Properties

Label 38.10.c.a
Level 38
Weight 10
Character orbit 38.c
Analytic conductor 19.571
Analytic rank 0
Dimension 14
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 38.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.5713617742\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - 3 x^{13} + 93094 x^{12} + 3763497 x^{11} + 6525212431 x^{10} + 244392759522 x^{9} + 198778066371639 x^{8} + 9216645483921129 x^{7} + 4530154246073222607 x^{6} + 142122715974381300066 x^{5} + 8185019559882294055671 x^{4} - 135327799885026117969495 x^{3} + 2044710563339570147369550 x^{2} - 9649457865335314261798875 x + 37683939484136051622500625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{6}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 16 - 16 \beta_{2} ) q^{2} + ( 24 - \beta_{1} - 24 \beta_{2} ) q^{3} -256 \beta_{2} q^{4} + ( 130 - \beta_{1} - 130 \beta_{2} + \beta_{5} + \beta_{7} ) q^{5} + ( -384 \beta_{2} - 16 \beta_{3} ) q^{6} + ( 261 + 7 \beta_{1} - 7 \beta_{3} + \beta_{8} ) q^{7} -4096 q^{8} + ( -7476 \beta_{2} - 16 \beta_{3} - \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{9} - \beta_{12} ) q^{9} +O(q^{10})\) \( q + ( 16 - 16 \beta_{2} ) q^{2} + ( 24 - \beta_{1} - 24 \beta_{2} ) q^{3} -256 \beta_{2} q^{4} + ( 130 - \beta_{1} - 130 \beta_{2} + \beta_{5} + \beta_{7} ) q^{5} + ( -384 \beta_{2} - 16 \beta_{3} ) q^{6} + ( 261 + 7 \beta_{1} - 7 \beta_{3} + \beta_{8} ) q^{7} -4096 q^{8} + ( -7476 \beta_{2} - 16 \beta_{3} - \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{9} - \beta_{12} ) q^{9} + ( -2080 \beta_{2} - 16 \beta_{3} + 16 \beta_{7} ) q^{10} + ( 1082 + 76 \beta_{1} - 76 \beta_{3} + \beta_{4} + 11 \beta_{5} - \beta_{6} - 2 \beta_{8} - \beta_{10} ) q^{11} + ( -6144 + 256 \beta_{1} - 256 \beta_{3} ) q^{12} + ( 913 \beta_{2} + 10 \beta_{3} + 3 \beta_{4} - 5 \beta_{6} - 29 \beta_{7} + 5 \beta_{9} + 5 \beta_{11} + 3 \beta_{12} ) q^{13} + ( 4176 + 112 \beta_{1} - 4176 \beta_{2} + 16 \beta_{8} + 16 \beta_{11} ) q^{14} + ( -32624 \beta_{2} - 273 \beta_{3} - \beta_{4} - 2 \beta_{6} + 53 \beta_{7} + 2 \beta_{9} - 22 \beta_{11} - \beta_{12} - 9 \beta_{13} ) q^{15} + ( -65536 + 65536 \beta_{2} ) q^{16} + ( 45184 - 831 \beta_{1} - 45184 \beta_{2} - 55 \beta_{5} - 55 \beta_{7} + 12 \beta_{8} - 4 \beta_{9} - 10 \beta_{10} + 12 \beta_{11} - 7 \beta_{12} - 10 \beta_{13} ) q^{17} + ( -119616 + 256 \beta_{1} - 256 \beta_{3} - 16 \beta_{4} - 32 \beta_{5} - 16 \beta_{6} ) q^{18} + ( 40603 - 760 \beta_{1} + 81624 \beta_{2} + 1178 \beta_{3} + 19 \beta_{5} + 19 \beta_{6} - 57 \beta_{7} + 19 \beta_{8} - 19 \beta_{10} + 38 \beta_{11} - 19 \beta_{13} ) q^{19} + ( -33280 + 256 \beta_{1} - 256 \beta_{3} - 256 \beta_{5} ) q^{20} + ( -175320 + 362 \beta_{1} + 175320 \beta_{2} - 412 \beta_{5} - 412 \beta_{7} + 54 \beta_{8} + 7 \beta_{9} + 54 \beta_{11} + 20 \beta_{12} ) q^{21} + ( 17312 + 1216 \beta_{1} - 17312 \beta_{2} + 176 \beta_{5} + 176 \beta_{7} - 32 \beta_{8} - 16 \beta_{9} - 16 \beta_{10} - 32 \beta_{11} - 16 \beta_{12} - 16 \beta_{13} ) q^{22} + ( 493275 \beta_{2} + 1262 \beta_{3} + 76 \beta_{4} - 24 \beta_{6} - 183 \beta_{7} + 24 \beta_{9} + 3 \beta_{11} + 76 \beta_{12} - 7 \beta_{13} ) q^{23} + ( -98304 + 4096 \beta_{1} + 98304 \beta_{2} ) q^{24} + ( -343700 \beta_{2} - 2660 \beta_{3} - 52 \beta_{4} - 2 \beta_{6} + 353 \beta_{7} + 2 \beta_{9} - \beta_{11} - 52 \beta_{12} - 62 \beta_{13} ) q^{25} + ( 14608 - 160 \beta_{1} + 160 \beta_{3} + 48 \beta_{4} + 464 \beta_{5} - 80 \beta_{6} - 80 \beta_{8} ) q^{26} + ( -137078 + 7423 \beta_{1} - 7423 \beta_{3} - \beta_{4} + 262 \beta_{5} - 107 \beta_{6} + 52 \beta_{8} + 36 \beta_{10} ) q^{27} + ( -66816 \beta_{2} + 1792 \beta_{3} + 256 \beta_{11} ) q^{28} + ( -1003803 \beta_{2} - 7164 \beta_{3} - 165 \beta_{4} - 110 \beta_{6} - 707 \beta_{7} + 110 \beta_{9} + 429 \beta_{11} - 165 \beta_{12} - 44 \beta_{13} ) q^{29} + ( -521984 + 4368 \beta_{1} - 4368 \beta_{3} - 16 \beta_{4} - 848 \beta_{5} - 32 \beta_{6} + 352 \beta_{8} + 144 \beta_{10} ) q^{30} + ( -1336580 + 8556 \beta_{1} - 8556 \beta_{3} + 65 \beta_{4} - 1100 \beta_{5} - 78 \beta_{6} + 132 \beta_{8} + 154 \beta_{10} ) q^{31} + 1048576 \beta_{2} q^{32} + ( -1976749 + 491 \beta_{1} + 1976749 \beta_{2} + 3339 \beta_{5} + 3339 \beta_{7} - 622 \beta_{8} - 299 \beta_{9} - 138 \beta_{10} - 622 \beta_{11} - 48 \beta_{12} - 138 \beta_{13} ) q^{33} + ( -722944 \beta_{2} - 13296 \beta_{3} - 112 \beta_{4} + 64 \beta_{6} - 880 \beta_{7} - 64 \beta_{9} + 192 \beta_{11} - 112 \beta_{12} - 160 \beta_{13} ) q^{34} + ( 354861 + 34561 \beta_{1} - 354861 \beta_{2} - 568 \beta_{5} - 568 \beta_{7} + 1099 \beta_{8} + 15 \beta_{9} - 28 \beta_{10} + 1099 \beta_{11} + 250 \beta_{12} - 28 \beta_{13} ) q^{35} + ( -1913856 + 4096 \beta_{1} + 1913856 \beta_{2} - 512 \beta_{5} - 512 \beta_{7} - 256 \beta_{9} + 256 \beta_{12} ) q^{36} + ( -480675 - 17933 \beta_{1} + 17933 \beta_{3} + 183 \beta_{4} - 5610 \beta_{5} - 149 \beta_{6} + 1381 \beta_{8} + 84 \beta_{10} ) q^{37} + ( 1955632 - 18848 \beta_{1} - 649648 \beta_{2} + 6688 \beta_{3} + 1216 \beta_{5} + 304 \beta_{7} - 304 \beta_{8} + 304 \beta_{9} + 304 \beta_{11} - 304 \beta_{13} ) q^{38} + ( 321530 + 7449 \beta_{1} - 7449 \beta_{3} + 158 \beta_{4} + 3421 \beta_{5} - 516 \beta_{6} - 1708 \beta_{8} - 603 \beta_{10} ) q^{39} + ( -532480 + 4096 \beta_{1} + 532480 \beta_{2} - 4096 \beta_{5} - 4096 \beta_{7} ) q^{40} + ( 513421 - 7946 \beta_{1} - 513421 \beta_{2} - 3128 \beta_{5} - 3128 \beta_{7} + 160 \beta_{8} + 657 \beta_{9} - 272 \beta_{10} + 160 \beta_{11} + 15 \beta_{12} - 272 \beta_{13} ) q^{41} + ( 2805120 \beta_{2} + 5792 \beta_{3} + 320 \beta_{4} - 112 \beta_{6} - 6592 \beta_{7} + 112 \beta_{9} + 864 \beta_{11} + 320 \beta_{12} ) q^{42} + ( 2838866 - 13531 \beta_{1} - 2838866 \beta_{2} + 1487 \beta_{5} + 1487 \beta_{7} - 1552 \beta_{8} + 942 \beta_{9} - 583 \beta_{10} - 1552 \beta_{11} - 596 \beta_{12} - 583 \beta_{13} ) q^{43} + ( -276992 \beta_{2} + 19456 \beta_{3} - 256 \beta_{4} + 256 \beta_{6} + 2816 \beta_{7} - 256 \beta_{9} - 512 \beta_{11} - 256 \beta_{12} - 256 \beta_{13} ) q^{44} + ( -5555829 + 8007 \beta_{1} - 8007 \beta_{3} - 1257 \beta_{4} - 12036 \beta_{5} + 762 \beta_{6} + 2895 \beta_{8} ) q^{45} + ( 7892400 - 20192 \beta_{1} + 20192 \beta_{3} + 1216 \beta_{4} + 2928 \beta_{5} - 384 \beta_{6} - 48 \beta_{8} + 112 \beta_{10} ) q^{46} + ( 8595314 \beta_{2} - 61623 \beta_{3} + 1131 \beta_{4} - 310 \beta_{6} - 1027 \beta_{7} + 310 \beta_{9} - 64 \beta_{11} + 1131 \beta_{12} + 1047 \beta_{13} ) q^{47} + ( 1572864 \beta_{2} + 65536 \beta_{3} ) q^{48} + ( -5673684 - 35761 \beta_{1} + 35761 \beta_{3} - 932 \beta_{4} + 16876 \beta_{5} - 175 \beta_{6} + 1661 \beta_{8} - 476 \beta_{10} ) q^{49} + ( -5499200 + 42560 \beta_{1} - 42560 \beta_{3} - 832 \beta_{4} - 5648 \beta_{5} - 32 \beta_{6} + 16 \beta_{8} + 992 \beta_{10} ) q^{50} + ( -23044307 \beta_{2} + 4558 \beta_{3} - 1556 \beta_{4} + 537 \beta_{6} + 15349 \beta_{7} - 537 \beta_{9} + 35 \beta_{11} - 1556 \beta_{12} + 483 \beta_{13} ) q^{51} + ( 233728 - 2560 \beta_{1} - 233728 \beta_{2} + 7424 \beta_{5} + 7424 \beta_{7} - 1280 \beta_{8} - 1280 \beta_{9} - 1280 \beta_{11} - 768 \beta_{12} ) q^{52} + ( 7753074 \beta_{2} - 154369 \beta_{3} + 2610 \beta_{4} + 2413 \beta_{6} - 8183 \beta_{7} - 2413 \beta_{9} - 1856 \beta_{11} + 2610 \beta_{12} + 1088 \beta_{13} ) q^{53} + ( -2193248 + 118768 \beta_{1} + 2193248 \beta_{2} + 4192 \beta_{5} + 4192 \beta_{7} + 832 \beta_{8} - 1712 \beta_{9} + 576 \beta_{10} + 832 \beta_{11} + 16 \beta_{12} + 576 \beta_{13} ) q^{54} + ( 20552305 - 233817 \beta_{1} - 20552305 \beta_{2} + 17892 \beta_{5} + 17892 \beta_{7} - 1019 \beta_{8} + 1560 \beta_{9} + 140 \beta_{10} - 1019 \beta_{11} - 1118 \beta_{12} + 140 \beta_{13} ) q^{55} + ( -1069056 - 28672 \beta_{1} + 28672 \beta_{3} - 4096 \beta_{8} ) q^{56} + ( 34444302 - 313937 \beta_{1} - 21413057 \beta_{2} + 227639 \beta_{3} - 627 \beta_{4} + 11134 \beta_{5} + 1976 \beta_{6} + 35017 \beta_{7} - 855 \beta_{8} + 798 \beta_{9} + 1121 \beta_{11} - 1748 \beta_{12} + 1482 \beta_{13} ) q^{57} + ( -16060848 + 114624 \beta_{1} - 114624 \beta_{3} - 2640 \beta_{4} + 11312 \beta_{5} - 1760 \beta_{6} - 6864 \beta_{8} + 704 \beta_{10} ) q^{58} + ( 23467378 + 53675 \beta_{1} - 23467378 \beta_{2} + 9972 \beta_{5} + 9972 \beta_{7} - 5128 \beta_{8} + 5207 \beta_{9} + 2286 \beta_{10} - 5128 \beta_{11} - 857 \beta_{12} + 2286 \beta_{13} ) q^{59} + ( -8351744 + 69888 \beta_{1} + 8351744 \beta_{2} - 13568 \beta_{5} - 13568 \beta_{7} + 5632 \beta_{8} - 512 \beta_{9} + 2304 \beta_{10} + 5632 \beta_{11} + 256 \beta_{12} + 2304 \beta_{13} ) q^{60} + ( 7071348 \beta_{2} + 68047 \beta_{3} + 2006 \beta_{4} + 2276 \beta_{6} - 32613 \beta_{7} - 2276 \beta_{9} - 14782 \beta_{11} + 2006 \beta_{12} + 968 \beta_{13} ) q^{61} + ( -21385280 + 136896 \beta_{1} + 21385280 \beta_{2} - 17600 \beta_{5} - 17600 \beta_{7} + 2112 \beta_{8} - 1248 \beta_{9} + 2464 \beta_{10} + 2112 \beta_{11} - 1040 \beta_{12} + 2464 \beta_{13} ) q^{62} + ( 19821532 \beta_{2} + 271460 \beta_{3} + 1196 \beta_{4} - 740 \beta_{6} - 39088 \beta_{7} + 740 \beta_{9} - 4924 \beta_{11} + 1196 \beta_{12} + 5040 \beta_{13} ) q^{63} + 16777216 q^{64} + ( 56357365 - 192398 \beta_{1} + 192398 \beta_{3} + 3097 \beta_{4} + 32011 \beta_{5} + 8397 \beta_{6} - 10771 \beta_{8} - 2022 \beta_{10} ) q^{65} + ( 31627984 \beta_{2} + 7856 \beta_{3} - 768 \beta_{4} + 4784 \beta_{6} + 53424 \beta_{7} - 4784 \beta_{9} - 9952 \beta_{11} - 768 \beta_{12} - 2208 \beta_{13} ) q^{66} + ( -24580445 \beta_{2} - 172348 \beta_{3} - 3422 \beta_{4} + 273 \beta_{6} - 17670 \beta_{7} - 273 \beta_{9} - 9633 \beta_{11} - 3422 \beta_{12} + 422 \beta_{13} ) q^{67} + ( -11567104 + 212736 \beta_{1} - 212736 \beta_{3} - 1792 \beta_{4} + 14080 \beta_{5} + 1024 \beta_{6} - 3072 \beta_{8} + 2560 \beta_{10} ) q^{68} + ( 45495766 - 1103555 \beta_{1} + 1103555 \beta_{3} + 1194 \beta_{4} - 18795 \beta_{5} - 484 \beta_{6} - 14006 \beta_{8} - 6972 \beta_{10} ) q^{69} + ( -5677776 \beta_{2} + 552976 \beta_{3} + 4000 \beta_{4} - 240 \beta_{6} - 9088 \beta_{7} + 240 \beta_{9} + 17584 \beta_{11} + 4000 \beta_{12} - 448 \beta_{13} ) q^{70} + ( -10578540 - 221949 \beta_{1} + 10578540 \beta_{2} + 46089 \beta_{5} + 46089 \beta_{7} - 11522 \beta_{8} - 344 \beta_{9} - 943 \beta_{10} - 11522 \beta_{11} + 1232 \beta_{12} - 943 \beta_{13} ) q^{71} + ( 30621696 \beta_{2} + 65536 \beta_{3} + 4096 \beta_{4} + 4096 \beta_{6} - 8192 \beta_{7} - 4096 \beta_{9} + 4096 \beta_{12} ) q^{72} + ( 8454772 - 135553 \beta_{1} - 8454772 \beta_{2} - 46468 \beta_{5} - 46468 \beta_{7} + 8289 \beta_{8} - 3371 \beta_{9} - 4172 \beta_{10} + 8289 \beta_{11} + 9856 \beta_{12} - 4172 \beta_{13} ) q^{73} + ( -7690800 - 286928 \beta_{1} + 7690800 \beta_{2} - 89760 \beta_{5} - 89760 \beta_{7} + 22096 \beta_{8} - 2384 \beta_{9} + 1344 \beta_{10} + 22096 \beta_{11} - 2928 \beta_{12} + 1344 \beta_{13} ) q^{74} + ( -79885426 + 547864 \beta_{1} - 547864 \beta_{3} - 7715 \beta_{4} - 149629 \beta_{5} + 3707 \beta_{6} + 17426 \beta_{8} + 2751 \beta_{10} ) q^{75} + ( 20895744 - 107008 \beta_{1} - 31290112 \beta_{2} - 194560 \beta_{3} + 14592 \beta_{5} - 4864 \beta_{6} + 19456 \beta_{7} - 9728 \beta_{8} + 4864 \beta_{9} + 4864 \beta_{10} - 4864 \beta_{11} ) q^{76} + ( -69234740 + 699230 \beta_{1} - 699230 \beta_{3} - 5294 \beta_{4} - 24416 \beta_{5} - 5421 \beta_{6} + 874 \beta_{8} + 3920 \beta_{10} ) q^{77} + ( 5144480 + 119184 \beta_{1} - 5144480 \beta_{2} + 54736 \beta_{5} + 54736 \beta_{7} - 27328 \beta_{8} - 8256 \beta_{9} - 9648 \beta_{10} - 27328 \beta_{11} - 2528 \beta_{12} - 9648 \beta_{13} ) q^{78} + ( 17475288 - 187385 \beta_{1} - 17475288 \beta_{2} + 110931 \beta_{5} + 110931 \beta_{7} + 58606 \beta_{8} + 4180 \beta_{9} - 237 \beta_{10} + 58606 \beta_{11} - 7684 \beta_{12} - 237 \beta_{13} ) q^{79} + ( 8519680 \beta_{2} + 65536 \beta_{3} - 65536 \beta_{7} ) q^{80} + ( -52863585 + 546286 \beta_{1} + 52863585 \beta_{2} - 48578 \beta_{5} - 48578 \beta_{7} - 10710 \beta_{8} + 7049 \beta_{9} - 7992 \beta_{10} - 10710 \beta_{11} - 10379 \beta_{12} - 7992 \beta_{13} ) q^{81} + ( -8214736 \beta_{2} - 127136 \beta_{3} + 240 \beta_{4} - 10512 \beta_{6} - 50048 \beta_{7} + 10512 \beta_{9} + 2560 \beta_{11} + 240 \beta_{12} - 4352 \beta_{13} ) q^{82} + ( -52777144 + 1036356 \beta_{1} - 1036356 \beta_{3} + 9144 \beta_{4} + 71445 \beta_{5} - 18570 \beta_{6} + 47434 \beta_{8} + 1211 \beta_{10} ) q^{83} + ( 44881920 - 92672 \beta_{1} + 92672 \beta_{3} + 5120 \beta_{4} + 105472 \beta_{5} - 1792 \beta_{6} - 13824 \beta_{8} ) q^{84} + ( 90138963 \beta_{2} - 1378662 \beta_{3} + 3435 \beta_{4} - 9825 \beta_{6} + 183963 \beta_{7} + 9825 \beta_{9} - 12147 \beta_{11} + 3435 \beta_{12} - 6036 \beta_{13} ) q^{85} + ( -45421856 \beta_{2} - 216496 \beta_{3} - 9536 \beta_{4} - 15072 \beta_{6} + 23792 \beta_{7} + 15072 \beta_{9} - 24832 \beta_{11} - 9536 \beta_{12} - 9328 \beta_{13} ) q^{86} + ( -214119311 + 2980818 \beta_{1} - 2980818 \beta_{3} - 3556 \beta_{4} + 151111 \beta_{5} - 17270 \beta_{6} - 28985 \beta_{8} - 3657 \beta_{10} ) q^{87} + ( -4431872 - 311296 \beta_{1} + 311296 \beta_{3} - 4096 \beta_{4} - 45056 \beta_{5} + 4096 \beta_{6} + 8192 \beta_{8} + 4096 \beta_{10} ) q^{88} + ( -234909380 \beta_{2} + 2682647 \beta_{3} - 3531 \beta_{4} - 21130 \beta_{6} + 89927 \beta_{7} + 21130 \beta_{9} + 69848 \beta_{11} - 3531 \beta_{12} - 8242 \beta_{13} ) q^{89} + ( -88893264 + 128112 \beta_{1} + 88893264 \beta_{2} - 192576 \beta_{5} - 192576 \beta_{7} + 46320 \beta_{8} + 12192 \beta_{9} + 46320 \beta_{11} + 20112 \beta_{12} ) q^{90} + ( -163911552 \beta_{2} - 2147360 \beta_{3} - 3398 \beta_{4} - 8504 \beta_{6} + 172948 \beta_{7} + 8504 \beta_{9} + 11120 \beta_{11} - 3398 \beta_{12} - 7224 \beta_{13} ) q^{91} + ( 126278400 - 323072 \beta_{1} - 126278400 \beta_{2} + 46848 \beta_{5} + 46848 \beta_{7} - 768 \beta_{8} - 6144 \beta_{9} + 1792 \beta_{10} - 768 \beta_{11} - 19456 \beta_{12} + 1792 \beta_{13} ) q^{92} + ( -262946619 + 657849 \beta_{1} + 262946619 \beta_{2} - 477968 \beta_{5} - 477968 \beta_{7} + 27717 \beta_{8} + 1013 \beta_{9} - 4272 \beta_{10} + 27717 \beta_{11} + 21433 \beta_{12} - 4272 \beta_{13} ) q^{93} + ( 137525024 + 985968 \beta_{1} - 985968 \beta_{3} + 18096 \beta_{4} + 16432 \beta_{5} - 4960 \beta_{6} + 1024 \beta_{8} - 16752 \beta_{10} ) q^{94} + ( 224752216 - 1261353 \beta_{1} - 107291974 \beta_{2} - 1715244 \beta_{3} + 17043 \beta_{4} + 157073 \beta_{5} + 2812 \beta_{6} + 203053 \beta_{7} - 45410 \beta_{8} - 26904 \beta_{9} - 9101 \beta_{10} + 21736 \beta_{11} + 190 \beta_{12} - 5643 \beta_{13} ) q^{95} + ( 25165824 - 1048576 \beta_{1} + 1048576 \beta_{3} ) q^{96} + ( 35089256 + 1240489 \beta_{1} - 35089256 \beta_{2} - 225124 \beta_{5} - 225124 \beta_{7} + 27797 \beta_{8} - 9256 \beta_{9} + 4852 \beta_{10} + 27797 \beta_{11} - 12553 \beta_{12} + 4852 \beta_{13} ) q^{97} + ( -90778944 - 572176 \beta_{1} + 90778944 \beta_{2} + 270016 \beta_{5} + 270016 \beta_{7} + 26576 \beta_{8} - 2800 \beta_{9} - 7616 \beta_{10} + 26576 \beta_{11} + 14912 \beta_{12} - 7616 \beta_{13} ) q^{98} + ( 75721781 \beta_{2} + 2679539 \beta_{3} - 3922 \beta_{4} + 29289 \beta_{6} + 540422 \beta_{7} - 29289 \beta_{9} - 111383 \beta_{11} - 3922 \beta_{12} - 16686 \beta_{13} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q + 112q^{2} + 165q^{3} - 1792q^{4} + 909q^{5} - 2640q^{6} + 3692q^{7} - 57344q^{8} - 52286q^{9} + O(q^{10}) \) \( 14q + 112q^{2} + 165q^{3} - 1792q^{4} + 909q^{5} - 2640q^{6} + 3692q^{7} - 57344q^{8} - 52286q^{9} - 14544q^{10} + 15656q^{11} - 84480q^{12} + 6423q^{13} + 29536q^{14} - 227715q^{15} - 458752q^{16} + 313667q^{17} - 1673152q^{18} + 1134224q^{19} - 465408q^{20} - 1227046q^{21} + 125248q^{22} + 3449345q^{23} - 675840q^{24} - 2398648q^{25} + 205536q^{26} - 1873854q^{27} - 472576q^{28} - 7002615q^{29} - 7286880q^{30} - 18666588q^{31} + 7340032q^{32} - 13827668q^{33} - 5018672q^{34} + 2584932q^{35} - 13385216q^{36} - 6866080q^{37} + 22760176q^{38} + 4568410q^{39} - 3723264q^{40} + 3564107q^{41} + 19632736q^{42} + 19837521q^{43} - 2003968q^{44} - 77788260q^{45} + 110379040q^{46} + 60353825q^{47} + 10813440q^{48} - 79579650q^{49} - 76756736q^{50} - 161350373q^{51} + 1644288q^{52} + 54744235q^{53} - 14990832q^{54} + 143199990q^{55} - 15122432q^{56} + 330686241q^{57} - 224083680q^{58} + 164456585q^{59} - 58295040q^{60} + 49328881q^{61} - 149332704q^{62} + 138012360q^{63} + 234881024q^{64} + 788015550q^{65} + 221242688q^{66} - 171522309q^{67} - 160597504q^{68} + 630323350q^{69} - 41358912q^{70} - 74596055q^{71} + 214163456q^{72} + 58695287q^{73} - 54928640q^{74} - 1115757144q^{75} + 73801472q^{76} - 965186644q^{77} + 36547280q^{78} + 121854617q^{79} + 59572224q^{80} - 368486747q^{81} - 57025712q^{82} - 732607256q^{83} + 628247552q^{84} + 634697565q^{85} - 317400336q^{86} - 2979036210q^{87} - 64126976q^{88} - 1652463181q^{89} - 622306080q^{90} - 1141270092q^{91} + 883032320q^{92} - 1839612746q^{93} + 1931322400q^{94} + 2397253503q^{95} + 346030080q^{96} + 248805607q^{97} - 636637200q^{98} + 520684712q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} - 3 x^{13} + 93094 x^{12} + 3763497 x^{11} + 6525212431 x^{10} + 244392759522 x^{9} + 198778066371639 x^{8} + 9216645483921129 x^{7} + 4530154246073222607 x^{6} + 142122715974381300066 x^{5} + 8185019559882294055671 x^{4} - 135327799885026117969495 x^{3} + 2044710563339570147369550 x^{2} - 9649457865335314261798875 x + 37683939484136051622500625\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(28\!\cdots\!82\)\( \nu^{13} + \)\(19\!\cdots\!36\)\( \nu^{12} - \)\(26\!\cdots\!78\)\( \nu^{11} - \)\(11\!\cdots\!94\)\( \nu^{10} - \)\(18\!\cdots\!12\)\( \nu^{9} - \)\(74\!\cdots\!89\)\( \nu^{8} - \)\(57\!\cdots\!18\)\( \nu^{7} - \)\(27\!\cdots\!18\)\( \nu^{6} - \)\(13\!\cdots\!64\)\( \nu^{5} - \)\(43\!\cdots\!82\)\( \nu^{4} - \)\(25\!\cdots\!82\)\( \nu^{3} + \)\(29\!\cdots\!30\)\( \nu^{2} - \)\(62\!\cdots\!50\)\( \nu + \)\(29\!\cdots\!50\)\(\)\()/ \)\(24\!\cdots\!75\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(13\!\cdots\!34\)\( \nu^{13} + \)\(26\!\cdots\!78\)\( \nu^{12} + \)\(12\!\cdots\!56\)\( \nu^{11} + \)\(77\!\cdots\!18\)\( \nu^{10} + \)\(87\!\cdots\!39\)\( \nu^{9} + \)\(52\!\cdots\!28\)\( \nu^{8} + \)\(27\!\cdots\!36\)\( \nu^{7} + \)\(18\!\cdots\!06\)\( \nu^{6} + \)\(63\!\cdots\!58\)\( \nu^{5} + \)\(32\!\cdots\!04\)\( \nu^{4} + \)\(17\!\cdots\!44\)\( \nu^{3} + \)\(83\!\cdots\!50\)\( \nu^{2} + \)\(45\!\cdots\!25\)\( \nu - \)\(21\!\cdots\!50\)\(\)\()/ \)\(49\!\cdots\!25\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(11\!\cdots\!68\)\( \nu^{13} - \)\(17\!\cdots\!81\)\( \nu^{12} - \)\(10\!\cdots\!12\)\( \nu^{11} - \)\(62\!\cdots\!61\)\( \nu^{10} - \)\(71\!\cdots\!28\)\( \nu^{9} - \)\(41\!\cdots\!06\)\( \nu^{8} - \)\(21\!\cdots\!72\)\( \nu^{7} - \)\(14\!\cdots\!37\)\( \nu^{6} - \)\(48\!\cdots\!16\)\( \nu^{5} - \)\(25\!\cdots\!58\)\( \nu^{4} - \)\(31\!\cdots\!88\)\( \nu^{3} - \)\(63\!\cdots\!25\)\( \nu^{2} + \)\(30\!\cdots\!00\)\( \nu - \)\(78\!\cdots\!25\)\(\)\()/ \)\(31\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(17\!\cdots\!24\)\( \nu^{13} + \)\(34\!\cdots\!83\)\( \nu^{12} + \)\(16\!\cdots\!16\)\( \nu^{11} + \)\(10\!\cdots\!23\)\( \nu^{10} + \)\(11\!\cdots\!04\)\( \nu^{9} + \)\(68\!\cdots\!58\)\( \nu^{8} + \)\(35\!\cdots\!96\)\( \nu^{7} + \)\(23\!\cdots\!91\)\( \nu^{6} + \)\(82\!\cdots\!88\)\( \nu^{5} + \)\(42\!\cdots\!94\)\( \nu^{4} + \)\(20\!\cdots\!84\)\( \nu^{3} + \)\(10\!\cdots\!75\)\( \nu^{2} - \)\(51\!\cdots\!00\)\( \nu + \)\(39\!\cdots\!75\)\(\)\()/ \)\(21\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(50\!\cdots\!63\)\( \nu^{13} - \)\(10\!\cdots\!71\)\( \nu^{12} - \)\(47\!\cdots\!42\)\( \nu^{11} - \)\(29\!\cdots\!26\)\( \nu^{10} - \)\(33\!\cdots\!73\)\( \nu^{9} - \)\(20\!\cdots\!96\)\( \nu^{8} - \)\(10\!\cdots\!02\)\( \nu^{7} - \)\(70\!\cdots\!92\)\( \nu^{6} - \)\(24\!\cdots\!31\)\( \nu^{5} - \)\(12\!\cdots\!28\)\( \nu^{4} - \)\(69\!\cdots\!08\)\( \nu^{3} - \)\(32\!\cdots\!50\)\( \nu^{2} + \)\(15\!\cdots\!00\)\( \nu - \)\(23\!\cdots\!75\)\(\)\()/ \)\(39\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(62\!\cdots\!63\)\( \nu^{13} - \)\(27\!\cdots\!69\)\( \nu^{12} + \)\(58\!\cdots\!87\)\( \nu^{11} + \)\(22\!\cdots\!91\)\( \nu^{10} + \)\(40\!\cdots\!18\)\( \nu^{9} + \)\(14\!\cdots\!06\)\( \nu^{8} + \)\(12\!\cdots\!47\)\( \nu^{7} + \)\(55\!\cdots\!07\)\( \nu^{6} + \)\(28\!\cdots\!46\)\( \nu^{5} + \)\(84\!\cdots\!98\)\( \nu^{4} + \)\(48\!\cdots\!43\)\( \nu^{3} - \)\(96\!\cdots\!65\)\( \nu^{2} + \)\(12\!\cdots\!75\)\( \nu - \)\(57\!\cdots\!25\)\(\)\()/ \)\(23\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(28\!\cdots\!24\)\( \nu^{13} + \)\(56\!\cdots\!33\)\( \nu^{12} + \)\(26\!\cdots\!16\)\( \nu^{11} + \)\(16\!\cdots\!73\)\( \nu^{10} + \)\(19\!\cdots\!04\)\( \nu^{9} + \)\(11\!\cdots\!58\)\( \nu^{8} + \)\(59\!\cdots\!96\)\( \nu^{7} + \)\(39\!\cdots\!41\)\( \nu^{6} + \)\(13\!\cdots\!88\)\( \nu^{5} + \)\(71\!\cdots\!94\)\( \nu^{4} + \)\(34\!\cdots\!84\)\( \nu^{3} + \)\(18\!\cdots\!25\)\( \nu^{2} - \)\(85\!\cdots\!00\)\( \nu + \)\(43\!\cdots\!25\)\(\)\()/ \)\(31\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(41\!\cdots\!97\)\( \nu^{13} + \)\(29\!\cdots\!56\)\( \nu^{12} - \)\(38\!\cdots\!38\)\( \nu^{11} - \)\(16\!\cdots\!24\)\( \nu^{10} - \)\(26\!\cdots\!52\)\( \nu^{9} - \)\(10\!\cdots\!69\)\( \nu^{8} - \)\(82\!\cdots\!28\)\( \nu^{7} - \)\(39\!\cdots\!78\)\( \nu^{6} - \)\(18\!\cdots\!44\)\( \nu^{5} - \)\(63\!\cdots\!47\)\( \nu^{4} - \)\(36\!\cdots\!22\)\( \nu^{3} + \)\(42\!\cdots\!30\)\( \nu^{2} - \)\(68\!\cdots\!25\)\( \nu + \)\(68\!\cdots\!75\)\(\)\()/ \)\(44\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(61\!\cdots\!28\)\( \nu^{13} - \)\(12\!\cdots\!51\)\( \nu^{12} - \)\(57\!\cdots\!52\)\( \nu^{11} - \)\(36\!\cdots\!31\)\( \nu^{10} - \)\(40\!\cdots\!88\)\( \nu^{9} - \)\(24\!\cdots\!26\)\( \nu^{8} - \)\(12\!\cdots\!12\)\( \nu^{7} - \)\(84\!\cdots\!27\)\( \nu^{6} - \)\(29\!\cdots\!36\)\( \nu^{5} - \)\(15\!\cdots\!18\)\( \nu^{4} - \)\(73\!\cdots\!48\)\( \nu^{3} - \)\(38\!\cdots\!75\)\( \nu^{2} + \)\(18\!\cdots\!00\)\( \nu - \)\(86\!\cdots\!75\)\(\)\()/ \)\(31\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(76\!\cdots\!93\)\( \nu^{13} - \)\(47\!\cdots\!59\)\( \nu^{12} + \)\(71\!\cdots\!57\)\( \nu^{11} + \)\(26\!\cdots\!01\)\( \nu^{10} + \)\(49\!\cdots\!98\)\( \nu^{9} + \)\(17\!\cdots\!66\)\( \nu^{8} + \)\(15\!\cdots\!17\)\( \nu^{7} + \)\(65\!\cdots\!77\)\( \nu^{6} + \)\(34\!\cdots\!06\)\( \nu^{5} + \)\(97\!\cdots\!78\)\( \nu^{4} + \)\(56\!\cdots\!73\)\( \nu^{3} - \)\(13\!\cdots\!15\)\( \nu^{2} + \)\(14\!\cdots\!25\)\( \nu - \)\(66\!\cdots\!75\)\(\)\()/ \)\(35\!\cdots\!00\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(10\!\cdots\!77\)\( \nu^{13} - \)\(16\!\cdots\!96\)\( \nu^{12} + \)\(94\!\cdots\!33\)\( \nu^{11} + \)\(40\!\cdots\!84\)\( \nu^{10} + \)\(66\!\cdots\!82\)\( \nu^{9} + \)\(26\!\cdots\!04\)\( \nu^{8} + \)\(20\!\cdots\!73\)\( \nu^{7} + \)\(99\!\cdots\!48\)\( \nu^{6} + \)\(46\!\cdots\!54\)\( \nu^{5} + \)\(15\!\cdots\!52\)\( \nu^{4} + \)\(89\!\cdots\!77\)\( \nu^{3} - \)\(10\!\cdots\!80\)\( \nu^{2} + \)\(22\!\cdots\!25\)\( \nu - \)\(16\!\cdots\!00\)\(\)\()/ \)\(35\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(15\!\cdots\!31\)\( \nu^{13} + \)\(10\!\cdots\!53\)\( \nu^{12} - \)\(14\!\cdots\!19\)\( \nu^{11} - \)\(53\!\cdots\!67\)\( \nu^{10} - \)\(10\!\cdots\!66\)\( \nu^{9} - \)\(34\!\cdots\!22\)\( \nu^{8} - \)\(30\!\cdots\!39\)\( \nu^{7} - \)\(13\!\cdots\!59\)\( \nu^{6} - \)\(69\!\cdots\!02\)\( \nu^{5} - \)\(19\!\cdots\!26\)\( \nu^{4} - \)\(11\!\cdots\!91\)\( \nu^{3} + \)\(27\!\cdots\!05\)\( \nu^{2} - \)\(28\!\cdots\!75\)\( \nu + \)\(13\!\cdots\!25\)\(\)\()/ \)\(35\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{12} + \beta_{9} + 2 \beta_{7} - \beta_{6} - \beta_{4} + 32 \beta_{3} - 26583 \beta_{2}\)
\(\nu^{3}\)\(=\)\(-36 \beta_{10} - 52 \beta_{8} + 35 \beta_{6} - 406 \beta_{5} - 71 \beta_{4} + 47365 \beta_{3} - 47365 \beta_{1} - 845938\)
\(\nu^{4}\)\(=\)\(-11448 \beta_{13} + 52030 \beta_{12} - 15702 \beta_{11} - 11448 \beta_{10} - 45184 \beta_{9} - 15702 \beta_{8} - 198740 \beta_{7} - 198740 \beta_{5} + 1258162311 \beta_{2} - 3000674 \beta_{1} - 1258162311\)
\(\nu^{5}\)\(=\)\(-3830256 \beta_{13} + 6051722 \beta_{12} - 4180162 \beta_{11} + 1545560 \beta_{9} - 52288204 \beta_{7} - 1545560 \beta_{6} + 6051722 \beta_{4} - 2391399703 \beta_{3} + 79216825858 \beta_{2}\)
\(\nu^{6}\)\(=\)\(1140296184 \beta_{10} + 1542167838 \beta_{8} + 2009199649 \beta_{6} + 14985660806 \beta_{5} + 2870453191 \beta_{4} - 231036642746 \beta_{3} + 231036642746 \beta_{1} + 63436713684549\)
\(\nu^{7}\)\(=\)\(303066446004 \beta_{13} - 441616433321 \beta_{12} + 310103988814 \beta_{11} + 303066446004 \beta_{10} - 53842768955 \beta_{9} + 310103988814 \beta_{8} + 4253033625010 \beta_{7} + 4253033625010 \beta_{5} - 6098966233032172 \beta_{2} + 126779676568195 \beta_{1} + 6098966233032172\)
\(\nu^{8}\)\(=\)\(86792913157104 \beta_{13} - 164655104212204 \beta_{12} + 114641442952860 \beta_{11} + 91734153370816 \beta_{9} + 1049564155594568 \beta_{7} - 91734153370816 \beta_{6} - 164655104212204 \beta_{4} + 16403947718430164 \beta_{3} - 3358973998434191805 \beta_{2}\)
\(\nu^{9}\)\(=\)\(-21544491993243552 \beta_{10} - 21966237278809684 \beta_{8} + 1135553561210672 \beta_{6} - 300375500284767736 \beta_{5} - 30179326293356516 \beta_{4} + 7014786199629432541 \beta_{3} - 7014786199629432541 \beta_{1} - 433058402539560590740\)
\(\nu^{10}\)\(=\)\(-5989779950101012080 \beta_{13} + 9721406783474515261 \beta_{12} - 7720137404613420396 \beta_{11} - 5989779950101012080 \beta_{10} - 4332008114915798785 \beta_{9} - 7720137404613420396 \beta_{8} - 70704033345864156458 \beta_{7} - 70704033345864156458 \beta_{5} + 185669641663948224365475 \beta_{2} - 1114164007994019899684 \beta_{1} - 185669641663948224365475\)
\(\nu^{11}\)\(=\)\(-1454573470541008381956 \beta_{13} + 1992602456070742974875 \beta_{12} - 1505142940081529604856 \beta_{11} - 39838062196026226477 \beta_{9} - 19964099963070813809326 \beta_{7} + 39838062196026226477 \beta_{6} + 1992602456070742974875 \beta_{4} - 401997508921039388957425 \beta_{3} + 29415434886674421320913334 \beta_{2}\)
\(\nu^{12}\)\(=\)\(394936272346976129420712 \beta_{10} + 497594733901654897316178 \beta_{8} + 212496394375571142831808 \beta_{6} + 4652485075170438320471516 \beta_{5} + 585852083740684460804746 \beta_{4} - 73673617936125002170717142 \beta_{3} + 73673617936125002170717142 \beta_{1} + 10632255945340624213017967395\)
\(\nu^{13}\)\(=\)\(95478728427033554332308240 \beta_{13} - 129006727772996463202723214 \beta_{12} + 100683609110753134601251126 \beta_{11} + 95478728427033554332308240 \beta_{10} + 7851138257618391745600120 \beta_{9} + 100683609110753134601251126 \beta_{8} + 1288107084366398291979612292 \beta_{7} + 1288107084366398291979612292 \beta_{5} - 1945194676776195593638651177270 \beta_{2} + 23668844752122450048904633555 \beta_{1} + 1945194676776195593638651177270\)

Character values

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

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

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
125.259 + 216.954i
97.2191 + 168.388i
5.46583 + 9.46709i
2.70706 + 4.68876i
−25.7605 44.6185i
−98.6185 170.812i
−104.772 181.470i
125.259 216.954i
97.2191 168.388i
5.46583 9.46709i
2.70706 4.68876i
−25.7605 + 44.6185i
−98.6185 + 170.812i
−104.772 + 181.470i
8.00000 + 13.8564i −113.259 196.170i −128.000 + 221.703i 563.006 + 975.155i 1812.14 3138.71i 3883.37 −4096.00 −15813.5 + 27389.8i −9008.10 + 15602.5i
7.2 8.00000 + 13.8564i −85.2191 147.604i −128.000 + 221.703i −867.647 1502.81i 1363.51 2361.66i −2874.94 −4096.00 −4683.10 + 8111.37i 13882.3 24044.9i
7.3 8.00000 + 13.8564i 6.53417 + 11.3175i −128.000 + 221.703i 80.6193 + 139.637i −104.547 + 181.080i −6270.17 −4096.00 9756.11 16898.1i −1289.91 + 2234.19i
7.4 8.00000 + 13.8564i 9.29294 + 16.0959i −128.000 + 221.703i 830.940 + 1439.23i −148.687 + 257.534i 10512.6 −4096.00 9668.78 16746.8i −13295.0 + 23027.7i
7.5 8.00000 + 13.8564i 37.7605 + 65.4031i −128.000 + 221.703i −802.961 1390.77i −604.168 + 1046.45i 1586.11 −4096.00 6989.79 12106.7i 12847.4 22252.3i
7.6 8.00000 + 13.8564i 110.618 + 191.597i −128.000 + 221.703i 1160.30 + 2009.69i −1769.90 + 3065.55i −7717.55 −4096.00 −14631.4 + 25342.3i −18564.7 + 32155.1i
7.7 8.00000 + 13.8564i 116.772 + 202.254i −128.000 + 221.703i −509.754 882.920i −1868.34 + 3236.07i 2726.62 −4096.00 −17429.7 + 30189.1i 8156.07 14126.7i
11.1 8.00000 13.8564i −113.259 + 196.170i −128.000 221.703i 563.006 975.155i 1812.14 + 3138.71i 3883.37 −4096.00 −15813.5 27389.8i −9008.10 15602.5i
11.2 8.00000 13.8564i −85.2191 + 147.604i −128.000 221.703i −867.647 + 1502.81i 1363.51 + 2361.66i −2874.94 −4096.00 −4683.10 8111.37i 13882.3 + 24044.9i
11.3 8.00000 13.8564i 6.53417 11.3175i −128.000 221.703i 80.6193 139.637i −104.547 181.080i −6270.17 −4096.00 9756.11 + 16898.1i −1289.91 2234.19i
11.4 8.00000 13.8564i 9.29294 16.0959i −128.000 221.703i 830.940 1439.23i −148.687 257.534i 10512.6 −4096.00 9668.78 + 16746.8i −13295.0 23027.7i
11.5 8.00000 13.8564i 37.7605 65.4031i −128.000 221.703i −802.961 + 1390.77i −604.168 1046.45i 1586.11 −4096.00 6989.79 + 12106.7i 12847.4 + 22252.3i
11.6 8.00000 13.8564i 110.618 191.597i −128.000 221.703i 1160.30 2009.69i −1769.90 3065.55i −7717.55 −4096.00 −14631.4 25342.3i −18564.7 32155.1i
11.7 8.00000 13.8564i 116.772 202.254i −128.000 221.703i −509.754 + 882.920i −1868.34 3236.07i 2726.62 −4096.00 −17429.7 30189.1i 8156.07 + 14126.7i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.7
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 38.10.c.a 14
19.c even 3 1 inner 38.10.c.a 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.10.c.a 14 1.a even 1 1 trivial
38.10.c.a 14 19.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{14} - \cdots\) acting on \(S_{10}^{\mathrm{new}}(38, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 16 T + 256 T^{2} )^{7} \)
$3$ \( 1 - 165 T - 29135 T^{2} + 4764138 T^{3} + 451677226 T^{4} - 62681097504 T^{5} + 14703036541461 T^{6} - 1601552642287833 T^{7} - 596851372361123640 T^{8} + 65626681026643355331 T^{9} + \)\(68\!\cdots\!55\)\( T^{10} - \)\(10\!\cdots\!28\)\( T^{11} + \)\(15\!\cdots\!85\)\( T^{12} + \)\(82\!\cdots\!39\)\( T^{13} - \)\(62\!\cdots\!54\)\( T^{14} + \)\(16\!\cdots\!37\)\( T^{15} + \)\(60\!\cdots\!65\)\( T^{16} - \)\(82\!\cdots\!36\)\( T^{17} + \)\(10\!\cdots\!55\)\( T^{18} + \)\(19\!\cdots\!33\)\( T^{19} - \)\(34\!\cdots\!60\)\( T^{20} - \)\(18\!\cdots\!91\)\( T^{21} + \)\(33\!\cdots\!01\)\( T^{22} - \)\(27\!\cdots\!12\)\( T^{23} + \)\(39\!\cdots\!74\)\( T^{24} + \)\(81\!\cdots\!46\)\( T^{25} - \)\(98\!\cdots\!35\)\( T^{26} - \)\(10\!\cdots\!95\)\( T^{27} + \)\(13\!\cdots\!29\)\( T^{28} \)
$5$ \( 1 - 909 T - 5223473 T^{2} + 6842219956 T^{3} + 7631944964557 T^{4} - 17998555332534719 T^{5} - 647245457166208194 T^{6} + \)\(24\!\cdots\!65\)\( T^{7} + \)\(79\!\cdots\!30\)\( T^{8} - \)\(42\!\cdots\!25\)\( T^{9} - \)\(40\!\cdots\!50\)\( T^{10} + \)\(79\!\cdots\!75\)\( T^{11} + \)\(30\!\cdots\!50\)\( T^{12} - \)\(59\!\cdots\!75\)\( T^{13} + \)\(24\!\cdots\!50\)\( T^{14} - \)\(11\!\cdots\!75\)\( T^{15} + \)\(11\!\cdots\!50\)\( T^{16} + \)\(59\!\cdots\!75\)\( T^{17} - \)\(59\!\cdots\!50\)\( T^{18} - \)\(12\!\cdots\!25\)\( T^{19} + \)\(44\!\cdots\!50\)\( T^{20} + \)\(26\!\cdots\!25\)\( T^{21} - \)\(13\!\cdots\!50\)\( T^{22} - \)\(74\!\cdots\!75\)\( T^{23} + \)\(61\!\cdots\!25\)\( T^{24} + \)\(10\!\cdots\!00\)\( T^{25} - \)\(16\!\cdots\!25\)\( T^{26} - \)\(54\!\cdots\!25\)\( T^{27} + \)\(11\!\cdots\!25\)\( T^{28} \)
$7$ \( ( 1 - 1846 T + 162836395 T^{2} - 395563390220 T^{3} + 13113902308594451 T^{4} - 40015938072154266698 T^{5} + \)\(70\!\cdots\!25\)\( T^{6} - \)\(21\!\cdots\!72\)\( T^{7} + \)\(28\!\cdots\!75\)\( T^{8} - \)\(65\!\cdots\!02\)\( T^{9} + \)\(86\!\cdots\!93\)\( T^{10} - \)\(10\!\cdots\!20\)\( T^{11} + \)\(17\!\cdots\!65\)\( T^{12} - \)\(79\!\cdots\!54\)\( T^{13} + \)\(17\!\cdots\!43\)\( T^{14} )^{2} \)
$11$ \( ( 1 - 7828 T + 10744399421 T^{2} + 73506903973280 T^{3} + 49953305391936986260 T^{4} + \)\(99\!\cdots\!40\)\( T^{5} + \)\(14\!\cdots\!48\)\( T^{6} + \)\(36\!\cdots\!96\)\( T^{7} + \)\(34\!\cdots\!68\)\( T^{8} + \)\(55\!\cdots\!40\)\( T^{9} + \)\(65\!\cdots\!60\)\( T^{10} + \)\(22\!\cdots\!80\)\( T^{11} + \)\(78\!\cdots\!71\)\( T^{12} - \)\(13\!\cdots\!48\)\( T^{13} + \)\(40\!\cdots\!31\)\( T^{14} )^{2} \)
$13$ \( 1 - 6423 T - 31740272629 T^{2} + 152266129981108 T^{3} + \)\(42\!\cdots\!71\)\( T^{4} + \)\(51\!\cdots\!53\)\( T^{5} - \)\(29\!\cdots\!88\)\( T^{6} - \)\(29\!\cdots\!57\)\( T^{7} + \)\(84\!\cdots\!86\)\( T^{8} + \)\(56\!\cdots\!57\)\( T^{9} + \)\(58\!\cdots\!56\)\( T^{10} - \)\(51\!\cdots\!37\)\( T^{11} - \)\(15\!\cdots\!08\)\( T^{12} + \)\(19\!\cdots\!87\)\( T^{13} + \)\(20\!\cdots\!86\)\( T^{14} + \)\(20\!\cdots\!51\)\( T^{15} - \)\(17\!\cdots\!32\)\( T^{16} - \)\(61\!\cdots\!29\)\( T^{17} + \)\(73\!\cdots\!96\)\( T^{18} + \)\(75\!\cdots\!01\)\( T^{19} + \)\(12\!\cdots\!54\)\( T^{20} - \)\(44\!\cdots\!29\)\( T^{21} - \)\(46\!\cdots\!28\)\( T^{22} + \)\(86\!\cdots\!89\)\( T^{23} + \)\(77\!\cdots\!79\)\( T^{24} + \)\(29\!\cdots\!16\)\( T^{25} - \)\(64\!\cdots\!09\)\( T^{26} - \)\(13\!\cdots\!59\)\( T^{27} + \)\(22\!\cdots\!09\)\( T^{28} \)
$17$ \( 1 - 313667 T - 468818776657 T^{2} + 96272340434591936 T^{3} + \)\(12\!\cdots\!27\)\( T^{4} - \)\(13\!\cdots\!11\)\( T^{5} - \)\(24\!\cdots\!56\)\( T^{6} + \)\(17\!\cdots\!15\)\( T^{7} + \)\(36\!\cdots\!46\)\( T^{8} - \)\(25\!\cdots\!43\)\( T^{9} - \)\(51\!\cdots\!04\)\( T^{10} + \)\(33\!\cdots\!91\)\( T^{11} + \)\(72\!\cdots\!76\)\( T^{12} - \)\(20\!\cdots\!29\)\( T^{13} - \)\(91\!\cdots\!86\)\( T^{14} - \)\(24\!\cdots\!13\)\( T^{15} + \)\(10\!\cdots\!84\)\( T^{16} + \)\(55\!\cdots\!43\)\( T^{17} - \)\(10\!\cdots\!24\)\( T^{18} - \)\(58\!\cdots\!51\)\( T^{19} + \)\(10\!\cdots\!34\)\( T^{20} + \)\(58\!\cdots\!95\)\( T^{21} - \)\(95\!\cdots\!16\)\( T^{22} - \)\(63\!\cdots\!87\)\( T^{23} + \)\(69\!\cdots\!23\)\( T^{24} + \)\(62\!\cdots\!08\)\( T^{25} - \)\(36\!\cdots\!37\)\( T^{26} - \)\(28\!\cdots\!59\)\( T^{27} + \)\(10\!\cdots\!69\)\( T^{28} \)
$19$ \( 1 - 1134224 T + 727494745261 T^{2} - 223621835454267072 T^{3} + \)\(30\!\cdots\!68\)\( T^{4} + \)\(54\!\cdots\!12\)\( T^{5} - \)\(77\!\cdots\!16\)\( T^{6} + \)\(60\!\cdots\!28\)\( T^{7} - \)\(25\!\cdots\!64\)\( T^{8} + \)\(56\!\cdots\!92\)\( T^{9} + \)\(10\!\cdots\!52\)\( T^{10} - \)\(24\!\cdots\!32\)\( T^{11} + \)\(25\!\cdots\!39\)\( T^{12} - \)\(12\!\cdots\!04\)\( T^{13} + \)\(36\!\cdots\!59\)\( T^{14} \)
$23$ \( 1 - 3449345 T + 1487353643681 T^{2} + 11807928096761327694 T^{3} - \)\(24\!\cdots\!09\)\( T^{4} + \)\(16\!\cdots\!87\)\( T^{5} + \)\(21\!\cdots\!82\)\( T^{6} - \)\(84\!\cdots\!91\)\( T^{7} + \)\(14\!\cdots\!92\)\( T^{8} - \)\(45\!\cdots\!03\)\( T^{9} - \)\(26\!\cdots\!26\)\( T^{10} + \)\(46\!\cdots\!79\)\( T^{11} - \)\(15\!\cdots\!40\)\( T^{12} - \)\(46\!\cdots\!33\)\( T^{13} + \)\(92\!\cdots\!54\)\( T^{14} - \)\(83\!\cdots\!79\)\( T^{15} - \)\(48\!\cdots\!60\)\( T^{16} + \)\(27\!\cdots\!13\)\( T^{17} - \)\(28\!\cdots\!86\)\( T^{18} - \)\(87\!\cdots\!29\)\( T^{19} + \)\(47\!\cdots\!28\)\( T^{20} - \)\(51\!\cdots\!97\)\( T^{21} + \)\(23\!\cdots\!22\)\( T^{22} + \)\(32\!\cdots\!01\)\( T^{23} - \)\(89\!\cdots\!41\)\( T^{24} + \)\(76\!\cdots\!78\)\( T^{25} + \)\(17\!\cdots\!61\)\( T^{26} - \)\(72\!\cdots\!35\)\( T^{27} + \)\(37\!\cdots\!89\)\( T^{28} \)
$29$ \( 1 + 7002615 T - 21892482805901 T^{2} - 99908308211445943024 T^{3} + \)\(15\!\cdots\!93\)\( T^{4} + \)\(53\!\cdots\!81\)\( T^{5} - \)\(18\!\cdots\!26\)\( T^{6} + \)\(13\!\cdots\!01\)\( T^{7} + \)\(57\!\cdots\!46\)\( T^{8} + \)\(72\!\cdots\!31\)\( T^{9} + \)\(26\!\cdots\!38\)\( T^{10} + \)\(24\!\cdots\!71\)\( T^{11} + \)\(25\!\cdots\!78\)\( T^{12} + \)\(19\!\cdots\!61\)\( T^{13} + \)\(24\!\cdots\!34\)\( T^{14} + \)\(28\!\cdots\!09\)\( T^{15} + \)\(53\!\cdots\!58\)\( T^{16} + \)\(74\!\cdots\!39\)\( T^{17} + \)\(11\!\cdots\!98\)\( T^{18} + \)\(46\!\cdots\!19\)\( T^{19} + \)\(53\!\cdots\!26\)\( T^{20} + \)\(18\!\cdots\!89\)\( T^{21} - \)\(35\!\cdots\!66\)\( T^{22} + \)\(15\!\cdots\!49\)\( T^{23} + \)\(65\!\cdots\!93\)\( T^{24} - \)\(59\!\cdots\!56\)\( T^{25} - \)\(19\!\cdots\!61\)\( T^{26} + \)\(88\!\cdots\!35\)\( T^{27} + \)\(18\!\cdots\!21\)\( T^{28} \)
$31$ \( ( 1 + 9333294 T + 148134985391023 T^{2} + \)\(10\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!35\)\( T^{4} + \)\(57\!\cdots\!94\)\( T^{5} + \)\(40\!\cdots\!61\)\( T^{6} + \)\(19\!\cdots\!04\)\( T^{7} + \)\(10\!\cdots\!31\)\( T^{8} + \)\(40\!\cdots\!54\)\( T^{9} + \)\(18\!\cdots\!85\)\( T^{10} + \)\(51\!\cdots\!40\)\( T^{11} + \)\(19\!\cdots\!73\)\( T^{12} + \)\(31\!\cdots\!74\)\( T^{13} + \)\(90\!\cdots\!91\)\( T^{14} )^{2} \)
$37$ \( ( 1 + 3433040 T + 360370797123145 T^{2} - \)\(96\!\cdots\!48\)\( T^{3} + \)\(63\!\cdots\!59\)\( T^{4} - \)\(35\!\cdots\!32\)\( T^{5} + \)\(11\!\cdots\!75\)\( T^{6} - \)\(43\!\cdots\!04\)\( T^{7} + \)\(14\!\cdots\!75\)\( T^{8} - \)\(59\!\cdots\!28\)\( T^{9} + \)\(14\!\cdots\!47\)\( T^{10} - \)\(27\!\cdots\!68\)\( T^{11} + \)\(13\!\cdots\!65\)\( T^{12} + \)\(16\!\cdots\!60\)\( T^{13} + \)\(62\!\cdots\!53\)\( T^{14} )^{2} \)
$41$ \( 1 - 3564107 T - 1773733816349839 T^{2} + \)\(90\!\cdots\!54\)\( T^{3} + \)\(16\!\cdots\!04\)\( T^{4} - \)\(94\!\cdots\!96\)\( T^{5} - \)\(11\!\cdots\!87\)\( T^{6} + \)\(55\!\cdots\!75\)\( T^{7} + \)\(60\!\cdots\!64\)\( T^{8} - \)\(21\!\cdots\!71\)\( T^{9} - \)\(27\!\cdots\!41\)\( T^{10} + \)\(53\!\cdots\!92\)\( T^{11} + \)\(11\!\cdots\!95\)\( T^{12} - \)\(62\!\cdots\!43\)\( T^{13} - \)\(39\!\cdots\!22\)\( T^{14} - \)\(20\!\cdots\!23\)\( T^{15} + \)\(11\!\cdots\!95\)\( T^{16} + \)\(18\!\cdots\!52\)\( T^{17} - \)\(31\!\cdots\!81\)\( T^{18} - \)\(79\!\cdots\!71\)\( T^{19} + \)\(74\!\cdots\!04\)\( T^{20} + \)\(22\!\cdots\!75\)\( T^{21} - \)\(14\!\cdots\!47\)\( T^{22} - \)\(40\!\cdots\!36\)\( T^{23} + \)\(23\!\cdots\!04\)\( T^{24} + \)\(41\!\cdots\!94\)\( T^{25} - \)\(26\!\cdots\!19\)\( T^{26} - \)\(17\!\cdots\!67\)\( T^{27} + \)\(16\!\cdots\!41\)\( T^{28} \)
$43$ \( 1 - 19837521 T - 1509371539523275 T^{2} + \)\(57\!\cdots\!50\)\( T^{3} + \)\(60\!\cdots\!57\)\( T^{4} - \)\(55\!\cdots\!51\)\( T^{5} + \)\(49\!\cdots\!16\)\( T^{6} + \)\(22\!\cdots\!33\)\( T^{7} - \)\(56\!\cdots\!60\)\( T^{8} + \)\(33\!\cdots\!57\)\( T^{9} + \)\(12\!\cdots\!68\)\( T^{10} - \)\(69\!\cdots\!55\)\( T^{11} + \)\(13\!\cdots\!30\)\( T^{12} + \)\(20\!\cdots\!03\)\( T^{13} - \)\(12\!\cdots\!90\)\( T^{14} + \)\(10\!\cdots\!29\)\( T^{15} + \)\(35\!\cdots\!70\)\( T^{16} - \)\(88\!\cdots\!85\)\( T^{17} + \)\(81\!\cdots\!68\)\( T^{18} + \)\(10\!\cdots\!51\)\( T^{19} - \)\(90\!\cdots\!40\)\( T^{20} + \)\(18\!\cdots\!31\)\( T^{21} + \)\(20\!\cdots\!16\)\( T^{22} - \)\(11\!\cdots\!93\)\( T^{23} + \)\(61\!\cdots\!93\)\( T^{24} + \)\(29\!\cdots\!50\)\( T^{25} - \)\(39\!\cdots\!75\)\( T^{26} - \)\(25\!\cdots\!03\)\( T^{27} + \)\(65\!\cdots\!49\)\( T^{28} \)
$47$ \( 1 - 60353825 T - 2620632466749919 T^{2} + \)\(14\!\cdots\!26\)\( T^{3} + \)\(66\!\cdots\!83\)\( T^{4} - \)\(17\!\cdots\!33\)\( T^{5} - \)\(14\!\cdots\!10\)\( T^{6} + \)\(24\!\cdots\!69\)\( T^{7} + \)\(20\!\cdots\!12\)\( T^{8} - \)\(23\!\cdots\!51\)\( T^{9} - \)\(28\!\cdots\!46\)\( T^{10} + \)\(25\!\cdots\!63\)\( T^{11} + \)\(40\!\cdots\!76\)\( T^{12} - \)\(18\!\cdots\!49\)\( T^{13} - \)\(46\!\cdots\!66\)\( T^{14} - \)\(21\!\cdots\!83\)\( T^{15} + \)\(50\!\cdots\!64\)\( T^{16} + \)\(35\!\cdots\!69\)\( T^{17} - \)\(45\!\cdots\!66\)\( T^{18} - \)\(41\!\cdots\!57\)\( T^{19} + \)\(39\!\cdots\!28\)\( T^{20} + \)\(53\!\cdots\!87\)\( T^{21} - \)\(36\!\cdots\!10\)\( T^{22} - \)\(49\!\cdots\!51\)\( T^{23} + \)\(20\!\cdots\!67\)\( T^{24} + \)\(49\!\cdots\!58\)\( T^{25} - \)\(10\!\cdots\!59\)\( T^{26} - \)\(26\!\cdots\!75\)\( T^{27} + \)\(48\!\cdots\!29\)\( T^{28} \)
$53$ \( 1 - 54744235 T - 8539409640633709 T^{2} + \)\(38\!\cdots\!48\)\( T^{3} + \)\(38\!\cdots\!91\)\( T^{4} - \)\(10\!\cdots\!91\)\( T^{5} - \)\(98\!\cdots\!36\)\( T^{6} - \)\(72\!\cdots\!37\)\( T^{7} + \)\(11\!\cdots\!22\)\( T^{8} + \)\(12\!\cdots\!09\)\( T^{9} + \)\(22\!\cdots\!08\)\( T^{10} - \)\(36\!\cdots\!77\)\( T^{11} - \)\(19\!\cdots\!20\)\( T^{12} + \)\(50\!\cdots\!79\)\( T^{13} + \)\(70\!\cdots\!90\)\( T^{14} + \)\(16\!\cdots\!07\)\( T^{15} - \)\(21\!\cdots\!80\)\( T^{16} - \)\(13\!\cdots\!49\)\( T^{17} + \)\(27\!\cdots\!68\)\( T^{18} + \)\(48\!\cdots\!37\)\( T^{19} + \)\(14\!\cdots\!18\)\( T^{20} - \)\(30\!\cdots\!49\)\( T^{21} - \)\(13\!\cdots\!76\)\( T^{22} - \)\(47\!\cdots\!23\)\( T^{23} + \)\(58\!\cdots\!59\)\( T^{24} + \)\(19\!\cdots\!16\)\( T^{25} - \)\(14\!\cdots\!49\)\( T^{26} - \)\(30\!\cdots\!55\)\( T^{27} + \)\(18\!\cdots\!29\)\( T^{28} \)
$59$ \( 1 - 164456585 T + 645894830577137 T^{2} - \)\(90\!\cdots\!30\)\( T^{3} + \)\(16\!\cdots\!66\)\( T^{4} - \)\(67\!\cdots\!40\)\( T^{5} + \)\(10\!\cdots\!93\)\( T^{6} - \)\(18\!\cdots\!05\)\( T^{7} - \)\(83\!\cdots\!48\)\( T^{8} + \)\(11\!\cdots\!75\)\( T^{9} + \)\(61\!\cdots\!19\)\( T^{10} + \)\(73\!\cdots\!60\)\( T^{11} - \)\(30\!\cdots\!83\)\( T^{12} + \)\(71\!\cdots\!75\)\( T^{13} - \)\(14\!\cdots\!02\)\( T^{14} + \)\(61\!\cdots\!25\)\( T^{15} - \)\(22\!\cdots\!43\)\( T^{16} + \)\(48\!\cdots\!40\)\( T^{17} + \)\(34\!\cdots\!79\)\( T^{18} + \)\(57\!\cdots\!25\)\( T^{19} - \)\(35\!\cdots\!28\)\( T^{20} - \)\(68\!\cdots\!95\)\( T^{21} + \)\(33\!\cdots\!33\)\( T^{22} - \)\(18\!\cdots\!60\)\( T^{23} + \)\(39\!\cdots\!66\)\( T^{24} - \)\(18\!\cdots\!70\)\( T^{25} + \)\(11\!\cdots\!77\)\( T^{26} - \)\(25\!\cdots\!15\)\( T^{27} + \)\(13\!\cdots\!41\)\( T^{28} \)
$61$ \( 1 - 49328881 T - 42356131219314273 T^{2} + \)\(13\!\cdots\!32\)\( T^{3} + \)\(90\!\cdots\!57\)\( T^{4} - \)\(16\!\cdots\!99\)\( T^{5} - \)\(13\!\cdots\!30\)\( T^{6} + \)\(17\!\cdots\!17\)\( T^{7} + \)\(14\!\cdots\!34\)\( T^{8} - \)\(19\!\cdots\!61\)\( T^{9} - \)\(93\!\cdots\!10\)\( T^{10} + \)\(55\!\cdots\!99\)\( T^{11} + \)\(63\!\cdots\!02\)\( T^{12} + \)\(45\!\cdots\!53\)\( T^{13} + \)\(46\!\cdots\!98\)\( T^{14} + \)\(53\!\cdots\!73\)\( T^{15} + \)\(86\!\cdots\!62\)\( T^{16} + \)\(88\!\cdots\!79\)\( T^{17} - \)\(17\!\cdots\!10\)\( T^{18} - \)\(43\!\cdots\!61\)\( T^{19} + \)\(36\!\cdots\!94\)\( T^{20} + \)\(52\!\cdots\!77\)\( T^{21} - \)\(45\!\cdots\!30\)\( T^{22} - \)\(68\!\cdots\!39\)\( T^{23} + \)\(43\!\cdots\!57\)\( T^{24} + \)\(78\!\cdots\!12\)\( T^{25} - \)\(27\!\cdots\!13\)\( T^{26} - \)\(37\!\cdots\!01\)\( T^{27} + \)\(89\!\cdots\!61\)\( T^{28} \)
$67$ \( 1 + 171522309 T - 140691473513169943 T^{2} - \)\(14\!\cdots\!30\)\( T^{3} + \)\(12\!\cdots\!98\)\( T^{4} + \)\(66\!\cdots\!04\)\( T^{5} - \)\(79\!\cdots\!11\)\( T^{6} - \)\(13\!\cdots\!59\)\( T^{7} + \)\(37\!\cdots\!04\)\( T^{8} - \)\(11\!\cdots\!71\)\( T^{9} - \)\(13\!\cdots\!53\)\( T^{10} + \)\(12\!\cdots\!96\)\( T^{11} + \)\(43\!\cdots\!57\)\( T^{12} - \)\(21\!\cdots\!47\)\( T^{13} - \)\(12\!\cdots\!18\)\( T^{14} - \)\(57\!\cdots\!09\)\( T^{15} + \)\(31\!\cdots\!13\)\( T^{16} + \)\(25\!\cdots\!08\)\( T^{17} - \)\(75\!\cdots\!93\)\( T^{18} - \)\(16\!\cdots\!97\)\( T^{19} + \)\(15\!\cdots\!16\)\( T^{20} - \)\(14\!\cdots\!17\)\( T^{21} - \)\(23\!\cdots\!71\)\( T^{22} + \)\(54\!\cdots\!68\)\( T^{23} + \)\(28\!\cdots\!02\)\( T^{24} - \)\(85\!\cdots\!90\)\( T^{25} - \)\(23\!\cdots\!63\)\( T^{26} + \)\(76\!\cdots\!43\)\( T^{27} + \)\(12\!\cdots\!69\)\( T^{28} \)
$71$ \( 1 + 74596055 T - 278659361842615555 T^{2} - \)\(10\!\cdots\!78\)\( T^{3} + \)\(43\!\cdots\!13\)\( T^{4} + \)\(68\!\cdots\!33\)\( T^{5} - \)\(48\!\cdots\!88\)\( T^{6} - \)\(15\!\cdots\!35\)\( T^{7} + \)\(40\!\cdots\!84\)\( T^{8} - \)\(95\!\cdots\!39\)\( T^{9} - \)\(28\!\cdots\!92\)\( T^{10} + \)\(86\!\cdots\!97\)\( T^{11} + \)\(16\!\cdots\!06\)\( T^{12} - \)\(20\!\cdots\!13\)\( T^{13} - \)\(80\!\cdots\!78\)\( T^{14} - \)\(95\!\cdots\!03\)\( T^{15} + \)\(34\!\cdots\!66\)\( T^{16} + \)\(83\!\cdots\!27\)\( T^{17} - \)\(12\!\cdots\!32\)\( T^{18} - \)\(19\!\cdots\!89\)\( T^{19} + \)\(37\!\cdots\!04\)\( T^{20} - \)\(67\!\cdots\!85\)\( T^{21} - \)\(93\!\cdots\!08\)\( T^{22} + \)\(61\!\cdots\!43\)\( T^{23} + \)\(17\!\cdots\!13\)\( T^{24} - \)\(19\!\cdots\!18\)\( T^{25} - \)\(24\!\cdots\!55\)\( T^{26} + \)\(29\!\cdots\!05\)\( T^{27} + \)\(18\!\cdots\!21\)\( T^{28} \)
$73$ \( 1 - 58695287 T - 215198725080579879 T^{2} - \)\(10\!\cdots\!94\)\( T^{3} + \)\(26\!\cdots\!56\)\( T^{4} + \)\(32\!\cdots\!48\)\( T^{5} - \)\(17\!\cdots\!27\)\( T^{6} - \)\(40\!\cdots\!77\)\( T^{7} + \)\(59\!\cdots\!52\)\( T^{8} + \)\(30\!\cdots\!25\)\( T^{9} + \)\(24\!\cdots\!11\)\( T^{10} - \)\(15\!\cdots\!56\)\( T^{11} - \)\(49\!\cdots\!25\)\( T^{12} + \)\(36\!\cdots\!81\)\( T^{13} + \)\(38\!\cdots\!22\)\( T^{14} + \)\(21\!\cdots\!53\)\( T^{15} - \)\(17\!\cdots\!25\)\( T^{16} - \)\(32\!\cdots\!32\)\( T^{17} + \)\(29\!\cdots\!71\)\( T^{18} + \)\(21\!\cdots\!25\)\( T^{19} + \)\(24\!\cdots\!68\)\( T^{20} - \)\(98\!\cdots\!09\)\( T^{21} - \)\(25\!\cdots\!67\)\( T^{22} + \)\(27\!\cdots\!04\)\( T^{23} + \)\(13\!\cdots\!44\)\( T^{24} - \)\(31\!\cdots\!78\)\( T^{25} - \)\(37\!\cdots\!99\)\( T^{26} - \)\(59\!\cdots\!11\)\( T^{27} + \)\(60\!\cdots\!89\)\( T^{28} \)
$79$ \( 1 - 121854617 T - 293022562823571531 T^{2} + \)\(92\!\cdots\!38\)\( T^{3} + \)\(37\!\cdots\!05\)\( T^{4} + \)\(20\!\cdots\!37\)\( T^{5} - \)\(56\!\cdots\!24\)\( T^{6} - \)\(49\!\cdots\!75\)\( T^{7} - \)\(42\!\cdots\!84\)\( T^{8} - \)\(56\!\cdots\!55\)\( T^{9} + \)\(56\!\cdots\!04\)\( T^{10} + \)\(21\!\cdots\!41\)\( T^{11} - \)\(18\!\cdots\!62\)\( T^{12} - \)\(15\!\cdots\!53\)\( T^{13} - \)\(83\!\cdots\!50\)\( T^{14} - \)\(18\!\cdots\!07\)\( T^{15} - \)\(26\!\cdots\!82\)\( T^{16} + \)\(36\!\cdots\!19\)\( T^{17} + \)\(11\!\cdots\!84\)\( T^{18} - \)\(13\!\cdots\!45\)\( T^{19} - \)\(12\!\cdots\!04\)\( T^{20} - \)\(17\!\cdots\!25\)\( T^{21} - \)\(24\!\cdots\!84\)\( T^{22} + \)\(10\!\cdots\!23\)\( T^{23} + \)\(23\!\cdots\!05\)\( T^{24} + \)\(67\!\cdots\!22\)\( T^{25} - \)\(25\!\cdots\!91\)\( T^{26} - \)\(12\!\cdots\!03\)\( T^{27} + \)\(12\!\cdots\!21\)\( T^{28} \)
$83$ \( ( 1 + 366303628 T + 534324485457307121 T^{2} + \)\(12\!\cdots\!36\)\( T^{3} + \)\(20\!\cdots\!56\)\( T^{4} + \)\(49\!\cdots\!04\)\( T^{5} + \)\(50\!\cdots\!16\)\( T^{6} + \)\(91\!\cdots\!24\)\( T^{7} + \)\(94\!\cdots\!48\)\( T^{8} + \)\(17\!\cdots\!36\)\( T^{9} + \)\(13\!\cdots\!12\)\( T^{10} + \)\(15\!\cdots\!16\)\( T^{11} + \)\(12\!\cdots\!03\)\( T^{12} + \)\(15\!\cdots\!12\)\( T^{13} + \)\(79\!\cdots\!87\)\( T^{14} )^{2} \)
$89$ \( 1 + 1652463181 T + 722928101622300623 T^{2} - \)\(87\!\cdots\!28\)\( T^{3} - \)\(13\!\cdots\!97\)\( T^{4} - \)\(62\!\cdots\!79\)\( T^{5} + \)\(20\!\cdots\!84\)\( T^{6} + \)\(45\!\cdots\!99\)\( T^{7} + \)\(26\!\cdots\!02\)\( T^{8} + \)\(79\!\cdots\!77\)\( T^{9} - \)\(90\!\cdots\!36\)\( T^{10} - \)\(64\!\cdots\!49\)\( T^{11} - \)\(11\!\cdots\!64\)\( T^{12} + \)\(12\!\cdots\!23\)\( T^{13} + \)\(12\!\cdots\!46\)\( T^{14} + \)\(44\!\cdots\!07\)\( T^{15} - \)\(14\!\cdots\!84\)\( T^{16} - \)\(27\!\cdots\!21\)\( T^{17} - \)\(13\!\cdots\!96\)\( T^{18} + \)\(41\!\cdots\!73\)\( T^{19} + \)\(48\!\cdots\!82\)\( T^{20} + \)\(29\!\cdots\!31\)\( T^{21} + \)\(46\!\cdots\!64\)\( T^{22} - \)\(49\!\cdots\!31\)\( T^{23} - \)\(37\!\cdots\!97\)\( T^{24} - \)\(85\!\cdots\!52\)\( T^{25} + \)\(24\!\cdots\!63\)\( T^{26} + \)\(19\!\cdots\!49\)\( T^{27} + \)\(41\!\cdots\!61\)\( T^{28} \)
$97$ \( 1 - 248805607 T - 4330873524811728051 T^{2} + \)\(91\!\cdots\!74\)\( T^{3} + \)\(10\!\cdots\!88\)\( T^{4} - \)\(18\!\cdots\!76\)\( T^{5} - \)\(17\!\cdots\!23\)\( T^{6} + \)\(24\!\cdots\!47\)\( T^{7} + \)\(22\!\cdots\!76\)\( T^{8} - \)\(23\!\cdots\!15\)\( T^{9} - \)\(24\!\cdots\!81\)\( T^{10} + \)\(15\!\cdots\!08\)\( T^{11} + \)\(23\!\cdots\!67\)\( T^{12} - \)\(45\!\cdots\!19\)\( T^{13} - \)\(18\!\cdots\!46\)\( T^{14} - \)\(34\!\cdots\!23\)\( T^{15} + \)\(13\!\cdots\!63\)\( T^{16} + \)\(66\!\cdots\!04\)\( T^{17} - \)\(83\!\cdots\!01\)\( T^{18} - \)\(58\!\cdots\!55\)\( T^{19} + \)\(43\!\cdots\!44\)\( T^{20} + \)\(35\!\cdots\!31\)\( T^{21} - \)\(19\!\cdots\!43\)\( T^{22} - \)\(15\!\cdots\!72\)\( T^{23} + \)\(66\!\cdots\!12\)\( T^{24} + \)\(45\!\cdots\!42\)\( T^{25} - \)\(16\!\cdots\!11\)\( T^{26} - \)\(70\!\cdots\!59\)\( T^{27} + \)\(21\!\cdots\!29\)\( T^{28} \)
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