# Properties

 Label 300.9.k.e Level $300$ Weight $9$ Character orbit 300.k Analytic conductor $122.214$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 300.k (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$122.213583018$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 4 x^{15} - 30946474 x^{14} - 8341419336 x^{13} + 380689791299777 x^{12} + 172233390406399084 x^{11} - 2382137738311270416524 x^{10} - 1303628014982980229183568 x^{9} + 8170157809851151257627202984 x^{8} + 4722131342312473670415721362432 x^{7} - 15187994345395678935434992322703872 x^{6} - 8501142216286271666986373720765290240 x^{5} + 13719409820001055356751562088159234867776 x^{4} + 6705290345628598892954879974291445910470400 x^{3} - 4432537206418276871411586926292645523148742400 x^{2} - 1274737393430152937401280967045971687758547840000 x + 575975308456587940916620472287127145276670888000000$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{26}\cdot 3^{20}\cdot 5^{18}$$ Twist minimal: no (minimal twist has level 60) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + ( -264 - 264 \beta_{1} - 5 \beta_{3} + \beta_{7} ) q^{7} -2187 \beta_{1} q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + ( -264 - 264 \beta_{1} - 5 \beta_{3} + \beta_{7} ) q^{7} -2187 \beta_{1} q^{9} + ( 1475 + 4 \beta_{2} - 4 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{7} + 2 \beta_{8} - \beta_{11} - \beta_{12} ) q^{11} + ( 1180 - 1181 \beta_{1} - 57 \beta_{2} - \beta_{4} + 3 \beta_{5} - \beta_{6} + 2 \beta_{8} - 3 \beta_{9} + \beta_{10} - 3 \beta_{11} - \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{13} + ( 2801 + 2812 \beta_{1} - 209 \beta_{3} - 5 \beta_{4} + 5 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 5 \beta_{9} + 2 \beta_{10} - 7 \beta_{12} + 5 \beta_{13} - 4 \beta_{14} ) q^{17} + ( 16271 \beta_{1} - 785 \beta_{2} - 785 \beta_{3} + 5 \beta_{7} - 5 \beta_{8} + \beta_{10} + 3 \beta_{11} - 3 \beta_{12} + 8 \beta_{13} + 3 \beta_{14} - 2 \beta_{15} ) q^{19} + ( 10242 - 269 \beta_{2} + 269 \beta_{3} + 9 \beta_{5} - 9 \beta_{6} - 9 \beta_{11} - 9 \beta_{12} ) q^{21} + ( -12274 + 12267 \beta_{1} - 751 \beta_{2} - 8 \beta_{4} + 2 \beta_{5} - 14 \beta_{6} - 14 \beta_{8} - 2 \beta_{9} + 14 \beta_{10} - 14 \beta_{11} - 8 \beta_{13} + 14 \beta_{14} - 5 \beta_{15} ) q^{23} + 2187 \beta_{3} q^{27} + ( 61838 \beta_{1} - 3212 \beta_{2} - 3212 \beta_{3} + 24 \beta_{7} - 24 \beta_{8} - 73 \beta_{9} - 34 \beta_{10} - 42 \beta_{11} + 42 \beta_{12} - 22 \beta_{13} - 3 \beta_{14} - 31 \beta_{15} ) q^{29} + ( 233991 - 216 \beta_{2} + 216 \beta_{3} + 118 \beta_{4} - 18 \beta_{5} - 65 \beta_{6} + 26 \beta_{7} + 26 \beta_{8} + 9 \beta_{11} + 9 \beta_{12} + 44 \beta_{14} + 44 \beta_{15} ) q^{31} + ( 8355 - 8377 \beta_{1} + 1430 \beta_{2} - 52 \beta_{4} + 33 \beta_{5} - 13 \beta_{6} + 98 \beta_{8} - 33 \beta_{9} + 13 \beta_{10} - 79 \beta_{11} - 52 \beta_{13} + 13 \beta_{14} + 11 \beta_{15} ) q^{33} + ( 133653 + 134020 \beta_{1} + 11207 \beta_{3} - 79 \beta_{4} + 213 \beta_{5} + 104 \beta_{6} - 306 \beta_{7} + 213 \beta_{9} + 104 \beta_{10} - 27 \beta_{12} + 79 \beta_{13} - 50 \beta_{14} ) q^{37} + ( 120159 \beta_{1} + 1176 \beta_{2} + 1176 \beta_{3} - 117 \beta_{7} + 117 \beta_{8} + 63 \beta_{9} - 81 \beta_{10} - 36 \beta_{11} + 36 \beta_{12} - 63 \beta_{13} - 9 \beta_{14} - 72 \beta_{15} ) q^{39} + ( 1021992 + 10208 \beta_{2} - 10208 \beta_{3} + 264 \beta_{4} - 372 \beta_{5} - 46 \beta_{6} - 178 \beta_{7} - 178 \beta_{8} + 59 \beta_{11} + 59 \beta_{12} - 23 \beta_{14} - 23 \beta_{15} ) q^{41} + ( -754831 + 754877 \beta_{1} + 4264 \beta_{2} - 32 \beta_{4} + 132 \beta_{5} - 101 \beta_{6} - 326 \beta_{8} - 132 \beta_{9} + 101 \beta_{10} + 126 \beta_{11} - 32 \beta_{13} + 101 \beta_{14} + 178 \beta_{15} ) q^{43} + ( 933620 + 933687 \beta_{1} + 8949 \beta_{3} - 128 \beta_{4} - 214 \beta_{5} + 284 \beta_{6} + 668 \beta_{7} - 214 \beta_{9} + 284 \beta_{10} + 250 \beta_{12} + 128 \beta_{13} + 3 \beta_{14} ) q^{47} + ( 2110565 \beta_{1} - 4307 \beta_{2} - 4307 \beta_{3} + 86 \beta_{7} - 86 \beta_{8} - 312 \beta_{9} - 416 \beta_{10} + 423 \beta_{11} - 423 \beta_{12} + 350 \beta_{13} - 234 \beta_{14} - 182 \beta_{15} ) q^{49} + ( 480576 + 2863 \beta_{2} - 2863 \beta_{3} - 103 \beta_{4} + 177 \beta_{5} - 19 \beta_{6} + 541 \beta_{7} + 541 \beta_{8} - 200 \beta_{11} - 200 \beta_{12} + 169 \beta_{14} + 169 \beta_{15} ) q^{51} + ( -1484969 + 1485163 \beta_{1} + 37856 \beta_{2} - 264 \beta_{4} - 383 \beta_{5} + 300 \beta_{6} - 102 \beta_{8} + 383 \beta_{9} - 300 \beta_{10} + 309 \beta_{11} - 264 \beta_{13} - 300 \beta_{14} - 189 \beta_{15} ) q^{53} + ( 1720674 + 1720539 \beta_{1} - 16015 \beta_{3} - 225 \beta_{4} - 198 \beta_{5} + 54 \beta_{6} - 252 \beta_{7} - 198 \beta_{9} + 54 \beta_{10} - 288 \beta_{12} + 225 \beta_{13} - 9 \beta_{14} ) q^{57} + ( 987422 \beta_{1} - 16213 \beta_{2} - 16213 \beta_{3} + 1646 \beta_{7} - 1646 \beta_{8} + 695 \beta_{9} - 50 \beta_{10} - 79 \beta_{11} + 79 \beta_{12} - 335 \beta_{13} - 241 \beta_{14} + 191 \beta_{15} ) q^{59} + ( 5336924 + 51723 \beta_{2} - 51723 \beta_{3} - 354 \beta_{4} + 210 \beta_{5} + 24 \beta_{6} + 914 \beta_{7} + 914 \beta_{8} + 285 \beta_{11} + 285 \beta_{12} - 264 \beta_{14} - 264 \beta_{15} ) q^{61} + ( -577368 + 577368 \beta_{1} + 10935 \beta_{2} + 2187 \beta_{8} ) q^{63} + ( 6217498 + 6216504 \beta_{1} - 76476 \beta_{3} + 856 \beta_{4} + 708 \beta_{5} - 578 \beta_{6} - 3138 \beta_{7} + 708 \beta_{9} - 578 \beta_{10} + 900 \beta_{12} - 856 \beta_{13} + 1124 \beta_{14} ) q^{67} + ( 1611725 \beta_{1} - 10439 \beta_{2} - 10439 \beta_{3} - 2890 \beta_{7} + 2890 \beta_{8} + 2262 \beta_{9} + 469 \beta_{10} + 214 \beta_{11} - 214 \beta_{12} + 410 \beta_{13} + 3 \beta_{14} + 466 \beta_{15} ) q^{69} + ( 4582348 - 61219 \beta_{2} + 61219 \beta_{3} - 1346 \beta_{4} + 1234 \beta_{5} + 1018 \beta_{6} - 4010 \beta_{7} - 4010 \beta_{8} - 30 \beta_{11} - 30 \beta_{12} + 79 \beta_{14} + 79 \beta_{15} ) q^{71} + ( -7755744 + 7756640 \beta_{1} - 21420 \beta_{2} + 1172 \beta_{4} - 1278 \beta_{5} + 995 \beta_{6} - 1360 \beta_{8} + 1278 \beta_{9} - 995 \beta_{10} + 192 \beta_{11} + 1172 \beta_{13} - 995 \beta_{14} - 382 \beta_{15} ) q^{73} + ( 11621387 + 11618626 \beta_{1} - 246466 \beta_{3} + 1204 \beta_{4} - 1585 \beta_{5} - 2216 \beta_{6} + 9264 \beta_{7} - 1585 \beta_{9} - 2216 \beta_{10} - 1492 \beta_{12} - 1204 \beta_{13} - 1040 \beta_{14} ) q^{77} + ( 5549627 \beta_{1} + 16356 \beta_{2} + 16356 \beta_{3} - 5092 \beta_{7} + 5092 \beta_{8} - 7278 \beta_{9} + 3213 \beta_{10} - 1947 \beta_{11} + 1947 \beta_{12} - 2106 \beta_{13} + 2199 \beta_{14} + 1014 \beta_{15} ) q^{79} -4782969 q^{81} + ( 1379099 - 1382054 \beta_{1} + 15728 \beta_{2} + 2415 \beta_{4} + 2963 \beta_{5} - 1924 \beta_{6} - 5420 \beta_{8} - 2963 \beta_{9} + 1924 \beta_{10} - 1502 \beta_{11} + 2415 \beta_{13} + 1924 \beta_{14} + 8 \beta_{15} ) q^{83} + ( 6897621 + 6894133 \beta_{1} - 63692 \beta_{3} + 839 \beta_{4} - 4287 \beta_{5} - 634 \beta_{6} - 5005 \beta_{7} - 4287 \beta_{9} - 634 \beta_{10} + 2000 \beta_{12} - 839 \beta_{13} - 1433 \beta_{14} ) q^{87} + ( -7737022 \beta_{1} - 139326 \beta_{2} - 139326 \beta_{3} + 13498 \beta_{7} - 13498 \beta_{8} + 7586 \beta_{9} + 3446 \beta_{10} + 227 \beta_{11} - 227 \beta_{12} + 2512 \beta_{13} + 2253 \beta_{14} + 1193 \beta_{15} ) q^{89} + ( 10679666 + 416656 \beta_{2} - 416656 \beta_{3} - 2394 \beta_{4} - 3618 \beta_{5} + 3492 \beta_{6} + 8414 \beta_{7} + 8414 \beta_{8} + 2778 \beta_{11} + 2778 \beta_{12} - 4914 \beta_{14} - 4914 \beta_{15} ) q^{91} + ( -626004 + 620226 \beta_{1} + 234295 \beta_{2} + 801 \beta_{4} + 2367 \beta_{5} + 459 \beta_{6} + 2232 \beta_{8} - 2367 \beta_{9} - 459 \beta_{10} + 3060 \beta_{11} + 801 \beta_{13} - 459 \beta_{14} - 3411 \beta_{15} ) q^{93} + ( -11583571 - 11574687 \beta_{1} + 137706 \beta_{3} - 370 \beta_{4} + 10248 \beta_{5} - 1276 \beta_{6} - 8252 \beta_{7} + 10248 \beta_{9} - 1276 \beta_{10} - 9870 \beta_{12} + 370 \beta_{13} + 88 \beta_{14} ) q^{97} + ( -3223638 \beta_{1} + 8748 \beta_{2} + 8748 \beta_{3} - 4374 \beta_{7} + 4374 \beta_{8} + 2187 \beta_{9} - 2187 \beta_{11} + 2187 \beta_{12} - 2187 \beta_{13} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 4220q^{7} + O(q^{10})$$ $$16q - 4220q^{7} + 23616q^{11} + 18900q^{13} + 44940q^{17} + 163944q^{21} - 196440q^{23} + 3742624q^{31} + 134460q^{33} + 2141100q^{37} + 16347000q^{41} - 12080280q^{43} + 14942400q^{47} + 7693704q^{51} - 23760300q^{53} + 27530280q^{57} + 85401912q^{61} - 9229140q^{63} + 99451240q^{67} + 73302480q^{71} - 124097320q^{73} + 185945400q^{77} - 76527504q^{81} + 22058160q^{83} + 110300940q^{87} + 170997360q^{91} - 9969480q^{93} - 185269800q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{15} - 30946474 x^{14} - 8341419336 x^{13} + 380689791299777 x^{12} + 172233390406399084 x^{11} - 2382137738311270416524 x^{10} - 1303628014982980229183568 x^{9} + 8170157809851151257627202984 x^{8} + 4722131342312473670415721362432 x^{7} - 15187994345395678935434992322703872 x^{6} - 8501142216286271666986373720765290240 x^{5} + 13719409820001055356751562088159234867776 x^{4} + 6705290345628598892954879974291445910470400 x^{3} - 4432537206418276871411586926292645523148742400 x^{2} - 1274737393430152937401280967045971687758547840000 x + 575975308456587940916620472287127145276670888000000$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$10\!\cdots\!83$$$$\nu^{15} -$$$$30\!\cdots\!55$$$$\nu^{14} -$$$$27\!\cdots\!62$$$$\nu^{13} -$$$$25\!\cdots\!16$$$$\nu^{12} +$$$$28\!\cdots\!87$$$$\nu^{11} +$$$$10\!\cdots\!25$$$$\nu^{10} -$$$$13\!\cdots\!92$$$$\nu^{9} -$$$$76\!\cdots\!42$$$$\nu^{8} +$$$$33\!\cdots\!24$$$$\nu^{7} +$$$$21\!\cdots\!12$$$$\nu^{6} -$$$$36\!\cdots\!48$$$$\nu^{5} -$$$$24\!\cdots\!32$$$$\nu^{4} +$$$$12\!\cdots\!00$$$$\nu^{3} +$$$$69\!\cdots\!00$$$$\nu^{2} -$$$$17\!\cdots\!00$$$$\nu -$$$$37\!\cdots\!00$$$$)/$$$$13\!\cdots\!00$$ $$\beta_{2}$$ $$=$$ $$($$$$13\!\cdots\!89$$$$\nu^{15} +$$$$39\!\cdots\!10$$$$\nu^{14} -$$$$38\!\cdots\!46$$$$\nu^{13} -$$$$12\!\cdots\!28$$$$\nu^{12} +$$$$37\!\cdots\!21$$$$\nu^{11} +$$$$13\!\cdots\!50$$$$\nu^{10} -$$$$14\!\cdots\!36$$$$\nu^{9} -$$$$68\!\cdots\!36$$$$\nu^{8} +$$$$16\!\cdots\!92$$$$\nu^{7} +$$$$16\!\cdots\!96$$$$\nu^{6} +$$$$31\!\cdots\!16$$$$\nu^{5} -$$$$17\!\cdots\!56$$$$\nu^{4} -$$$$70\!\cdots\!00$$$$\nu^{3} +$$$$52\!\cdots\!00$$$$\nu^{2} +$$$$15\!\cdots\!00$$$$\nu -$$$$66\!\cdots\!00$$$$)/$$$$16\!\cdots\!00$$ $$\beta_{3}$$ $$=$$ $$($$$$14\!\cdots\!11$$$$\nu^{15} +$$$$38\!\cdots\!40$$$$\nu^{14} -$$$$40\!\cdots\!54$$$$\nu^{13} -$$$$11\!\cdots\!72$$$$\nu^{12} +$$$$39\!\cdots\!79$$$$\nu^{11} +$$$$13\!\cdots\!00$$$$\nu^{10} -$$$$15\!\cdots\!64$$$$\nu^{9} -$$$$67\!\cdots\!64$$$$\nu^{8} +$$$$19\!\cdots\!08$$$$\nu^{7} +$$$$16\!\cdots\!04$$$$\nu^{6} +$$$$27\!\cdots\!84$$$$\nu^{5} -$$$$17\!\cdots\!44$$$$\nu^{4} -$$$$68\!\cdots\!00$$$$\nu^{3} +$$$$51\!\cdots\!00$$$$\nu^{2} +$$$$15\!\cdots\!00$$$$\nu -$$$$65\!\cdots\!00$$$$)/$$$$16\!\cdots\!00$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$88\!\cdots\!51$$$$\nu^{15} -$$$$82\!\cdots\!40$$$$\nu^{14} +$$$$29\!\cdots\!14$$$$\nu^{13} +$$$$22\!\cdots\!77$$$$\nu^{12} -$$$$29\!\cdots\!39$$$$\nu^{11} -$$$$23\!\cdots\!50$$$$\nu^{10} +$$$$93\!\cdots\!24$$$$\nu^{9} +$$$$11\!\cdots\!49$$$$\nu^{8} +$$$$53\!\cdots\!72$$$$\nu^{7} -$$$$28\!\cdots\!64$$$$\nu^{6} -$$$$77\!\cdots\!44$$$$\nu^{5} +$$$$30\!\cdots\!04$$$$\nu^{4} +$$$$11\!\cdots\!00$$$$\nu^{3} -$$$$98\!\cdots\!00$$$$\nu^{2} -$$$$23\!\cdots\!00$$$$\nu +$$$$12\!\cdots\!00$$$$)/$$$$25\!\cdots\!00$$ $$\beta_{5}$$ $$=$$ $$($$$$17\!\cdots\!38$$$$\nu^{15} +$$$$40\!\cdots\!95$$$$\nu^{14} -$$$$46\!\cdots\!82$$$$\nu^{13} -$$$$29\!\cdots\!01$$$$\nu^{12} +$$$$47\!\cdots\!82$$$$\nu^{11} +$$$$45\!\cdots\!25$$$$\nu^{10} -$$$$21\!\cdots\!62$$$$\nu^{9} -$$$$26\!\cdots\!87$$$$\nu^{8} +$$$$47\!\cdots\!64$$$$\nu^{7} +$$$$67\!\cdots\!32$$$$\nu^{6} -$$$$37\!\cdots\!28$$$$\nu^{5} -$$$$73\!\cdots\!52$$$$\nu^{4} -$$$$27\!\cdots\!00$$$$\nu^{3} +$$$$21\!\cdots\!00$$$$\nu^{2} +$$$$21\!\cdots\!00$$$$\nu -$$$$20\!\cdots\!00$$$$)/$$$$25\!\cdots\!00$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$29\!\cdots\!93$$$$\nu^{15} -$$$$38\!\cdots\!15$$$$\nu^{14} +$$$$79\!\cdots\!72$$$$\nu^{13} +$$$$42\!\cdots\!46$$$$\nu^{12} -$$$$80\!\cdots\!97$$$$\nu^{11} -$$$$68\!\cdots\!35$$$$\nu^{10} +$$$$37\!\cdots\!42$$$$\nu^{9} +$$$$40\!\cdots\!52$$$$\nu^{8} -$$$$82\!\cdots\!44$$$$\nu^{7} -$$$$10\!\cdots\!72$$$$\nu^{6} +$$$$69\!\cdots\!48$$$$\nu^{5} +$$$$11\!\cdots\!52$$$$\nu^{4} -$$$$18\!\cdots\!00$$$$\nu^{3} -$$$$33\!\cdots\!00$$$$\nu^{2} -$$$$15\!\cdots\!00$$$$\nu +$$$$33\!\cdots\!00$$$$)/$$$$20\!\cdots\!00$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$28\!\cdots\!49$$$$\nu^{15} +$$$$48\!\cdots\!40$$$$\nu^{14} +$$$$49\!\cdots\!86$$$$\nu^{13} -$$$$12\!\cdots\!52$$$$\nu^{12} -$$$$57\!\cdots\!61$$$$\nu^{11} +$$$$12\!\cdots\!00$$$$\nu^{10} +$$$$56\!\cdots\!76$$$$\nu^{9} -$$$$61\!\cdots\!24$$$$\nu^{8} -$$$$29\!\cdots\!72$$$$\nu^{7} +$$$$14\!\cdots\!64$$$$\nu^{6} +$$$$73\!\cdots\!44$$$$\nu^{5} -$$$$15\!\cdots\!04$$$$\nu^{4} -$$$$74\!\cdots\!00$$$$\nu^{3} +$$$$53\!\cdots\!00$$$$\nu^{2} +$$$$15\!\cdots\!00$$$$\nu -$$$$71\!\cdots\!00$$$$)/$$$$16\!\cdots\!00$$ $$\beta_{8}$$ $$=$$ $$($$$$57\!\cdots\!23$$$$\nu^{15} -$$$$97\!\cdots\!30$$$$\nu^{14} -$$$$98\!\cdots\!22$$$$\nu^{13} +$$$$25\!\cdots\!04$$$$\nu^{12} +$$$$11\!\cdots\!47$$$$\nu^{11} -$$$$25\!\cdots\!50$$$$\nu^{10} -$$$$11\!\cdots\!52$$$$\nu^{9} +$$$$12\!\cdots\!48$$$$\nu^{8} +$$$$59\!\cdots\!44$$$$\nu^{7} -$$$$29\!\cdots\!28$$$$\nu^{6} -$$$$14\!\cdots\!88$$$$\nu^{5} +$$$$31\!\cdots\!08$$$$\nu^{4} +$$$$14\!\cdots\!00$$$$\nu^{3} -$$$$10\!\cdots\!00$$$$\nu^{2} -$$$$30\!\cdots\!00$$$$\nu +$$$$14\!\cdots\!00$$$$)/$$$$32\!\cdots\!00$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$28\!\cdots\!83$$$$\nu^{15} +$$$$13\!\cdots\!55$$$$\nu^{14} +$$$$67\!\cdots\!62$$$$\nu^{13} -$$$$31\!\cdots\!84$$$$\nu^{12} -$$$$65\!\cdots\!87$$$$\nu^{11} +$$$$31\!\cdots\!75$$$$\nu^{10} +$$$$34\!\cdots\!92$$$$\nu^{9} -$$$$14\!\cdots\!58$$$$\nu^{8} -$$$$10\!\cdots\!24$$$$\nu^{7} +$$$$37\!\cdots\!88$$$$\nu^{6} +$$$$19\!\cdots\!48$$$$\nu^{5} -$$$$42\!\cdots\!68$$$$\nu^{4} -$$$$16\!\cdots\!00$$$$\nu^{3} +$$$$17\!\cdots\!00$$$$\nu^{2} +$$$$36\!\cdots\!00$$$$\nu -$$$$23\!\cdots\!00$$$$)/$$$$51\!\cdots\!00$$ $$\beta_{10}$$ $$=$$ $$($$$$-$$$$18\!\cdots\!99$$$$\nu^{15} +$$$$86\!\cdots\!90$$$$\nu^{14} +$$$$42\!\cdots\!86$$$$\nu^{13} -$$$$20\!\cdots\!52$$$$\nu^{12} -$$$$42\!\cdots\!11$$$$\nu^{11} +$$$$18\!\cdots\!50$$$$\nu^{10} +$$$$23\!\cdots\!76$$$$\nu^{9} -$$$$84\!\cdots\!24$$$$\nu^{8} -$$$$74\!\cdots\!72$$$$\nu^{7} +$$$$19\!\cdots\!64$$$$\nu^{6} +$$$$12\!\cdots\!44$$$$\nu^{5} -$$$$22\!\cdots\!04$$$$\nu^{4} -$$$$10\!\cdots\!00$$$$\nu^{3} +$$$$11\!\cdots\!00$$$$\nu^{2} +$$$$27\!\cdots\!00$$$$\nu -$$$$18\!\cdots\!00$$$$)/$$$$20\!\cdots\!00$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$12\!\cdots\!99$$$$\nu^{15} -$$$$25\!\cdots\!10$$$$\nu^{14} +$$$$36\!\cdots\!86$$$$\nu^{13} +$$$$80\!\cdots\!48$$$$\nu^{12} -$$$$37\!\cdots\!11$$$$\nu^{11} -$$$$92\!\cdots\!50$$$$\nu^{10} +$$$$17\!\cdots\!76$$$$\nu^{9} +$$$$48\!\cdots\!76$$$$\nu^{8} -$$$$34\!\cdots\!72$$$$\nu^{7} -$$$$12\!\cdots\!36$$$$\nu^{6} +$$$$15\!\cdots\!44$$$$\nu^{5} +$$$$12\!\cdots\!96$$$$\nu^{4} +$$$$20\!\cdots\!00$$$$\nu^{3} -$$$$31\!\cdots\!00$$$$\nu^{2} -$$$$32\!\cdots\!00$$$$\nu +$$$$21\!\cdots\!00$$$$)/$$$$98\!\cdots\!00$$ $$\beta_{12}$$ $$=$$ $$($$$$13\!\cdots\!57$$$$\nu^{15} +$$$$28\!\cdots\!80$$$$\nu^{14} -$$$$38\!\cdots\!98$$$$\nu^{13} -$$$$89\!\cdots\!64$$$$\nu^{12} +$$$$40\!\cdots\!73$$$$\nu^{11} +$$$$10\!\cdots\!00$$$$\nu^{10} -$$$$18\!\cdots\!68$$$$\nu^{9} -$$$$53\!\cdots\!68$$$$\nu^{8} +$$$$35\!\cdots\!96$$$$\nu^{7} +$$$$13\!\cdots\!48$$$$\nu^{6} -$$$$14\!\cdots\!92$$$$\nu^{5} -$$$$13\!\cdots\!28$$$$\nu^{4} -$$$$24\!\cdots\!00$$$$\nu^{3} +$$$$35\!\cdots\!00$$$$\nu^{2} +$$$$41\!\cdots\!00$$$$\nu -$$$$26\!\cdots\!00$$$$)/$$$$98\!\cdots\!00$$ $$\beta_{13}$$ $$=$$ $$($$$$35\!\cdots\!47$$$$\nu^{15} -$$$$59\!\cdots\!45$$$$\nu^{14} -$$$$93\!\cdots\!58$$$$\nu^{13} +$$$$11\!\cdots\!56$$$$\nu^{12} +$$$$95\!\cdots\!83$$$$\nu^{11} -$$$$90\!\cdots\!25$$$$\nu^{10} -$$$$47\!\cdots\!28$$$$\nu^{9} +$$$$34\!\cdots\!22$$$$\nu^{8} +$$$$12\!\cdots\!16$$$$\nu^{7} -$$$$69\!\cdots\!92$$$$\nu^{6} -$$$$14\!\cdots\!32$$$$\nu^{5} +$$$$76\!\cdots\!12$$$$\nu^{4} +$$$$64\!\cdots\!00$$$$\nu^{3} -$$$$42\!\cdots\!00$$$$\nu^{2} -$$$$11\!\cdots\!00$$$$\nu +$$$$67\!\cdots\!00$$$$)/$$$$10\!\cdots\!00$$ $$\beta_{14}$$ $$=$$ $$($$$$81\!\cdots\!23$$$$\nu^{15} -$$$$43\!\cdots\!80$$$$\nu^{14} -$$$$21\!\cdots\!22$$$$\nu^{13} +$$$$20\!\cdots\!04$$$$\nu^{12} +$$$$21\!\cdots\!47$$$$\nu^{11} +$$$$53\!\cdots\!00$$$$\nu^{10} -$$$$99\!\cdots\!52$$$$\nu^{9} -$$$$48\!\cdots\!52$$$$\nu^{8} +$$$$22\!\cdots\!44$$$$\nu^{7} +$$$$14\!\cdots\!72$$$$\nu^{6} -$$$$20\!\cdots\!88$$$$\nu^{5} -$$$$17\!\cdots\!92$$$$\nu^{4} +$$$$34\!\cdots\!00$$$$\nu^{3} +$$$$50\!\cdots\!00$$$$\nu^{2} +$$$$59\!\cdots\!00$$$$\nu -$$$$47\!\cdots\!00$$$$)/$$$$20\!\cdots\!00$$ $$\beta_{15}$$ $$=$$ $$($$$$-$$$$82\!\cdots\!29$$$$\nu^{15} +$$$$43\!\cdots\!90$$$$\nu^{14} +$$$$21\!\cdots\!06$$$$\nu^{13} -$$$$18\!\cdots\!92$$$$\nu^{12} -$$$$21\!\cdots\!81$$$$\nu^{11} -$$$$55\!\cdots\!50$$$$\nu^{10} +$$$$10\!\cdots\!96$$$$\nu^{9} +$$$$50\!\cdots\!96$$$$\nu^{8} -$$$$22\!\cdots\!12$$$$\nu^{7} -$$$$15\!\cdots\!56$$$$\nu^{6} +$$$$20\!\cdots\!24$$$$\nu^{5} +$$$$18\!\cdots\!16$$$$\nu^{4} -$$$$35\!\cdots\!00$$$$\nu^{3} -$$$$51\!\cdots\!00$$$$\nu^{2} -$$$$63\!\cdots\!00$$$$\nu +$$$$48\!\cdots\!00$$$$)/$$$$20\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{8} + \beta_{7} - 5 \beta_{3} + 5 \beta_{2} + 2 \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$182 \beta_{15} + 182 \beta_{14} - 423 \beta_{12} - 423 \beta_{11} + 612 \beta_{8} + 616 \beta_{7} - 416 \beta_{6} - 312 \beta_{5} - 350 \beta_{4} - 6957 \beta_{3} + 6937 \beta_{2} + 7736648$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-122304 \beta_{15} - 122460 \beta_{14} + 1050 \beta_{13} - 434283 \beta_{12} - 431745 \beta_{11} - 1248 \beta_{10} - 936 \beta_{9} + 6319326 \beta_{8} + 6323010 \beta_{7} - 686022 \beta_{6} - 649116 \beta_{5} - 180180 \beta_{4} - 111940931 \beta_{3} + 111899249 \beta_{2} + 23207920 \beta_{1} + 3171620690$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$1581974898 \beta_{15} + 1578256746 \beta_{14} + 720720 \beta_{13} - 4371133965 \beta_{12} - 4367669853 \beta_{11} - 2744088 \beta_{10} - 2596464 \beta_{9} + 8362070498 \beta_{8} + 8412639858 \beta_{7} - 4584516018 \beta_{6} - 2024420316 \beta_{5} - 2868382230 \beta_{4} - 296881435855 \beta_{3} + 295986075055 \beta_{2} + 12684860360 \beta_{1} + 49091605596266$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-553114356830 \beta_{15} - 560222059300 \beta_{14} + 14341918150 \beta_{13} - 7933449354825 \beta_{12} - 7889755318815 \beta_{11} - 22922588410 \beta_{10} - 10122107820 \beta_{9} + 50318087359510 \beta_{8} + 50401960935850 \beta_{7} - 9584255630180 \beta_{6} - 6184563553980 \beta_{5} - 4322319824900 \beta_{4} - 1317490003229187 \beta_{3} + 1314525665396757 \beta_{2} + 245432245720562 \beta_{1} + 55110853069709528$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$12465155101708470 \beta_{15} + 12401107047779550 \beta_{14} + 25933926156600 \beta_{13} - 42248016756725643 \beta_{12} - 42153077494042683 \beta_{11} - 57505561221960 \beta_{10} - 37107407288520 \beta_{9} + 101308684769915598 \beta_{8} + 101913005565381390 \beta_{7} - 45912882233997798 \beta_{6} - 17237120246424612 \beta_{5} - 25291808566459050 \beta_{4} - 3852387056713212537 \beta_{3} + 3836594953747848713 \beta_{2} + 330634680368503208 \beta_{1} + 387195121776623134302$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$3360350739761114022 \beta_{15} + 3213426838590261804 \beta_{14} + 177042860752087050 \beta_{13} - 99540149815743369711 \beta_{12} - 98949341544271439913 \beta_{11} - 321390496554245622 \beta_{10} - 120659983434499236 \beta_{9} + 456197436616720802978 \beta_{8} + 457619989623298019422 \beta_{7} - 113213427027517965588 \beta_{6} - 59925449541719421948 \beta_{5} - 57241436500778944260 \beta_{4} - 14053938630048276133357 \beta_{3} + 14000115714474318272523 \beta_{2} + 2710074162272535300366 \beta_{1} + 717907804241079216208696$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$100888426644207867689794 \beta_{15} + 100037879377023156785826 \beta_{14} + 457931976106213384160 \beta_{13} - 414699431792991213742389 \beta_{12} - 413111514089911395856629 \beta_{11} - 905708489657388980576 \beta_{10} - 479404289005455029280 \beta_{9} + 1146727399028578524102290 \beta_{8} + 1154037949719155411299218 \beta_{7} - 457086138078636360944586 \beta_{6} - 169449362643595846095516 \beta_{5} - 240675948869808442598790 \beta_{4} - 42719171344691150328817079 \beta_{3} + 42494738615149014126641271 \beta_{2} + 5742734040835541176169056 \beta_{1} + 3469776983347456518428417394$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$101707166765096391227254722 \beta_{15} + 99409876126812553961780028 \beta_{14} + 2166087788857897810512390 \beta_{13} - 1123495485595075110379305393 \beta_{12} - 1116045172902727535327054343 \beta_{11} - 4113782956081184948497098 \beta_{10} - 1525047159632645248606620 \beta_{9} + 4422760114169446456400985006 \beta_{8} + 4443467036449453855977594834 \beta_{7} - 1255789018780199289717664644 \beta_{6} - 597973389890611725453534012 \beta_{5} - 654834940196921335188866340 \beta_{4} - 146468018129588486148258625587 \beta_{3} + 145701091648199751689663165829 \beta_{2} + 31224716359529105898250035602 \beta_{1} + 8351924563014709643528946872936$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$868295582189427794234515588710 \beta_{15} + 857789974743680513714157294030 \beta_{14} + 6548363139931401138250357800 \beta_{13} - 4155407722646011069110615010635 \beta_{12} - 4133012268423491317060485833115 \beta_{11} - 12557917359063123242581031400 \beta_{10} - 5979748281038943448633401240 \beta_{9} + 12489135603661969721132483453310 \beta_{8} + 12577798094484747125508159524990 \beta_{7} - 4603509338076924271080340619430 \beta_{6} - 1752582665242427607242012657700 \beta_{5} - 2391190636493015843630058088650 \beta_{4} - 455014389367911493518377754899545 \beta_{3} + 452092691537149956160093835499177 \beta_{2} + 83511385870047966571229270816936 \beta_{1} + 33358337033079224660375451819429598$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$14\!\cdots\!10$$$$\beta_{15} +$$$$14\!\cdots\!44$$$$\beta_{14} +$$$$26\!\cdots\!50$$$$\beta_{13} -$$$$12\!\cdots\!75$$$$\beta_{12} -$$$$12\!\cdots\!33$$$$\beta_{11} -$$$$50\!\cdots\!86$$$$\beta_{10} -$$$$19\!\cdots\!04$$$$\beta_{9} +$$$$44\!\cdots\!26$$$$\beta_{8} +$$$$44\!\cdots\!62$$$$\beta_{7} -$$$$13\!\cdots\!84$$$$\beta_{6} -$$$$60\!\cdots\!96$$$$\beta_{5} -$$$$71\!\cdots\!00$$$$\beta_{4} -$$$$15\!\cdots\!17$$$$\beta_{3} +$$$$15\!\cdots\!55$$$$\beta_{2} +$$$$36\!\cdots\!42$$$$\beta_{1} +$$$$92\!\cdots\!36$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$($$$$79\!\cdots\!58$$$$\beta_{15} +$$$$78\!\cdots\!50$$$$\beta_{14} +$$$$85\!\cdots\!20$$$$\beta_{13} -$$$$42\!\cdots\!57$$$$\beta_{12} -$$$$41\!\cdots\!73$$$$\beta_{11} -$$$$16\!\cdots\!64$$$$\beta_{10} -$$$$72\!\cdots\!28$$$$\beta_{9} +$$$$13\!\cdots\!86$$$$\beta_{8} +$$$$13\!\cdots\!10$$$$\beta_{7} -$$$$46\!\cdots\!54$$$$\beta_{6} -$$$$18\!\cdots\!60$$$$\beta_{5} -$$$$24\!\cdots\!30$$$$\beta_{4} -$$$$47\!\cdots\!59$$$$\beta_{3} +$$$$47\!\cdots\!99$$$$\beta_{2} +$$$$11\!\cdots\!72$$$$\beta_{1} +$$$$33\!\cdots\!38$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$17\!\cdots\!86$$$$\beta_{15} +$$$$17\!\cdots\!32$$$$\beta_{14} +$$$$31\!\cdots\!70$$$$\beta_{13} -$$$$12\!\cdots\!17$$$$\beta_{12} -$$$$12\!\cdots\!03$$$$\beta_{11} -$$$$60\!\cdots\!06$$$$\beta_{10} -$$$$23\!\cdots\!28$$$$\beta_{9} +$$$$45\!\cdots\!18$$$$\beta_{8} +$$$$45\!\cdots\!46$$$$\beta_{7} -$$$$14\!\cdots\!48$$$$\beta_{6} -$$$$62\!\cdots\!80$$$$\beta_{5} -$$$$75\!\cdots\!20$$$$\beta_{4} -$$$$15\!\cdots\!91$$$$\beta_{3} +$$$$15\!\cdots\!09$$$$\beta_{2} +$$$$43\!\cdots\!22$$$$\beta_{1} +$$$$98\!\cdots\!28$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$76\!\cdots\!98$$$$\beta_{15} +$$$$74\!\cdots\!54$$$$\beta_{14} +$$$$10\!\cdots\!40$$$$\beta_{13} -$$$$43\!\cdots\!31$$$$\beta_{12} -$$$$43\!\cdots\!35$$$$\beta_{11} -$$$$20\!\cdots\!80$$$$\beta_{10} -$$$$87\!\cdots\!52$$$$\beta_{9} +$$$$14\!\cdots\!26$$$$\beta_{8} +$$$$14\!\cdots\!22$$$$\beta_{7} -$$$$48\!\cdots\!42$$$$\beta_{6} -$$$$19\!\cdots\!52$$$$\beta_{5} -$$$$24\!\cdots\!70$$$$\beta_{4} -$$$$50\!\cdots\!69$$$$\beta_{3} +$$$$49\!\cdots\!73$$$$\beta_{2} +$$$$13\!\cdots\!84$$$$\beta_{1} +$$$$33\!\cdots\!50$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$20\!\cdots\!10$$$$\beta_{15} +$$$$19\!\cdots\!40$$$$\beta_{14} +$$$$37\!\cdots\!50$$$$\beta_{13} -$$$$13\!\cdots\!55$$$$\beta_{12} -$$$$13\!\cdots\!05$$$$\beta_{11} -$$$$72\!\cdots\!90$$$$\beta_{10} -$$$$28\!\cdots\!00$$$$\beta_{9} +$$$$46\!\cdots\!70$$$$\beta_{8} +$$$$46\!\cdots\!70$$$$\beta_{7} -$$$$15\!\cdots\!60$$$$\beta_{6} -$$$$64\!\cdots\!20$$$$\beta_{5} -$$$$79\!\cdots\!00$$$$\beta_{4} -$$$$16\!\cdots\!81$$$$\beta_{3} +$$$$16\!\cdots\!95$$$$\beta_{2} +$$$$50\!\cdots\!98$$$$\beta_{1} +$$$$10\!\cdots\!00$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
157.1
 −2402.08 + 1.00000i −1675.46 + 1.00000i 1685.41 + 1.00000i 1773.40 + 1.00000i −2315.27 + 1.00000i −681.621 + 1.00000i 391.920 + 1.00000i 3225.69 + 1.00000i −2402.08 − 1.00000i −1675.46 − 1.00000i 1685.41 − 1.00000i 1773.40 − 1.00000i −2315.27 − 1.00000i −681.621 − 1.00000i 391.920 − 1.00000i 3225.69 − 1.00000i
0 −33.0681 + 33.0681i 0 0 0 −2666.08 2666.08i 0 2187.00i 0
157.2 0 −33.0681 + 33.0681i 0 0 0 −1939.46 1939.46i 0 2187.00i 0
157.3 0 −33.0681 + 33.0681i 0 0 0 1421.41 + 1421.41i 0 2187.00i 0
157.4 0 −33.0681 + 33.0681i 0 0 0 1509.40 + 1509.40i 0 2187.00i 0
157.5 0 33.0681 33.0681i 0 0 0 −2579.27 2579.27i 0 2187.00i 0
157.6 0 33.0681 33.0681i 0 0 0 −945.621 945.621i 0 2187.00i 0
157.7 0 33.0681 33.0681i 0 0 0 127.920 + 127.920i 0 2187.00i 0
157.8 0 33.0681 33.0681i 0 0 0 2961.69 + 2961.69i 0 2187.00i 0
193.1 0 −33.0681 33.0681i 0 0 0 −2666.08 + 2666.08i 0 2187.00i 0
193.2 0 −33.0681 33.0681i 0 0 0 −1939.46 + 1939.46i 0 2187.00i 0
193.3 0 −33.0681 33.0681i 0 0 0 1421.41 1421.41i 0 2187.00i 0
193.4 0 −33.0681 33.0681i 0 0 0 1509.40 1509.40i 0 2187.00i 0
193.5 0 33.0681 + 33.0681i 0 0 0 −2579.27 + 2579.27i 0 2187.00i 0
193.6 0 33.0681 + 33.0681i 0 0 0 −945.621 + 945.621i 0 2187.00i 0
193.7 0 33.0681 + 33.0681i 0 0 0 127.920 127.920i 0 2187.00i 0
193.8 0 33.0681 + 33.0681i 0 0 0 2961.69 2961.69i 0 2187.00i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 193.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.9.k.e 16
5.b even 2 1 60.9.k.a 16
5.c odd 4 1 60.9.k.a 16
5.c odd 4 1 inner 300.9.k.e 16
15.d odd 2 1 180.9.l.c 16
15.e even 4 1 180.9.l.c 16
20.d odd 2 1 240.9.bg.c 16
20.e even 4 1 240.9.bg.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.9.k.a 16 5.b even 2 1
60.9.k.a 16 5.c odd 4 1
180.9.l.c 16 15.d odd 2 1
180.9.l.c 16 15.e even 4 1
240.9.bg.c 16 20.d odd 2 1
240.9.bg.c 16 20.e even 4 1
300.9.k.e 16 1.a even 1 1 trivial
300.9.k.e 16 5.c odd 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$33\!\cdots\!76$$$$T_{7}^{12} +$$$$14\!\cdots\!40$$$$T_{7}^{11} +$$$$31\!\cdots\!00$$$$T_{7}^{10} -$$$$19\!\cdots\!20$$$$T_{7}^{9} +$$$$12\!\cdots\!16$$$$T_{7}^{8} +$$$$50\!\cdots\!60$$$$T_{7}^{7} +$$$$11\!\cdots\!00$$$$T_{7}^{6} -$$$$99\!\cdots\!80$$$$T_{7}^{5} +$$$$18\!\cdots\!36$$$$T_{7}^{4} +$$$$47\!\cdots\!80$$$$T_{7}^{3} +$$$$69\!\cdots\!00$$$$T_{7}^{2} -$$$$19\!\cdots\!40$$$$T_{7} +$$$$26\!\cdots\!96$$">$$T_{7}^{16} + \cdots$$ acting on $$S_{9}^{\mathrm{new}}(300, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$( 4782969 + T^{4} )^{4}$$
$5$ $$T^{16}$$
$7$ $$26\!\cdots\!96$$$$-$$$$19\!\cdots\!40$$$$T +$$$$69\!\cdots\!00$$$$T^{2} +$$$$47\!\cdots\!80$$$$T^{3} +$$$$18\!\cdots\!36$$$$T^{4} -$$$$99\!\cdots\!80$$$$T^{5} +$$$$11\!\cdots\!00$$$$T^{6} +$$$$50\!\cdots\!60$$$$T^{7} +$$$$12\!\cdots\!16$$$$T^{8} -$$$$19\!\cdots\!20$$$$T^{9} +$$$$31\!\cdots\!00$$$$T^{10} + 1466083349566961440 T^{11} + 337814088968876 T^{12} - 3419547960 T^{13} + 8904200 T^{14} + 4220 T^{15} + T^{16}$$
$11$ $$($$$$28\!\cdots\!76$$$$+$$$$54\!\cdots\!92$$$$T -$$$$34\!\cdots\!72$$$$T^{2} -$$$$62\!\cdots\!56$$$$T^{3} + 78407113061855820 T^{4} + 9407276642544 T^{5} - 828889522 T^{6} - 11808 T^{7} + T^{8} )^{2}$$
$13$ $$13\!\cdots\!00$$$$+$$$$27\!\cdots\!00$$$$T +$$$$27\!\cdots\!00$$$$T^{2} -$$$$26\!\cdots\!00$$$$T^{3} +$$$$12\!\cdots\!00$$$$T^{4} -$$$$32\!\cdots\!00$$$$T^{5} +$$$$84\!\cdots\!00$$$$T^{6} -$$$$78\!\cdots\!00$$$$T^{7} +$$$$36\!\cdots\!00$$$$T^{8} -$$$$24\!\cdots\!00$$$$T^{9} +$$$$43\!\cdots\!00$$$$T^{10} -$$$$65\!\cdots\!00$$$$T^{11} + 4569427860557872500 T^{12} + 5493339999000 T^{13} + 178605000 T^{14} - 18900 T^{15} + T^{16}$$
$17$ $$66\!\cdots\!36$$$$+$$$$47\!\cdots\!80$$$$T +$$$$16\!\cdots\!00$$$$T^{2} +$$$$19\!\cdots\!40$$$$T^{3} +$$$$12\!\cdots\!16$$$$T^{4} +$$$$13\!\cdots\!60$$$$T^{5} +$$$$70\!\cdots\!00$$$$T^{6} -$$$$45\!\cdots\!20$$$$T^{7} +$$$$17\!\cdots\!36$$$$T^{8} +$$$$24\!\cdots\!40$$$$T^{9} +$$$$98\!\cdots\!00$$$$T^{10} -$$$$70\!\cdots\!80$$$$T^{11} +$$$$32\!\cdots\!36$$$$T^{12} + 466494350184120 T^{13} + 1009801800 T^{14} - 44940 T^{15} + T^{16}$$
$19$ $$16\!\cdots\!00$$$$+$$$$83\!\cdots\!00$$$$T^{2} +$$$$15\!\cdots\!00$$$$T^{4} +$$$$14\!\cdots\!00$$$$T^{6} +$$$$67\!\cdots\!00$$$$T^{8} +$$$$15\!\cdots\!00$$$$T^{10} +$$$$16\!\cdots\!00$$$$T^{12} + 74925194600 T^{14} + T^{16}$$
$23$ $$38\!\cdots\!96$$$$+$$$$88\!\cdots\!80$$$$T +$$$$10\!\cdots\!00$$$$T^{2} +$$$$37\!\cdots\!60$$$$T^{3} +$$$$70\!\cdots\!36$$$$T^{4} +$$$$16\!\cdots\!60$$$$T^{5} -$$$$14\!\cdots\!00$$$$T^{6} -$$$$34\!\cdots\!80$$$$T^{7} +$$$$10\!\cdots\!16$$$$T^{8} -$$$$11\!\cdots\!60$$$$T^{9} +$$$$51\!\cdots\!00$$$$T^{10} +$$$$44\!\cdots\!80$$$$T^{11} +$$$$60\!\cdots\!76$$$$T^{12} - 40178973026491680 T^{13} + 19294336800 T^{14} + 196440 T^{15} + T^{16}$$
$29$ $$11\!\cdots\!56$$$$+$$$$12\!\cdots\!32$$$$T^{2} +$$$$78\!\cdots\!08$$$$T^{4} +$$$$28\!\cdots\!44$$$$T^{6} +$$$$20\!\cdots\!20$$$$T^{8} +$$$$60\!\cdots\!64$$$$T^{10} +$$$$78\!\cdots\!88$$$$T^{12} + 4625041638812 T^{14} + T^{16}$$
$31$ $$( -$$$$35\!\cdots\!44$$$$+$$$$11\!\cdots\!68$$$$T +$$$$10\!\cdots\!08$$$$T^{2} -$$$$55\!\cdots\!44$$$$T^{3} +$$$$87\!\cdots\!20$$$$T^{4} + 6337851820155065536 T^{5} - 2743810029512 T^{6} - 1871312 T^{7} + T^{8} )^{2}$$
$37$ $$20\!\cdots\!36$$$$-$$$$57\!\cdots\!00$$$$T +$$$$80\!\cdots\!00$$$$T^{2} +$$$$12\!\cdots\!00$$$$T^{3} +$$$$78\!\cdots\!16$$$$T^{4} +$$$$17\!\cdots\!00$$$$T^{5} +$$$$25\!\cdots\!00$$$$T^{6} +$$$$49\!\cdots\!00$$$$T^{7} +$$$$24\!\cdots\!36$$$$T^{8} +$$$$76\!\cdots\!00$$$$T^{9} +$$$$10\!\cdots\!00$$$$T^{10} -$$$$72\!\cdots\!00$$$$T^{11} +$$$$85\!\cdots\!36$$$$T^{12} + 16759728505785177000 T^{13} + 2292154605000 T^{14} - 2141100 T^{15} + T^{16}$$
$41$ $$($$$$67\!\cdots\!00$$$$-$$$$10\!\cdots\!00$$$$T -$$$$22\!\cdots\!00$$$$T^{2} +$$$$61\!\cdots\!00$$$$T^{3} -$$$$49\!\cdots\!00$$$$T^{4} +$$$$14\!\cdots\!00$$$$T^{5} + 2231531398100 T^{6} - 8173500 T^{7} + T^{8} )^{2}$$
$43$ $$25\!\cdots\!56$$$$-$$$$38\!\cdots\!40$$$$T +$$$$28\!\cdots\!00$$$$T^{2} -$$$$72\!\cdots\!80$$$$T^{3} +$$$$17\!\cdots\!56$$$$T^{4} +$$$$45\!\cdots\!20$$$$T^{5} +$$$$77\!\cdots\!00$$$$T^{6} -$$$$14\!\cdots\!60$$$$T^{7} -$$$$33\!\cdots\!04$$$$T^{8} +$$$$19\!\cdots\!80$$$$T^{9} +$$$$11\!\cdots\!00$$$$T^{10} +$$$$34\!\cdots\!60$$$$T^{11} +$$$$66\!\cdots\!16$$$$T^{12} +$$$$15\!\cdots\!40$$$$T^{13} + 72966582439200 T^{14} + 12080280 T^{15} + T^{16}$$
$47$ $$50\!\cdots\!96$$$$+$$$$17\!\cdots\!00$$$$T +$$$$31\!\cdots\!00$$$$T^{2} -$$$$36\!\cdots\!00$$$$T^{3} +$$$$20\!\cdots\!36$$$$T^{4} -$$$$56\!\cdots\!00$$$$T^{5} +$$$$12\!\cdots\!00$$$$T^{6} -$$$$13\!\cdots\!00$$$$T^{7} +$$$$71\!\cdots\!16$$$$T^{8} -$$$$23\!\cdots\!00$$$$T^{9} +$$$$49\!\cdots\!00$$$$T^{10} -$$$$64\!\cdots\!00$$$$T^{11} +$$$$58\!\cdots\!76$$$$T^{12} -$$$$58\!\cdots\!00$$$$T^{13} + 111637658880000 T^{14} - 14942400 T^{15} + T^{16}$$
$53$ $$68\!\cdots\!36$$$$-$$$$19\!\cdots\!00$$$$T +$$$$28\!\cdots\!00$$$$T^{2} +$$$$15\!\cdots\!00$$$$T^{3} +$$$$24\!\cdots\!16$$$$T^{4} +$$$$21\!\cdots\!00$$$$T^{5} +$$$$12\!\cdots\!00$$$$T^{6} +$$$$51\!\cdots\!00$$$$T^{7} +$$$$15\!\cdots\!36$$$$T^{8} +$$$$30\!\cdots\!00$$$$T^{9} +$$$$43\!\cdots\!00$$$$T^{10} +$$$$40\!\cdots\!00$$$$T^{11} +$$$$25\!\cdots\!36$$$$T^{12} +$$$$19\!\cdots\!00$$$$T^{13} + 282275928045000 T^{14} + 23760300 T^{15} + T^{16}$$
$59$ $$22\!\cdots\!56$$$$+$$$$12\!\cdots\!32$$$$T^{2} +$$$$44\!\cdots\!08$$$$T^{4} +$$$$37\!\cdots\!44$$$$T^{6} +$$$$13\!\cdots\!20$$$$T^{8} +$$$$23\!\cdots\!64$$$$T^{10} +$$$$19\!\cdots\!88$$$$T^{12} + 757587401147012 T^{14} + T^{16}$$
$61$ $$($$$$82\!\cdots\!76$$$$-$$$$76\!\cdots\!44$$$$T +$$$$13\!\cdots\!72$$$$T^{2} +$$$$19\!\cdots\!08$$$$T^{3} -$$$$61\!\cdots\!80$$$$T^{4} +$$$$18\!\cdots\!92$$$$T^{5} + 489871424599572 T^{6} - 42700956 T^{7} + T^{8} )^{2}$$
$67$ $$14\!\cdots\!16$$$$-$$$$13\!\cdots\!20$$$$T +$$$$64\!\cdots\!00$$$$T^{2} -$$$$12\!\cdots\!60$$$$T^{3} +$$$$13\!\cdots\!76$$$$T^{4} -$$$$89\!\cdots\!40$$$$T^{5} +$$$$42\!\cdots\!00$$$$T^{6} -$$$$23\!\cdots\!20$$$$T^{7} +$$$$18\!\cdots\!76$$$$T^{8} -$$$$11\!\cdots\!60$$$$T^{9} +$$$$46\!\cdots\!00$$$$T^{10} -$$$$10\!\cdots\!80$$$$T^{11} +$$$$28\!\cdots\!56$$$$T^{12} -$$$$12\!\cdots\!80$$$$T^{13} + 4945274568768800 T^{14} - 99451240 T^{15} + T^{16}$$
$71$ $$( -$$$$31\!\cdots\!00$$$$+$$$$57\!\cdots\!00$$$$T +$$$$48\!\cdots\!00$$$$T^{2} -$$$$12\!\cdots\!00$$$$T^{3} +$$$$17\!\cdots\!00$$$$T^{4} +$$$$61\!\cdots\!00$$$$T^{5} - 1691518641343600 T^{6} - 36651240 T^{7} + T^{8} )^{2}$$
$73$ $$35\!\cdots\!16$$$$+$$$$14\!\cdots\!60$$$$T +$$$$28\!\cdots\!00$$$$T^{2} +$$$$33\!\cdots\!80$$$$T^{3} +$$$$24\!\cdots\!76$$$$T^{4} +$$$$12\!\cdots\!20$$$$T^{5} +$$$$48\!\cdots\!00$$$$T^{6} +$$$$20\!\cdots\!60$$$$T^{7} +$$$$11\!\cdots\!76$$$$T^{8} +$$$$51\!\cdots\!80$$$$T^{9} +$$$$16\!\cdots\!00$$$$T^{10} +$$$$32\!\cdots\!40$$$$T^{11} +$$$$73\!\cdots\!56$$$$T^{12} +$$$$25\!\cdots\!40$$$$T^{13} + 7700072415591200 T^{14} + 124097320 T^{15} + T^{16}$$
$79$ $$28\!\cdots\!36$$$$+$$$$26\!\cdots\!28$$$$T^{2} +$$$$90\!\cdots\!88$$$$T^{4} +$$$$14\!\cdots\!56$$$$T^{6} +$$$$12\!\cdots\!20$$$$T^{8} +$$$$58\!\cdots\!16$$$$T^{10} +$$$$14\!\cdots\!48$$$$T^{12} + 19448961959977968 T^{14} + T^{16}$$
$83$ $$44\!\cdots\!16$$$$-$$$$68\!\cdots\!20$$$$T +$$$$51\!\cdots\!00$$$$T^{2} -$$$$19\!\cdots\!40$$$$T^{3} +$$$$40\!\cdots\!76$$$$T^{4} -$$$$71\!\cdots\!40$$$$T^{5} +$$$$39\!\cdots\!00$$$$T^{6} -$$$$14\!\cdots\!80$$$$T^{7} +$$$$29\!\cdots\!76$$$$T^{8} -$$$$22\!\cdots\!60$$$$T^{9} +$$$$51\!\cdots\!00$$$$T^{10} -$$$$20\!\cdots\!20$$$$T^{11} +$$$$45\!\cdots\!56$$$$T^{12} -$$$$18\!\cdots\!80$$$$T^{13} + 243281211292800 T^{14} - 22058160 T^{15} + T^{16}$$
$89$ $$10\!\cdots\!00$$$$+$$$$67\!\cdots\!00$$$$T^{2} +$$$$98\!\cdots\!00$$$$T^{4} +$$$$65\!\cdots\!00$$$$T^{6} +$$$$23\!\cdots\!00$$$$T^{8} +$$$$49\!\cdots\!00$$$$T^{10} +$$$$60\!\cdots\!00$$$$T^{12} + 38354286008864600 T^{14} + T^{16}$$
$97$ $$36\!\cdots\!36$$$$+$$$$14\!\cdots\!00$$$$T +$$$$27\!\cdots\!00$$$$T^{2} +$$$$28\!\cdots\!00$$$$T^{3} +$$$$17\!\cdots\!16$$$$T^{4} +$$$$52\!\cdots\!00$$$$T^{5} +$$$$92\!\cdots\!00$$$$T^{6} +$$$$94\!\cdots\!00$$$$T^{7} +$$$$63\!\cdots\!36$$$$T^{8} +$$$$50\!\cdots\!00$$$$T^{9} +$$$$79\!\cdots\!00$$$$T^{10} +$$$$83\!\cdots\!00$$$$T^{11} +$$$$47\!\cdots\!36$$$$T^{12} +$$$$11\!\cdots\!00$$$$T^{13} + 17162449396020000 T^{14} + 185269800 T^{15} + T^{16}$$