Properties

Label 10.19.c.b
Level 10
Weight 19
Character orbit 10.c
Analytic conductor 20.539
Analytic rank 0
Dimension 10
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 19 \)
Character orbit: \([\chi]\) = 10.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(20.5386137710\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{8}\cdot 5^{17} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -256 - 256 \beta_{1} ) q^{2} + ( 323 - 323 \beta_{1} - \beta_{3} ) q^{3} + 131072 \beta_{1} q^{4} + ( -143613 - 34238 \beta_{1} - 26 \beta_{2} - 7 \beta_{3} + \beta_{5} ) q^{5} + ( -165376 + 256 \beta_{2} + 256 \beta_{3} ) q^{6} + ( -7182853 - 7182853 \beta_{1} - 518 \beta_{2} + \beta_{3} + 4 \beta_{4} - \beta_{6} - \beta_{7} + \beta_{9} ) q^{7} + ( 33554432 - 33554432 \beta_{1} ) q^{8} + ( -122787829 \beta_{1} - 1860 \beta_{2} + 1854 \beta_{3} - 28 \beta_{4} - 40 \beta_{5} - 9 \beta_{6} - 4 \beta_{7} + 11 \beta_{8} ) q^{9} +O(q^{10})\) \( q + ( -256 - 256 \beta_{1} ) q^{2} + ( 323 - 323 \beta_{1} - \beta_{3} ) q^{3} + 131072 \beta_{1} q^{4} + ( -143613 - 34238 \beta_{1} - 26 \beta_{2} - 7 \beta_{3} + \beta_{5} ) q^{5} + ( -165376 + 256 \beta_{2} + 256 \beta_{3} ) q^{6} + ( -7182853 - 7182853 \beta_{1} - 518 \beta_{2} + \beta_{3} + 4 \beta_{4} - \beta_{6} - \beta_{7} + \beta_{9} ) q^{7} + ( 33554432 - 33554432 \beta_{1} ) q^{8} + ( -122787829 \beta_{1} - 1860 \beta_{2} + 1854 \beta_{3} - 28 \beta_{4} - 40 \beta_{5} - 9 \beta_{6} - 4 \beta_{7} + 11 \beta_{8} ) q^{9} + ( 28000000 + 45529856 \beta_{1} + 8448 \beta_{2} - 4864 \beta_{3} + 256 \beta_{4} - 256 \beta_{5} ) q^{10} + ( -468461738 + 17633 \beta_{2} + 17467 \beta_{3} + 116 \beta_{4} + 926 \beta_{5} + 61 \beta_{6} - 19 \beta_{7} + 81 \beta_{8} + 43 \beta_{9} ) q^{11} + ( 42336256 + 42336256 \beta_{1} - 131072 \beta_{2} ) q^{12} + ( -1185852867 + 1185852867 \beta_{1} - 347 \beta_{2} - 259245 \beta_{3} + 935 \beta_{4} - 2493 \beta_{5} - 23 \beta_{6} - 122 \beta_{7} + 38 \beta_{8} - 225 \beta_{9} ) q^{13} + ( 3677620736 \beta_{1} + 132352 \beta_{2} - 132864 \beta_{3} - 1024 \beta_{4} - 1024 \beta_{5} + 512 \beta_{6} - 256 \beta_{8} - 256 \beta_{9} ) q^{14} + ( 13258021187 - 3507630115 \beta_{1} - 225337 \beta_{2} + 389196 \beta_{3} + 1938 \beta_{4} + 408 \beta_{5} + 240 \beta_{6} + 1320 \beta_{7} + 270 \beta_{8} - 765 \beta_{9} ) q^{15} -17179869184 q^{16} + ( 5742509677 + 5742509677 \beta_{1} + 496394 \beta_{2} - 730 \beta_{3} - 13906 \beta_{4} - 12238 \beta_{5} - 1269 \beta_{6} - 2451 \beta_{7} + 1999 \beta_{8} + 2903 \beta_{9} ) q^{17} + ( -31433684224 + 31433684224 \beta_{1} + 1536 \beta_{2} - 950784 \beta_{3} - 3072 \beta_{4} + 17408 \beta_{5} + 3328 \beta_{6} - 1280 \beta_{7} - 2816 \beta_{8} + 2816 \beta_{9} ) q^{18} + ( -104693582720 \beta_{1} + 2696184 \beta_{2} - 2671126 \beta_{3} + 85882 \beta_{4} + 37524 \beta_{5} - 10323 \beta_{6} + 4519 \beta_{7} + 1135 \beta_{8} + 11351 \beta_{9} ) q^{19} + ( 4487643136 - 18823643136 \beta_{1} - 917504 \beta_{2} + 3407872 \beta_{3} - 131072 \beta_{4} ) q^{20} + ( 259994989502 + 26903956 \beta_{2} + 26897764 \beta_{3} - 354180 \beta_{4} - 186024 \beta_{5} - 9072 \beta_{6} + 19947 \beta_{7} - 10365 \beta_{8} + 5682 \beta_{9} ) q^{21} + ( 119926204928 + 119926204928 \beta_{1} - 8985600 \beta_{2} + 42496 \beta_{3} + 207360 \beta_{4} - 266752 \beta_{5} - 10752 \beta_{6} + 20480 \beta_{7} - 31744 \beta_{8} + 9728 \beta_{9} ) q^{22} + ( -240984937997 + 240984937997 \beta_{1} + 43083 \beta_{2} - 40659794 \beta_{3} - 518126 \beta_{4} + 457694 \beta_{5} + 5679 \beta_{6} + 7355 \beta_{7} - 38033 \beta_{8} + 35728 \beta_{9} ) q^{23} + ( -21676163072 \beta_{1} + 33554432 \beta_{2} - 33554432 \beta_{3} ) q^{24} + ( -962870126875 + 1255801096440 \beta_{1} - 3805300 \beta_{2} + 818880 \beta_{3} + 38530 \beta_{4} - 201060 \beta_{5} + 33165 \beta_{6} + 85220 \beta_{7} - 60205 \beta_{8} + 101560 \beta_{9} ) q^{25} + ( 607156667904 + 66455552 \beta_{2} + 66277888 \beta_{3} - 877568 \beta_{4} + 398848 \beta_{5} + 37120 \beta_{6} + 25344 \beta_{7} + 47872 \beta_{8} + 67328 \beta_{9} ) q^{26} + ( 1031946809720 + 1031946809720 \beta_{1} + 36915096 \beta_{2} - 96360 \beta_{3} - 1346798 \beta_{4} - 2666514 \beta_{5} + 50488 \beta_{6} + 87842 \beta_{7} + 45872 \beta_{8} + 115964 \beta_{9} ) q^{27} + ( 941470908416 - 941470908416 \beta_{1} + 131072 \beta_{2} + 67895296 \beta_{3} + 524288 \beta_{5} - 131072 \beta_{6} + 131072 \beta_{7} + 131072 \beta_{8} ) q^{28} + ( -4070056481000 \beta_{1} + 216841962 \beta_{2} - 216447532 \beta_{3} + 2723622 \beta_{4} - 1252004 \beta_{5} + 62735 \beta_{6} + 164473 \beta_{7} - 59223 \beta_{8} + 233469 \beta_{9} ) q^{29} + ( -4292006733312 - 2496100114432 \beta_{1} - 41947904 \beta_{2} - 157320448 \beta_{3} - 391680 \beta_{4} - 600576 \beta_{5} - 399360 \beta_{6} - 276480 \beta_{7} + 126720 \beta_{8} + 264960 \beta_{9} ) q^{30} + ( 4407689167222 + 32984561 \beta_{2} + 33171103 \beta_{3} - 6569396 \beta_{4} - 3774044 \beta_{5} - 91760 \beta_{6} - 221682 \beta_{7} - 403691 \beta_{8} + 530591 \beta_{9} ) q^{31} + ( 4398046511104 + 4398046511104 \beta_{1} ) q^{32} + ( -9102820772554 + 9102820772554 \beta_{1} + 1651068 \beta_{2} + 1616891408 \beta_{3} - 1425912 \beta_{4} + 10029936 \beta_{5} - 254949 \beta_{6} + 834417 \beta_{7} + 360051 \beta_{8} + 816651 \beta_{9} ) q^{33} + ( -2940164954624 \beta_{1} - 126889984 \beta_{2} + 127263744 \beta_{3} + 427008 \beta_{4} + 6692864 \beta_{5} + 952320 \beta_{6} + 302592 \beta_{7} - 1254912 \beta_{8} - 231424 \beta_{9} ) q^{34} + ( -9268024280165 + 7456112731883 \beta_{1} + 2195462439 \beta_{2} - 2953348242 \beta_{3} - 6590772 \beta_{4} - 3226558 \beta_{5} + 705505 \beta_{6} + 542465 \beta_{7} - 354135 \beta_{8} - 1033555 \beta_{9} ) q^{35} + ( 16094046322688 + 243007488 \beta_{2} + 243793920 \beta_{3} + 5242880 \beta_{4} - 3670016 \beta_{5} - 524288 \beta_{6} + 1179648 \beta_{7} - 1441792 \beta_{9} ) q^{36} + ( 28512654807177 + 28512654807177 \beta_{1} - 4627305679 \beta_{2} + 652497 \beta_{3} + 4225541 \beta_{4} - 1817041 \beta_{5} - 66005 \beta_{6} + 426936 \beta_{7} - 586492 \beta_{8} - 121097 \beta_{9} ) q^{37} + ( -26801557176320 + 26801557176320 \beta_{1} - 6414848 \beta_{2} + 1374031360 \beta_{3} - 12379648 \beta_{4} - 31591936 \beta_{5} + 1485824 \beta_{6} - 3799552 \beta_{7} - 3196416 \beta_{8} - 2615296 \beta_{9} ) q^{38} + ( -131608362398838 \beta_{1} + 230097549 \beta_{2} - 233038119 \beta_{3} - 18250020 \beta_{4} - 11411700 \beta_{5} - 4626510 \beta_{6} - 2144280 \beta_{7} + 5086755 \beta_{8} - 336045 \beta_{9} ) q^{39} + ( -5967689285632 + 3670016000000 \beta_{1} - 637534208 \beta_{2} - 1107296256 \beta_{3} + 33554432 \beta_{4} + 33554432 \beta_{5} ) q^{40} + ( 22300124545582 + 5482767114 \beta_{2} + 5479783420 \beta_{3} + 88908400 \beta_{4} + 66972192 \beta_{5} + 3390366 \beta_{6} - 3799761 \beta_{7} + 4879490 \beta_{8} - 1486401 \beta_{9} ) q^{41} + ( -66558717312512 - 66558717312512 \beta_{1} - 13773240320 \beta_{2} + 1585152 \beta_{3} + 43047936 \beta_{4} + 138292224 \beta_{5} - 2784000 \beta_{6} - 7428864 \beta_{7} + 1198848 \beta_{8} - 4108032 \beta_{9} ) q^{42} + ( -67409866380237 + 67409866380237 \beta_{1} - 9403414 \beta_{2} + 7401429881 \beta_{3} + 135040336 \beta_{4} - 98624168 \beta_{5} + 3081682 \beta_{6} - 3727346 \beta_{7} + 6333462 \beta_{8} - 5676068 \beta_{9} ) q^{43} + ( -61402216923136 \beta_{1} + 2289434624 \beta_{2} - 2311192576 \beta_{3} - 121372672 \beta_{4} + 15204352 \beta_{5} - 2490368 \beta_{6} - 7995392 \beta_{7} + 5636096 \beta_{8} - 10616832 \beta_{9} ) q^{44} + ( 131319200677798 + 137690729010597 \beta_{1} - 5970941397 \beta_{2} - 29635114764 \beta_{3} + 17777419 \beta_{4} - 44883880 \beta_{5} + 10137245 \beta_{6} - 5079840 \beta_{7} + 9413760 \beta_{8} - 6428945 \beta_{9} ) q^{45} + ( 123384288254464 + 10397878016 \beta_{2} + 10419936512 \beta_{3} + 249809920 \beta_{4} + 15470592 \beta_{5} - 3336704 \beta_{6} - 429056 \beta_{7} + 590080 \beta_{8} - 18882816 \beta_{9} ) q^{46} + ( -333224284498413 - 333224284498413 \beta_{1} - 50001844372 \beta_{2} - 26688393 \beta_{3} - 168503722 \beta_{4} - 44817850 \beta_{5} + 13513693 \beta_{6} + 4754943 \beta_{7} + 13174700 \beta_{8} - 4681793 \beta_{9} ) q^{47} + ( -5549097746432 + 5549097746432 \beta_{1} + 17179869184 \beta_{3} ) q^{48} + ( 577895670954829 \beta_{1} + 40371851112 \beta_{2} - 40345313374 \beta_{3} + 330403324 \beta_{4} - 223323360 \beta_{5} + 43026477 \beta_{6} + 24524752 \beta_{7} - 27760553 \beta_{8} + 17278910 \beta_{9} ) q^{49} + ( 567979833168640 - 74990328208640 \beta_{1} + 764523520 \beta_{2} - 1183790080 \beta_{3} - 61335040 \beta_{4} + 41607680 \beta_{5} - 30306560 \beta_{6} - 13326080 \beta_{7} - 10586880 \beta_{8} - 41411840 \beta_{9} ) q^{50} + ( -249985331581898 + 55393021235 \beta_{2} + 55361139635 \beta_{3} - 911173008 \beta_{4} - 306894204 \beta_{5} - 4660806 \beta_{6} + 5357850 \beta_{7} - 24565368 \beta_{8} + 55749924 \beta_{9} ) q^{51} + ( -155432106983424 - 155432106983424 \beta_{1} - 33979760640 \beta_{2} + 45481984 \beta_{3} + 326762496 \beta_{4} + 122552320 \beta_{5} - 15990784 \beta_{6} + 3014656 \beta_{7} - 29491200 \beta_{8} - 4980736 \beta_{9} ) q^{52} + ( -758744494373167 + 758744494373167 \beta_{1} + 124491405 \beta_{2} + 16057045323 \beta_{3} - 498760369 \beta_{4} + 842523363 \beta_{5} - 44162252 \beta_{6} + 71092343 \beta_{7} + 18154401 \beta_{8} + 53399062 \beta_{9} ) q^{53} + ( -528356766576640 \beta_{1} - 9425596416 \beta_{2} + 9474932736 \beta_{3} - 337847296 \beta_{4} + 1027407872 \beta_{5} - 35412480 \beta_{6} - 9562624 \beta_{7} - 41430016 \beta_{8} - 17943552 \beta_{9} ) q^{54} + ( 2789715961827534 + 251240460028024 \beta_{1} + 52464654853 \beta_{2} - 53705424939 \beta_{3} - 141427340 \beta_{4} - 206430818 \beta_{5} - 35674355 \beta_{6} + 63418235 \beta_{7} - 21181040 \beta_{8} + 223062030 \beta_{9} ) q^{55} + ( -482033105108992 - 17414750208 \beta_{2} - 17347641344 \beta_{3} + 134217728 \beta_{4} - 134217728 \beta_{5} - 67108864 \beta_{7} - 33554432 \beta_{8} + 33554432 \beta_{9} ) q^{56} + ( -1398908900924920 - 1398908900924920 \beta_{1} + 195538308878 \beta_{2} + 33125382 \beta_{3} - 683007756 \beta_{4} - 2970962412 \beta_{5} - 14693448 \beta_{6} + 79551324 \beta_{7} - 18431934 \beta_{8} + 166319124 \beta_{9} ) q^{57} + ( -1041934459136000 + 1041934459136000 \beta_{1} - 100974080 \beta_{2} + 110922110464 \beta_{3} - 1017760256 \beta_{4} - 376734208 \beta_{5} - 58165248 \beta_{6} - 26044928 \beta_{7} - 44606976 \beta_{8} - 74929152 \beta_{9} ) q^{58} + ( -3372598217174000 \beta_{1} - 31800511448 \beta_{2} + 31914525806 \beta_{3} + 1218599194 \beta_{4} - 1708510340 \beta_{5} - 102863883 \beta_{6} + 31758067 \beta_{7} + 161738497 \beta_{8} + 141130905 \beta_{9} ) q^{59} + ( 459752094433280 + 1737755353022464 \beta_{1} + 51012698112 \beta_{2} + 29535371264 \beta_{3} - 53477376 \beta_{4} + 254017536 \beta_{5} + 173015040 \beta_{6} - 31457280 \beta_{7} - 100270080 \beta_{8} - 35389440 \beta_{9} ) q^{60} + ( 6470478021108342 - 116960951728 \beta_{2} - 117545859830 \beta_{3} - 2534337168 \beta_{4} + 874444182 \beta_{5} + 36828877 \beta_{6} + 391463872 \beta_{7} + 209496452 \beta_{8} - 52881099 \beta_{9} ) q^{61} + ( -1128368426808832 - 1128368426808832 \beta_{1} - 16935849984 \beta_{2} - 47754752 \beta_{3} + 715610112 \beta_{4} + 2647920640 \beta_{5} + 80241152 \beta_{6} + 33260032 \beta_{7} - 32486400 \beta_{8} - 239176192 \beta_{9} ) q^{62} + ( -10867432394447797 + 10867432394447797 \beta_{1} + 95849943 \beta_{2} - 346187442678 \beta_{3} + 2182782714 \beta_{4} + 827804054 \beta_{5} + 485696239 \beta_{6} - 201602765 \beta_{7} - 228321233 \beta_{8} + 297452708 \beta_{9} ) q^{63} -2251799813685248 \beta_{1} q^{64} + ( 7222890655353417 + 7332698026740195 \beta_{1} - 78088421332 \beta_{2} + 144432077936 \beta_{3} - 1248485342 \beta_{4} - 1102751152 \beta_{5} - 17887670 \beta_{6} + 112636565 \beta_{7} + 292303090 \beta_{8} - 589461255 \beta_{9} ) q^{65} + ( 4660644235547648 - 414346873856 \beta_{2} - 413501527040 \beta_{3} + 2932697088 \beta_{4} - 2202630144 \beta_{5} - 148343808 \beta_{6} - 278877696 \beta_{7} - 301235712 \beta_{8} - 116889600 \beta_{9} ) q^{66} + ( -10840390609781413 - 10840390609781413 \beta_{1} + 731733274305 \beta_{2} + 54359022 \beta_{3} + 391735548 \beta_{4} + 2568862780 \beta_{5} - 214714982 \beta_{6} - 405494162 \beta_{7} + 160355960 \beta_{8} + 153868542 \beta_{9} ) q^{67} + ( -752682228383744 + 752682228383744 \beta_{1} - 95682560 \beta_{2} - 65063354368 \beta_{3} + 1604059136 \beta_{4} - 1822687232 \beta_{5} - 321257472 \beta_{6} + 166330368 \beta_{7} + 380502016 \beta_{8} - 262012928 \beta_{9} ) q^{68} + ( -20543274265065598 \beta_{1} - 947973413216 \beta_{2} + 947564220944 \beta_{3} - 2853079584 \beta_{4} + 3040422996 \beta_{5} + 356546637 \beta_{6} - 72231180 \beta_{7} - 388229586 \beta_{8} - 368644041 \beta_{9} ) q^{69} + ( 4281379075084288 + 463849356360192 \beta_{1} + 194018765568 \beta_{2} + 1318095534336 \beta_{3} + 861238784 \beta_{4} + 2513236480 \beta_{5} - 319480320 \beta_{6} + 41738240 \beta_{7} + 355248640 \beta_{8} + 173931520 \beta_{9} ) q^{70} + ( 19438042455117542 - 77881626663 \beta_{2} - 76805757241 \beta_{3} + 11537365164 \beta_{4} + 2527967476 \beta_{5} + 136096440 \beta_{6} - 1069316002 \beta_{7} - 123091971 \beta_{8} - 19557889 \beta_{9} ) q^{71} + ( -4120075858608128 - 4120075858608128 \beta_{1} - 124621160448 \beta_{2} - 201326592 \beta_{3} - 2281701376 \beta_{4} - 402653184 \beta_{5} - 167772160 \beta_{6} - 436207616 \beta_{7} + 369098752 \beta_{8} + 369098752 \beta_{9} ) q^{72} + ( -23376744402309407 + 23376744402309407 \beta_{1} - 1468590194 \beta_{2} - 1841009184198 \beta_{3} - 1002831520 \beta_{4} - 10847969496 \beta_{5} - 1078098716 \beta_{6} - 34364684 \beta_{7} + 609159536 \beta_{8} - 1434225510 \beta_{9} ) q^{73} + ( -14598479261274624 \beta_{1} + 1184423214592 \beta_{2} - 1184757293056 \beta_{3} - 1546900992 \beta_{4} - 616576000 \beta_{5} - 92398336 \beta_{6} - 126192896 \beta_{7} + 181142784 \beta_{8} - 119141120 \beta_{9} ) q^{74} + ( 2055525217949995 + 1145026464697445 \beta_{1} - 2820593818890 \beta_{2} + 1239077892865 \beta_{3} + 12150508710 \beta_{4} + 4596075630 \beta_{5} + 64807560 \beta_{6} - 1231252170 \beta_{7} - 1372774620 \beta_{8} + 385383090 \beta_{9} ) q^{75} + ( 13722397274275840 - 350109827072 \beta_{2} - 353394229248 \beta_{3} - 4918345728 \beta_{4} + 11256725504 \beta_{5} + 592314368 \beta_{6} + 1353056256 \beta_{7} + 1487798272 \beta_{8} - 148766720 \beta_{9} ) q^{76} + ( 5284738595111894 + 5284738595111894 \beta_{1} + 2467502061430 \beta_{2} - 882932314 \beta_{3} + 6258521200 \beta_{4} + 25592389648 \beta_{5} + 1476483643 \beta_{6} + 1233971415 \beta_{7} - 593551329 \beta_{8} - 3148610757 \beta_{9} ) q^{77} + ( -33691740774102528 + 33691740774102528 \beta_{1} + 752785920 \beta_{2} + 118562731008 \beta_{3} + 1750609920 \beta_{4} + 7593400320 \beta_{5} + 1733322240 \beta_{6} - 635450880 \beta_{7} - 1216181760 \beta_{8} + 1388236800 \beta_{9} ) q^{78} + ( -20272194266936160 \beta_{1} - 1803632305052 \beta_{2} + 1800483361444 \beta_{3} - 18443301456 \beta_{4} + 4050291744 \beta_{5} - 336718772 \beta_{6} - 1177373920 \beta_{7} + 687357276 \beta_{8} - 1620931184 \beta_{9} ) q^{79} + ( 2467252553121792 + 588204361121792 \beta_{1} + 446676598784 \beta_{2} + 120259084288 \beta_{3} - 17179869184 \beta_{5} ) q^{80} + ( 29247961571645341 + 67788369954 \beta_{2} + 73175635104 \beta_{3} + 23437695976 \beta_{4} - 12547772212 \beta_{5} - 1005662968 \beta_{6} - 1334852775 \beta_{7} - 1658209104 \beta_{8} - 1388540303 \beta_{9} ) q^{81} + ( -5708831883668992 - 5708831883668992 \beta_{1} - 2806412936704 \beta_{2} + 763825664 \beta_{3} - 5615669248 \beta_{4} - 39905431552 \beta_{5} + 104805120 \beta_{6} + 1840672512 \beta_{7} - 868630784 \beta_{8} + 1629668096 \beta_{9} ) q^{82} + ( -11401323777950277 + 11401323777950277 \beta_{1} + 1460215024 \beta_{2} + 2280747907271 \beta_{3} - 4101115158 \beta_{4} + 6690309802 \beta_{5} - 1747798978 \beta_{6} + 1560822580 \beta_{7} + 1388084896 \beta_{8} - 100607556 \beta_{9} ) q^{83} + ( 34078063264006144 \beta_{1} + 3525543723008 \beta_{2} - 3526355320832 \beta_{3} + 24382537728 \beta_{4} - 46423080960 \beta_{5} + 2614493184 \beta_{6} + 1189085184 \beta_{7} + 744751104 \beta_{8} + 1358561280 \beta_{9} ) q^{84} + ( 48408810467289965 - 54595083415440647 \beta_{1} + 3834109024349 \beta_{2} - 7305252642267 \beta_{3} - 37225845717 \beta_{4} + 16978227097 \beta_{5} + 1569258795 \beta_{6} + 678473685 \beta_{7} + 4058929285 \beta_{8} - 333958745 \beta_{9} ) q^{85} + ( 34513851586681344 - 1892358775552 \beta_{2} - 1897173323520 \beta_{3} - 59818113024 \beta_{4} - 9322539008 \beta_{5} + 165289984 \beta_{6} + 1743111168 \beta_{7} - 168292864 \beta_{8} + 3074439680 \beta_{9} ) q^{86} + ( -111836615897165320 - 111836615897165320 \beta_{1} + 610288439954 \beta_{2} + 3355744662 \beta_{3} + 8738006172 \beta_{4} - 3301504548 \beta_{5} - 4363299306 \beta_{6} - 5036543238 \beta_{7} + 1007554644 \beta_{8} + 5031920730 \beta_{9} ) q^{87} + ( -15718967532322816 + 15718967532322816 \beta_{1} + 5570035712 \beta_{2} + 1177760563200 \beta_{3} + 34963718144 \beta_{4} + 27179089920 \beta_{5} + 2684354560 \beta_{6} + 1409286144 \beta_{7} + 1275068416 \beta_{8} + 4160749568 \beta_{9} ) q^{88} + ( 208959694965127280 \beta_{1} - 6232396159768 \beta_{2} + 6253531104460 \beta_{3} + 63068507440 \beta_{4} + 59767477032 \beta_{5} - 6361238752 \beta_{6} + 4087890164 \beta_{7} - 2628378878 \beta_{8} + 8057436898 \beta_{9} ) q^{89} + ( 1631111253196544 - 68866542000229120 \beta_{1} + 9115150377216 \beta_{2} + 6058028381952 \beta_{3} - 16041292544 \beta_{4} + 6939254016 \beta_{5} - 1294695680 \beta_{6} + 3895573760 \beta_{7} - 764112640 \beta_{8} + 4055732480 \beta_{9} ) q^{90} + ( 133766358149482182 + 10943015151279 \beta_{2} + 10938254606411 \beta_{3} - 99749243192 \beta_{4} - 34796387272 \beta_{5} - 1209737592 \beta_{6} + 4362245008 \beta_{7} - 1647240202 \beta_{8} + 3255277654 \beta_{9} ) q^{91} + ( -31586377793142784 - 31586377793142784 \beta_{1} - 5329360519168 \beta_{2} - 5646974976 \beta_{3} - 59990867968 \beta_{4} - 67911811072 \beta_{5} + 964034560 \beta_{6} - 744357888 \beta_{7} + 4682940416 \beta_{8} + 4985061376 \beta_{9} ) q^{92} + ( -15638523818497954 + 15638523818497954 \beta_{1} - 4231483308 \beta_{2} + 5794585972772 \beta_{3} - 120543174564 \beta_{4} + 25890088596 \beta_{5} + 4309045101 \beta_{6} - 6130144245 \beta_{7} - 13569664407 \beta_{8} + 1898660937 \beta_{9} ) q^{93} + ( 170610833663187456 \beta_{1} + 12807304387840 \beta_{2} - 12793639930624 \beta_{3} + 31663583232 \beta_{4} + 54610322432 \beta_{5} - 4676770816 \beta_{6} + 2242240000 \beta_{7} - 2174184192 \beta_{8} + 4571262208 \beta_{9} ) q^{94} + ( -93027556013227960 - 306113798979843720 \beta_{1} - 4289699539860 \beta_{2} - 24644545422230 \beta_{3} + 38294037060 \beta_{4} - 58179187830 \beta_{5} - 4984036205 \beta_{6} + 9231859935 \beta_{7} - 17827543715 \beta_{8} - 6991603495 \beta_{9} ) q^{95} + ( 2841138046173184 - 4398046511104 \beta_{2} - 4398046511104 \beta_{3} ) q^{96} + ( 85354010574075537 + 85354010574075537 \beta_{1} - 12218665838304 \beta_{2} + 10782674656 \beta_{3} + 58059002562 \beta_{4} - 108343291666 \beta_{5} + 2041762506 \beta_{6} + 14674939052 \beta_{7} - 12824437162 \beta_{8} - 2424283738 \beta_{9} ) q^{97} + ( 147941291764436224 - 147941291764436224 \beta_{1} - 6793660928 \beta_{2} + 20663594108416 \beta_{3} - 141754031104 \beta_{4} - 27412470784 \beta_{5} - 17293114624 \beta_{6} + 4736441600 \beta_{7} + 2683300608 \beta_{8} - 11530102528 \beta_{9} ) q^{98} + ( 649666215480416762 \beta_{1} + 1705155427353 \beta_{2} - 1755735293115 \beta_{3} - 198971035594 \beta_{4} - 33395499604 \beta_{5} + 18615271707 \beta_{6} - 10397884015 \beta_{7} - 2952280831 \beta_{8} - 24518990871 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 2560q^{2} + 3230q^{3} - 1436130q^{5} - 1653760q^{6} - 71828530q^{7} + 335544320q^{8} + O(q^{10}) \) \( 10q - 2560q^{2} + 3230q^{3} - 1436130q^{5} - 1653760q^{6} - 71828530q^{7} + 335544320q^{8} + 280000000q^{10} - 4684617380q^{11} + 423362560q^{12} - 11858528670q^{13} + 132580211870q^{15} - 171798691840q^{16} + 57425096770q^{17} - 314336842240q^{18} + 44876431360q^{20} + 2599949895020q^{21} + 1199262049280q^{22} - 2409849379970q^{23} - 9628701268750q^{25} + 6071566679040q^{26} + 10319468097200q^{27} + 9414709084160q^{28} - 42920067333120q^{30} + 44076891672220q^{31} + 43980465111040q^{32} - 91028207725540q^{33} - 92680242801650q^{35} + 160940463226880q^{36} + 285126548071770q^{37} - 268015571763200q^{38} - 59676892856320q^{40} + 223001245455820q^{41} - 665587173125120q^{42} - 674098663802370q^{43} + 1313192006777980q^{45} + 1233842882544640q^{46} - 3332242844984130q^{47} - 55490977464320q^{48} + 5679798331686400q^{50} - 2499853315818980q^{51} - 1554321069834240q^{52} - 7587444943731670q^{53} + 27897159618275340q^{55} - 4820331051089920q^{56} - 13989089009249200q^{57} - 10419344591360000q^{58} + 4597520944332800q^{60} + 64704780211083420q^{61} - 11283684268088320q^{62} - 108674323944477970q^{63} + 72228906553534170q^{65} + 46606442355476480q^{66} - 108403906097814130q^{67} - 7526822283837440q^{68} + 42813790750842880q^{70} + 194380424551175420q^{71} - 41200758586081280q^{72} - 233767444023094070q^{73} + 20555252179499950q^{75} + 137223972742758400q^{76} + 52847385951118940q^{77} - 336917407741025280q^{78} + 24672525531217920q^{80} + 292479615716453410q^{81} - 57088318836689920q^{82} - 114013237779502770q^{83} + 484088104672899650q^{85} + 345138515866813440q^{86} - 1118366158971653200q^{87} - 157189675323228160q^{88} + 16311112531965440q^{90} + 1337663581494821820q^{91} - 315863777931427840q^{92} - 156385238184979540q^{93} - 930275560132279600q^{95} + 28411380461731840q^{96} + 853540105740755370q^{97} + 1479412917644362240q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} + 15200484 x^{8} + 52963214026118 x^{6} + 31233027106270421412 x^{4} + 3470052491961139195356481 x^{2} + 20667093555813840346094397504\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(4240228015797793237 \nu^{9} + 64527456960583686051740367 \nu^{7} + 225701258493109901998279256900399 \nu^{5} + 136539371251285164830406333633322886757 \nu^{3} + 18378248053675851593147151559929698471066032 \nu\)\()/ \)\(13\!\cdots\!56\)\( \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(60\!\cdots\!23\)\( \nu^{9} - \)\(98\!\cdots\!02\)\( \nu^{8} - \)\(91\!\cdots\!89\)\( \nu^{7} - \)\(14\!\cdots\!46\)\( \nu^{6} - \)\(31\!\cdots\!45\)\( \nu^{5} - \)\(50\!\cdots\!70\)\( \nu^{4} - \)\(17\!\cdots\!51\)\( \nu^{3} - \)\(25\!\cdots\!14\)\( \nu^{2} - \)\(14\!\cdots\!92\)\( \nu - \)\(12\!\cdots\!68\)\(\)\()/ \)\(10\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(60\!\cdots\!23\)\( \nu^{9} - \)\(98\!\cdots\!02\)\( \nu^{8} + \)\(91\!\cdots\!89\)\( \nu^{7} - \)\(14\!\cdots\!46\)\( \nu^{6} + \)\(31\!\cdots\!45\)\( \nu^{5} - \)\(50\!\cdots\!70\)\( \nu^{4} + \)\(17\!\cdots\!51\)\( \nu^{3} - \)\(25\!\cdots\!14\)\( \nu^{2} + \)\(14\!\cdots\!92\)\( \nu - \)\(12\!\cdots\!68\)\(\)\()/ \)\(10\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(12\!\cdots\!79\)\( \nu^{9} - \)\(27\!\cdots\!18\)\( \nu^{8} + \)\(18\!\cdots\!77\)\( \nu^{7} - \)\(41\!\cdots\!54\)\( \nu^{6} + \)\(65\!\cdots\!05\)\( \nu^{5} - \)\(13\!\cdots\!90\)\( \nu^{4} + \)\(38\!\cdots\!43\)\( \nu^{3} - \)\(66\!\cdots\!86\)\( \nu^{2} + \)\(38\!\cdots\!96\)\( \nu - \)\(28\!\cdots\!52\)\(\)\()/ \)\(16\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(22\!\cdots\!13\)\( \nu^{9} - \)\(14\!\cdots\!84\)\( \nu^{8} - \)\(33\!\cdots\!79\)\( \nu^{7} - \)\(22\!\cdots\!42\)\( \nu^{6} - \)\(11\!\cdots\!75\)\( \nu^{5} - \)\(80\!\cdots\!80\)\( \nu^{4} - \)\(62\!\cdots\!61\)\( \nu^{3} - \)\(53\!\cdots\!78\)\( \nu^{2} - \)\(49\!\cdots\!72\)\( \nu - \)\(46\!\cdots\!16\)\(\)\()/ \)\(25\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(25\!\cdots\!20\)\( \nu^{9} - \)\(46\!\cdots\!79\)\( \nu^{8} - \)\(39\!\cdots\!24\)\( \nu^{7} - \)\(72\!\cdots\!01\)\( \nu^{6} - \)\(14\!\cdots\!16\)\( \nu^{5} - \)\(27\!\cdots\!61\)\( \nu^{4} - \)\(12\!\cdots\!56\)\( \nu^{3} - \)\(23\!\cdots\!59\)\( \nu^{2} - \)\(29\!\cdots\!04\)\( \nu - \)\(20\!\cdots\!20\)\(\)\()/ \)\(16\!\cdots\!60\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(23\!\cdots\!01\)\( \nu^{9} + \)\(53\!\cdots\!81\)\( \nu^{8} + \)\(36\!\cdots\!67\)\( \nu^{7} + \)\(79\!\cdots\!47\)\( \nu^{6} + \)\(12\!\cdots\!71\)\( \nu^{5} + \)\(26\!\cdots\!31\)\( \nu^{4} + \)\(76\!\cdots\!93\)\( \nu^{3} + \)\(11\!\cdots\!73\)\( \nu^{2} + \)\(76\!\cdots\!88\)\( \nu - \)\(62\!\cdots\!12\)\(\)\()/ \)\(50\!\cdots\!80\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(79\!\cdots\!13\)\( \nu^{9} - \)\(59\!\cdots\!62\)\( \nu^{8} + \)\(12\!\cdots\!09\)\( \nu^{7} - \)\(91\!\cdots\!71\)\( \nu^{6} + \)\(41\!\cdots\!95\)\( \nu^{5} - \)\(33\!\cdots\!00\)\( \nu^{4} + \)\(23\!\cdots\!31\)\( \nu^{3} - \)\(26\!\cdots\!39\)\( \nu^{2} + \)\(23\!\cdots\!52\)\( \nu - \)\(32\!\cdots\!28\)\(\)\()/ \)\(12\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(47\!\cdots\!97\)\( \nu^{9} + \)\(40\!\cdots\!98\)\( \nu^{8} + \)\(71\!\cdots\!31\)\( \nu^{7} + \)\(61\!\cdots\!54\)\( \nu^{6} + \)\(25\!\cdots\!95\)\( \nu^{5} + \)\(20\!\cdots\!30\)\( \nu^{4} + \)\(15\!\cdots\!29\)\( \nu^{3} + \)\(86\!\cdots\!86\)\( \nu^{2} + \)\(16\!\cdots\!48\)\( \nu + \)\(76\!\cdots\!32\)\(\)\()/ \)\(50\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{9} - \beta_{7} - 2 \beta_{6} + 28 \beta_{5} - 18 \beta_{4} + 4087 \beta_{3} - 4087 \beta_{2} + 36000 \beta_{1}\)\()/90000\)
\(\nu^{2}\)\(=\)\((\)\(6160 \beta_{9} - 1417 \beta_{8} - 9856 \beta_{7} + 2353 \beta_{6} + 17714 \beta_{5} + 23604 \beta_{4} - 5240941 \beta_{3} - 5243701 \beta_{2} - 273608712000\)\()/90000\)
\(\nu^{3}\)\(=\)\((\)\(2413097 \beta_{9} - 4309000 \beta_{8} + 6891097 \beta_{7} + 22907194 \beta_{6} - 214918716 \beta_{5} + 106803746 \beta_{4} - 33249534439 \beta_{3} + 33240240439 \beta_{2} + 444372046668000 \beta_{1}\)\()/90000\)
\(\nu^{4}\)\(=\)\((\)\(-9531833704 \beta_{9} + 826991637 \beta_{8} + 18469220712 \beta_{7} - 5104015901 \beta_{6} - 53744951402 \beta_{5} - 80216647396 \beta_{4} + 14812762570049 \beta_{3} + 14817422932793 \beta_{2} + 450461828815272000\)\()/18000\)
\(\nu^{5}\)\(=\)\((\)\(-9082901147321 \beta_{9} + 83439080564240 \beta_{8} - 74138523758601 \beta_{7} - 260004834786802 \beta_{6} + 1894898164814908 \beta_{5} - 1000737105397858 \beta_{4} + 305664052829029647 \beta_{3} - 305570708499713007 \beta_{2} - 5641396746053180364000 \beta_{1}\)\()/90000\)
\(\nu^{6}\)\(=\)\((\)\(412465889919285440 \beta_{9} - 14831226089191337 \beta_{8} - 914293177202451536 \beta_{7} + 256453077770832593 \beta_{6} + 2966291840197844914 \beta_{5} + 4877465327601132084 \beta_{4} - 849587900932015465661 \beta_{3} - 849848106367616990501 \beta_{2} - 21431575798258527374472000\)\()/90000\)
\(\nu^{7}\)\(=\)\((\)\(40600040232937432297 \beta_{9} - 1037548332201953369560 \beta_{8} + 790345198594269934617 \beta_{7} + 2792377540070584006634 \beta_{6} - 18221370021909987364796 \beta_{5} + 10076020245889045344866 \beta_{4} - 3013325961789949285650519 \beta_{3} + 3012402077820907862380359 \beta_{2} + 62515096957982540351564748000 \beta_{1}\)\()/90000\)
\(\nu^{8}\)\(=\)\((\)\(-787292031096830741243608 \beta_{9} + 9513174280977547697589 \beta_{8} + 1862658084118980857332344 \beta_{7} - 524282118474575791096637 \beta_{6} - 6287910008155041274765034 \beta_{5} - 10731158768119677847097572 \beta_{4} + 1830807322437455694702000593 \beta_{3} + 1831367919546284246574690281 \beta_{2} + 42955315660900722215473229736000\)\()/18000\)
\(\nu^{9}\)\(=\)\((\)\(-207748174112971900442790841 \beta_{9} + 11486740295353602575164688480 \beta_{8} - 8298689443967425004547723401 \beta_{7} - 29383462402144606746245556002 \beta_{6} + 183227892898526125238321843388 \beta_{5} - 103429448809999239781200267298 \beta_{4} + 30639451003781376754781073244607 \beta_{3} - 30630060719292666147514982891327 \beta_{2} - 665346183217094546364878877970764000 \beta_{1}\)\()/90000\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
3223.13i
369.775i
749.210i
79.4383i
2026.71i
3223.13i
369.775i
749.210i
79.4383i
2026.71i
−256.000 + 256.000i −22485.0 22485.0i 131072.i −1.41140e6 + 1.35005e6i 1.15123e7 −5.59623e7 + 5.59623e7i 3.35544e7 + 3.35544e7i 6.23730e8i 1.57043e7 7.06932e8i
3.2 −256.000 + 256.000i −12815.7 12815.7i 131072.i −120054. 1.94943e6i 6.56166e6 2.37564e7 2.37564e7i 3.35544e7 + 3.35544e7i 5.89343e7i 5.29788e8 + 4.68321e8i
3.3 −256.000 + 256.000i 4473.85 + 4473.85i 131072.i 1.47317e6 + 1.28237e6i −2.29061e6 −6.84961e6 + 6.84961e6i 3.35544e7 + 3.35544e7i 3.47390e8i −7.05418e8 + 4.88445e7i
3.4 −256.000 + 256.000i 10768.9 + 10768.9i 131072.i −1.49934e6 + 1.25167e6i −5.51369e6 3.13009e7 3.13009e7i 3.35544e7 + 3.35544e7i 1.55481e8i 6.34033e7 7.04258e8i
3.5 −256.000 + 256.000i 21673.0 + 21673.0i 131072.i 839560. 1.76347e6i −1.10966e7 −2.81596e7 + 2.81596e7i 3.35544e7 + 3.35544e7i 5.52014e8i 2.36522e8 + 6.66376e8i
7.1 −256.000 256.000i −22485.0 + 22485.0i 131072.i −1.41140e6 1.35005e6i 1.15123e7 −5.59623e7 5.59623e7i 3.35544e7 3.35544e7i 6.23730e8i 1.57043e7 + 7.06932e8i
7.2 −256.000 256.000i −12815.7 + 12815.7i 131072.i −120054. + 1.94943e6i 6.56166e6 2.37564e7 + 2.37564e7i 3.35544e7 3.35544e7i 5.89343e7i 5.29788e8 4.68321e8i
7.3 −256.000 256.000i 4473.85 4473.85i 131072.i 1.47317e6 1.28237e6i −2.29061e6 −6.84961e6 6.84961e6i 3.35544e7 3.35544e7i 3.47390e8i −7.05418e8 4.88445e7i
7.4 −256.000 256.000i 10768.9 10768.9i 131072.i −1.49934e6 1.25167e6i −5.51369e6 3.13009e7 + 3.13009e7i 3.35544e7 3.35544e7i 1.55481e8i 6.34033e7 + 7.04258e8i
7.5 −256.000 256.000i 21673.0 21673.0i 131072.i 839560. + 1.76347e6i −1.10966e7 −2.81596e7 2.81596e7i 3.35544e7 3.35544e7i 5.52014e8i 2.36522e8 6.66376e8i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.5
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 10.19.c.b 10
3.b odd 2 1 90.19.g.b 10
5.b even 2 1 50.19.c.d 10
5.c odd 4 1 inner 10.19.c.b 10
5.c odd 4 1 50.19.c.d 10
15.e even 4 1 90.19.g.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.19.c.b 10 1.a even 1 1 trivial
10.19.c.b 10 5.c odd 4 1 inner
50.19.c.d 10 5.b even 2 1
50.19.c.d 10 5.c odd 4 1
90.19.g.b 10 3.b odd 2 1
90.19.g.b 10 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{10} - \cdots\) acting on \(S_{19}^{\mathrm{new}}(10, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 512 T + 131072 T^{2} )^{5} \)
$3$ \( 1 - 3230 T + 5216450 T^{2} - 3862561803390 T^{3} - 179583736865535855 T^{4} + \)\(22\!\cdots\!64\)\( T^{5} + \)\(11\!\cdots\!80\)\( T^{6} + \)\(12\!\cdots\!40\)\( T^{7} + \)\(13\!\cdots\!50\)\( T^{8} - \)\(60\!\cdots\!40\)\( T^{9} - \)\(20\!\cdots\!52\)\( T^{10} - \)\(23\!\cdots\!60\)\( T^{11} + \)\(20\!\cdots\!50\)\( T^{12} + \)\(72\!\cdots\!60\)\( T^{13} + \)\(25\!\cdots\!80\)\( T^{14} + \)\(19\!\cdots\!36\)\( T^{15} - \)\(60\!\cdots\!55\)\( T^{16} - \)\(50\!\cdots\!10\)\( T^{17} + \)\(26\!\cdots\!50\)\( T^{18} - \)\(63\!\cdots\!70\)\( T^{19} + \)\(76\!\cdots\!01\)\( T^{20} \)
$5$ \( 1 + 1436130 T + 5845585322825 T^{2} + 8315883141295500000 T^{3} + \)\(33\!\cdots\!50\)\( T^{4} + \)\(31\!\cdots\!00\)\( T^{5} + \)\(12\!\cdots\!50\)\( T^{6} + \)\(12\!\cdots\!00\)\( T^{7} + \)\(32\!\cdots\!25\)\( T^{8} + \)\(30\!\cdots\!50\)\( T^{9} + \)\(80\!\cdots\!25\)\( T^{10} \)
$7$ \( 1 + 71828530 T + 2579668860980450 T^{2} - \)\(26\!\cdots\!10\)\( T^{3} - \)\(44\!\cdots\!55\)\( T^{4} - \)\(19\!\cdots\!64\)\( T^{5} - \)\(18\!\cdots\!20\)\( T^{6} + \)\(19\!\cdots\!60\)\( T^{7} + \)\(55\!\cdots\!50\)\( T^{8} + \)\(34\!\cdots\!40\)\( T^{9} + \)\(20\!\cdots\!48\)\( T^{10} + \)\(56\!\cdots\!60\)\( T^{11} + \)\(14\!\cdots\!50\)\( T^{12} + \)\(83\!\cdots\!40\)\( T^{13} - \)\(13\!\cdots\!20\)\( T^{14} - \)\(21\!\cdots\!36\)\( T^{15} - \)\(83\!\cdots\!55\)\( T^{16} - \)\(80\!\cdots\!90\)\( T^{17} + \)\(12\!\cdots\!50\)\( T^{18} + \)\(57\!\cdots\!70\)\( T^{19} + \)\(13\!\cdots\!01\)\( T^{20} \)
$11$ \( ( 1 + 2342308690 T + 6231128958562799145 T^{2} - \)\(21\!\cdots\!20\)\( T^{3} - \)\(20\!\cdots\!90\)\( T^{4} - \)\(69\!\cdots\!52\)\( T^{5} - \)\(11\!\cdots\!90\)\( T^{6} - \)\(66\!\cdots\!20\)\( T^{7} + \)\(10\!\cdots\!45\)\( T^{8} + \)\(22\!\cdots\!90\)\( T^{9} + \)\(53\!\cdots\!01\)\( T^{10} )^{2} \)
$13$ \( 1 + 11858528670 T + 70312351108605984450 T^{2} + \)\(60\!\cdots\!10\)\( T^{3} - \)\(94\!\cdots\!55\)\( T^{4} + \)\(30\!\cdots\!64\)\( T^{5} + \)\(60\!\cdots\!80\)\( T^{6} + \)\(13\!\cdots\!40\)\( T^{7} + \)\(13\!\cdots\!50\)\( T^{8} - \)\(17\!\cdots\!40\)\( T^{9} - \)\(13\!\cdots\!52\)\( T^{10} - \)\(20\!\cdots\!60\)\( T^{11} + \)\(16\!\cdots\!50\)\( T^{12} + \)\(18\!\cdots\!60\)\( T^{13} + \)\(97\!\cdots\!80\)\( T^{14} + \)\(54\!\cdots\!36\)\( T^{15} - \)\(19\!\cdots\!55\)\( T^{16} + \)\(13\!\cdots\!90\)\( T^{17} + \)\(17\!\cdots\!50\)\( T^{18} + \)\(34\!\cdots\!30\)\( T^{19} + \)\(32\!\cdots\!01\)\( T^{20} \)
$17$ \( 1 - 57425096770 T + \)\(16\!\cdots\!50\)\( T^{2} - \)\(19\!\cdots\!10\)\( T^{3} - \)\(18\!\cdots\!55\)\( T^{4} + \)\(38\!\cdots\!36\)\( T^{5} + \)\(34\!\cdots\!80\)\( T^{6} + \)\(50\!\cdots\!60\)\( T^{7} - \)\(14\!\cdots\!50\)\( T^{8} - \)\(64\!\cdots\!60\)\( T^{9} + \)\(12\!\cdots\!48\)\( T^{10} - \)\(90\!\cdots\!40\)\( T^{11} - \)\(27\!\cdots\!50\)\( T^{12} + \)\(14\!\cdots\!40\)\( T^{13} + \)\(13\!\cdots\!80\)\( T^{14} + \)\(21\!\cdots\!64\)\( T^{15} - \)\(14\!\cdots\!55\)\( T^{16} - \)\(21\!\cdots\!90\)\( T^{17} + \)\(25\!\cdots\!50\)\( T^{18} - \)\(12\!\cdots\!30\)\( T^{19} + \)\(30\!\cdots\!01\)\( T^{20} \)
$19$ \( 1 - \)\(13\!\cdots\!10\)\( T^{2} + \)\(41\!\cdots\!45\)\( T^{4} - \)\(37\!\cdots\!20\)\( T^{6} + \)\(70\!\cdots\!10\)\( T^{8} - \)\(50\!\cdots\!52\)\( T^{10} + \)\(76\!\cdots\!10\)\( T^{12} - \)\(43\!\cdots\!20\)\( T^{14} + \)\(52\!\cdots\!45\)\( T^{16} - \)\(17\!\cdots\!10\)\( T^{18} + \)\(14\!\cdots\!01\)\( T^{20} \)
$23$ \( 1 + 2409849379970 T + \)\(29\!\cdots\!50\)\( T^{2} + \)\(52\!\cdots\!10\)\( T^{3} - \)\(21\!\cdots\!55\)\( T^{4} - \)\(67\!\cdots\!36\)\( T^{5} - \)\(86\!\cdots\!20\)\( T^{6} - \)\(19\!\cdots\!60\)\( T^{7} + \)\(12\!\cdots\!50\)\( T^{8} + \)\(95\!\cdots\!60\)\( T^{9} + \)\(13\!\cdots\!48\)\( T^{10} + \)\(31\!\cdots\!40\)\( T^{11} + \)\(13\!\cdots\!50\)\( T^{12} - \)\(66\!\cdots\!40\)\( T^{13} - \)\(96\!\cdots\!20\)\( T^{14} - \)\(24\!\cdots\!64\)\( T^{15} - \)\(24\!\cdots\!55\)\( T^{16} + \)\(20\!\cdots\!90\)\( T^{17} + \)\(35\!\cdots\!50\)\( T^{18} + \)\(95\!\cdots\!30\)\( T^{19} + \)\(12\!\cdots\!01\)\( T^{20} \)
$29$ \( 1 - \)\(14\!\cdots\!10\)\( T^{2} + \)\(98\!\cdots\!45\)\( T^{4} - \)\(44\!\cdots\!20\)\( T^{6} + \)\(14\!\cdots\!10\)\( T^{8} - \)\(35\!\cdots\!52\)\( T^{10} + \)\(64\!\cdots\!10\)\( T^{12} - \)\(87\!\cdots\!20\)\( T^{14} + \)\(85\!\cdots\!45\)\( T^{16} - \)\(54\!\cdots\!10\)\( T^{18} + \)\(17\!\cdots\!01\)\( T^{20} \)
$31$ \( ( 1 - 22038445836110 T + \)\(25\!\cdots\!45\)\( T^{2} - \)\(52\!\cdots\!20\)\( T^{3} + \)\(28\!\cdots\!10\)\( T^{4} - \)\(51\!\cdots\!52\)\( T^{5} + \)\(19\!\cdots\!10\)\( T^{6} - \)\(25\!\cdots\!20\)\( T^{7} + \)\(85\!\cdots\!45\)\( T^{8} - \)\(52\!\cdots\!10\)\( T^{9} + \)\(16\!\cdots\!01\)\( T^{10} )^{2} \)
$37$ \( 1 - 285126548071770 T + \)\(40\!\cdots\!50\)\( T^{2} - \)\(47\!\cdots\!10\)\( T^{3} + \)\(65\!\cdots\!45\)\( T^{4} - \)\(11\!\cdots\!64\)\( T^{5} + \)\(18\!\cdots\!80\)\( T^{6} - \)\(24\!\cdots\!40\)\( T^{7} + \)\(35\!\cdots\!50\)\( T^{8} - \)\(49\!\cdots\!60\)\( T^{9} + \)\(66\!\cdots\!48\)\( T^{10} - \)\(83\!\cdots\!40\)\( T^{11} + \)\(10\!\cdots\!50\)\( T^{12} - \)\(12\!\cdots\!60\)\( T^{13} + \)\(15\!\cdots\!80\)\( T^{14} - \)\(16\!\cdots\!36\)\( T^{15} + \)\(15\!\cdots\!45\)\( T^{16} - \)\(18\!\cdots\!90\)\( T^{17} + \)\(26\!\cdots\!50\)\( T^{18} - \)\(31\!\cdots\!30\)\( T^{19} + \)\(18\!\cdots\!01\)\( T^{20} \)
$41$ \( ( 1 - 111500622727910 T + \)\(28\!\cdots\!45\)\( T^{2} - \)\(37\!\cdots\!20\)\( T^{3} + \)\(37\!\cdots\!10\)\( T^{4} - \)\(56\!\cdots\!52\)\( T^{5} + \)\(39\!\cdots\!10\)\( T^{6} - \)\(43\!\cdots\!20\)\( T^{7} + \)\(35\!\cdots\!45\)\( T^{8} - \)\(14\!\cdots\!10\)\( T^{9} + \)\(14\!\cdots\!01\)\( T^{10} )^{2} \)
$43$ \( 1 + 674098663802370 T + \)\(22\!\cdots\!50\)\( T^{2} + \)\(18\!\cdots\!10\)\( T^{3} + \)\(93\!\cdots\!45\)\( T^{4} + \)\(13\!\cdots\!64\)\( T^{5} + \)\(43\!\cdots\!80\)\( T^{6} + \)\(36\!\cdots\!40\)\( T^{7} - \)\(36\!\cdots\!50\)\( T^{8} - \)\(34\!\cdots\!40\)\( T^{9} - \)\(11\!\cdots\!52\)\( T^{10} - \)\(87\!\cdots\!60\)\( T^{11} - \)\(23\!\cdots\!50\)\( T^{12} + \)\(59\!\cdots\!60\)\( T^{13} + \)\(17\!\cdots\!80\)\( T^{14} + \)\(13\!\cdots\!36\)\( T^{15} + \)\(24\!\cdots\!45\)\( T^{16} + \)\(11\!\cdots\!90\)\( T^{17} + \)\(37\!\cdots\!50\)\( T^{18} + \)\(28\!\cdots\!30\)\( T^{19} + \)\(10\!\cdots\!01\)\( T^{20} \)
$47$ \( 1 + 3332242844984130 T + \)\(55\!\cdots\!50\)\( T^{2} + \)\(38\!\cdots\!90\)\( T^{3} - \)\(16\!\cdots\!55\)\( T^{4} - \)\(41\!\cdots\!64\)\( T^{5} + \)\(27\!\cdots\!80\)\( T^{6} + \)\(13\!\cdots\!60\)\( T^{7} + \)\(15\!\cdots\!50\)\( T^{8} + \)\(32\!\cdots\!40\)\( T^{9} - \)\(61\!\cdots\!52\)\( T^{10} + \)\(40\!\cdots\!60\)\( T^{11} + \)\(23\!\cdots\!50\)\( T^{12} + \)\(27\!\cdots\!40\)\( T^{13} + \)\(67\!\cdots\!80\)\( T^{14} - \)\(12\!\cdots\!36\)\( T^{15} - \)\(64\!\cdots\!55\)\( T^{16} + \)\(18\!\cdots\!10\)\( T^{17} + \)\(33\!\cdots\!50\)\( T^{18} + \)\(25\!\cdots\!70\)\( T^{19} + \)\(94\!\cdots\!01\)\( T^{20} \)
$53$ \( 1 + 7587444943731670 T + \)\(28\!\cdots\!50\)\( T^{2} + \)\(73\!\cdots\!10\)\( T^{3} + \)\(17\!\cdots\!45\)\( T^{4} + \)\(67\!\cdots\!64\)\( T^{5} + \)\(29\!\cdots\!80\)\( T^{6} + \)\(79\!\cdots\!40\)\( T^{7} + \)\(37\!\cdots\!50\)\( T^{8} + \)\(21\!\cdots\!60\)\( T^{9} + \)\(84\!\cdots\!48\)\( T^{10} + \)\(23\!\cdots\!40\)\( T^{11} + \)\(44\!\cdots\!50\)\( T^{12} + \)\(10\!\cdots\!60\)\( T^{13} + \)\(40\!\cdots\!80\)\( T^{14} + \)\(10\!\cdots\!36\)\( T^{15} + \)\(28\!\cdots\!45\)\( T^{16} + \)\(13\!\cdots\!90\)\( T^{17} + \)\(56\!\cdots\!50\)\( T^{18} + \)\(16\!\cdots\!30\)\( T^{19} + \)\(23\!\cdots\!01\)\( T^{20} \)
$59$ \( 1 - \)\(45\!\cdots\!10\)\( T^{2} + \)\(10\!\cdots\!45\)\( T^{4} - \)\(16\!\cdots\!20\)\( T^{6} + \)\(18\!\cdots\!10\)\( T^{8} - \)\(15\!\cdots\!52\)\( T^{10} + \)\(10\!\cdots\!10\)\( T^{12} - \)\(51\!\cdots\!20\)\( T^{14} + \)\(18\!\cdots\!45\)\( T^{16} - \)\(45\!\cdots\!10\)\( T^{18} + \)\(56\!\cdots\!01\)\( T^{20} \)
$61$ \( ( 1 - 32352390105541710 T + \)\(70\!\cdots\!45\)\( T^{2} - \)\(99\!\cdots\!20\)\( T^{3} + \)\(13\!\cdots\!10\)\( T^{4} - \)\(15\!\cdots\!52\)\( T^{5} + \)\(18\!\cdots\!10\)\( T^{6} - \)\(18\!\cdots\!20\)\( T^{7} + \)\(17\!\cdots\!45\)\( T^{8} - \)\(11\!\cdots\!10\)\( T^{9} + \)\(47\!\cdots\!01\)\( T^{10} )^{2} \)
$67$ \( 1 + 108403906097814130 T + \)\(58\!\cdots\!50\)\( T^{2} + \)\(23\!\cdots\!90\)\( T^{3} + \)\(85\!\cdots\!45\)\( T^{4} + \)\(29\!\cdots\!36\)\( T^{5} + \)\(92\!\cdots\!80\)\( T^{6} + \)\(25\!\cdots\!60\)\( T^{7} + \)\(64\!\cdots\!50\)\( T^{8} + \)\(17\!\cdots\!40\)\( T^{9} + \)\(49\!\cdots\!48\)\( T^{10} + \)\(13\!\cdots\!60\)\( T^{11} + \)\(35\!\cdots\!50\)\( T^{12} + \)\(10\!\cdots\!40\)\( T^{13} + \)\(27\!\cdots\!80\)\( T^{14} + \)\(66\!\cdots\!64\)\( T^{15} + \)\(14\!\cdots\!45\)\( T^{16} + \)\(28\!\cdots\!10\)\( T^{17} + \)\(52\!\cdots\!50\)\( T^{18} + \)\(72\!\cdots\!70\)\( T^{19} + \)\(49\!\cdots\!01\)\( T^{20} \)
$71$ \( ( 1 - 97190212275587710 T + \)\(11\!\cdots\!45\)\( T^{2} - \)\(74\!\cdots\!20\)\( T^{3} + \)\(46\!\cdots\!10\)\( T^{4} - \)\(22\!\cdots\!52\)\( T^{5} + \)\(98\!\cdots\!10\)\( T^{6} - \)\(33\!\cdots\!20\)\( T^{7} + \)\(10\!\cdots\!45\)\( T^{8} - \)\(18\!\cdots\!10\)\( T^{9} + \)\(41\!\cdots\!01\)\( T^{10} )^{2} \)
$73$ \( 1 + 233767444023094070 T + \)\(27\!\cdots\!50\)\( T^{2} + \)\(22\!\cdots\!10\)\( T^{3} + \)\(14\!\cdots\!45\)\( T^{4} + \)\(65\!\cdots\!64\)\( T^{5} + \)\(11\!\cdots\!80\)\( T^{6} - \)\(12\!\cdots\!60\)\( T^{7} - \)\(18\!\cdots\!50\)\( T^{8} - \)\(15\!\cdots\!40\)\( T^{9} - \)\(10\!\cdots\!52\)\( T^{10} - \)\(54\!\cdots\!60\)\( T^{11} - \)\(22\!\cdots\!50\)\( T^{12} - \)\(51\!\cdots\!40\)\( T^{13} + \)\(16\!\cdots\!80\)\( T^{14} + \)\(32\!\cdots\!36\)\( T^{15} + \)\(25\!\cdots\!45\)\( T^{16} + \)\(13\!\cdots\!90\)\( T^{17} + \)\(56\!\cdots\!50\)\( T^{18} + \)\(16\!\cdots\!30\)\( T^{19} + \)\(25\!\cdots\!01\)\( T^{20} \)
$79$ \( 1 - \)\(10\!\cdots\!10\)\( T^{2} + \)\(55\!\cdots\!45\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{6} + \)\(40\!\cdots\!10\)\( T^{8} - \)\(67\!\cdots\!52\)\( T^{10} + \)\(84\!\cdots\!10\)\( T^{12} - \)\(76\!\cdots\!20\)\( T^{14} + \)\(48\!\cdots\!45\)\( T^{16} - \)\(19\!\cdots\!10\)\( T^{18} + \)\(37\!\cdots\!01\)\( T^{20} \)
$83$ \( 1 + 114013237779502770 T + \)\(64\!\cdots\!50\)\( T^{2} + \)\(49\!\cdots\!10\)\( T^{3} + \)\(41\!\cdots\!45\)\( T^{4} + \)\(24\!\cdots\!64\)\( T^{5} + \)\(13\!\cdots\!80\)\( T^{6} + \)\(10\!\cdots\!40\)\( T^{7} + \)\(67\!\cdots\!50\)\( T^{8} + \)\(25\!\cdots\!60\)\( T^{9} + \)\(11\!\cdots\!48\)\( T^{10} + \)\(89\!\cdots\!40\)\( T^{11} + \)\(82\!\cdots\!50\)\( T^{12} + \)\(43\!\cdots\!60\)\( T^{13} + \)\(19\!\cdots\!80\)\( T^{14} + \)\(12\!\cdots\!36\)\( T^{15} + \)\(76\!\cdots\!45\)\( T^{16} + \)\(31\!\cdots\!90\)\( T^{17} + \)\(14\!\cdots\!50\)\( T^{18} + \)\(88\!\cdots\!30\)\( T^{19} + \)\(27\!\cdots\!01\)\( T^{20} \)
$89$ \( 1 - \)\(36\!\cdots\!10\)\( T^{2} + \)\(81\!\cdots\!45\)\( T^{4} - \)\(11\!\cdots\!20\)\( T^{6} + \)\(13\!\cdots\!10\)\( T^{8} - \)\(14\!\cdots\!52\)\( T^{10} + \)\(20\!\cdots\!10\)\( T^{12} - \)\(26\!\cdots\!20\)\( T^{14} + \)\(27\!\cdots\!45\)\( T^{16} - \)\(19\!\cdots\!10\)\( T^{18} + \)\(77\!\cdots\!01\)\( T^{20} \)
$97$ \( 1 - 853540105740755370 T + \)\(36\!\cdots\!50\)\( T^{2} - \)\(37\!\cdots\!10\)\( T^{3} + \)\(19\!\cdots\!45\)\( T^{4} - \)\(26\!\cdots\!64\)\( T^{5} + \)\(22\!\cdots\!80\)\( T^{6} - \)\(19\!\cdots\!40\)\( T^{7} + \)\(15\!\cdots\!50\)\( T^{8} - \)\(16\!\cdots\!60\)\( T^{9} + \)\(82\!\cdots\!48\)\( T^{10} - \)\(96\!\cdots\!40\)\( T^{11} + \)\(51\!\cdots\!50\)\( T^{12} - \)\(38\!\cdots\!60\)\( T^{13} + \)\(24\!\cdots\!80\)\( T^{14} - \)\(17\!\cdots\!36\)\( T^{15} + \)\(71\!\cdots\!45\)\( T^{16} - \)\(80\!\cdots\!90\)\( T^{17} + \)\(45\!\cdots\!50\)\( T^{18} - \)\(61\!\cdots\!30\)\( T^{19} + \)\(41\!\cdots\!01\)\( T^{20} \)
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