Properties

Label 38.8.c.a
Level $38$
Weight $8$
Character orbit 38.c
Analytic conductor $11.871$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 38.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.8706309684\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 10165 x^{10} - 228496 x^{9} + 79937698 x^{8} - 1362528496 x^{7} + 222057453949 x^{6} - 1418140039384 x^{5} + 423936076682002 x^{4} - 1421321152056072 x^{3} + 376701330885210661 x^{2} + 2892184322477465880 x + 206636219229734030025\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{3}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 8 \beta_{2} q^{2} + ( \beta_{1} - 2 \beta_{2} ) q^{3} + ( -64 - 64 \beta_{2} ) q^{4} + ( -21 \beta_{2} - \beta_{6} ) q^{5} + ( 16 + 16 \beta_{2} + 8 \beta_{3} ) q^{6} + ( -87 + 3 \beta_{1} + 3 \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{7} + 512 q^{8} + ( -1206 - 1206 \beta_{2} + 13 \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{10} ) q^{9} +O(q^{10})\) \( q + 8 \beta_{2} q^{2} + ( \beta_{1} - 2 \beta_{2} ) q^{3} + ( -64 - 64 \beta_{2} ) q^{4} + ( -21 \beta_{2} - \beta_{6} ) q^{5} + ( 16 + 16 \beta_{2} + 8 \beta_{3} ) q^{6} + ( -87 + 3 \beta_{1} + 3 \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{7} + 512 q^{8} + ( -1206 - 1206 \beta_{2} + 13 \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{10} ) q^{9} + ( 168 + 168 \beta_{2} + 8 \beta_{4} ) q^{10} + ( -981 - 2 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} + 3 \beta_{6} + \beta_{7} - \beta_{9} - \beta_{11} ) q^{11} + ( -128 - 64 \beta_{1} - 64 \beta_{3} ) q^{12} + ( 2902 + 2902 \beta_{2} - 72 \beta_{3} + 3 \beta_{4} + \beta_{5} - 7 \beta_{9} + \beta_{10} ) q^{13} + ( -24 \beta_{1} - 696 \beta_{2} - 8 \beta_{6} + 8 \beta_{7} ) q^{14} + ( -678 - 678 \beta_{2} - 105 \beta_{3} + 9 \beta_{4} - 3 \beta_{5} - 3 \beta_{8} + 6 \beta_{9} - 3 \beta_{10} ) q^{15} + 4096 \beta_{2} q^{16} + ( -71 \beta_{1} - 7532 \beta_{2} + 6 \beta_{5} + 26 \beta_{6} - \beta_{7} - 2 \beta_{8} + 2 \beta_{11} ) q^{17} + ( 9648 - 104 \beta_{1} - 104 \beta_{3} + 24 \beta_{4} - 24 \beta_{6} - 8 \beta_{10} ) q^{18} + ( -5050 + 157 \beta_{1} - 5727 \beta_{2} + 120 \beta_{3} + 7 \beta_{4} + 9 \beta_{5} + 5 \beta_{6} + \beta_{7} - 5 \beta_{8} + 7 \beta_{10} + 5 \beta_{11} ) q^{19} + ( -1344 - 64 \beta_{4} + 64 \beta_{6} ) q^{20} + ( -142 \beta_{1} - 7647 \beta_{2} + 19 \beta_{5} + 30 \beta_{6} - 27 \beta_{7} ) q^{21} + ( 16 \beta_{1} - 7848 \beta_{2} - 24 \beta_{6} - 8 \beta_{7} - 8 \beta_{8} + 8 \beta_{11} ) q^{22} + ( 12716 + 12716 \beta_{2} - 373 \beta_{3} - 117 \beta_{4} + \beta_{5} - 5 \beta_{8} + 8 \beta_{9} + \beta_{10} ) q^{23} + ( 512 \beta_{1} - 1024 \beta_{2} ) q^{24} + ( -38931 - 38931 \beta_{2} + 337 \beta_{3} + 26 \beta_{4} - 22 \beta_{8} + 63 \beta_{9} ) q^{25} + ( -23216 + 576 \beta_{1} + 576 \beta_{3} - 24 \beta_{4} + 24 \beta_{6} - 56 \beta_{7} + 56 \beta_{9} - 8 \beta_{10} ) q^{26} + ( 45652 - 1192 \beta_{1} - 1192 \beta_{3} + 303 \beta_{4} - 303 \beta_{6} + 90 \beta_{7} - 90 \beta_{9} - 27 \beta_{10} - 12 \beta_{11} ) q^{27} + ( 5568 + 5568 \beta_{2} - 192 \beta_{3} + 64 \beta_{4} - 64 \beta_{9} ) q^{28} + ( 18255 + 18255 \beta_{2} + 400 \beta_{3} - 159 \beta_{4} + 34 \beta_{5} + 20 \beta_{8} + 34 \beta_{9} + 34 \beta_{10} ) q^{29} + ( 5424 + 840 \beta_{1} + 840 \beta_{3} - 72 \beta_{4} + 72 \beta_{6} + 48 \beta_{7} - 48 \beta_{9} + 24 \beta_{10} + 24 \beta_{11} ) q^{30} + ( -30533 - 1561 \beta_{1} - 1561 \beta_{3} + 165 \beta_{4} - 165 \beta_{6} - 7 \beta_{7} + 7 \beta_{9} + 8 \beta_{10} - 14 \beta_{11} ) q^{31} + ( -32768 - 32768 \beta_{2} ) q^{32} + ( -630 \beta_{1} + 8579 \beta_{2} - 25 \beta_{5} + 513 \beta_{6} + 21 \beta_{7} + 30 \beta_{8} - 30 \beta_{11} ) q^{33} + ( 60256 + 60256 \beta_{2} - 568 \beta_{3} - 208 \beta_{4} - 48 \beta_{5} + 16 \beta_{8} + 8 \beta_{9} - 48 \beta_{10} ) q^{34} + ( -905 \beta_{1} + 88756 \beta_{2} - 47 \beta_{5} + 509 \beta_{6} - 56 \beta_{7} + 28 \beta_{8} - 28 \beta_{11} ) q^{35} + ( 832 \beta_{1} + 77184 \beta_{2} - 64 \beta_{5} + 192 \beta_{6} ) q^{36} + ( 76877 + 1662 \beta_{1} + 1662 \beta_{3} + 22 \beta_{4} - 22 \beta_{6} + \beta_{7} - \beta_{9} + 61 \beta_{10} + 12 \beta_{11} ) q^{37} + ( 45816 - 960 \beta_{1} + 5416 \beta_{2} + 296 \beta_{3} - 96 \beta_{4} - 16 \beta_{5} + 56 \beta_{6} + 40 \beta_{8} - 8 \beta_{9} - 72 \beta_{10} ) q^{38} + ( -222785 + 1990 \beta_{1} + 1990 \beta_{3} + 330 \beta_{4} - 330 \beta_{6} + 273 \beta_{7} - 273 \beta_{9} + 167 \beta_{10} - 15 \beta_{11} ) q^{39} + ( -10752 \beta_{2} - 512 \beta_{6} ) q^{40} + ( -4225 \beta_{1} - 81753 \beta_{2} + 33 \beta_{5} - 1105 \beta_{6} - 400 \beta_{7} - 16 \beta_{8} + 16 \beta_{11} ) q^{41} + ( 61176 + 61176 \beta_{2} - 1136 \beta_{3} - 240 \beta_{4} - 152 \beta_{5} + 216 \beta_{9} - 152 \beta_{10} ) q^{42} + ( 2194 \beta_{1} - 296941 \beta_{2} + 151 \beta_{5} - 378 \beta_{6} + 105 \beta_{7} + 55 \beta_{8} - 55 \beta_{11} ) q^{43} + ( 62784 + 62784 \beta_{2} + 128 \beta_{3} + 192 \beta_{4} + 64 \beta_{8} + 64 \beta_{9} ) q^{44} + ( -319356 + 9216 \beta_{1} + 9216 \beta_{3} - 954 \beta_{4} + 954 \beta_{6} - 342 \beta_{7} + 342 \beta_{9} + 126 \beta_{10} ) q^{45} + ( -101728 + 2984 \beta_{1} + 2984 \beta_{3} + 936 \beta_{4} - 936 \beta_{6} + 64 \beta_{7} - 64 \beta_{9} - 8 \beta_{10} + 40 \beta_{11} ) q^{46} + ( 29372 + 29372 \beta_{2} + 801 \beta_{3} - 629 \beta_{4} - 151 \beta_{5} - 3 \beta_{8} + 576 \beta_{9} - 151 \beta_{10} ) q^{47} + ( 8192 + 8192 \beta_{2} + 4096 \beta_{3} ) q^{48} + ( 8087 + 6041 \beta_{1} + 6041 \beta_{3} + 501 \beta_{4} - 501 \beta_{6} - 282 \beta_{7} + 282 \beta_{9} - 21 \beta_{10} - 20 \beta_{11} ) q^{49} + ( 311448 - 2696 \beta_{1} - 2696 \beta_{3} - 208 \beta_{4} + 208 \beta_{6} + 504 \beta_{7} - 504 \beta_{9} + 176 \beta_{11} ) q^{50} + ( 253765 + 253765 \beta_{2} - 15549 \beta_{3} + 285 \beta_{4} + 28 \beta_{5} + 21 \beta_{8} - 861 \beta_{9} + 28 \beta_{10} ) q^{51} + ( -4608 \beta_{1} - 185728 \beta_{2} - 64 \beta_{5} - 192 \beta_{6} + 448 \beta_{7} ) q^{52} + ( 156270 + 156270 \beta_{2} + 782 \beta_{3} - 891 \beta_{4} - 209 \beta_{5} - 32 \beta_{8} + 9 \beta_{9} - 209 \beta_{10} ) q^{53} + ( 9536 \beta_{1} + 365216 \beta_{2} - 216 \beta_{5} + 2424 \beta_{6} - 720 \beta_{7} - 96 \beta_{8} + 96 \beta_{11} ) q^{54} + ( -16679 \beta_{1} + 455449 \beta_{2} + 148 \beta_{5} + 4291 \beta_{6} - 635 \beta_{7} + 4 \beta_{8} - 4 \beta_{11} ) q^{55} + ( -44544 + 1536 \beta_{1} + 1536 \beta_{3} - 512 \beta_{4} + 512 \beta_{6} - 512 \beta_{7} + 512 \beta_{9} ) q^{56} + ( -115261 - 18344 \beta_{1} - 527080 \beta_{2} - 16389 \beta_{3} - 1044 \beta_{4} + 244 \beta_{5} - 1956 \beta_{6} + 798 \beta_{7} - 78 \beta_{8} - 1113 \beta_{9} + 110 \beta_{10} ) q^{57} + ( -146040 - 3200 \beta_{1} - 3200 \beta_{3} + 1272 \beta_{4} - 1272 \beta_{6} + 272 \beta_{7} - 272 \beta_{9} - 272 \beta_{10} - 160 \beta_{11} ) q^{58} + ( 456 \beta_{1} - 313266 \beta_{2} - 253 \beta_{5} - 93 \beta_{6} + 2140 \beta_{7} - 30 \beta_{8} + 30 \beta_{11} ) q^{59} + ( -6720 \beta_{1} + 43392 \beta_{2} + 192 \beta_{5} - 576 \beta_{6} - 384 \beta_{7} + 192 \beta_{8} - 192 \beta_{11} ) q^{60} + ( -299251 - 299251 \beta_{2} + 5384 \beta_{3} - 1135 \beta_{4} + 292 \beta_{5} + 40 \beta_{8} + 1036 \beta_{9} + 292 \beta_{10} ) q^{61} + ( 12488 \beta_{1} - 244264 \beta_{2} + 64 \beta_{5} + 1320 \beta_{6} + 56 \beta_{7} - 112 \beta_{8} + 112 \beta_{11} ) q^{62} + ( 370204 + 370204 \beta_{2} - 35488 \beta_{3} + 3768 \beta_{4} + 204 \beta_{5} - 48 \beta_{8} - 612 \beta_{9} + 204 \beta_{10} ) q^{63} + 262144 q^{64} + ( 218096 + 9168 \beta_{1} + 9168 \beta_{3} + 6859 \beta_{4} - 6859 \beta_{6} - 397 \beta_{7} + 397 \beta_{9} - 395 \beta_{10} - 18 \beta_{11} ) q^{65} + ( -68632 - 68632 \beta_{2} - 5040 \beta_{3} - 4104 \beta_{4} + 200 \beta_{5} - 240 \beta_{8} - 168 \beta_{9} + 200 \beta_{10} ) q^{66} + ( 895080 + 895080 \beta_{2} + 27578 \beta_{3} + 3441 \beta_{4} + 139 \beta_{5} - 182 \beta_{8} + 146 \beta_{9} + 139 \beta_{10} ) q^{67} + ( -482048 + 4544 \beta_{1} + 4544 \beta_{3} + 1664 \beta_{4} - 1664 \beta_{6} + 64 \beta_{7} - 64 \beta_{9} + 384 \beta_{10} - 128 \beta_{11} ) q^{68} + ( -1323421 + 10584 \beta_{1} + 10584 \beta_{3} - 3123 \beta_{4} + 3123 \beta_{6} - 672 \beta_{7} + 672 \beta_{9} + 8 \beta_{10} - 420 \beta_{11} ) q^{69} + ( -710048 - 710048 \beta_{2} - 7240 \beta_{3} - 4072 \beta_{4} + 376 \beta_{5} - 224 \beta_{8} + 448 \beta_{9} + 376 \beta_{10} ) q^{70} + ( -11102 \beta_{1} - 663115 \beta_{2} + 217 \beta_{5} - 854 \beta_{6} - 1759 \beta_{7} - 365 \beta_{8} + 365 \beta_{11} ) q^{71} + ( -617472 - 617472 \beta_{2} + 6656 \beta_{3} - 1536 \beta_{4} + 512 \beta_{5} + 512 \beta_{10} ) q^{72} + ( -10003 \beta_{1} - 2547191 \beta_{2} + 753 \beta_{5} - 1893 \beta_{6} + 490 \beta_{7} - 28 \beta_{8} + 28 \beta_{11} ) q^{73} + ( -13296 \beta_{1} + 615016 \beta_{2} + 488 \beta_{5} + 176 \beta_{6} - 8 \beta_{7} + 96 \beta_{8} - 96 \beta_{11} ) q^{74} + ( 980935 - 22906 \beta_{1} - 22906 \beta_{3} - 15051 \beta_{4} + 15051 \beta_{6} - 771 \beta_{7} + 771 \beta_{9} - 660 \beta_{10} - 129 \beta_{11} ) q^{75} + ( -43328 - 2368 \beta_{1} + 323200 \beta_{2} - 10048 \beta_{3} + 320 \beta_{4} - 448 \beta_{5} - 768 \beta_{6} - 64 \beta_{7} + 64 \beta_{9} + 128 \beta_{10} - 320 \beta_{11} ) q^{76} + ( -198979 + 7408 \beta_{1} + 7408 \beta_{3} + 6184 \beta_{4} - 6184 \beta_{6} + 303 \beta_{7} - 303 \beta_{9} - 215 \beta_{10} + 848 \beta_{11} ) q^{77} + ( -15920 \beta_{1} - 1782280 \beta_{2} + 1336 \beta_{5} + 2640 \beta_{6} - 2184 \beta_{7} - 120 \beta_{8} + 120 \beta_{11} ) q^{78} + ( -52932 \beta_{1} + 1191935 \beta_{2} - 761 \beta_{5} - 5778 \beta_{6} - 481 \beta_{7} - 255 \beta_{8} + 255 \beta_{11} ) q^{79} + ( 86016 + 86016 \beta_{2} + 4096 \beta_{4} ) q^{80} + ( 108259 \beta_{1} + 1001553 \beta_{2} - 1447 \beta_{5} + 8805 \beta_{6} + 1008 \beta_{7} - 504 \beta_{8} + 504 \beta_{11} ) q^{81} + ( 654024 + 654024 \beta_{2} - 33800 \beta_{3} + 8840 \beta_{4} - 264 \beta_{5} + 128 \beta_{8} + 3200 \beta_{9} - 264 \beta_{10} ) q^{82} + ( -1077613 + 89977 \beta_{1} + 89977 \beta_{3} - 3608 \beta_{4} + 3608 \beta_{6} + 833 \beta_{7} - 833 \beta_{9} - 897 \beta_{10} + 35 \beta_{11} ) q^{83} + ( -489408 + 9088 \beta_{1} + 9088 \beta_{3} + 1920 \beta_{4} - 1920 \beta_{6} + 1728 \beta_{7} - 1728 \beta_{9} + 1216 \beta_{10} ) q^{84} + ( 3064572 + 3064572 \beta_{2} + 3216 \beta_{3} - 15393 \beta_{4} + 489 \beta_{5} + 1068 \beta_{8} + 201 \beta_{9} + 489 \beta_{10} ) q^{85} + ( 2375528 + 2375528 \beta_{2} + 17552 \beta_{3} + 3024 \beta_{4} - 1208 \beta_{5} - 440 \beta_{8} - 840 \beta_{9} - 1208 \beta_{10} ) q^{86} + ( 1278060 - 70209 \beta_{1} - 70209 \beta_{3} + 20697 \beta_{4} - 20697 \beta_{6} + 1644 \beta_{7} - 1644 \beta_{9} - 1869 \beta_{10} - 117 \beta_{11} ) q^{87} + ( -502272 - 1024 \beta_{1} - 1024 \beta_{3} - 1536 \beta_{4} + 1536 \beta_{6} + 512 \beta_{7} - 512 \beta_{9} - 512 \beta_{11} ) q^{88} + ( 828604 + 828604 \beta_{2} - 35401 \beta_{3} + 2804 \beta_{4} - 1972 \beta_{5} + 550 \beta_{8} - 1419 \beta_{9} - 1972 \beta_{10} ) q^{89} + ( -73728 \beta_{1} - 2554848 \beta_{2} + 1008 \beta_{5} - 7632 \beta_{6} + 2736 \beta_{7} ) q^{90} + ( -6169234 - 6169234 \beta_{2} - 56838 \beta_{3} - 958 \beta_{4} + 1520 \beta_{5} + 120 \beta_{8} - 722 \beta_{9} + 1520 \beta_{10} ) q^{91} + ( -23872 \beta_{1} - 813824 \beta_{2} - 64 \beta_{5} + 7488 \beta_{6} - 512 \beta_{7} + 320 \beta_{8} - 320 \beta_{11} ) q^{92} + ( 2510 \beta_{1} + 5240971 \beta_{2} - 2191 \beta_{5} + 12756 \beta_{6} + 1959 \beta_{7} - 240 \beta_{8} + 240 \beta_{11} ) q^{93} + ( -234976 - 6408 \beta_{1} - 6408 \beta_{3} + 5032 \beta_{4} - 5032 \beta_{6} + 4608 \beta_{7} - 4608 \beta_{9} + 1208 \beta_{10} + 24 \beta_{11} ) q^{94} + ( 1431517 + 58121 \beta_{1} + 312113 \beta_{2} - 43111 \beta_{3} - 23579 \beta_{4} - 779 \beta_{5} + 4921 \beta_{6} - 456 \beta_{7} - 57 \beta_{8} + 3781 \beta_{9} + 190 \beta_{10} + 703 \beta_{11} ) q^{95} + ( -65536 - 32768 \beta_{1} - 32768 \beta_{3} ) q^{96} + ( -30640 \beta_{1} + 1714671 \beta_{2} - 2796 \beta_{5} + 1358 \beta_{6} - 5256 \beta_{7} + 1220 \beta_{8} - 1220 \beta_{11} ) q^{97} + ( -48328 \beta_{1} + 64696 \beta_{2} - 168 \beta_{5} + 4008 \beta_{6} + 2256 \beta_{7} - 160 \beta_{8} + 160 \beta_{11} ) q^{98} + ( 247844 + 247844 \beta_{2} + 63141 \beta_{3} + 1239 \beta_{4} + 191 \beta_{5} + 30 \beta_{8} - 1584 \beta_{9} + 191 \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 48q^{2} + 12q^{3} - 384q^{4} + 124q^{5} + 96q^{6} - 1036q^{7} + 6144q^{8} - 7232q^{9} + O(q^{10}) \) \( 12q - 48q^{2} + 12q^{3} - 384q^{4} + 124q^{5} + 96q^{6} - 1036q^{7} + 6144q^{8} - 7232q^{9} + 992q^{10} - 11760q^{11} - 1536q^{12} + 17390q^{13} + 4144q^{14} - 4062q^{15} - 24576q^{16} + 45254q^{17} + 115712q^{18} - 26264q^{19} - 15872q^{20} + 46034q^{21} + 47040q^{22} + 76554q^{23} + 6144q^{24} - 233468q^{25} - 278240q^{26} + 546408q^{27} + 33152q^{28} + 109808q^{29} + 64992q^{30} - 367004q^{31} - 196608q^{32} - 50480q^{33} + 362032q^{34} - 531444q^{35} - 462848q^{36} + 922140q^{37} + 517760q^{38} - 2676440q^{39} + 63488q^{40} + 489142q^{41} + 368272q^{42} + 1781092q^{43} + 376320q^{44} - 3827592q^{45} - 1224864q^{46} + 178950q^{47} + 49152q^{48} + 96332q^{49} + 3735488q^{50} + 1520200q^{51} + 1112960q^{52} + 939902q^{53} - 2185632q^{54} - 2722538q^{55} - 530432q^{56} + 1773906q^{57} - 1756928q^{58} + 1874564q^{59} - 259968q^{60} - 1791828q^{61} + 1468016q^{62} + 2212152q^{63} + 3145728q^{64} + 2592956q^{65} - 403840q^{66} + 5363976q^{67} - 5792512q^{68} - 15864224q^{69} - 4251552q^{70} + 3980204q^{71} - 3702784q^{72} + 15279830q^{73} - 3688560q^{74} + 11837664q^{75} - 2461184q^{76} - 2416228q^{77} + 10705760q^{78} - 7164236q^{79} + 507904q^{80} - 5997626q^{81} + 3913136q^{82} - 12916808q^{83} - 5892352q^{84} + 18415506q^{85} + 14248736q^{86} + 15255300q^{87} - 6021120q^{88} + 4966022q^{89} + 15310368q^{90} - 37018212q^{91} + 4899456q^{92} - 31429094q^{93} - 2863200q^{94} + 15365984q^{95} - 786432q^{96} - 10277950q^{97} - 385328q^{98} + 1480976q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 10165 x^{10} - 228496 x^{9} + 79937698 x^{8} - 1362528496 x^{7} + 222057453949 x^{6} - 1418140039384 x^{5} + 423936076682002 x^{4} - 1421321152056072 x^{3} + 376701330885210661 x^{2} + 2892184322477465880 x + 206636219229734030025\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(\)\(14\!\cdots\!36\)\( \nu^{11} - \)\(62\!\cdots\!34\)\( \nu^{10} + \)\(13\!\cdots\!08\)\( \nu^{9} - \)\(91\!\cdots\!77\)\( \nu^{8} + \)\(11\!\cdots\!84\)\( \nu^{7} - \)\(63\!\cdots\!50\)\( \nu^{6} + \)\(30\!\cdots\!16\)\( \nu^{5} - \)\(12\!\cdots\!54\)\( \nu^{4} + \)\(48\!\cdots\!88\)\( \nu^{3} - \)\(27\!\cdots\!42\)\( \nu^{2} + \)\(30\!\cdots\!24\)\( \nu - \)\(21\!\cdots\!25\)\(\)\()/ \)\(18\!\cdots\!95\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(19\!\cdots\!06\)\( \nu^{11} - \)\(37\!\cdots\!88\)\( \nu^{10} - \)\(18\!\cdots\!39\)\( \nu^{9} + \)\(11\!\cdots\!04\)\( \nu^{8} - \)\(13\!\cdots\!46\)\( \nu^{7} - \)\(22\!\cdots\!32\)\( \nu^{6} - \)\(31\!\cdots\!70\)\( \nu^{5} - \)\(34\!\cdots\!56\)\( \nu^{4} - \)\(81\!\cdots\!50\)\( \nu^{3} - \)\(68\!\cdots\!48\)\( \nu^{2} - \)\(79\!\cdots\!95\)\( \nu - \)\(90\!\cdots\!00\)\(\)\()/ \)\(58\!\cdots\!05\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(22\!\cdots\!62\)\( \nu^{11} + \)\(62\!\cdots\!52\)\( \nu^{10} + \)\(38\!\cdots\!51\)\( \nu^{9} + \)\(34\!\cdots\!96\)\( \nu^{8} + \)\(28\!\cdots\!38\)\( \nu^{7} - \)\(57\!\cdots\!80\)\( \nu^{6} + \)\(12\!\cdots\!47\)\( \nu^{5} + \)\(47\!\cdots\!92\)\( \nu^{4} + \)\(16\!\cdots\!46\)\( \nu^{3} + \)\(25\!\cdots\!56\)\( \nu^{2} + \)\(15\!\cdots\!83\)\( \nu + \)\(17\!\cdots\!40\)\(\)\()/ \)\(12\!\cdots\!60\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(91\!\cdots\!95\)\( \nu^{11} - \)\(49\!\cdots\!52\)\( \nu^{10} + \)\(20\!\cdots\!94\)\( \nu^{9} - \)\(47\!\cdots\!88\)\( \nu^{8} + \)\(30\!\cdots\!63\)\( \nu^{7} - \)\(39\!\cdots\!52\)\( \nu^{6} + \)\(16\!\cdots\!06\)\( \nu^{5} - \)\(10\!\cdots\!20\)\( \nu^{4} + \)\(27\!\cdots\!83\)\( \nu^{3} - \)\(18\!\cdots\!60\)\( \nu^{2} + \)\(33\!\cdots\!34\)\( \nu - \)\(13\!\cdots\!00\)\(\)\()/ \)\(41\!\cdots\!20\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(70\!\cdots\!57\)\( \nu^{11} - \)\(12\!\cdots\!60\)\( \nu^{10} - \)\(54\!\cdots\!70\)\( \nu^{9} + \)\(81\!\cdots\!52\)\( \nu^{8} - \)\(36\!\cdots\!61\)\( \nu^{7} - \)\(66\!\cdots\!28\)\( \nu^{6} - \)\(25\!\cdots\!38\)\( \nu^{5} - \)\(11\!\cdots\!12\)\( \nu^{4} - \)\(18\!\cdots\!09\)\( \nu^{3} - \)\(68\!\cdots\!76\)\( \nu^{2} + \)\(59\!\cdots\!98\)\( \nu - \)\(36\!\cdots\!00\)\(\)\()/ \)\(12\!\cdots\!60\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(97\!\cdots\!03\)\( \nu^{11} - \)\(41\!\cdots\!36\)\( \nu^{10} - \)\(67\!\cdots\!98\)\( \nu^{9} - \)\(21\!\cdots\!46\)\( \nu^{8} - \)\(39\!\cdots\!95\)\( \nu^{7} - \)\(27\!\cdots\!08\)\( \nu^{6} + \)\(75\!\cdots\!26\)\( \nu^{5} - \)\(11\!\cdots\!38\)\( \nu^{4} + \)\(16\!\cdots\!93\)\( \nu^{3} - \)\(13\!\cdots\!64\)\( \nu^{2} + \)\(58\!\cdots\!34\)\( \nu - \)\(10\!\cdots\!50\)\(\)\()/ \)\(13\!\cdots\!40\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(22\!\cdots\!68\)\( \nu^{11} + \)\(68\!\cdots\!45\)\( \nu^{10} - \)\(20\!\cdots\!60\)\( \nu^{9} + \)\(74\!\cdots\!43\)\( \nu^{8} - \)\(15\!\cdots\!24\)\( \nu^{7} + \)\(50\!\cdots\!33\)\( \nu^{6} - \)\(13\!\cdots\!92\)\( \nu^{5} + \)\(11\!\cdots\!27\)\( \nu^{4} + \)\(57\!\cdots\!64\)\( \nu^{3} + \)\(16\!\cdots\!81\)\( \nu^{2} + \)\(18\!\cdots\!52\)\( \nu + \)\(68\!\cdots\!35\)\(\)\()/ \)\(12\!\cdots\!60\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(25\!\cdots\!90\)\( \nu^{11} - \)\(18\!\cdots\!22\)\( \nu^{10} + \)\(27\!\cdots\!69\)\( \nu^{9} - \)\(77\!\cdots\!18\)\( \nu^{8} + \)\(21\!\cdots\!38\)\( \nu^{7} - \)\(49\!\cdots\!82\)\( \nu^{6} + \)\(65\!\cdots\!41\)\( \nu^{5} - \)\(67\!\cdots\!10\)\( \nu^{4} + \)\(11\!\cdots\!38\)\( \nu^{3} - \)\(28\!\cdots\!90\)\( \nu^{2} + \)\(95\!\cdots\!29\)\( \nu + \)\(58\!\cdots\!70\)\(\)\()/ \)\(13\!\cdots\!40\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(44\!\cdots\!61\)\( \nu^{11} + \)\(40\!\cdots\!28\)\( \nu^{10} + \)\(43\!\cdots\!99\)\( \nu^{9} - \)\(61\!\cdots\!64\)\( \nu^{8} + \)\(32\!\cdots\!81\)\( \nu^{7} - \)\(28\!\cdots\!48\)\( \nu^{6} + \)\(72\!\cdots\!15\)\( \nu^{5} + \)\(60\!\cdots\!76\)\( \nu^{4} + \)\(10\!\cdots\!25\)\( \nu^{3} + \)\(13\!\cdots\!08\)\( \nu^{2} + \)\(46\!\cdots\!15\)\( \nu + \)\(44\!\cdots\!80\)\(\)\()/ \)\(12\!\cdots\!80\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(54\!\cdots\!16\)\( \nu^{11} + \)\(38\!\cdots\!53\)\( \nu^{10} + \)\(58\!\cdots\!92\)\( \nu^{9} + \)\(18\!\cdots\!58\)\( \nu^{8} + \)\(34\!\cdots\!52\)\( \nu^{7} + \)\(14\!\cdots\!85\)\( \nu^{6} + \)\(83\!\cdots\!04\)\( \nu^{5} + \)\(21\!\cdots\!14\)\( \nu^{4} + \)\(17\!\cdots\!48\)\( \nu^{3} + \)\(29\!\cdots\!17\)\( \nu^{2} + \)\(66\!\cdots\!60\)\( \nu - \)\(12\!\cdots\!82\)\(\)\()/ \)\(77\!\cdots\!08\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{10} + \beta_{5} - 3 \beta_{4} + 17 \beta_{3} - 3389 \beta_{2} - 3389\)
\(\nu^{3}\)\(=\)\(-12 \beta_{11} - 33 \beta_{10} - 90 \beta_{9} + 90 \beta_{7} - 321 \beta_{6} + 321 \beta_{4} - 5656 \beta_{3} - 5656 \beta_{1} + 57246\)
\(\nu^{4}\)\(=\)\(600 \beta_{11} - 600 \beta_{8} + 288 \beta_{7} + 30984 \beta_{6} - 8248 \beta_{5} + 18856673 \beta_{2} + 238424 \beta_{1}\)
\(\nu^{5}\)\(=\)\(396768 \beta_{10} + 699552 \beta_{9} + 108960 \beta_{8} + 396768 \beta_{5} - 3373896 \beta_{4} + 40158097 \beta_{3} - 801300360 \beta_{2} - 801300360\)
\(\nu^{6}\)\(=\)\(-7469976 \beta_{11} - 64221577 \beta_{10} - 7354800 \beta_{9} + 7354800 \beta_{7} - 281457387 \beta_{6} + 281457387 \beta_{4} - 2470947113 \beta_{3} - 2470947113 \beta_{1} + 133326061805\)
\(\nu^{7}\)\(=\)\(895452876 \beta_{11} - 895452876 \beta_{8} - 4972985226 \beta_{7} + 29723881977 \beta_{6} - 4002412257 \beta_{5} + 8278739888598 \beta_{2} + 314109844408 \beta_{1}\)
\(\nu^{8}\)\(=\)\(512960788720 \beta_{10} + 143312360832 \beta_{9} + 71932681008 \beta_{8} + 512960788720 \beta_{5} - 2500314222864 \beta_{4} + 23234933989232 \beta_{3} - 1042956318730433 \beta_{2} - 1042956318730433\)
\(\nu^{9}\)\(=\)\(-7458965056128 \beta_{11} - 37556383808832 \beta_{10} - 36380156730624 \beta_{9} + 36380156730624 \beta_{7} - 253765792372560 \beta_{6} + 253765792372560 \beta_{4} - 2569282242509281 \beta_{3} - 2569282242509281 \beta_{1} + 77672576049678672\)
\(\nu^{10}\)\(=\)\(645482617454256 \beta_{11} - 645482617454256 \beta_{8} - 1708010348226336 \beta_{7} + 21995003545915731 \beta_{6} - 4208136679853329 \beta_{5} + 8538889026899898269 \beta_{2} + 209622837978924161 \beta_{1}\)
\(\nu^{11}\)\(=\)\(339533109634217889 \beta_{10} + 278437856418188154 \beta_{9} + 63157667753360076 \beta_{8} + 339533109634217889 \beta_{5} - 2161828332710597361 \beta_{4} + 21504535967549531032 \beta_{3} - 699755288323892444238 \beta_{2} - 699755288323892444238\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).

\(n\) \(21\)
\(\chi(n)\) \(-1 - \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−46.4369 80.4310i
−21.5062 37.2499i
−12.7080 22.0110i
19.2647 + 33.3674i
25.8545 + 44.7813i
35.5320 + 61.5431i
−46.4369 + 80.4310i
−21.5062 + 37.2499i
−12.7080 + 22.0110i
19.2647 33.3674i
25.8545 44.7813i
35.5320 61.5431i
−4.00000 6.92820i −45.4369 78.6990i −32.0000 + 55.4256i 149.770 + 259.410i −363.495 + 629.592i −708.303 512.000 −3035.52 + 5257.68i 1198.16 2075.28i
7.2 −4.00000 6.92820i −20.5062 35.5178i −32.0000 + 55.4256i −273.130 473.076i −164.050 + 284.143i −154.247 512.000 252.489 437.324i −2185.04 + 3784.60i
7.3 −4.00000 6.92820i −11.7080 20.2789i −32.0000 + 55.4256i 68.7885 + 119.145i −93.6643 + 162.231i 1024.63 512.000 819.344 1419.14i 550.308 953.162i
7.4 −4.00000 6.92820i 20.2647 + 35.0995i −32.0000 + 55.4256i −75.0460 129.983i 162.117 280.796i −892.194 512.000 272.185 471.439i −600.368 + 1039.87i
7.5 −4.00000 6.92820i 26.8545 + 46.5134i −32.0000 + 55.4256i 253.607 + 439.261i 214.836 372.107i −1033.15 512.000 −348.830 + 604.192i 2028.86 3514.09i
7.6 −4.00000 6.92820i 36.5320 + 63.2752i −32.0000 + 55.4256i −61.9899 107.370i 292.256 506.202i 1245.26 512.000 −1575.67 + 2729.14i −495.919 + 858.957i
11.1 −4.00000 + 6.92820i −45.4369 + 78.6990i −32.0000 55.4256i 149.770 259.410i −363.495 629.592i −708.303 512.000 −3035.52 5257.68i 1198.16 + 2075.28i
11.2 −4.00000 + 6.92820i −20.5062 + 35.5178i −32.0000 55.4256i −273.130 + 473.076i −164.050 284.143i −154.247 512.000 252.489 + 437.324i −2185.04 3784.60i
11.3 −4.00000 + 6.92820i −11.7080 + 20.2789i −32.0000 55.4256i 68.7885 119.145i −93.6643 162.231i 1024.63 512.000 819.344 + 1419.14i 550.308 + 953.162i
11.4 −4.00000 + 6.92820i 20.2647 35.0995i −32.0000 55.4256i −75.0460 + 129.983i 162.117 + 280.796i −892.194 512.000 272.185 + 471.439i −600.368 1039.87i
11.5 −4.00000 + 6.92820i 26.8545 46.5134i −32.0000 55.4256i 253.607 439.261i 214.836 + 372.107i −1033.15 512.000 −348.830 604.192i 2028.86 + 3514.09i
11.6 −4.00000 + 6.92820i 36.5320 63.2752i −32.0000 55.4256i −61.9899 + 107.370i 292.256 + 506.202i 1245.26 512.000 −1575.67 2729.14i −495.919 858.957i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.8.c.a 12
19.c even 3 1 inner 38.8.c.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.8.c.a 12 1.a even 1 1 trivial
38.8.c.a 12 19.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{12} - \cdots\) acting on \(S_{8}^{\mathrm{new}}(38, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 8 T + 64 T^{2} )^{6} \)
$3$ \( 1 - 12 T - 2873 T^{2} - 164580 T^{3} + 4066231 T^{4} + 748062720 T^{5} - 1262640456 T^{6} - 1266950128896 T^{7} - 53277377598855 T^{8} + 3013227548238348 T^{9} + 199847003133469617 T^{10} - 4192018663187042460 T^{11} - \)\(43\!\cdots\!26\)\( T^{12} - \)\(91\!\cdots\!20\)\( T^{13} + \)\(95\!\cdots\!73\)\( T^{14} + \)\(31\!\cdots\!44\)\( T^{15} - \)\(12\!\cdots\!55\)\( T^{16} - \)\(63\!\cdots\!72\)\( T^{17} - \)\(13\!\cdots\!04\)\( T^{18} + \)\(17\!\cdots\!60\)\( T^{19} + \)\(21\!\cdots\!51\)\( T^{20} - \)\(18\!\cdots\!60\)\( T^{21} - \)\(71\!\cdots\!77\)\( T^{22} - \)\(65\!\cdots\!56\)\( T^{23} + \)\(11\!\cdots\!81\)\( T^{24} \)
$5$ \( 1 - 124 T - 109953 T^{2} - 10653200 T^{3} + 13660736644 T^{4} + 2491824506468 T^{5} - 951717995179701 T^{6} - 348405214989849360 T^{7} + 17046000230360559550 T^{8} + \)\(25\!\cdots\!00\)\( T^{9} + \)\(48\!\cdots\!75\)\( T^{10} - \)\(12\!\cdots\!00\)\( T^{11} - \)\(35\!\cdots\!00\)\( T^{12} - \)\(96\!\cdots\!00\)\( T^{13} + \)\(29\!\cdots\!75\)\( T^{14} + \)\(12\!\cdots\!00\)\( T^{15} + \)\(63\!\cdots\!50\)\( T^{16} - \)\(10\!\cdots\!00\)\( T^{17} - \)\(21\!\cdots\!25\)\( T^{18} + \)\(44\!\cdots\!00\)\( T^{19} + \)\(18\!\cdots\!00\)\( T^{20} - \)\(11\!\cdots\!00\)\( T^{21} - \)\(93\!\cdots\!25\)\( T^{22} - \)\(82\!\cdots\!00\)\( T^{23} + \)\(51\!\cdots\!25\)\( T^{24} \)
$7$ \( ( 1 + 518 T + 2580708 T^{2} + 585636002 T^{3} + 3650249323703 T^{4} + 744544606791020 T^{5} + 3757269319629230376 T^{6} + \)\(61\!\cdots\!60\)\( T^{7} + \)\(24\!\cdots\!47\)\( T^{8} + \)\(32\!\cdots\!14\)\( T^{9} + \)\(11\!\cdots\!08\)\( T^{10} + \)\(19\!\cdots\!74\)\( T^{11} + \)\(31\!\cdots\!49\)\( T^{12} )^{2} \)
$11$ \( ( 1 + 5880 T + 63552354 T^{2} + 235904599808 T^{3} + 1242892621867374 T^{4} + 2953394425115410992 T^{5} + \)\(15\!\cdots\!62\)\( T^{6} + \)\(57\!\cdots\!32\)\( T^{7} + \)\(47\!\cdots\!34\)\( T^{8} + \)\(17\!\cdots\!88\)\( T^{9} + \)\(91\!\cdots\!74\)\( T^{10} + \)\(16\!\cdots\!80\)\( T^{11} + \)\(54\!\cdots\!21\)\( T^{12} )^{2} \)
$13$ \( 1 - 17390 T - 24450759 T^{2} - 147352515886 T^{3} + 30076752641644770 T^{4} - 55347895095666983378 T^{5} - \)\(11\!\cdots\!65\)\( T^{6} - \)\(19\!\cdots\!22\)\( T^{7} + \)\(11\!\cdots\!12\)\( T^{8} + \)\(73\!\cdots\!42\)\( T^{9} + \)\(87\!\cdots\!73\)\( T^{10} - \)\(76\!\cdots\!22\)\( T^{11} - \)\(24\!\cdots\!48\)\( T^{12} - \)\(47\!\cdots\!74\)\( T^{13} + \)\(34\!\cdots\!97\)\( T^{14} + \)\(18\!\cdots\!46\)\( T^{15} + \)\(18\!\cdots\!52\)\( T^{16} - \)\(19\!\cdots\!54\)\( T^{17} - \)\(70\!\cdots\!85\)\( T^{18} - \)\(21\!\cdots\!94\)\( T^{19} + \)\(72\!\cdots\!70\)\( T^{20} - \)\(22\!\cdots\!42\)\( T^{21} - \)\(23\!\cdots\!91\)\( T^{22} - \)\(10\!\cdots\!70\)\( T^{23} + \)\(37\!\cdots\!61\)\( T^{24} \)
$17$ \( 1 - 45254 T - 234580727 T^{2} + 38700949741762 T^{3} - 254353422176226298 T^{4} - \)\(10\!\cdots\!22\)\( T^{5} - \)\(52\!\cdots\!89\)\( T^{6} + \)\(26\!\cdots\!34\)\( T^{7} + \)\(17\!\cdots\!92\)\( T^{8} - \)\(27\!\cdots\!38\)\( T^{9} - \)\(62\!\cdots\!79\)\( T^{10} + \)\(80\!\cdots\!26\)\( T^{11} + \)\(12\!\cdots\!00\)\( T^{12} + \)\(33\!\cdots\!98\)\( T^{13} - \)\(10\!\cdots\!91\)\( T^{14} - \)\(19\!\cdots\!46\)\( T^{15} + \)\(48\!\cdots\!72\)\( T^{16} + \)\(30\!\cdots\!62\)\( T^{17} - \)\(25\!\cdots\!21\)\( T^{18} - \)\(20\!\cdots\!34\)\( T^{19} - \)\(20\!\cdots\!38\)\( T^{20} + \)\(12\!\cdots\!06\)\( T^{21} - \)\(31\!\cdots\!23\)\( T^{22} - \)\(25\!\cdots\!58\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \)
$19$ \( 1 + 26264 T + 2570714186 T^{2} + 64689568823400 T^{3} + 2655238247551933174 T^{4} + \)\(74\!\cdots\!56\)\( T^{5} + \)\(20\!\cdots\!54\)\( T^{6} + \)\(66\!\cdots\!84\)\( T^{7} + \)\(21\!\cdots\!54\)\( T^{8} + \)\(46\!\cdots\!00\)\( T^{9} + \)\(16\!\cdots\!26\)\( T^{10} + \)\(14\!\cdots\!36\)\( T^{11} + \)\(51\!\cdots\!61\)\( T^{12} \)
$23$ \( 1 - 76554 T - 8951711043 T^{2} - 29916559914278 T^{3} + \)\(10\!\cdots\!96\)\( T^{4} + \)\(16\!\cdots\!42\)\( T^{5} - \)\(37\!\cdots\!47\)\( T^{6} - \)\(33\!\cdots\!30\)\( T^{7} + \)\(94\!\cdots\!50\)\( T^{8} + \)\(11\!\cdots\!58\)\( T^{9} + \)\(50\!\cdots\!57\)\( T^{10} - \)\(29\!\cdots\!42\)\( T^{11} - \)\(20\!\cdots\!96\)\( T^{12} - \)\(99\!\cdots\!74\)\( T^{13} + \)\(58\!\cdots\!13\)\( T^{14} + \)\(46\!\cdots\!34\)\( T^{15} + \)\(12\!\cdots\!50\)\( T^{16} - \)\(15\!\cdots\!10\)\( T^{17} - \)\(57\!\cdots\!63\)\( T^{18} + \)\(89\!\cdots\!46\)\( T^{19} + \)\(19\!\cdots\!56\)\( T^{20} - \)\(18\!\cdots\!26\)\( T^{21} - \)\(18\!\cdots\!07\)\( T^{22} - \)\(54\!\cdots\!62\)\( T^{23} + \)\(24\!\cdots\!41\)\( T^{24} \)
$29$ \( 1 - 109808 T - 35456391561 T^{2} + 1610106457469900 T^{3} + \)\(72\!\cdots\!04\)\( T^{4} + \)\(33\!\cdots\!76\)\( T^{5} - \)\(89\!\cdots\!33\)\( T^{6} - \)\(16\!\cdots\!28\)\( T^{7} + \)\(49\!\cdots\!30\)\( T^{8} + \)\(20\!\cdots\!32\)\( T^{9} + \)\(28\!\cdots\!19\)\( T^{10} - \)\(10\!\cdots\!72\)\( T^{11} - \)\(86\!\cdots\!04\)\( T^{12} - \)\(18\!\cdots\!48\)\( T^{13} + \)\(85\!\cdots\!39\)\( T^{14} + \)\(10\!\cdots\!28\)\( T^{15} + \)\(44\!\cdots\!30\)\( T^{16} - \)\(25\!\cdots\!72\)\( T^{17} - \)\(23\!\cdots\!53\)\( T^{18} + \)\(15\!\cdots\!44\)\( T^{19} + \)\(56\!\cdots\!84\)\( T^{20} + \)\(21\!\cdots\!00\)\( T^{21} - \)\(82\!\cdots\!61\)\( T^{22} - \)\(44\!\cdots\!72\)\( T^{23} + \)\(69\!\cdots\!81\)\( T^{24} \)
$31$ \( ( 1 + 183502 T + 134104602712 T^{2} + 18526572014620050 T^{3} + \)\(79\!\cdots\!15\)\( T^{4} + \)\(87\!\cdots\!08\)\( T^{5} + \)\(27\!\cdots\!04\)\( T^{6} + \)\(23\!\cdots\!88\)\( T^{7} + \)\(60\!\cdots\!15\)\( T^{8} + \)\(38\!\cdots\!50\)\( T^{9} + \)\(76\!\cdots\!92\)\( T^{10} + \)\(28\!\cdots\!02\)\( T^{11} + \)\(43\!\cdots\!61\)\( T^{12} )^{2} \)
$37$ \( ( 1 - 461070 T + 547569013512 T^{2} - 185773420236593742 T^{3} + \)\(12\!\cdots\!07\)\( T^{4} - \)\(32\!\cdots\!92\)\( T^{5} + \)\(15\!\cdots\!20\)\( T^{6} - \)\(30\!\cdots\!36\)\( T^{7} + \)\(11\!\cdots\!23\)\( T^{8} - \)\(15\!\cdots\!54\)\( T^{9} + \)\(44\!\cdots\!52\)\( T^{10} - \)\(35\!\cdots\!10\)\( T^{11} + \)\(73\!\cdots\!69\)\( T^{12} )^{2} \)
$41$ \( 1 - 489142 T + 35305801619 T^{2} - 54701641104991538 T^{3} + \)\(37\!\cdots\!19\)\( T^{4} - \)\(34\!\cdots\!28\)\( T^{5} + \)\(95\!\cdots\!60\)\( T^{6} - \)\(24\!\cdots\!40\)\( T^{7} - \)\(13\!\cdots\!51\)\( T^{8} + \)\(79\!\cdots\!50\)\( T^{9} - \)\(37\!\cdots\!99\)\( T^{10} + \)\(15\!\cdots\!62\)\( T^{11} - \)\(18\!\cdots\!46\)\( T^{12} + \)\(30\!\cdots\!22\)\( T^{13} - \)\(14\!\cdots\!39\)\( T^{14} + \)\(58\!\cdots\!50\)\( T^{15} - \)\(19\!\cdots\!71\)\( T^{16} - \)\(68\!\cdots\!40\)\( T^{17} + \)\(51\!\cdots\!60\)\( T^{18} - \)\(37\!\cdots\!08\)\( T^{19} + \)\(76\!\cdots\!79\)\( T^{20} - \)\(22\!\cdots\!98\)\( T^{21} + \)\(27\!\cdots\!19\)\( T^{22} - \)\(74\!\cdots\!02\)\( T^{23} + \)\(29\!\cdots\!61\)\( T^{24} \)
$43$ \( 1 - 1781092 T + 981909160659 T^{2} + 58766133031559564 T^{3} - \)\(27\!\cdots\!74\)\( T^{4} + \)\(17\!\cdots\!64\)\( T^{5} - \)\(10\!\cdots\!07\)\( T^{6} + \)\(25\!\cdots\!32\)\( T^{7} + \)\(31\!\cdots\!36\)\( T^{8} - \)\(21\!\cdots\!96\)\( T^{9} + \)\(11\!\cdots\!95\)\( T^{10} + \)\(18\!\cdots\!88\)\( T^{11} - \)\(55\!\cdots\!24\)\( T^{12} + \)\(50\!\cdots\!16\)\( T^{13} + \)\(84\!\cdots\!55\)\( T^{14} - \)\(43\!\cdots\!28\)\( T^{15} + \)\(17\!\cdots\!36\)\( T^{16} + \)\(37\!\cdots\!24\)\( T^{17} - \)\(43\!\cdots\!43\)\( T^{18} + \)\(18\!\cdots\!52\)\( T^{19} - \)\(81\!\cdots\!74\)\( T^{20} + \)\(47\!\cdots\!48\)\( T^{21} + \)\(21\!\cdots\!91\)\( T^{22} - \)\(10\!\cdots\!56\)\( T^{23} + \)\(16\!\cdots\!01\)\( T^{24} \)
$47$ \( 1 - 178950 T - 1751674983303 T^{2} + 955597726565802046 T^{3} + \)\(16\!\cdots\!96\)\( T^{4} - \)\(13\!\cdots\!86\)\( T^{5} - \)\(68\!\cdots\!99\)\( T^{6} + \)\(12\!\cdots\!82\)\( T^{7} - \)\(62\!\cdots\!98\)\( T^{8} - \)\(62\!\cdots\!94\)\( T^{9} + \)\(33\!\cdots\!21\)\( T^{10} + \)\(14\!\cdots\!66\)\( T^{11} - \)\(23\!\cdots\!24\)\( T^{12} + \)\(72\!\cdots\!58\)\( T^{13} + \)\(86\!\cdots\!49\)\( T^{14} - \)\(81\!\cdots\!18\)\( T^{15} - \)\(41\!\cdots\!78\)\( T^{16} + \)\(40\!\cdots\!26\)\( T^{17} - \)\(11\!\cdots\!91\)\( T^{18} - \)\(11\!\cdots\!62\)\( T^{19} + \)\(70\!\cdots\!16\)\( T^{20} + \)\(21\!\cdots\!58\)\( T^{21} - \)\(19\!\cdots\!47\)\( T^{22} - \)\(10\!\cdots\!50\)\( T^{23} + \)\(28\!\cdots\!81\)\( T^{24} \)
$53$ \( 1 - 939902 T - 5305961360983 T^{2} + 3598919025982818242 T^{3} + \)\(17\!\cdots\!94\)\( T^{4} - \)\(79\!\cdots\!54\)\( T^{5} - \)\(40\!\cdots\!05\)\( T^{6} + \)\(12\!\cdots\!54\)\( T^{7} + \)\(73\!\cdots\!08\)\( T^{8} - \)\(12\!\cdots\!10\)\( T^{9} - \)\(11\!\cdots\!23\)\( T^{10} + \)\(59\!\cdots\!18\)\( T^{11} + \)\(14\!\cdots\!20\)\( T^{12} + \)\(70\!\cdots\!66\)\( T^{13} - \)\(15\!\cdots\!87\)\( T^{14} - \)\(21\!\cdots\!30\)\( T^{15} + \)\(14\!\cdots\!88\)\( T^{16} + \)\(27\!\cdots\!78\)\( T^{17} - \)\(10\!\cdots\!45\)\( T^{18} - \)\(24\!\cdots\!82\)\( T^{19} + \)\(62\!\cdots\!74\)\( T^{20} + \)\(15\!\cdots\!34\)\( T^{21} - \)\(26\!\cdots\!67\)\( T^{22} - \)\(55\!\cdots\!26\)\( T^{23} + \)\(69\!\cdots\!81\)\( T^{24} \)
$59$ \( 1 - 1874564 T - 1173604027217 T^{2} + 2408576674749842068 T^{3} - \)\(47\!\cdots\!81\)\( T^{4} + \)\(17\!\cdots\!04\)\( T^{5} + \)\(15\!\cdots\!48\)\( T^{6} - \)\(45\!\cdots\!72\)\( T^{7} + \)\(43\!\cdots\!49\)\( T^{8} - \)\(58\!\cdots\!12\)\( T^{9} + \)\(19\!\cdots\!93\)\( T^{10} + \)\(24\!\cdots\!92\)\( T^{11} - \)\(56\!\cdots\!54\)\( T^{12} + \)\(61\!\cdots\!48\)\( T^{13} + \)\(12\!\cdots\!73\)\( T^{14} - \)\(89\!\cdots\!08\)\( T^{15} + \)\(16\!\cdots\!29\)\( T^{16} - \)\(43\!\cdots\!28\)\( T^{17} + \)\(37\!\cdots\!88\)\( T^{18} + \)\(10\!\cdots\!56\)\( T^{19} - \)\(70\!\cdots\!21\)\( T^{20} + \)\(88\!\cdots\!72\)\( T^{21} - \)\(10\!\cdots\!17\)\( T^{22} - \)\(42\!\cdots\!16\)\( T^{23} + \)\(56\!\cdots\!61\)\( T^{24} \)
$61$ \( 1 + 1791828 T - 12651490872897 T^{2} - 16844355957694144496 T^{3} + \)\(10\!\cdots\!12\)\( T^{4} + \)\(92\!\cdots\!92\)\( T^{5} - \)\(57\!\cdots\!25\)\( T^{6} - \)\(32\!\cdots\!20\)\( T^{7} + \)\(25\!\cdots\!34\)\( T^{8} + \)\(74\!\cdots\!76\)\( T^{9} - \)\(95\!\cdots\!85\)\( T^{10} - \)\(90\!\cdots\!76\)\( T^{11} + \)\(31\!\cdots\!48\)\( T^{12} - \)\(28\!\cdots\!96\)\( T^{13} - \)\(94\!\cdots\!85\)\( T^{14} + \)\(23\!\cdots\!36\)\( T^{15} + \)\(24\!\cdots\!54\)\( T^{16} - \)\(99\!\cdots\!20\)\( T^{17} - \)\(55\!\cdots\!25\)\( T^{18} + \)\(27\!\cdots\!72\)\( T^{19} + \)\(96\!\cdots\!32\)\( T^{20} - \)\(50\!\cdots\!76\)\( T^{21} - \)\(11\!\cdots\!97\)\( T^{22} + \)\(52\!\cdots\!88\)\( T^{23} + \)\(92\!\cdots\!41\)\( T^{24} \)
$67$ \( 1 - 5363976 T - 5204027168013 T^{2} + 30379245409182544184 T^{3} + \)\(17\!\cdots\!27\)\( T^{4} - \)\(22\!\cdots\!52\)\( T^{5} - \)\(17\!\cdots\!56\)\( T^{6} + \)\(18\!\cdots\!48\)\( T^{7} + \)\(10\!\cdots\!49\)\( T^{8} + \)\(56\!\cdots\!80\)\( T^{9} - \)\(47\!\cdots\!83\)\( T^{10} - \)\(38\!\cdots\!44\)\( T^{11} + \)\(29\!\cdots\!34\)\( T^{12} - \)\(23\!\cdots\!12\)\( T^{13} - \)\(17\!\cdots\!07\)\( T^{14} + \)\(12\!\cdots\!60\)\( T^{15} + \)\(14\!\cdots\!09\)\( T^{16} + \)\(15\!\cdots\!64\)\( T^{17} - \)\(85\!\cdots\!84\)\( T^{18} - \)\(67\!\cdots\!44\)\( T^{19} + \)\(32\!\cdots\!87\)\( T^{20} + \)\(33\!\cdots\!92\)\( T^{21} - \)\(34\!\cdots\!37\)\( T^{22} - \)\(21\!\cdots\!52\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \)
$71$ \( 1 - 3980204 T - 28730106281593 T^{2} + 18053900623633878380 T^{3} + \)\(90\!\cdots\!22\)\( T^{4} + \)\(33\!\cdots\!52\)\( T^{5} - \)\(11\!\cdots\!99\)\( T^{6} - \)\(26\!\cdots\!28\)\( T^{7} + \)\(10\!\cdots\!16\)\( T^{8} + \)\(29\!\cdots\!40\)\( T^{9} - \)\(18\!\cdots\!65\)\( T^{10} - \)\(16\!\cdots\!40\)\( T^{11} - \)\(42\!\cdots\!84\)\( T^{12} - \)\(15\!\cdots\!40\)\( T^{13} - \)\(15\!\cdots\!65\)\( T^{14} + \)\(22\!\cdots\!40\)\( T^{15} + \)\(72\!\cdots\!76\)\( T^{16} - \)\(16\!\cdots\!28\)\( T^{17} - \)\(64\!\cdots\!59\)\( T^{18} + \)\(17\!\cdots\!12\)\( T^{19} + \)\(42\!\cdots\!62\)\( T^{20} + \)\(76\!\cdots\!80\)\( T^{21} - \)\(11\!\cdots\!93\)\( T^{22} - \)\(14\!\cdots\!64\)\( T^{23} + \)\(32\!\cdots\!81\)\( T^{24} \)
$73$ \( 1 - 15279830 T + 81389604693739 T^{2} - \)\(19\!\cdots\!18\)\( T^{3} + \)\(99\!\cdots\!03\)\( T^{4} - \)\(85\!\cdots\!92\)\( T^{5} + \)\(33\!\cdots\!88\)\( T^{6} - \)\(76\!\cdots\!28\)\( T^{7} + \)\(34\!\cdots\!61\)\( T^{8} - \)\(17\!\cdots\!14\)\( T^{9} + \)\(57\!\cdots\!37\)\( T^{10} - \)\(17\!\cdots\!38\)\( T^{11} + \)\(59\!\cdots\!50\)\( T^{12} - \)\(19\!\cdots\!86\)\( T^{13} + \)\(70\!\cdots\!33\)\( T^{14} - \)\(24\!\cdots\!22\)\( T^{15} + \)\(51\!\cdots\!41\)\( T^{16} - \)\(12\!\cdots\!96\)\( T^{17} + \)\(60\!\cdots\!52\)\( T^{18} - \)\(17\!\cdots\!96\)\( T^{19} + \)\(22\!\cdots\!83\)\( T^{20} - \)\(47\!\cdots\!06\)\( T^{21} + \)\(22\!\cdots\!11\)\( T^{22} - \)\(45\!\cdots\!90\)\( T^{23} + \)\(33\!\cdots\!41\)\( T^{24} \)
$79$ \( 1 + 7164236 T - 27621575053969 T^{2} - \)\(48\!\cdots\!92\)\( T^{3} - \)\(86\!\cdots\!38\)\( T^{4} + \)\(11\!\cdots\!08\)\( T^{5} + \)\(59\!\cdots\!37\)\( T^{6} - \)\(15\!\cdots\!96\)\( T^{7} - \)\(10\!\cdots\!64\)\( T^{8} - \)\(39\!\cdots\!40\)\( T^{9} - \)\(28\!\cdots\!21\)\( T^{10} + \)\(49\!\cdots\!92\)\( T^{11} + \)\(33\!\cdots\!32\)\( T^{12} + \)\(94\!\cdots\!28\)\( T^{13} - \)\(10\!\cdots\!01\)\( T^{14} - \)\(28\!\cdots\!60\)\( T^{15} - \)\(14\!\cdots\!04\)\( T^{16} - \)\(39\!\cdots\!04\)\( T^{17} + \)\(29\!\cdots\!17\)\( T^{18} + \)\(10\!\cdots\!52\)\( T^{19} - \)\(15\!\cdots\!98\)\( T^{20} - \)\(17\!\cdots\!88\)\( T^{21} - \)\(18\!\cdots\!69\)\( T^{22} + \)\(93\!\cdots\!24\)\( T^{23} + \)\(25\!\cdots\!81\)\( T^{24} \)
$83$ \( ( 1 + 6458404 T + 78509530003758 T^{2} + \)\(41\!\cdots\!04\)\( T^{3} + \)\(33\!\cdots\!78\)\( T^{4} + \)\(13\!\cdots\!32\)\( T^{5} + \)\(98\!\cdots\!98\)\( T^{6} + \)\(36\!\cdots\!64\)\( T^{7} + \)\(24\!\cdots\!62\)\( T^{8} + \)\(82\!\cdots\!32\)\( T^{9} + \)\(42\!\cdots\!78\)\( T^{10} + \)\(95\!\cdots\!28\)\( T^{11} + \)\(39\!\cdots\!89\)\( T^{12} )^{2} \)
$89$ \( 1 - 4966022 T - 162235647182159 T^{2} + \)\(63\!\cdots\!98\)\( T^{3} + \)\(14\!\cdots\!38\)\( T^{4} - \)\(42\!\cdots\!38\)\( T^{5} - \)\(95\!\cdots\!37\)\( T^{6} + \)\(25\!\cdots\!66\)\( T^{7} + \)\(48\!\cdots\!92\)\( T^{8} - \)\(10\!\cdots\!58\)\( T^{9} - \)\(22\!\cdots\!47\)\( T^{10} + \)\(18\!\cdots\!06\)\( T^{11} + \)\(10\!\cdots\!20\)\( T^{12} + \)\(81\!\cdots\!74\)\( T^{13} - \)\(43\!\cdots\!27\)\( T^{14} - \)\(91\!\cdots\!62\)\( T^{15} + \)\(18\!\cdots\!52\)\( T^{16} + \)\(43\!\cdots\!34\)\( T^{17} - \)\(71\!\cdots\!77\)\( T^{18} - \)\(14\!\cdots\!42\)\( T^{19} + \)\(21\!\cdots\!18\)\( T^{20} + \)\(40\!\cdots\!62\)\( T^{21} - \)\(46\!\cdots\!59\)\( T^{22} - \)\(62\!\cdots\!38\)\( T^{23} + \)\(56\!\cdots\!41\)\( T^{24} \)
$97$ \( 1 + 10277950 T - 169949822062069 T^{2} + \)\(13\!\cdots\!38\)\( T^{3} + \)\(51\!\cdots\!35\)\( T^{4} - \)\(29\!\cdots\!64\)\( T^{5} - \)\(20\!\cdots\!12\)\( T^{6} + \)\(85\!\cdots\!56\)\( T^{7} - \)\(41\!\cdots\!87\)\( T^{8} - \)\(59\!\cdots\!34\)\( T^{9} + \)\(70\!\cdots\!33\)\( T^{10} + \)\(34\!\cdots\!06\)\( T^{11} - \)\(60\!\cdots\!26\)\( T^{12} + \)\(27\!\cdots\!78\)\( T^{13} + \)\(46\!\cdots\!77\)\( T^{14} - \)\(31\!\cdots\!98\)\( T^{15} - \)\(17\!\cdots\!07\)\( T^{16} + \)\(29\!\cdots\!08\)\( T^{17} - \)\(56\!\cdots\!08\)\( T^{18} - \)\(66\!\cdots\!88\)\( T^{19} + \)\(92\!\cdots\!35\)\( T^{20} + \)\(19\!\cdots\!74\)\( T^{21} - \)\(20\!\cdots\!81\)\( T^{22} + \)\(98\!\cdots\!50\)\( T^{23} + \)\(77\!\cdots\!81\)\( T^{24} \)
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