_{16} - 5096896541520 \beta_{15} - 15593726881505 \beta_{14} - 15593726881505 \beta_{13} - 18496362809160 \beta_{12} - 18496362809160 \beta_{11} - 20558837655112 \beta_{9} + 20558837655112 \beta_{8} - 51807147285088 \beta_{7} - 32373269611680 \beta_{6} + 32373269611680 \beta_{5} + 6400568462744112 \beta_{4} + 20558837655112 \beta_{1}$$$$)/3840$$ $$\nu^{12}$$ $$=$$ $$($$$$12410747120880 \beta_{16} - 12410747120880 \beta_{15} - 1628446716390 \beta_{14} + 1628446716390 \beta_{13} + 18156210673345 \beta_{12} - 18156210673345 \beta_{11} - 6741746542040 \beta_{10} + 1106879061279 \beta_{9} + 7848625603319 \beta_{8} - 124121284715660 \beta_{6} - 124121284715660 \beta_{5} - 13367968300 \beta_{3} - 4734521643440 \beta_{2} + 19348158111407 \beta_{1} - 2382801661432934$$$$)/160$$ $$\nu^{13}$$ $$=$$ $$($$$$440456221542876 \beta_{19} + 2622145591363600 \beta_{18} - 5054280616403728 \beta_{17} - 7163056313422848 \beta_{16} - 7163056313422848 \beta_{15} - 7963850873682997 \beta_{14} - 7963850873682997 \beta_{13} + 3119829047575740 \beta_{12} + 3119829047575740 \beta_{11} - 3053392107546092 \beta_{9} + 3053392107546092 \beta_{8} - 108773241188168 \beta_{7} + 29972870644361916 \beta_{6} - 29972870644361916 \beta_{5} - 262370419783884648 \beta_{4} + 3053392107546092 \beta_{1}$$$$)/3840$$ $$\nu^{14}$$ $$=$$ $$($$$$-84205040731630992 \beta_{16} + 84205040731630992 \beta_{15} - 120547493546657123 \beta_{14} + 120547493546657123 \beta_{13} + 7030156399557600 \beta_{12} - 7030156399557600 \beta_{11} + 3043671214528560 \beta_{10} + 21702646541893632 \beta_{9} + 18658975327365072 \beta_{8} + 209968187928002184 \beta_{6} + 209968187928002184 \beta_{5} + 133874082937224 \beta_{3} + 97456376417952992 \beta_{2} + 73374875750432016 \beta_{1} - 15023226309335919552$$$$)/3840$$ $$\nu^{15}$$ $$=$$ $$($$$$3487660655181351 \beta_{19} + 47398577306303160 \beta_{18} + 86133028754561272 \beta_{17} + 34977364257281352 \beta_{16} + 34977364257281352 \beta_{15} + 110700788835411238 \beta_{14} + 110700788835411238 \beta_{13} + 173080483479383190 \beta_{12} + 173080483479383190 \beta_{11} + 117038481840317358 \beta_{9} - 117038481840317358 \beta_{8} + 239597181295702452 \beta_{7} + 219161334218589831 \beta_{6} - 219161334218589831 \beta_{5} - 25226294182487728068 \beta_{4} - 117038481840317358 \beta_{1}$$$$)/480$$ $$\nu^{16}$$ $$=$$ $$($$$$-12251622931276270848 \beta_{16} + 12251622931276270848 \beta_{15} - 8118701567081188407 \beta_{14} + 8118701567081188407 \beta_{13} - 18517924754091579060 \beta_{12} + 18517924754091579060 \beta_{11} + 9380517320458882960 \beta_{10} + 9153526402022194068 \beta_{9} - 226990918436688892 \beta_{8} + 75078278917671317316 \beta_{6} + 75078278917671317316 \beta_{5} + 113158507977357156 \beta_{3} + 8727103804645430928 \beta_{2} - 30369662179190557276 \beta_{1} + 3395023935851246219832$$$$)/3840$$ $$\nu^{17}$$ $$=$$ $$($$$$-15236107113943399536 \beta_{19} - 63969575176698141200 \beta_{18} + 293392227530733222208 \beta_{17} + 317220726629521897488 \beta_{16} + 317220726629521897488 \beta_{15} + 441689693809038574417 \beta_{14} + 441689693809038574417 \beta_{13} - 137634910590601483560 \beta_{12} - 137634910590601483560 \beta_{11} - 18832414496397422792 \beta_{9} + 18832414496397422792 \beta_{8} - 59209619368296750848 \beta_{7} - 902856014155155059376 \beta_{6} + 902856014155155059376 \beta_{5} + 5040531846945107486832 \beta_{4} + 18832414496397422792 \beta_{1}$$$$)/3840$$ $$\nu^{18}$$ $$=$$ $$($$$$526163551596594027960 \beta_{16} - 526163551596594027960 \beta_{15} + 659755447141189572475 \beta_{14} - 659755447141189572475 \beta_{13} - 259147863864720893985 \beta_{12} + 259147863864720893985 \beta_{11} + 83532478440403840200 \beta_{10} - 60828125318245908111 \beta_{9} - 144360603758649748311 \beta_{8} - 1708504331750013376470 \beta_{6} - 1708504331750013376470 \beta_{5} + 6152838546166078170 \beta_{3} - 585750067488518263000 \beta_{2} - 354408868139717865903 \beta_{1} + 52416086050617997130406$$$$)/480$$ $$\nu^{19}$$ $$=$$ $$($$$$1804110465624400481844 \beta_{19} - 25647352402046987464400 \beta_{18} - 17259974752170871319792 \beta_{17} - 18206281846999046970432 \beta_{16} - 18206281846999046970432 \beta_{15} - 34101527580668764092083 \beta_{14} - 34101527580668764092083 \beta_{13} - 68841569159995305234060 \beta_{12} - 68841569159995305234060 \beta_{11} - 26295881740381091576132 \beta_{9} + 26295881740381091576132 \beta_{8} - 88330204944137178981848 \beta_{7} + 14103553879394172364884 \beta_{6} - 14103553879394172364884 \beta_{5} + 11160908773514238066455112 \beta_{4} + 26295881740381091576132 \beta_{1}$$$$)/3840$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$351$$ $$\chi(n)$$ $$1$$ $$\beta_{4}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
143.1
 −11.4741 − 7.80740i 3.75557 − 3.81117i −10.8505 − 10.2794i 5.50401 + 11.9953i −1.99079 + 10.4027i 1.99079 + 10.4027i −5.50401 + 11.9953i 10.8505 − 10.2794i −3.75557 − 3.81117i 11.4741 − 7.80740i −11.4741 + 7.80740i 3.75557 + 3.81117i −10.8505 + 10.2794i 5.50401 − 11.9953i −1.99079 − 10.4027i 1.99079 − 10.4027i −5.50401 − 11.9953i 10.8505 + 10.2794i −3.75557 + 3.81117i 11.4741 + 7.80740i
0 −20.3843 + 20.3843i 0 0 0 −76.9082 76.9082i 0 588.037i 0
143.2 0 −17.2921 + 17.2921i 0 0 0 154.079 + 154.079i 0 355.037i 0
143.3 0 −9.68301 + 9.68301i 0 0 0 −48.6629 48.6629i 0 55.4787i 0
143.4 0 −7.48311 + 7.48311i 0 0 0 −19.2260 19.2260i 0 131.006i 0
143.5 0 −0.839817 + 0.839817i 0 0 0 99.3589 + 99.3589i 0 241.589i 0
143.6 0 0.839817 0.839817i 0 0 0 −99.3589 99.3589i 0 241.589i 0
143.7 0 7.48311 7.48311i 0 0 0 19.2260 + 19.2260i 0 131.006i 0
143.8 0 9.68301 9.68301i 0 0 0 48.6629 + 48.6629i 0 55.4787i 0
143.9 0 17.2921 17.2921i 0 0 0 −154.079 154.079i 0 355.037i 0
143.10 0 20.3843 20.3843i 0 0 0 76.9082 + 76.9082i 0 588.037i 0
207.1 0 −20.3843 20.3843i 0 0 0 −76.9082 + 76.9082i 0 588.037i 0
207.2 0 −17.2921 17.2921i 0 0 0 154.079 154.079i 0 355.037i 0
207.3 0 −9.68301 9.68301i 0 0 0 −48.6629 + 48.6629i 0 55.4787i 0
207.4 0 −7.48311 7.48311i 0 0 0 −19.2260 + 19.2260i 0 131.006i 0
207.5 0 −0.839817 0.839817i 0 0 0 99.3589 99.3589i 0 241.589i 0
207.6 0 0.839817 + 0.839817i 0 0 0 −99.3589 + 99.3589i 0 241.589i 0
207.7 0 7.48311 + 7.48311i 0 0 0 19.2260 19.2260i 0 131.006i 0
207.8 0 9.68301 + 9.68301i 0 0 0 48.6629 48.6629i 0 55.4787i 0
207.9 0 17.2921 + 17.2921i 0 0 0 −154.079 + 154.079i 0 355.037i 0
207.10 0 20.3843 + 20.3843i 0 0 0 76.9082 76.9082i 0 588.037i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 207.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.6.n.g 20
4.b odd 2 1 inner 400.6.n.g 20
5.b even 2 1 80.6.n.d 20
5.c odd 4 1 80.6.n.d 20
5.c odd 4 1 inner 400.6.n.g 20
20.d odd 2 1 80.6.n.d 20
20.e even 4 1 80.6.n.d 20
20.e even 4 1 inner 400.6.n.g 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.6.n.d 20 5.b even 2 1
80.6.n.d 20 5.c odd 4 1
80.6.n.d 20 20.d odd 2 1
80.6.n.d 20 20.e even 4 1
400.6.n.g 20 1.a even 1 1 trivial
400.6.n.g 20 4.b odd 2 1 inner
400.6.n.g 20 5.c odd 4 1 inner
400.6.n.g 20 20.e even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{20} + 1095980 T_{3}^{16} + 297452922160 T_{3}^{12} +$$12246534533136960

'>$$12\!\cdots\!60$$$$T_{3}^{8} +$$108964362492176302080
'>$$10\!\cdots\!80$$$$T_{3}^{4} +$$216763542247808434176'>$$21\!\cdots\!76$$ acting on $$S_{6}^{\mathrm{new}}(400, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 4910 T^{4} + 4404762445 T^{8} + 138021311000520 T^{12} - 6849972091873491390 T^{16} +$$$$85\!\cdots\!52$$$$T^{20} -$$$$23\!\cdots\!90$$$$T^{24} +$$$$16\!\cdots\!20$$$$T^{28} +$$$$18\!\cdots\!45$$$$T^{32} +$$$$72\!\cdots\!10$$$$T^{36} +$$$$51\!\cdots\!01$$$$T^{40}$$
$5$ 1
$7$ $$1 - 50878370 T^{4} + 88119764714863965 T^{8} +$$$$17\!\cdots\!80$$$$T^{12} +$$$$80\!\cdots\!70$$$$T^{16} +$$$$16\!\cdots\!76$$$$T^{20} +$$$$63\!\cdots\!70$$$$T^{24} +$$$$10\!\cdots\!80$$$$T^{28} +$$$$44\!\cdots\!65$$$$T^{32} -$$$$20\!\cdots\!70$$$$T^{36} +$$$$32\!\cdots\!01$$$$T^{40}$$
$11$ $$( 1 - 630250 T^{2} + 230184799605 T^{4} - 63113737617167000 T^{6} +$$$$13\!\cdots\!10$$$$T^{8} -$$$$24\!\cdots\!00$$$$T^{10} +$$$$35\!\cdots\!10$$$$T^{12} -$$$$42\!\cdots\!00$$$$T^{14} +$$$$40\!\cdots\!05$$$$T^{16} -$$$$28\!\cdots\!50$$$$T^{18} +$$$$11\!\cdots\!01$$$$T^{20} )^{2}$$
$13$ $$( 1 + 402 T + 80802 T^{2} + 412195466 T^{3} + 274076716325 T^{4} - 98802411059208 T^{5} + 23088035017184312 T^{6} + 33522742681396790296 T^{7} -$$$$55\!\cdots\!50$$$$T^{8} -$$$$34\!\cdots\!68$$$$T^{9} +$$$$30\!\cdots\!32$$$$T^{10} -$$$$12\!\cdots\!24$$$$T^{11} -$$$$75\!\cdots\!50$$$$T^{12} +$$$$17\!\cdots\!72$$$$T^{13} +$$$$43\!\cdots\!12$$$$T^{14} -$$$$69\!\cdots\!44$$$$T^{15} +$$$$71\!\cdots\!25$$$$T^{16} +$$$$40\!\cdots\!62$$$$T^{17} +$$$$29\!\cdots\!02$$$$T^{18} +$$$$53\!\cdots\!86$$$$T^{19} +$$$$49\!\cdots\!49$$$$T^{20} )^{2}$$
$17$ $$( 1 - 1118 T + 624962 T^{2} - 1188702526 T^{3} - 1739300659939 T^{4} + 3459207301051384 T^{5} - 2073890095879259656 T^{6} +$$$$50\!\cdots\!88$$$$T^{7} +$$$$54\!\cdots\!26$$$$T^{8} -$$$$82\!\cdots\!84$$$$T^{9} +$$$$52\!\cdots\!56$$$$T^{10} -$$$$11\!\cdots\!88$$$$T^{11} +$$$$10\!\cdots\!74$$$$T^{12} +$$$$14\!\cdots\!84$$$$T^{13} -$$$$84\!\cdots\!56$$$$T^{14} +$$$$19\!\cdots\!88$$$$T^{15} -$$$$14\!\cdots\!11$$$$T^{16} -$$$$13\!\cdots\!18$$$$T^{17} +$$$$10\!\cdots\!62$$$$T^{18} -$$$$26\!\cdots\!26$$$$T^{19} +$$$$33\!\cdots\!49$$$$T^{20} )^{2}$$
$19$ $$( 1 + 8249550 T^{2} + 36376777568405 T^{4} +$$$$13\!\cdots\!00$$$$T^{6} +$$$$42\!\cdots\!10$$$$T^{8} +$$$$11\!\cdots\!00$$$$T^{10} +$$$$26\!\cdots\!10$$$$T^{12} +$$$$49\!\cdots\!00$$$$T^{14} +$$$$83\!\cdots\!05$$$$T^{16} +$$$$11\!\cdots\!50$$$$T^{18} +$$$$86\!\cdots\!01$$$$T^{20} )^{2}$$
$23$ $$1 - 207859623483490 T^{4} +$$$$17\!\cdots\!45$$$$T^{8} -$$$$53\!\cdots\!80$$$$T^{12} -$$$$11\!\cdots\!90$$$$T^{16} +$$$$13\!\cdots\!52$$$$T^{20} -$$$$19\!\cdots\!90$$$$T^{24} -$$$$15\!\cdots\!80$$$$T^{28} +$$$$86\!\cdots\!45$$$$T^{32} -$$$$18\!\cdots\!90$$$$T^{36} +$$$$14\!\cdots\!01$$$$T^{40}$$
$29$ $$( 1 - 99799410 T^{2} + 5177263888867445 T^{4} -$$$$18\!\cdots\!20$$$$T^{6} +$$$$49\!\cdots\!10$$$$T^{8} -$$$$10\!\cdots\!52$$$$T^{10} +$$$$20\!\cdots\!10$$$$T^{12} -$$$$32\!\cdots\!20$$$$T^{14} +$$$$38\!\cdots\!45$$$$T^{16} -$$$$31\!\cdots\!10$$$$T^{18} +$$$$13\!\cdots\!01$$$$T^{20} )^{2}$$
$31$ $$( 1 - 101180050 T^{2} + 5842226806223805 T^{4} -$$$$25\!\cdots\!00$$$$T^{6} +$$$$95\!\cdots\!10$$$$T^{8} -$$$$29\!\cdots\!00$$$$T^{10} +$$$$77\!\cdots\!10$$$$T^{12} -$$$$17\!\cdots\!00$$$$T^{14} +$$$$32\!\cdots\!05$$$$T^{16} -$$$$45\!\cdots\!50$$$$T^{18} +$$$$36\!\cdots\!01$$$$T^{20} )^{2}$$
$37$ $$( 1 + 22130 T + 244868450 T^{2} + 1308308732410 T^{3} + 13809729158765845 T^{4} +$$$$24\!\cdots\!80$$$$T^{5} +$$$$28\!\cdots\!00$$$$T^{6} +$$$$15\!\cdots\!60$$$$T^{7} +$$$$10\!\cdots\!10$$$$T^{8} +$$$$15\!\cdots\!80$$$$T^{9} +$$$$18\!\cdots\!00$$$$T^{10} +$$$$10\!\cdots\!60$$$$T^{11} +$$$$50\!\cdots\!90$$$$T^{12} +$$$$51\!\cdots\!80$$$$T^{13} +$$$$64\!\cdots\!00$$$$T^{14} +$$$$38\!\cdots\!60$$$$T^{15} +$$$$15\!\cdots\!05$$$$T^{16} +$$$$10\!\cdots\!30$$$$T^{17} +$$$$13\!\cdots\!50$$$$T^{18} +$$$$82\!\cdots\!10$$$$T^{19} +$$$$25\!\cdots\!49$$$$T^{20} )^{2}$$
$41$ $$( 1 + 1690 T + 164119545 T^{2} - 1028151329120 T^{3} + 25291870422416110 T^{4} - 93379183076463398452 T^{5} +$$$$29\!\cdots\!10$$$$T^{6} -$$$$13\!\cdots\!20$$$$T^{7} +$$$$25\!\cdots\!45$$$$T^{8} +$$$$30\!\cdots\!90$$$$T^{9} +$$$$20\!\cdots\!01$$$$T^{10} )^{4}$$
$43$ $$1 + 22292551011039310 T^{4} +$$$$40\!\cdots\!45$$$$T^{8} +$$$$23\!\cdots\!20$$$$T^{12} +$$$$67\!\cdots\!10$$$$T^{16} +$$$$21\!\cdots\!52$$$$T^{20} +$$$$31\!\cdots\!10$$$$T^{24} +$$$$51\!\cdots\!20$$$$T^{28} +$$$$41\!\cdots\!45$$$$T^{32} +$$$$10\!\cdots\!10$$$$T^{36} +$$$$22\!\cdots\!01$$$$T^{40}$$
$47$ $$1 + 74279974827999230 T^{4} +$$$$48\!\cdots\!65$$$$T^{8} +$$$$19\!\cdots\!80$$$$T^{12} +$$$$14\!\cdots\!70$$$$T^{16} -$$$$25\!\cdots\!24$$$$T^{20} +$$$$39\!\cdots\!70$$$$T^{24} +$$$$15\!\cdots\!80$$$$T^{28} +$$$$10\!\cdots\!65$$$$T^{32} +$$$$43\!\cdots\!30$$$$T^{36} +$$$$16\!\cdots\!01$$$$T^{40}$$
$53$ $$( 1 + 91226 T + 4161091538 T^{2} + 141622957033378 T^{3} + 3893612180777232821 T^{4} +$$$$81\!\cdots\!12$$$$T^{5} +$$$$12\!\cdots\!56$$$$T^{6} +$$$$10\!\cdots\!76$$$$T^{7} -$$$$19\!\cdots\!54$$$$T^{8} -$$$$10\!\cdots\!72$$$$T^{9} -$$$$26\!\cdots\!96$$$$T^{10} -$$$$45\!\cdots\!96$$$$T^{11} -$$$$34\!\cdots\!46$$$$T^{12} +$$$$80\!\cdots\!32$$$$T^{13} +$$$$39\!\cdots\!56$$$$T^{14} +$$$$10\!\cdots\!16$$$$T^{15} +$$$$20\!\cdots\!29$$$$T^{16} +$$$$31\!\cdots\!46$$$$T^{17} +$$$$38\!\cdots\!38$$$$T^{18} +$$$$35\!\cdots\!18$$$$T^{19} +$$$$16\!\cdots\!49$$$$T^{20} )^{2}$$
$59$ $$( 1 + 4265422750 T^{2} + 8278744468566020805 T^{4} +$$$$98\!\cdots\!00$$$$T^{6} +$$$$85\!\cdots\!10$$$$T^{8} +$$$$63\!\cdots\!00$$$$T^{10} +$$$$43\!\cdots\!10$$$$T^{12} +$$$$25\!\cdots\!00$$$$T^{14} +$$$$11\!\cdots\!05$$$$T^{16} +$$$$29\!\cdots\!50$$$$T^{18} +$$$$34\!\cdots\!01$$$$T^{20} )^{2}$$
$61$ $$( 1 + 10270 T + 3306972765 T^{2} + 29698468018720 T^{3} + 4981733552475787870 T^{4} +$$$$35\!\cdots\!24$$$$T^{5} +$$$$42\!\cdots\!70$$$$T^{6} +$$$$21\!\cdots\!20$$$$T^{7} +$$$$19\!\cdots\!65$$$$T^{8} +$$$$52\!\cdots\!70$$$$T^{9} +$$$$42\!\cdots\!01$$$$T^{10} )^{4}$$
$67$ $$1 - 2038648142805486290 T^{4} -$$$$25\!\cdots\!55$$$$T^{8} +$$$$14\!\cdots\!20$$$$T^{12} +$$$$73\!\cdots\!10$$$$T^{16} -$$$$57\!\cdots\!48$$$$T^{20} +$$$$24\!\cdots\!10$$$$T^{24} +$$$$15\!\cdots\!20$$$$T^{28} -$$$$93\!\cdots\!55$$$$T^{32} -$$$$24\!\cdots\!90$$$$T^{36} +$$$$40\!\cdots\!01$$$$T^{40}$$
$71$ $$( 1 - 14421599170 T^{2} + 96751044929093627565 T^{4} -$$$$40\!\cdots\!20$$$$T^{6} +$$$$11\!\cdots\!70$$$$T^{8} -$$$$24\!\cdots\!24$$$$T^{10} +$$$$37\!\cdots\!70$$$$T^{12} -$$$$42\!\cdots\!20$$$$T^{14} +$$$$33\!\cdots\!65$$$$T^{16} -$$$$16\!\cdots\!70$$$$T^{18} +$$$$36\!\cdots\!01$$$$T^{20} )^{2}$$
$73$ $$( 1 + 132186 T + 8736569298 T^{2} + 431249337673738 T^{3} + 10750357541220477581 T^{4} -$$$$12\!\cdots\!88$$$$T^{5} -$$$$17\!\cdots\!84$$$$T^{6} -$$$$58\!\cdots\!44$$$$T^{7} +$$$$47\!\cdots\!06$$$$T^{8} +$$$$60\!\cdots\!08$$$$T^{9} +$$$$32\!\cdots\!04$$$$T^{10} +$$$$12\!\cdots\!44$$$$T^{11} +$$$$20\!\cdots\!94$$$$T^{12} -$$$$52\!\cdots\!08$$$$T^{13} -$$$$32\!\cdots\!84$$$$T^{14} -$$$$47\!\cdots\!84$$$$T^{15} +$$$$85\!\cdots\!69$$$$T^{16} +$$$$70\!\cdots\!66$$$$T^{17} +$$$$29\!\cdots\!98$$$$T^{18} +$$$$93\!\cdots\!98$$$$T^{19} +$$$$14\!\cdots\!49$$$$T^{20} )^{2}$$
$79$ $$( 1 + 18614864790 T^{2} +$$$$16\!\cdots\!45$$$$T^{4} +$$$$10\!\cdots\!80$$$$T^{6} +$$$$44\!\cdots\!10$$$$T^{8} +$$$$15\!\cdots\!48$$$$T^{10} +$$$$42\!\cdots\!10$$$$T^{12} +$$$$90\!\cdots\!80$$$$T^{14} +$$$$14\!\cdots\!45$$$$T^{16} +$$$$14\!\cdots\!90$$$$T^{18} +$$$$76\!\cdots\!01$$$$T^{20} )^{2}$$
$83$ $$1 - 18609684090616711570 T^{4} -$$$$46\!\cdots\!35$$$$T^{8} +$$$$34\!\cdots\!80$$$$T^{12} +$$$$25\!\cdots\!70$$$$T^{16} -$$$$17\!\cdots\!24$$$$T^{20} +$$$$60\!\cdots\!70$$$$T^{24} +$$$$20\!\cdots\!80$$$$T^{28} -$$$$64\!\cdots\!35$$$$T^{32} -$$$$62\!\cdots\!70$$$$T^{36} +$$$$80\!\cdots\!01$$$$T^{40}$$
$89$ $$( 1 - 33120686970 T^{2} +$$$$41\!\cdots\!65$$$$T^{4} -$$$$20\!\cdots\!20$$$$T^{6} -$$$$32\!\cdots\!30$$$$T^{8} +$$$$73\!\cdots\!76$$$$T^{10} -$$$$10\!\cdots\!30$$$$T^{12} -$$$$20\!\cdots\!20$$$$T^{14} +$$$$12\!\cdots\!65$$$$T^{16} -$$$$31\!\cdots\!70$$$$T^{18} +$$$$29\!\cdots\!01$$$$T^{20} )^{2}$$
$97$ $$( 1 + 187386 T + 17556756498 T^{2} + 497237634071002 T^{3} -$$$$23\!\cdots\!35$$$$T^{4} -$$$$42\!\cdots\!24$$$$T^{5} -$$$$37\!\cdots\!32$$$$T^{6} -$$$$18\!\cdots\!68$$$$T^{7} +$$$$11\!\cdots\!50$$$$T^{8} +$$$$33\!\cdots\!76$$$$T^{9} +$$$$36\!\cdots\!68$$$$T^{10} +$$$$29\!\cdots\!32$$$$T^{11} +$$$$87\!\cdots\!50$$$$T^{12} -$$$$11\!\cdots\!24$$$$T^{13} -$$$$20\!\cdots\!32$$$$T^{14} -$$$$19\!\cdots\!68$$$$T^{15} -$$$$93\!\cdots\!15$$$$T^{16} +$$$$17\!\cdots\!86$$$$T^{17} +$$$$51\!\cdots\!98$$$$T^{18} +$$$$47\!\cdots\!02$$$$T^{19} +$$$$21\!\cdots\!49$$$$T^{20} )^{2}$$