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 $2$

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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:

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

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