# Properties

 Label 160.6.n.b Level 160 Weight 6 Character orbit 160.n Analytic conductor 25.661 Analytic rank 0 Dimension 14 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$160 = 2^{5} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 160.n (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.6614111701$$ Analytic rank: $$0$$ Dimension: $$14$$ Relative dimension: $$7$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ Defining polynomial: $$x^{14} - 4 x^{13} + 8 x^{12} - 4626 x^{11} + 149441 x^{10} - 2113414 x^{9} + 17958066 x^{8} - 97717112 x^{7} + 355171384 x^{6} - 910571904 x^{5} + 2428303248 x^{4} - 9166992192 x^{3} + 32237484304 x^{2} - 66916821408 x + 69451154208$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{31}\cdot 5^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{1} - \beta_{2} ) q^{3} + ( 3 - 4 \beta_{1} + \beta_{2} - \beta_{5} ) q^{5} + ( 5 - 5 \beta_{1} + \beta_{6} + \beta_{8} ) q^{7} + ( 58 \beta_{1} - \beta_{2} - \beta_{6} + \beta_{7} + \beta_{11} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{1} - \beta_{2} ) q^{3} + ( 3 - 4 \beta_{1} + \beta_{2} - \beta_{5} ) q^{5} + ( 5 - 5 \beta_{1} + \beta_{6} + \beta_{8} ) q^{7} + ( 58 \beta_{1} - \beta_{2} - \beta_{6} + \beta_{7} + \beta_{11} ) q^{9} + ( 36 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{12} ) q^{11} + ( -30 + 31 \beta_{1} + \beta_{2} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{10} - \beta_{12} ) q^{13} + ( 19 - 163 \beta_{1} + 6 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + 5 \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} + 2 \beta_{10} - 3 \beta_{11} - \beta_{12} - \beta_{13} ) q^{15} + ( 90 + 90 \beta_{1} - 20 \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + \beta_{7} - 2 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} + 3 \beta_{12} - 3 \beta_{13} ) q^{17} + ( 402 + 4 \beta_{1} + 6 \beta_{2} - \beta_{3} - 4 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - \beta_{10} - 6 \beta_{13} ) q^{19} + ( 426 + \beta_{1} - 7 \beta_{2} + 6 \beta_{3} - \beta_{4} + 8 \beta_{5} + 12 \beta_{6} - 11 \beta_{7} + 6 \beta_{8} + 6 \beta_{9} - 2 \beta_{10} - 4 \beta_{13} ) q^{21} + ( 217 + 221 \beta_{1} - 39 \beta_{2} + 3 \beta_{3} - 7 \beta_{4} + 10 \beta_{5} + 4 \beta_{6} - 8 \beta_{7} + 13 \beta_{9} - 3 \beta_{10} - 3 \beta_{12} ) q^{23} + ( -305 + 77 \beta_{1} - 13 \beta_{2} + 7 \beta_{3} - \beta_{4} - 6 \beta_{5} + 21 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} + 16 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} + \beta_{12} + 6 \beta_{13} ) q^{25} + ( -150 + 158 \beta_{1} - 6 \beta_{2} - 7 \beta_{3} - 15 \beta_{4} - 4 \beta_{5} - 18 \beta_{6} + 38 \beta_{7} - 14 \beta_{8} + \beta_{10} + 2 \beta_{11} - \beta_{12} + 2 \beta_{13} ) q^{27} + ( 17 - 1235 \beta_{1} + 25 \beta_{2} + 17 \beta_{3} - 7 \beta_{4} - 7 \beta_{5} + 11 \beta_{6} + 23 \beta_{7} + 8 \beta_{8} - 8 \beta_{9} - 18 \beta_{11} - 4 \beta_{12} ) q^{29} + ( 358 \beta_{1} - 95 \beta_{2} - 8 \beta_{4} + 26 \beta_{5} - 77 \beta_{6} + 14 \beta_{7} + 9 \beta_{8} - 9 \beta_{9} + 6 \beta_{11} + 10 \beta_{12} ) q^{31} + ( -185 + 211 \beta_{1} + 20 \beta_{2} + 25 \beta_{3} - \beta_{4} - 53 \beta_{5} - 100 \beta_{6} - 11 \beta_{7} - 18 \beta_{8} - 5 \beta_{10} + 7 \beta_{11} + 5 \beta_{12} + 7 \beta_{13} ) q^{33} + ( -249 + 191 \beta_{1} + 124 \beta_{2} - 16 \beta_{3} - 27 \beta_{4} - 7 \beta_{5} - 79 \beta_{6} - 3 \beta_{7} + 26 \beta_{8} - 8 \beta_{9} - 4 \beta_{10} + 6 \beta_{11} + 7 \beta_{12} + 12 \beta_{13} ) q^{35} + ( -52 - 83 \beta_{1} - 225 \beta_{2} + 47 \beta_{3} - 16 \beta_{4} + 88 \beta_{5} + 25 \beta_{7} - 32 \beta_{9} - 16 \beta_{10} - 6 \beta_{11} - 16 \beta_{12} + 6 \beta_{13} ) q^{37} + ( -846 + 28 \beta_{1} + 129 \beta_{2} + 4 \beta_{3} - 28 \beta_{4} - 64 \beta_{5} - 93 \beta_{6} - 40 \beta_{7} - 25 \beta_{8} - 25 \beta_{9} + 4 \beta_{10} + 12 \beta_{13} ) q^{39} + ( 805 + 10 \beta_{1} + 49 \beta_{2} + 46 \beta_{3} - 10 \beta_{4} + 23 \beta_{5} + 5 \beta_{6} - 100 \beta_{7} - 18 \beta_{8} - 18 \beta_{9} - 2 \beta_{10} + 3 \beta_{13} ) q^{41} + ( -259 - 163 \beta_{1} - 219 \beta_{2} - 40 \beta_{3} - 56 \beta_{4} - 32 \beta_{5} + 66 \beta_{6} - 180 \beta_{7} + 26 \beta_{9} + 10 \beta_{10} - 18 \beta_{11} + 10 \beta_{12} + 18 \beta_{13} ) q^{43} + ( 1235 - 2651 \beta_{1} - 36 \beta_{2} + 64 \beta_{3} - 7 \beta_{4} + 15 \beta_{5} + 516 \beta_{6} - 19 \beta_{7} + 40 \beta_{8} - 10 \beta_{9} - 15 \beta_{10} - 10 \beta_{11} + 5 \beta_{12} ) q^{45} + ( 51 + \beta_{1} + 14 \beta_{2} + 19 \beta_{3} - 33 \beta_{4} - 84 \beta_{5} + 331 \beta_{6} + 70 \beta_{7} - 31 \beta_{8} - 5 \beta_{10} + 18 \beta_{11} + 5 \beta_{12} + 18 \beta_{13} ) q^{47} + ( 58 - 624 \beta_{1} + 287 \beta_{2} + 58 \beta_{3} - 8 \beta_{4} - 32 \beta_{5} + 247 \beta_{6} + 139 \beta_{7} + 46 \beta_{8} - 46 \beta_{9} + 15 \beta_{11} + 10 \beta_{12} ) q^{49} + ( -60 + 5994 \beta_{1} - 609 \beta_{2} - 60 \beta_{3} - 106 \beta_{4} + 260 \beta_{5} - 455 \beta_{6} + 4 \beta_{8} - 4 \beta_{9} + 14 \beta_{11} - 12 \beta_{12} ) q^{51} + ( 381 - 282 \beta_{1} + 78 \beta_{2} + 71 \beta_{3} - 28 \beta_{4} - 159 \beta_{5} - 283 \beta_{6} - 40 \beta_{7} - 86 \beta_{8} + 7 \beta_{10} - 18 \beta_{11} - 7 \beta_{12} - 18 \beta_{13} ) q^{53} + ( 4522 - 1802 \beta_{1} + 556 \beta_{2} + 64 \beta_{3} - 47 \beta_{4} + 45 \beta_{5} - 96 \beta_{6} - 13 \beta_{7} + 9 \beta_{8} + 23 \beta_{9} - 21 \beta_{10} - 21 \beta_{11} - 12 \beta_{12} - 27 \beta_{13} ) q^{55} + ( -606 - 676 \beta_{1} - 1504 \beta_{2} + 75 \beta_{3} - 5 \beta_{4} + 156 \beta_{5} + 18 \beta_{6} + 40 \beta_{7} + 58 \beta_{9} + 13 \beta_{10} - 12 \beta_{11} + 13 \beta_{12} + 12 \beta_{13} ) q^{57} + ( -886 + 124 \beta_{1} + 638 \beta_{2} - 97 \beta_{3} - 124 \beta_{4} - 339 \beta_{5} - 620 \beta_{6} + 79 \beta_{7} + 16 \beta_{8} + 16 \beta_{9} - 9 \beta_{10} - 6 \beta_{13} ) q^{59} + ( 1113 + 18 \beta_{1} + 618 \beta_{2} + 43 \beta_{3} - 18 \beta_{4} - 23 \beta_{5} - 511 \beta_{6} - 150 \beta_{7} + 90 \beta_{8} + 90 \beta_{9} + 46 \beta_{10} + 30 \beta_{13} ) q^{61} + ( 1693 + 1641 \beta_{1} - 1213 \beta_{2} + 59 \beta_{3} - 7 \beta_{4} + 218 \beta_{5} + 26 \beta_{6} + 100 \beta_{7} - 39 \beta_{9} + 19 \beta_{10} + 74 \beta_{11} + 19 \beta_{12} - 74 \beta_{13} ) q^{63} + ( 3010 - 4063 \beta_{1} + 357 \beta_{2} + 62 \beta_{3} - 66 \beta_{4} + 57 \beta_{5} + 647 \beta_{6} + 38 \beta_{7} - 86 \beta_{8} + 78 \beta_{9} + 4 \beta_{10} - 6 \beta_{11} - 42 \beta_{12} - 57 \beta_{13} ) q^{65} + ( 8043 - 8019 \beta_{1} - 124 \beta_{2} - 130 \beta_{3} - 154 \beta_{4} + 184 \beta_{5} + 1043 \beta_{6} + 348 \beta_{7} + 70 \beta_{8} + 6 \beta_{10} - 84 \beta_{11} - 6 \beta_{12} - 84 \beta_{13} ) q^{67} + ( 49 + 11890 \beta_{1} - 424 \beta_{2} + 49 \beta_{3} + 26 \beta_{4} - 57 \beta_{5} - 455 \beta_{6} + 218 \beta_{7} - 152 \beta_{8} + 152 \beta_{9} + 146 \beta_{11} + 44 \beta_{12} ) q^{69} + ( 148 - 774 \beta_{1} - 615 \beta_{2} + 148 \beta_{3} - 12 \beta_{4} - 202 \beta_{5} - 829 \beta_{6} + 170 \beta_{7} - 27 \beta_{8} + 27 \beta_{9} - 138 \beta_{11} - 78 \beta_{12} ) q^{71} + ( -1479 + 1525 \beta_{1} + 58 \beta_{2} + 49 \beta_{3} + 3 \beta_{4} - 68 \beta_{5} - 1060 \beta_{6} - 100 \beta_{7} + 330 \beta_{8} + 9 \beta_{10} - 36 \beta_{11} - 9 \beta_{12} - 36 \beta_{13} ) q^{73} + ( 4003 + 5427 \beta_{1} + 1501 \beta_{2} - 209 \beta_{3} - 163 \beta_{4} - 24 \beta_{5} - 792 \beta_{6} + 122 \beta_{7} - 148 \beta_{8} + 4 \beta_{9} + 47 \beta_{10} + 92 \beta_{11} - 31 \beta_{12} - 26 \beta_{13} ) q^{75} + ( -657 - 593 \beta_{1} - 1299 \beta_{2} - 74 \beta_{3} + 10 \beta_{4} - 41 \beta_{5} + 59 \beta_{6} - 75 \beta_{7} + 14 \beta_{9} + 69 \beta_{10} + 48 \beta_{11} + 69 \beta_{12} - 48 \beta_{13} ) q^{77} + ( -10544 - 136 \beta_{1} - 18 \beta_{2} + 208 \beta_{3} + 136 \beta_{4} + 414 \beta_{5} + 108 \beta_{6} - 298 \beta_{7} + 150 \beta_{8} + 150 \beta_{9} + 18 \beta_{10} + 66 \beta_{13} ) q^{79} + ( 12022 + 4 \beta_{1} + 711 \beta_{2} - 28 \beta_{3} - 4 \beta_{4} - 15 \beta_{5} - 773 \beta_{6} + 90 \beta_{7} - 230 \beta_{8} - 230 \beta_{9} - 38 \beta_{10} - 21 \beta_{13} ) q^{81} + ( -2429 - 2133 \beta_{1} - 669 \beta_{2} - 163 \beta_{3} - 133 \beta_{4} - 318 \beta_{5} + 44 \beta_{6} - 376 \beta_{7} - 170 \beta_{9} - 89 \beta_{10} - 36 \beta_{11} - 89 \beta_{12} + 36 \beta_{13} ) q^{83} + ( -3872 - 13966 \beta_{1} + 10 \beta_{2} - 122 \beta_{3} + 56 \beta_{4} - 96 \beta_{5} + 3272 \beta_{6} + 44 \beta_{7} - 18 \beta_{8} - 356 \beta_{9} + 87 \beta_{10} + 72 \beta_{11} + 29 \beta_{12} - 6 \beta_{13} ) q^{85} + ( 9948 - 9916 \beta_{1} + 290 \beta_{2} + 274 \beta_{3} + 242 \beta_{4} - 300 \beta_{5} + 4110 \beta_{6} - 796 \beta_{7} + 246 \beta_{8} + 16 \beta_{10} - 22 \beta_{11} - 16 \beta_{12} - 22 \beta_{13} ) q^{87} + ( -54 + 3182 \beta_{1} + 2670 \beta_{2} - 54 \beta_{3} - 80 \beta_{4} + 74 \beta_{5} + 2664 \beta_{6} - 190 \beta_{7} - 4 \beta_{8} + 4 \beta_{9} - 162 \beta_{11} - 140 \beta_{12} ) q^{89} + ( -252 + 33838 \beta_{1} - 1685 \beta_{2} - 252 \beta_{3} + 64 \beta_{4} + 262 \beta_{5} - 1359 \beta_{6} - 502 \beta_{7} - 56 \beta_{8} + 56 \beta_{9} + 66 \beta_{11} + 138 \beta_{12} ) q^{91} + ( -22289 + 22023 \beta_{1} - 227 \beta_{2} - 175 \beta_{3} + 91 \beta_{4} + 353 \beta_{5} - 1161 \beta_{6} + 185 \beta_{7} + 24 \beta_{8} - 52 \beta_{10} + 140 \beta_{11} + 52 \beta_{12} + 140 \beta_{13} ) q^{93} + ( 11762 - 10444 \beta_{1} + 2417 \beta_{2} + 242 \beta_{3} + 469 \beta_{4} - 159 \beta_{5} - 217 \beta_{6} - 85 \beta_{7} - 128 \beta_{8} - 116 \beta_{9} + 67 \beta_{10} + 27 \beta_{11} + 124 \beta_{12} + 99 \beta_{13} ) q^{95} + ( -3079 - 2919 \beta_{1} - 3976 \beta_{2} - 133 \beta_{3} - 27 \beta_{4} - 444 \beta_{5} - 112 \beta_{6} - 114 \beta_{7} + 106 \beta_{9} - 139 \beta_{10} - 66 \beta_{11} - 139 \beta_{12} + 66 \beta_{13} ) q^{97} + ( -28504 - 12 \beta_{1} + 1313 \beta_{2} - 287 \beta_{3} + 12 \beta_{4} - 253 \beta_{5} - 1555 \beta_{6} + 529 \beta_{7} - 46 \beta_{8} - 46 \beta_{9} + 57 \beta_{10} - 10 \beta_{13} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14q + 10q^{3} + 42q^{5} + 66q^{7} + O(q^{10})$$ $$14q + 10q^{3} + 42q^{5} + 66q^{7} - 414q^{13} + 278q^{15} + 1222q^{17} + 5672q^{19} + 5924q^{21} + 2902q^{23} - 4466q^{25} - 2168q^{27} - 2444q^{33} - 2618q^{35} - 1790q^{37} - 11076q^{39} + 11644q^{41} - 3982q^{43} + 14704q^{45} - 1278q^{47} + 5882q^{53} + 65608q^{55} - 14552q^{57} - 8504q^{59} + 20564q^{61} + 19422q^{63} + 40798q^{65} + 107926q^{67} - 16418q^{73} + 66586q^{75} - 13348q^{77} - 146544q^{79} + 173806q^{81} - 36398q^{83} - 66262q^{85} + 124384q^{87} - 306620q^{93} + 173768q^{95} - 60314q^{97} - 388628q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} - 4 x^{13} + 8 x^{12} - 4626 x^{11} + 149441 x^{10} - 2113414 x^{9} + 17958066 x^{8} - 97717112 x^{7} + 355171384 x^{6} - 910571904 x^{5} + 2428303248 x^{4} - 9166992192 x^{3} + 32237484304 x^{2} - 66916821408 x + 69451154208$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-17698243422788399786322792428 \nu^{13} + 33339598605867540664447340809 \nu^{12} - 111566481532271037945221541610 \nu^{11} + 81574043330479363757248078257118 \nu^{10} - 2472391595710102013111236607934330 \nu^{9} + 32355518440523178255983518692998557 \nu^{8} - 254388696751243387855192637041212756 \nu^{7} + 1246643719825304584828348337878941040 \nu^{6} - 3992608702077023462626800882531182472 \nu^{5} + 8586609210797225701138502778301522360 \nu^{4} - 23885609292141888818043668122534492304 \nu^{3} + 93524178076331793639201391781753566912 \nu^{2} - 355219515211437437156695767539235567040 \nu + 518482390237866563439088670755099656144$$$$)/$$$$22\!\cdots\!00$$ $$\beta_{2}$$ $$=$$ $$($$$$53\!\cdots\!57$$$$\nu^{13} +$$$$13\!\cdots\!59$$$$\nu^{12} +$$$$55\!\cdots\!00$$$$\nu^{11} -$$$$25\!\cdots\!32$$$$\nu^{10} +$$$$64\!\cdots\!75$$$$\nu^{9} -$$$$65\!\cdots\!23$$$$\nu^{8} +$$$$37\!\cdots\!44$$$$\nu^{7} -$$$$10\!\cdots\!80$$$$\nu^{6} +$$$$14\!\cdots\!08$$$$\nu^{5} -$$$$26\!\cdots\!00$$$$\nu^{4} +$$$$34\!\cdots\!36$$$$\nu^{3} -$$$$97\!\cdots\!68$$$$\nu^{2} +$$$$74\!\cdots\!40$$$$\nu +$$$$29\!\cdots\!84$$$$)/$$$$68\!\cdots\!00$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$11\!\cdots\!89$$$$\nu^{13} +$$$$18\!\cdots\!32$$$$\nu^{12} -$$$$29\!\cdots\!50$$$$\nu^{11} +$$$$54\!\cdots\!14$$$$\nu^{10} -$$$$16\!\cdots\!25$$$$\nu^{9} +$$$$22\!\cdots\!46$$$$\nu^{8} -$$$$19\!\cdots\!88$$$$\nu^{7} +$$$$10\!\cdots\!60$$$$\nu^{6} -$$$$37\!\cdots\!16$$$$\nu^{5} +$$$$87\!\cdots\!00$$$$\nu^{4} -$$$$21\!\cdots\!72$$$$\nu^{3} +$$$$83\!\cdots\!36$$$$\nu^{2} -$$$$30\!\cdots\!80$$$$\nu +$$$$60\!\cdots\!32$$$$)/$$$$13\!\cdots\!00$$ $$\beta_{4}$$ $$=$$ $$($$$$12\!\cdots\!53$$$$\nu^{13} -$$$$46\!\cdots\!14$$$$\nu^{12} +$$$$52\!\cdots\!50$$$$\nu^{11} -$$$$54\!\cdots\!78$$$$\nu^{10} +$$$$18\!\cdots\!25$$$$\nu^{9} -$$$$25\!\cdots\!92$$$$\nu^{8} +$$$$20\!\cdots\!76$$$$\nu^{7} -$$$$10\!\cdots\!20$$$$\nu^{6} +$$$$31\!\cdots\!32$$$$\nu^{5} -$$$$63\!\cdots\!00$$$$\nu^{4} +$$$$22\!\cdots\!44$$$$\nu^{3} -$$$$97\!\cdots\!72$$$$\nu^{2} +$$$$31\!\cdots\!60$$$$\nu -$$$$24\!\cdots\!64$$$$)/$$$$13\!\cdots\!00$$ $$\beta_{5}$$ $$=$$ $$($$$$13\!\cdots\!41$$$$\nu^{13} +$$$$13\!\cdots\!42$$$$\nu^{12} +$$$$37\!\cdots\!50$$$$\nu^{11} -$$$$63\!\cdots\!66$$$$\nu^{10} +$$$$17\!\cdots\!25$$$$\nu^{9} -$$$$19\!\cdots\!24$$$$\nu^{8} +$$$$13\!\cdots\!72$$$$\nu^{7} -$$$$50\!\cdots\!40$$$$\nu^{6} +$$$$11\!\cdots\!04$$$$\nu^{5} -$$$$19\!\cdots\!00$$$$\nu^{4} +$$$$10\!\cdots\!68$$$$\nu^{3} -$$$$43\!\cdots\!84$$$$\nu^{2} +$$$$89\!\cdots\!20$$$$\nu +$$$$32\!\cdots\!92$$$$)/$$$$13\!\cdots\!00$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$69\!\cdots\!36$$$$\nu^{13} +$$$$25\!\cdots\!43$$$$\nu^{12} +$$$$59\!\cdots\!50$$$$\nu^{11} +$$$$32\!\cdots\!86$$$$\nu^{10} -$$$$10\!\cdots\!50$$$$\nu^{9} +$$$$13\!\cdots\!79$$$$\nu^{8} -$$$$10\!\cdots\!12$$$$\nu^{7} +$$$$50\!\cdots\!40$$$$\nu^{6} -$$$$14\!\cdots\!84$$$$\nu^{5} +$$$$29\!\cdots\!00$$$$\nu^{4} -$$$$10\!\cdots\!28$$$$\nu^{3} +$$$$47\!\cdots\!64$$$$\nu^{2} -$$$$12\!\cdots\!20$$$$\nu +$$$$15\!\cdots\!68$$$$)/$$$$68\!\cdots\!00$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$15\!\cdots\!33$$$$\nu^{13} +$$$$55\!\cdots\!04$$$$\nu^{12} +$$$$75\!\cdots\!50$$$$\nu^{11} +$$$$72\!\cdots\!58$$$$\nu^{10} -$$$$22\!\cdots\!25$$$$\nu^{9} +$$$$31\!\cdots\!62$$$$\nu^{8} -$$$$24\!\cdots\!36$$$$\nu^{7} +$$$$11\!\cdots\!20$$$$\nu^{6} -$$$$36\!\cdots\!52$$$$\nu^{5} +$$$$75\!\cdots\!00$$$$\nu^{4} -$$$$22\!\cdots\!84$$$$\nu^{3} +$$$$10\!\cdots\!92$$$$\nu^{2} -$$$$34\!\cdots\!60$$$$\nu +$$$$46\!\cdots\!04$$$$)/$$$$13\!\cdots\!00$$ $$\beta_{8}$$ $$=$$ $$($$$$-$$$$95\!\cdots\!18$$$$\nu^{13} +$$$$36\!\cdots\!09$$$$\nu^{12} +$$$$86\!\cdots\!50$$$$\nu^{11} +$$$$44\!\cdots\!18$$$$\nu^{10} -$$$$14\!\cdots\!00$$$$\nu^{9} +$$$$19\!\cdots\!77$$$$\nu^{8} -$$$$14\!\cdots\!56$$$$\nu^{7} +$$$$70\!\cdots\!20$$$$\nu^{6} -$$$$20\!\cdots\!92$$$$\nu^{5} +$$$$41\!\cdots\!00$$$$\nu^{4} -$$$$14\!\cdots\!64$$$$\nu^{3} +$$$$65\!\cdots\!32$$$$\nu^{2} -$$$$16\!\cdots\!60$$$$\nu +$$$$21\!\cdots\!84$$$$)/$$$$34\!\cdots\!00$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$96\!\cdots\!91$$$$\nu^{13} -$$$$18\!\cdots\!17$$$$\nu^{12} -$$$$31\!\cdots\!00$$$$\nu^{11} +$$$$44\!\cdots\!16$$$$\nu^{10} -$$$$11\!\cdots\!25$$$$\nu^{9} +$$$$12\!\cdots\!49$$$$\nu^{8} -$$$$78\!\cdots\!72$$$$\nu^{7} +$$$$26\!\cdots\!40$$$$\nu^{6} -$$$$59\!\cdots\!04$$$$\nu^{5} +$$$$14\!\cdots\!00$$$$\nu^{4} -$$$$84\!\cdots\!68$$$$\nu^{3} +$$$$25\!\cdots\!84$$$$\nu^{2} -$$$$44\!\cdots\!20$$$$\nu +$$$$16\!\cdots\!08$$$$)/$$$$34\!\cdots\!00$$ $$\beta_{10}$$ $$=$$ $$($$$$67\!\cdots\!51$$$$\nu^{13} +$$$$63\!\cdots\!62$$$$\nu^{12} +$$$$95\!\cdots\!50$$$$\nu^{11} -$$$$30\!\cdots\!26$$$$\nu^{10} +$$$$85\!\cdots\!75$$$$\nu^{9} -$$$$10\!\cdots\!64$$$$\nu^{8} +$$$$71\!\cdots\!92$$$$\nu^{7} -$$$$28\!\cdots\!40$$$$\nu^{6} +$$$$63\!\cdots\!44$$$$\nu^{5} -$$$$68\!\cdots\!00$$$$\nu^{4} +$$$$55\!\cdots\!48$$$$\nu^{3} -$$$$27\!\cdots\!24$$$$\nu^{2} +$$$$61\!\cdots\!20$$$$\nu +$$$$24\!\cdots\!12$$$$)/$$$$13\!\cdots\!00$$ $$\beta_{11}$$ $$=$$ $$($$$$96\!\cdots\!11$$$$\nu^{13} -$$$$69\!\cdots\!68$$$$\nu^{12} -$$$$64\!\cdots\!50$$$$\nu^{11} -$$$$44\!\cdots\!86$$$$\nu^{10} +$$$$15\!\cdots\!75$$$$\nu^{9} -$$$$23\!\cdots\!54$$$$\nu^{8} +$$$$20\!\cdots\!12$$$$\nu^{7} -$$$$10\!\cdots\!40$$$$\nu^{6} +$$$$34\!\cdots\!84$$$$\nu^{5} -$$$$72\!\cdots\!00$$$$\nu^{4} +$$$$19\!\cdots\!28$$$$\nu^{3} -$$$$99\!\cdots\!64$$$$\nu^{2} +$$$$31\!\cdots\!20$$$$\nu -$$$$47\!\cdots\!68$$$$)/$$$$13\!\cdots\!00$$ $$\beta_{12}$$ $$=$$ $$($$$$-$$$$12\!\cdots\!43$$$$\nu^{13} +$$$$19\!\cdots\!84$$$$\nu^{12} -$$$$72\!\cdots\!50$$$$\nu^{11} +$$$$56\!\cdots\!18$$$$\nu^{10} -$$$$16\!\cdots\!75$$$$\nu^{9} +$$$$21\!\cdots\!02$$$$\nu^{8} -$$$$17\!\cdots\!56$$$$\nu^{7} +$$$$84\!\cdots\!20$$$$\nu^{6} -$$$$26\!\cdots\!92$$$$\nu^{5} +$$$$58\!\cdots\!00$$$$\nu^{4} -$$$$17\!\cdots\!64$$$$\nu^{3} +$$$$79\!\cdots\!32$$$$\nu^{2} -$$$$24\!\cdots\!60$$$$\nu +$$$$35\!\cdots\!84$$$$)/$$$$13\!\cdots\!00$$ $$\beta_{13}$$ $$=$$ $$($$$$22\!\cdots\!67$$$$\nu^{13} -$$$$12\!\cdots\!46$$$$\nu^{12} -$$$$64\!\cdots\!50$$$$\nu^{11} -$$$$10\!\cdots\!42$$$$\nu^{10} +$$$$30\!\cdots\!75$$$$\nu^{9} -$$$$37\!\cdots\!88$$$$\nu^{8} +$$$$26\!\cdots\!64$$$$\nu^{7} -$$$$11\!\cdots\!80$$$$\nu^{6} +$$$$28\!\cdots\!48$$$$\nu^{5} -$$$$56\!\cdots\!00$$$$\nu^{4} +$$$$24\!\cdots\!16$$$$\nu^{3} -$$$$10\!\cdots\!08$$$$\nu^{2} +$$$$24\!\cdots\!40$$$$\nu -$$$$18\!\cdots\!96$$$$)/$$$$13\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$3 \beta_{13} + \beta_{12} + 3 \beta_{11} - \beta_{10} + 2 \beta_{8} - 6 \beta_{7} + 47 \beta_{6} - 12 \beta_{5} + 2 \beta_{4} + 6 \beta_{3} + 5 \beta_{2} - 58 \beta_{1} + 62$$$$)/160$$ $$\nu^{2}$$ $$=$$ $$($$$$-17 \beta_{12} - 36 \beta_{11} - 8 \beta_{9} + 8 \beta_{8} + 34 \beta_{7} - 343 \beta_{6} + 78 \beta_{5} - 52 \beta_{4} + 9 \beta_{3} - 369 \beta_{2} - 7771 \beta_{1} + 9$$$$)/80$$ $$\nu^{3}$$ $$=$$ $$($$$$-1347 \beta_{13} + 589 \beta_{12} + 1347 \beta_{11} + 589 \beta_{10} - 98 \beta_{9} + 2788 \beta_{7} - 1141 \beta_{6} - 2654 \beta_{5} + 1730 \beta_{4} - 1430 \beta_{3} + 23031 \beta_{2} + 151146 \beta_{1} + 151446$$$$)/160$$ $$\nu^{4}$$ $$=$$ $$($$$$3975 \beta_{13} - 1802 \beta_{10} - 351 \beta_{9} - 351 \beta_{8} - 8407 \beta_{7} + 38099 \beta_{6} + 1367 \beta_{5} + 95 \beta_{4} + 5152 \beta_{3} - 34844 \beta_{2} - 95 \beta_{1} - 577119$$$$)/16$$ $$\nu^{5}$$ $$=$$ $$($$$$-649107 \beta_{13} - 290279 \beta_{12} - 649107 \beta_{11} + 290279 \beta_{10} + 36622 \beta_{8} + 1443134 \beta_{7} - 11517663 \beta_{6} + 1218028 \beta_{5} - 781168 \beta_{4} - 820184 \beta_{3} - 529905 \beta_{2} - 85301178 \beta_{1} + 85262162$$$$)/160$$ $$\nu^{6}$$ $$=$$ $$($$$$4550703 \beta_{12} + 10121334 \beta_{11} + 561642 \beta_{9} - 561642 \beta_{8} - 2468176 \beta_{7} + 87011117 \beta_{6} - 21064612 \beta_{5} + 12764028 \beta_{4} + 87259 \beta_{3} + 95311701 \beta_{2} + 1385752939 \beta_{1} + 87259$$$$)/80$$ $$\nu^{7}$$ $$=$$ $$($$$$322070793 \beta_{13} - 144504171 \beta_{12} - 322070793 \beta_{11} - 144504171 \beta_{10} - 27779418 \beta_{9} - 596632972 \beta_{7} + 263310439 \beta_{6} + 734365386 \beta_{5} - 407814610 \beta_{4} + 396562870 \beta_{3} - 5762721989 \beta_{2} - 43334847614 \beta_{1} - 43346099354$$$$)/160$$ $$\nu^{8}$$ $$=$$ $$($$$$-1015819605 \beta_{13} + 456147646 \beta_{10} + 49378493 \beta_{9} + 49378493 \beta_{8} + 2105820101 \beta_{7} - 9530822297 \beta_{6} - 230167781 \beta_{5} + 13998595 \beta_{4} - 1273984576 \beta_{3} + 8698986772 \beta_{2} - 13998595 \beta_{1} + 137706097181$$$$)/16$$ $$\nu^{9}$$ $$=$$ $$($$$$160815790533 \beta_{13} + 72189394841 \beta_{12} + 160815790533 \beta_{11} - 72189394841 \beta_{10} - 14855380258 \beta_{8} - 368450766226 \beta_{7} + 2882226273177 \beta_{6} - 297336337332 \beta_{5} + 198840392312 \beta_{4} + 203775166976 \beta_{3} + 131585772135 \beta_{2} + 21734308994502 \beta_{1} - 21729374219838$$$$)/160$$ $$\nu^{10}$$ $$=$$ $$($$$$-1141073051377 \beta_{12} - 2541616650546 \beta_{11} - 120090369758 \beta_{9} + 120090369758 \beta_{8} + 567954791864 \beta_{7} - 21748890401923 \beta_{6} + 5264577690148 \beta_{5} - 3184174585092 \beta_{4} - 37301571341 \beta_{3} - 23829293506979 \beta_{2} - 343928025881101 \beta_{1} - 37301571341$$$$)/80$$ $$\nu^{11}$$ $$=$$ $$($$$$-80396898736167 \beta_{13} + 36092621151669 \beta_{12} + 80396898736167 \beta_{11} + 36092621151669 \beta_{10} + 7522750781062 \beta_{9} + 148599757458868 \beta_{7} - 65796954147081 \beta_{6} - 184367396857174 \beta_{5} + 101889575298750 \beta_{4} - 99483670723130 \beta_{3} + 1441378962706091 \beta_{2} + 10871613161477026 \beta_{1} + 10874019066052646$$$$)/160$$ $$\nu^{12}$$ $$=$$ $$($$$$254232041165115 \beta_{13} - 114135193791106 \beta_{10} - 11945785122003 \beta_{9} - 11945785122003 \beta_{8} - 526520672238731 \beta_{7} + 2383259696234647 \beta_{6} + 56664374886891 \beta_{5} - 3772606785165 \beta_{4} + 318441629622336 \beta_{3} - 2175180653618252 \beta_{2} + 3772606785165 \beta_{1} - 34390883573292819$$$$)/16$$ $$\nu^{13}$$ $$=$$ $$($$$$-40202344576531227 \beta_{13} - 18048302994489639 \beta_{12} - 40202344576531227 \beta_{11} + 18048302994489639 \beta_{10} + 3770934495970462 \beta_{8} + 92208397241437614 \beta_{7} - 720803456067542983 \beta_{6} + 74302779885282508 \beta_{5} - 49753870264335208 \beta_{4} - 50951304283788064 \beta_{3} - 32903001289298425 \beta_{2} - 5438315933376697018 \beta_{1} + 5437118499357244162$$$$)/160$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$97$$ $$101$$ $$\chi(n)$$ $$-1$$ $$\beta_{1}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
63.1
 −15.8126 − 15.8126i 3.77108 + 3.77108i 2.04998 + 2.04998i 6.86993 + 6.86993i 4.57273 + 4.57273i −2.20370 − 2.20370i 2.75256 + 2.75256i −15.8126 + 15.8126i 3.77108 − 3.77108i 2.04998 − 2.04998i 6.86993 − 6.86993i 4.57273 − 4.57273i −2.20370 + 2.20370i 2.75256 − 2.75256i
0 −16.5519 + 16.5519i 0 13.9288 54.1386i 0 −2.15894 2.15894i 0 304.928i 0
63.2 0 −13.4331 + 13.4331i 0 15.0352 + 53.8418i 0 −76.4413 76.4413i 0 117.896i 0
63.3 0 −5.65790 + 5.65790i 0 −54.5514 + 12.2124i 0 23.8754 + 23.8754i 0 178.976i 0
63.4 0 1.47740 1.47740i 0 55.2474 8.52775i 0 156.265 + 156.265i 0 238.635i 0
63.5 0 9.28112 9.28112i 0 42.6946 + 36.0856i 0 −105.819 105.819i 0 70.7216i 0
63.6 0 10.9598 10.9598i 0 −14.9228 53.8731i 0 −75.2427 75.2427i 0 2.76340i 0
63.7 0 18.9245 18.9245i 0 −36.4318 + 42.3996i 0 112.521 + 112.521i 0 473.272i 0
127.1 0 −16.5519 16.5519i 0 13.9288 + 54.1386i 0 −2.15894 + 2.15894i 0 304.928i 0
127.2 0 −13.4331 13.4331i 0 15.0352 53.8418i 0 −76.4413 + 76.4413i 0 117.896i 0
127.3 0 −5.65790 5.65790i 0 −54.5514 12.2124i 0 23.8754 23.8754i 0 178.976i 0
127.4 0 1.47740 + 1.47740i 0 55.2474 + 8.52775i 0 156.265 156.265i 0 238.635i 0
127.5 0 9.28112 + 9.28112i 0 42.6946 36.0856i 0 −105.819 + 105.819i 0 70.7216i 0
127.6 0 10.9598 + 10.9598i 0 −14.9228 + 53.8731i 0 −75.2427 + 75.2427i 0 2.76340i 0
127.7 0 18.9245 + 18.9245i 0 −36.4318 42.3996i 0 112.521 112.521i 0 473.272i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 127.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 160.6.n.b yes 14
4.b odd 2 1 160.6.n.a 14
5.c odd 4 1 160.6.n.a 14
20.e even 4 1 inner 160.6.n.b yes 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.n.a 14 4.b odd 2 1
160.6.n.a 14 5.c odd 4 1
160.6.n.b yes 14 1.a even 1 1 trivial
160.6.n.b yes 14 20.e even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{14} - \cdots$$ acting on $$S_{6}^{\mathrm{new}}(160, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 10 T + 50 T^{2} - 254 T^{3} - 91369 T^{4} + 1354100 T^{5} - 8940292 T^{6} + 156709276 T^{7} + 3134936461 T^{8} - 38721912918 T^{9} + 290596015278 T^{10} - 2245195781154 T^{11} + 9414782746155 T^{12} - 473122978927848 T^{13} + 19255041373064328 T^{14} - 114968883879467064 T^{15} + 555933506377706595 T^{16} - 32216105460571098678 T^{17} +$$$$10\!\cdots\!78$$$$T^{18} -$$$$32\!\cdots\!74$$$$T^{19} +$$$$64\!\cdots\!89$$$$T^{20} +$$$$78\!\cdots\!32$$$$T^{21} -$$$$10\!\cdots\!92$$$$T^{22} +$$$$40\!\cdots\!00$$$$T^{23} -$$$$65\!\cdots\!81$$$$T^{24} -$$$$44\!\cdots\!78$$$$T^{25} +$$$$21\!\cdots\!50$$$$T^{26} -$$$$10\!\cdots\!30$$$$T^{27} +$$$$25\!\cdots\!49$$$$T^{28}$$
$5$ $$1 - 42 T + 3115 T^{2} - 77660 T^{3} - 1600075 T^{4} + 15516250 T^{5} - 16330865625 T^{6} + 925536875000 T^{7} - 51033955078125 T^{8} + 151525878906250 T^{9} - 48830413818359375 T^{10} - 7406234741210937500 T^{11} +$$$$92\!\cdots\!75$$$$T^{12} -$$$$39\!\cdots\!50$$$$T^{13} +$$$$29\!\cdots\!25$$$$T^{14}$$
$7$ $$1 - 66 T + 2178 T^{2} + 3104786 T^{3} - 390660977 T^{4} - 28598040732 T^{5} + 7558178349116 T^{6} - 730117525814980 T^{7} - 100452887411613603 T^{8} + 13091475719468548578 T^{9} +$$$$44\!\cdots\!50$$$$T^{10} -$$$$38\!\cdots\!06$$$$T^{11} +$$$$37\!\cdots\!31$$$$T^{12} +$$$$47\!\cdots\!84$$$$T^{13} -$$$$77\!\cdots\!76$$$$T^{14} +$$$$79\!\cdots\!88$$$$T^{15} +$$$$10\!\cdots\!19$$$$T^{16} -$$$$18\!\cdots\!58$$$$T^{17} +$$$$35\!\cdots\!50$$$$T^{18} +$$$$17\!\cdots\!46$$$$T^{19} -$$$$22\!\cdots\!47$$$$T^{20} -$$$$27\!\cdots\!40$$$$T^{21} +$$$$48\!\cdots\!16$$$$T^{22} -$$$$30\!\cdots\!24$$$$T^{23} -$$$$70\!\cdots\!73$$$$T^{24} +$$$$93\!\cdots\!98$$$$T^{25} +$$$$11\!\cdots\!78$$$$T^{26} -$$$$56\!\cdots\!62$$$$T^{27} +$$$$14\!\cdots\!49$$$$T^{28}$$
$11$ $$1 - 1517670 T^{2} + 1118279059571 T^{4} - 531239593921187868 T^{6} +$$$$18\!\cdots\!61$$$$T^{8} -$$$$48\!\cdots\!18$$$$T^{10} +$$$$10\!\cdots\!67$$$$T^{12} -$$$$18\!\cdots\!88$$$$T^{14} +$$$$27\!\cdots\!67$$$$T^{16} -$$$$32\!\cdots\!18$$$$T^{18} +$$$$31\!\cdots\!61$$$$T^{20} -$$$$24\!\cdots\!68$$$$T^{22} +$$$$13\!\cdots\!71$$$$T^{24} -$$$$46\!\cdots\!70$$$$T^{26} +$$$$78\!\cdots\!01$$$$T^{28}$$
$13$ $$1 + 414 T + 85698 T^{2} + 485371686 T^{3} + 407510115667 T^{4} - 25330305055892 T^{5} + 72383288599671444 T^{6} +$$$$13\!\cdots\!64$$$$T^{7} +$$$$19\!\cdots\!85$$$$T^{8} -$$$$26\!\cdots\!74$$$$T^{9} +$$$$24\!\cdots\!22$$$$T^{10} +$$$$12\!\cdots\!02$$$$T^{11} -$$$$99\!\cdots\!17$$$$T^{12} -$$$$27\!\cdots\!44$$$$T^{13} +$$$$44\!\cdots\!64$$$$T^{14} -$$$$10\!\cdots\!92$$$$T^{15} -$$$$13\!\cdots\!33$$$$T^{16} +$$$$63\!\cdots\!14$$$$T^{17} +$$$$45\!\cdots\!22$$$$T^{18} -$$$$18\!\cdots\!82$$$$T^{19} +$$$$49\!\cdots\!65$$$$T^{20} +$$$$13\!\cdots\!48$$$$T^{21} +$$$$26\!\cdots\!44$$$$T^{22} -$$$$33\!\cdots\!56$$$$T^{23} +$$$$20\!\cdots\!83$$$$T^{24} +$$$$89\!\cdots\!02$$$$T^{25} +$$$$58\!\cdots\!98$$$$T^{26} +$$$$10\!\cdots\!02$$$$T^{27} +$$$$94\!\cdots\!49$$$$T^{28}$$
$17$ $$1 - 1222 T + 746642 T^{2} - 1023656774 T^{3} - 7517818418661 T^{4} + 9805314295428420 T^{5} - 5845038493789599340 T^{6} +$$$$85\!\cdots\!80$$$$T^{7} +$$$$21\!\cdots\!05$$$$T^{8} -$$$$32\!\cdots\!70$$$$T^{9} +$$$$19\!\cdots\!70$$$$T^{10} -$$$$33\!\cdots\!50$$$$T^{11} -$$$$20\!\cdots\!37$$$$T^{12} +$$$$66\!\cdots\!24$$$$T^{13} -$$$$44\!\cdots\!04$$$$T^{14} +$$$$93\!\cdots\!68$$$$T^{15} -$$$$41\!\cdots\!13$$$$T^{16} -$$$$96\!\cdots\!50$$$$T^{17} +$$$$80\!\cdots\!70$$$$T^{18} -$$$$19\!\cdots\!90$$$$T^{19} +$$$$17\!\cdots\!45$$$$T^{20} +$$$$99\!\cdots\!40$$$$T^{21} -$$$$96\!\cdots\!40$$$$T^{22} +$$$$22\!\cdots\!40$$$$T^{23} -$$$$25\!\cdots\!89$$$$T^{24} -$$$$48\!\cdots\!82$$$$T^{25} +$$$$50\!\cdots\!42$$$$T^{26} -$$$$11\!\cdots\!54$$$$T^{27} +$$$$13\!\cdots\!49$$$$T^{28}$$
$19$ $$( 1 - 2836 T + 12192485 T^{2} - 21773970856 T^{3} + 58437600122669 T^{4} - 77435325554759820 T^{5} +$$$$17\!\cdots\!85$$$$T^{6} -$$$$20\!\cdots\!00$$$$T^{7} +$$$$43\!\cdots\!15$$$$T^{8} -$$$$47\!\cdots\!20$$$$T^{9} +$$$$88\!\cdots\!31$$$$T^{10} -$$$$81\!\cdots\!56$$$$T^{11} +$$$$11\!\cdots\!15$$$$T^{12} -$$$$65\!\cdots\!36$$$$T^{13} +$$$$57\!\cdots\!99$$$$T^{14} )^{2}$$
$23$ $$1 - 2902 T + 4210802 T^{2} + 8585545830 T^{3} - 77137172452561 T^{4} + 5298422918089420 T^{5} +$$$$34\!\cdots\!32$$$$T^{6} -$$$$17\!\cdots\!48$$$$T^{7} +$$$$43\!\cdots\!33$$$$T^{8} +$$$$14\!\cdots\!50$$$$T^{9} -$$$$10\!\cdots\!54$$$$T^{10} +$$$$57\!\cdots\!98$$$$T^{11} -$$$$19\!\cdots\!21$$$$T^{12} -$$$$51\!\cdots\!28$$$$T^{13} +$$$$13\!\cdots\!40$$$$T^{14} -$$$$33\!\cdots\!04$$$$T^{15} -$$$$79\!\cdots\!29$$$$T^{16} +$$$$15\!\cdots\!86$$$$T^{17} -$$$$18\!\cdots\!54$$$$T^{18} +$$$$15\!\cdots\!50$$$$T^{19} +$$$$30\!\cdots\!17$$$$T^{20} -$$$$81\!\cdots\!36$$$$T^{21} +$$$$10\!\cdots\!32$$$$T^{22} +$$$$10\!\cdots\!60$$$$T^{23} -$$$$94\!\cdots\!89$$$$T^{24} +$$$$67\!\cdots\!10$$$$T^{25} +$$$$21\!\cdots\!02$$$$T^{26} -$$$$94\!\cdots\!86$$$$T^{27} +$$$$20\!\cdots\!49$$$$T^{28}$$
$29$ $$1 - 127109206 T^{2} + 8400089254893139 T^{4} -$$$$38\!\cdots\!76$$$$T^{6} +$$$$13\!\cdots\!57$$$$T^{8} -$$$$39\!\cdots\!02$$$$T^{10} +$$$$98\!\cdots\!23$$$$T^{12} -$$$$21\!\cdots\!92$$$$T^{14} +$$$$41\!\cdots\!23$$$$T^{16} -$$$$69\!\cdots\!02$$$$T^{18} +$$$$10\!\cdots\!57$$$$T^{20} -$$$$12\!\cdots\!76$$$$T^{22} +$$$$11\!\cdots\!39$$$$T^{24} -$$$$70\!\cdots\!06$$$$T^{26} +$$$$23\!\cdots\!01$$$$T^{28}$$
$31$ $$1 - 265679454 T^{2} + 34933996110970539 T^{4} -$$$$30\!\cdots\!64$$$$T^{6} +$$$$19\!\cdots\!17$$$$T^{8} -$$$$92\!\cdots\!38$$$$T^{10} +$$$$36\!\cdots\!43$$$$T^{12} -$$$$11\!\cdots\!88$$$$T^{14} +$$$$29\!\cdots\!43$$$$T^{16} -$$$$62\!\cdots\!38$$$$T^{18} +$$$$10\!\cdots\!17$$$$T^{20} -$$$$13\!\cdots\!64$$$$T^{22} +$$$$12\!\cdots\!39$$$$T^{24} -$$$$80\!\cdots\!54$$$$T^{26} +$$$$24\!\cdots\!01$$$$T^{28}$$
$37$ $$1 + 1790 T + 1602050 T^{2} - 1085215470730 T^{3} - 5589943426343837 T^{4} - 5189725886098796788 T^{5} +$$$$58\!\cdots\!80$$$$T^{6} +$$$$37\!\cdots\!60$$$$T^{7} +$$$$20\!\cdots\!01$$$$T^{8} -$$$$14\!\cdots\!14$$$$T^{9} -$$$$11\!\cdots\!78$$$$T^{10} -$$$$12\!\cdots\!50$$$$T^{11} -$$$$40\!\cdots\!45$$$$T^{12} +$$$$67\!\cdots\!72$$$$T^{13} +$$$$47\!\cdots\!96$$$$T^{14} +$$$$47\!\cdots\!04$$$$T^{15} -$$$$19\!\cdots\!05$$$$T^{16} -$$$$41\!\cdots\!50$$$$T^{17} -$$$$25\!\cdots\!78$$$$T^{18} -$$$$23\!\cdots\!98$$$$T^{19} +$$$$22\!\cdots\!49$$$$T^{20} +$$$$29\!\cdots\!80$$$$T^{21} +$$$$31\!\cdots\!80$$$$T^{22} -$$$$19\!\cdots\!16$$$$T^{23} -$$$$14\!\cdots\!13$$$$T^{24} -$$$$19\!\cdots\!90$$$$T^{25} +$$$$19\!\cdots\!50$$$$T^{26} +$$$$15\!\cdots\!30$$$$T^{27} +$$$$59\!\cdots\!49$$$$T^{28}$$
$41$ $$( 1 - 5822 T + 527400583 T^{2} - 3511555483012 T^{3} + 138605008918596701 T^{4} -$$$$95\!\cdots\!50$$$$T^{5} +$$$$23\!\cdots\!95$$$$T^{6} -$$$$14\!\cdots\!20$$$$T^{7} +$$$$26\!\cdots\!95$$$$T^{8} -$$$$12\!\cdots\!50$$$$T^{9} +$$$$21\!\cdots\!01$$$$T^{10} -$$$$63\!\cdots\!12$$$$T^{11} +$$$$11\!\cdots\!83$$$$T^{12} -$$$$14\!\cdots\!22$$$$T^{13} +$$$$28\!\cdots\!01$$$$T^{14} )^{2}$$
$43$ $$1 + 3982 T + 7928162 T^{2} + 5090675647610 T^{3} + 44126507369726119 T^{4} + 95654814508201483332 T^{5} +$$$$12\!\cdots\!96$$$$T^{6} +$$$$15\!\cdots\!12$$$$T^{7} +$$$$46\!\cdots\!93$$$$T^{8} +$$$$33\!\cdots\!78$$$$T^{9} +$$$$35\!\cdots\!38$$$$T^{10} +$$$$10\!\cdots\!70$$$$T^{11} +$$$$78\!\cdots\!59$$$$T^{12} +$$$$84\!\cdots\!64$$$$T^{13} +$$$$18\!\cdots\!08$$$$T^{14} +$$$$12\!\cdots\!52$$$$T^{15} +$$$$16\!\cdots\!91$$$$T^{16} +$$$$34\!\cdots\!90$$$$T^{17} +$$$$16\!\cdots\!38$$$$T^{18} +$$$$22\!\cdots\!54$$$$T^{19} +$$$$47\!\cdots\!57$$$$T^{20} +$$$$23\!\cdots\!84$$$$T^{21} +$$$$28\!\cdots\!96$$$$T^{22} +$$$$30\!\cdots\!76$$$$T^{23} +$$$$20\!\cdots\!31$$$$T^{24} +$$$$35\!\cdots\!70$$$$T^{25} +$$$$80\!\cdots\!62$$$$T^{26} +$$$$59\!\cdots\!26$$$$T^{27} +$$$$22\!\cdots\!49$$$$T^{28}$$
$47$ $$1 + 1278 T + 816642 T^{2} + 351784473730 T^{3} + 13143771115043551 T^{4} - 55735540634425394844 T^{5} -$$$$20\!\cdots\!24$$$$T^{6} -$$$$35\!\cdots\!24$$$$T^{7} +$$$$21\!\cdots\!21$$$$T^{8} +$$$$27\!\cdots\!06$$$$T^{9} +$$$$23\!\cdots\!58$$$$T^{10} +$$$$14\!\cdots\!22$$$$T^{11} -$$$$33\!\cdots\!25$$$$T^{12} -$$$$57\!\cdots\!44$$$$T^{13} -$$$$35\!\cdots\!52$$$$T^{14} -$$$$13\!\cdots\!08$$$$T^{15} -$$$$17\!\cdots\!25$$$$T^{16} +$$$$16\!\cdots\!46$$$$T^{17} +$$$$64\!\cdots\!58$$$$T^{18} +$$$$17\!\cdots\!42$$$$T^{19} +$$$$31\!\cdots\!29$$$$T^{20} -$$$$11\!\cdots\!32$$$$T^{21} -$$$$15\!\cdots\!24$$$$T^{22} -$$$$97\!\cdots\!08$$$$T^{23} +$$$$52\!\cdots\!99$$$$T^{24} +$$$$32\!\cdots\!90$$$$T^{25} +$$$$17\!\cdots\!42$$$$T^{26} +$$$$62\!\cdots\!46$$$$T^{27} +$$$$11\!\cdots\!49$$$$T^{28}$$
$53$ $$1 - 5882 T + 17298962 T^{2} + 13844528488830 T^{3} - 174311180941592381 T^{4} -$$$$81\!\cdots\!52$$$$T^{5} +$$$$14\!\cdots\!36$$$$T^{6} -$$$$35\!\cdots\!12$$$$T^{7} -$$$$10\!\cdots\!67$$$$T^{8} +$$$$15\!\cdots\!82$$$$T^{9} +$$$$26\!\cdots\!38$$$$T^{10} -$$$$97\!\cdots\!30$$$$T^{11} +$$$$23\!\cdots\!39$$$$T^{12} +$$$$29\!\cdots\!36$$$$T^{13} -$$$$56\!\cdots\!72$$$$T^{14} +$$$$12\!\cdots\!48$$$$T^{15} +$$$$41\!\cdots\!11$$$$T^{16} -$$$$71\!\cdots\!10$$$$T^{17} +$$$$82\!\cdots\!38$$$$T^{18} +$$$$19\!\cdots\!26$$$$T^{19} -$$$$56\!\cdots\!83$$$$T^{20} -$$$$78\!\cdots\!84$$$$T^{21} +$$$$13\!\cdots\!36$$$$T^{22} -$$$$31\!\cdots\!36$$$$T^{23} -$$$$28\!\cdots\!69$$$$T^{24} +$$$$94\!\cdots\!10$$$$T^{25} +$$$$49\!\cdots\!62$$$$T^{26} -$$$$70\!\cdots\!26$$$$T^{27} +$$$$50\!\cdots\!49$$$$T^{28}$$
$59$ $$( 1 + 4252 T + 2005910877 T^{2} + 20518241068280 T^{3} + 2395790088022200413 T^{4} +$$$$22\!\cdots\!48$$$$T^{5} +$$$$23\!\cdots\!29$$$$T^{6} +$$$$16\!\cdots\!28$$$$T^{7} +$$$$16\!\cdots\!71$$$$T^{8} +$$$$11\!\cdots\!48$$$$T^{9} +$$$$87\!\cdots\!87$$$$T^{10} +$$$$53\!\cdots\!80$$$$T^{11} +$$$$37\!\cdots\!23$$$$T^{12} +$$$$56\!\cdots\!52$$$$T^{13} +$$$$95\!\cdots\!99$$$$T^{14} )^{2}$$
$61$ $$( 1 - 10282 T + 2916253203 T^{2} + 3083320715268 T^{3} + 3046127491767085861 T^{4} +$$$$70\!\cdots\!38$$$$T^{5} +$$$$15\!\cdots\!39$$$$T^{6} +$$$$10\!\cdots\!44$$$$T^{7} +$$$$12\!\cdots\!39$$$$T^{8} +$$$$50\!\cdots\!38$$$$T^{9} +$$$$18\!\cdots\!61$$$$T^{10} +$$$$15\!\cdots\!68$$$$T^{11} +$$$$12\!\cdots\!03$$$$T^{12} -$$$$37\!\cdots\!82$$$$T^{13} +$$$$30\!\cdots\!01$$$$T^{14} )^{2}$$
$67$ $$1 - 107926 T + 5824010738 T^{2} - 286091118212770 T^{3} + 7910959270847152663 T^{4} +$$$$10\!\cdots\!32$$$$T^{5} -$$$$16\!\cdots\!76$$$$T^{6} +$$$$10\!\cdots\!52$$$$T^{7} -$$$$49\!\cdots\!51$$$$T^{8} +$$$$12\!\cdots\!38$$$$T^{9} -$$$$20\!\cdots\!90$$$$T^{10} -$$$$24\!\cdots\!26$$$$T^{11} +$$$$60\!\cdots\!11$$$$T^{12} -$$$$31\!\cdots\!32$$$$T^{13} +$$$$11\!\cdots\!96$$$$T^{14} -$$$$43\!\cdots\!24$$$$T^{15} +$$$$10\!\cdots\!39$$$$T^{16} -$$$$61\!\cdots\!18$$$$T^{17} -$$$$68\!\cdots\!90$$$$T^{18} +$$$$57\!\cdots\!66$$$$T^{19} -$$$$30\!\cdots\!99$$$$T^{20} +$$$$86\!\cdots\!36$$$$T^{21} -$$$$18\!\cdots\!76$$$$T^{22} +$$$$15\!\cdots\!24$$$$T^{23} +$$$$15\!\cdots\!87$$$$T^{24} -$$$$77\!\cdots\!10$$$$T^{25} +$$$$21\!\cdots\!38$$$$T^{26} -$$$$53\!\cdots\!82$$$$T^{27} +$$$$66\!\cdots\!49$$$$T^{28}$$
$71$ $$1 - 12494857902 T^{2} + 77600272039468829243 T^{4} -$$$$32\!\cdots\!32$$$$T^{6} +$$$$99\!\cdots\!41$$$$T^{8} -$$$$25\!\cdots\!58$$$$T^{10} +$$$$55\!\cdots\!91$$$$T^{12} -$$$$10\!\cdots\!64$$$$T^{14} +$$$$17\!\cdots\!91$$$$T^{16} -$$$$26\!\cdots\!58$$$$T^{18} +$$$$34\!\cdots\!41$$$$T^{20} -$$$$35\!\cdots\!32$$$$T^{22} +$$$$28\!\cdots\!43$$$$T^{24} -$$$$14\!\cdots\!02$$$$T^{26} +$$$$38\!\cdots\!01$$$$T^{28}$$
$73$ $$1 + 16418 T + 134775362 T^{2} - 257655666175822 T^{3} - 4365114300655601877 T^{4} +$$$$39\!\cdots\!72$$$$T^{5} +$$$$40\!\cdots\!12$$$$T^{6} +$$$$72\!\cdots\!32$$$$T^{7} -$$$$10\!\cdots\!35$$$$T^{8} -$$$$45\!\cdots\!66$$$$T^{9} +$$$$74\!\cdots\!94$$$$T^{10} +$$$$14\!\cdots\!50$$$$T^{11} +$$$$46\!\cdots\!07$$$$T^{12} -$$$$13\!\cdots\!16$$$$T^{13} -$$$$14\!\cdots\!04$$$$T^{14} -$$$$27\!\cdots\!88$$$$T^{15} +$$$$20\!\cdots\!43$$$$T^{16} +$$$$12\!\cdots\!50$$$$T^{17} +$$$$13\!\cdots\!94$$$$T^{18} -$$$$17\!\cdots\!38$$$$T^{19} -$$$$80\!\cdots\!15$$$$T^{20} +$$$$11\!\cdots\!24$$$$T^{21} +$$$$13\!\cdots\!12$$$$T^{22} +$$$$28\!\cdots\!96$$$$T^{23} -$$$$63\!\cdots\!73$$$$T^{24} -$$$$78\!\cdots\!54$$$$T^{25} +$$$$84\!\cdots\!62$$$$T^{26} +$$$$21\!\cdots\!74$$$$T^{27} +$$$$27\!\cdots\!49$$$$T^{28}$$
$79$ $$( 1 + 73272 T + 14910422889 T^{2} + 713614712327088 T^{3} + 91878682859930077941 T^{4} +$$$$30\!\cdots\!92$$$$T^{5} +$$$$35\!\cdots\!29$$$$T^{6} +$$$$97\!\cdots\!08$$$$T^{7} +$$$$10\!\cdots\!71$$$$T^{8} +$$$$29\!\cdots\!92$$$$T^{9} +$$$$26\!\cdots\!59$$$$T^{10} +$$$$63\!\cdots\!88$$$$T^{11} +$$$$41\!\cdots\!11$$$$T^{12} +$$$$62\!\cdots\!72$$$$T^{13} +$$$$26\!\cdots\!99$$$$T^{14} )^{2}$$
$83$ $$1 + 36398 T + 662407202 T^{2} - 351755925637942 T^{3} - 18959358682913357257 T^{4} +$$$$10\!\cdots\!44$$$$T^{5} +$$$$11\!\cdots\!08$$$$T^{6} +$$$$60\!\cdots\!36$$$$T^{7} +$$$$11\!\cdots\!21$$$$T^{8} -$$$$15\!\cdots\!90$$$$T^{9} -$$$$10\!\cdots\!50$$$$T^{10} -$$$$86\!\cdots\!90$$$$T^{11} -$$$$70\!\cdots\!85$$$$T^{12} +$$$$38\!\cdots\!20$$$$T^{13} +$$$$31\!\cdots\!80$$$$T^{14} +$$$$15\!\cdots\!60$$$$T^{15} -$$$$10\!\cdots\!65$$$$T^{16} -$$$$52\!\cdots\!30$$$$T^{17} -$$$$25\!\cdots\!50$$$$T^{18} -$$$$14\!\cdots\!70$$$$T^{19} +$$$$43\!\cdots\!29$$$$T^{20} +$$$$88\!\cdots\!52$$$$T^{21} +$$$$64\!\cdots\!08$$$$T^{22} +$$$$23\!\cdots\!92$$$$T^{23} -$$$$17\!\cdots\!93$$$$T^{24} -$$$$12\!\cdots\!94$$$$T^{25} +$$$$92\!\cdots\!02$$$$T^{26} +$$$$20\!\cdots\!14$$$$T^{27} +$$$$21\!\cdots\!49$$$$T^{28}$$
$89$ $$1 - 30334038558 T^{2} +$$$$36\!\cdots\!35$$$$T^{4} -$$$$20\!\cdots\!52$$$$T^{6} +$$$$39\!\cdots\!69$$$$T^{8} -$$$$12\!\cdots\!10$$$$T^{10} +$$$$42\!\cdots\!95$$$$T^{12} -$$$$38\!\cdots\!60$$$$T^{14} +$$$$13\!\cdots\!95$$$$T^{16} -$$$$11\!\cdots\!10$$$$T^{18} +$$$$12\!\cdots\!69$$$$T^{20} -$$$$19\!\cdots\!52$$$$T^{22} +$$$$10\!\cdots\!35$$$$T^{24} -$$$$27\!\cdots\!58$$$$T^{26} +$$$$28\!\cdots\!01$$$$T^{28}$$
$97$ $$1 + 60314 T + 1818889298 T^{2} - 623535693824102 T^{3} -$$$$16\!\cdots\!77$$$$T^{4} -$$$$33\!\cdots\!64$$$$T^{5} +$$$$29\!\cdots\!52$$$$T^{6} +$$$$51\!\cdots\!52$$$$T^{7} +$$$$12\!\cdots\!01$$$$T^{8} -$$$$44\!\cdots\!50$$$$T^{9} +$$$$18\!\cdots\!50$$$$T^{10} +$$$$91\!\cdots\!50$$$$T^{11} +$$$$44\!\cdots\!75$$$$T^{12} -$$$$82\!\cdots\!00$$$$T^{13} -$$$$73\!\cdots\!00$$$$T^{14} -$$$$70\!\cdots\!00$$$$T^{15} +$$$$32\!\cdots\!75$$$$T^{16} +$$$$57\!\cdots\!50$$$$T^{17} +$$$$99\!\cdots\!50$$$$T^{18} -$$$$20\!\cdots\!50$$$$T^{19} +$$$$51\!\cdots\!49$$$$T^{20} +$$$$17\!\cdots\!36$$$$T^{21} +$$$$88\!\cdots\!52$$$$T^{22} -$$$$84\!\cdots\!48$$$$T^{23} -$$$$36\!\cdots\!73$$$$T^{24} -$$$$11\!\cdots\!86$$$$T^{25} +$$$$29\!\cdots\!98$$$$T^{26} +$$$$83\!\cdots\!98$$$$T^{27} +$$$$11\!\cdots\!49$$$$T^{28}$$