Properties

Label 37.8.b.a
Level $37$
Weight $8$
Character orbit 37.b
Analytic conductor $11.558$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 37.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.5582459429\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial: \(x^{20} + 1702 x^{18} + 1194509 x^{16} + 450999516 x^{14} + 100204783492 x^{12} + 13461378480848 x^{10} + 1081011973644416 x^{8} + 49304995250225664 x^{6} + 1131877572418003968 x^{4} + 9402469145336696832 x^{2} + 130757963535876096\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{19}\cdot 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 4 + \beta_{4} ) q^{3} + ( -42 + \beta_{2} ) q^{4} + \beta_{12} q^{5} + ( 12 \beta_{1} - \beta_{11} ) q^{6} + ( -87 + 2 \beta_{4} + \beta_{8} ) q^{7} + ( -42 \beta_{1} + \beta_{3} ) q^{8} + ( 618 + 2 \beta_{2} - 2 \beta_{4} + \beta_{6} - \beta_{8} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 4 + \beta_{4} ) q^{3} + ( -42 + \beta_{2} ) q^{4} + \beta_{12} q^{5} + ( 12 \beta_{1} - \beta_{11} ) q^{6} + ( -87 + 2 \beta_{4} + \beta_{8} ) q^{7} + ( -42 \beta_{1} + \beta_{3} ) q^{8} + ( 618 + 2 \beta_{2} - 2 \beta_{4} + \beta_{6} - \beta_{8} ) q^{9} + ( -43 + 8 \beta_{4} - \beta_{7} + \beta_{8} ) q^{10} + ( 176 + 2 \beta_{2} + 6 \beta_{4} + \beta_{10} ) q^{11} + ( -1518 + 11 \beta_{2} - 17 \beta_{4} - \beta_{5} + \beta_{8} ) q^{12} + ( -14 \beta_{1} - \beta_{11} - 2 \beta_{12} + \beta_{17} ) q^{13} + ( -50 \beta_{1} + \beta_{3} - \beta_{11} - 13 \beta_{12} + \beta_{15} ) q^{14} + ( -126 \beta_{1} - 2 \beta_{3} - \beta_{11} - 4 \beta_{12} - \beta_{16} + \beta_{17} ) q^{15} + ( 1792 - 64 \beta_{2} - 6 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} ) q^{16} + ( 280 \beta_{1} - 3 \beta_{3} - \beta_{11} + 3 \beta_{12} + \beta_{19} ) q^{17} + ( 322 \beta_{1} + 3 \beta_{3} - \beta_{11} + 7 \beta_{12} - \beta_{13} - \beta_{15} ) q^{18} + ( -301 \beta_{1} + 2 \beta_{11} - 30 \beta_{12} + \beta_{15} - \beta_{16} + \beta_{18} ) q^{19} + ( 119 \beta_{1} + 6 \beta_{3} + 3 \beta_{11} - 16 \beta_{12} - \beta_{13} - 2 \beta_{17} + \beta_{18} ) q^{20} + ( 5649 + 17 \beta_{2} - 282 \beta_{4} + 3 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{14} ) q^{21} + ( -56 \beta_{1} + 13 \beta_{3} - 32 \beta_{12} - 2 \beta_{13} + \beta_{18} - 2 \beta_{19} ) q^{22} + ( 848 \beta_{1} - \beta_{3} - 13 \beta_{11} + 23 \beta_{12} - \beta_{13} + \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{23} + ( -1564 \beta_{1} + 19 \beta_{3} + 7 \beta_{11} - 67 \beta_{12} - 2 \beta_{13} - \beta_{15} - 4 \beta_{16} + 3 \beta_{18} - 4 \beta_{19} ) q^{24} + ( -5403 - 101 \beta_{2} + 227 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 7 \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{14} ) q^{25} + ( 2430 - 57 \beta_{2} - 248 \beta_{4} + \beta_{5} - 3 \beta_{6} - 6 \beta_{7} + 15 \beta_{8} + \beta_{9} - 6 \beta_{10} - \beta_{14} ) q^{26} + ( -15168 + 65 \beta_{2} + 215 \beta_{4} - 6 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} + 11 \beta_{8} + 6 \beta_{10} + \beta_{14} ) q^{27} + ( -2081 - 144 \beta_{2} + 114 \beta_{4} - \beta_{5} + 3 \beta_{6} + 7 \beta_{7} - 16 \beta_{8} + 3 \beta_{9} - 8 \beta_{10} + \beta_{14} ) q^{28} + ( -105 \beta_{1} - 3 \beta_{3} - 15 \beta_{11} - 64 \beta_{12} + 5 \beta_{13} + 7 \beta_{15} + 6 \beta_{16} + \beta_{17} + \beta_{18} + 5 \beta_{19} ) q^{29} + ( 21493 + 128 \beta_{2} - 237 \beta_{4} + 8 \beta_{5} + 7 \beta_{6} - 16 \beta_{7} + 10 \beta_{8} - 3 \beta_{9} + 9 \beta_{10} - 2 \beta_{14} ) q^{30} + ( -1113 \beta_{1} + \beta_{3} - 7 \beta_{11} + 49 \beta_{12} + 4 \beta_{13} + 7 \beta_{15} + 2 \beta_{16} + 12 \beta_{17} - 7 \beta_{18} + \beta_{19} ) q^{31} + ( 4536 \beta_{1} - 69 \beta_{3} - 41 \beta_{11} + 250 \beta_{12} + 2 \beta_{13} - 8 \beta_{15} - 2 \beta_{17} - 5 \beta_{18} + 10 \beta_{19} ) q^{32} + ( 15436 + 260 \beta_{2} + 1015 \beta_{4} + 8 \beta_{5} + 23 \beta_{6} - 14 \beta_{7} - 2 \beta_{8} - 6 \beta_{9} + 24 \beta_{10} + 2 \beta_{14} ) q^{33} + ( -47895 + 656 \beta_{2} - 171 \beta_{4} - 23 \beta_{5} - 16 \beta_{6} + 17 \beta_{7} + 40 \beta_{8} - 8 \beta_{9} + 26 \beta_{10} ) q^{34} + ( -6363 \beta_{1} - 36 \beta_{3} + 17 \beta_{11} + 8 \beta_{12} + 9 \beta_{13} - \beta_{15} - 15 \beta_{16} + 12 \beta_{17} + 3 \beta_{18} - 4 \beta_{19} ) q^{35} + ( 24195 + 125 \beta_{2} - 501 \beta_{4} + 2 \beta_{5} + 41 \beta_{6} + 25 \beta_{7} - 17 \beta_{8} - 3 \beta_{9} - 24 \beta_{10} - \beta_{14} ) q^{36} + ( 5398 + 521 \beta_{1} - 93 \beta_{2} - 49 \beta_{3} - 241 \beta_{4} - 8 \beta_{5} + 18 \beta_{6} - 16 \beta_{7} + 32 \beta_{8} + 6 \beta_{9} + 8 \beta_{10} + 62 \beta_{11} + 217 \beta_{12} + 8 \beta_{13} - \beta_{14} - 7 \beta_{15} + 10 \beta_{16} - 13 \beta_{17} - 5 \beta_{18} - \beta_{19} ) q^{37} + ( 52438 - 233 \beta_{2} + 197 \beta_{4} + 23 \beta_{5} + 24 \beta_{6} + 39 \beta_{7} - 162 \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{14} ) q^{38} + ( 3497 \beta_{1} + 41 \beta_{3} - 60 \beta_{11} + 440 \beta_{12} - 5 \beta_{13} - 27 \beta_{15} + 19 \beta_{16} + 12 \beta_{17} - 3 \beta_{18} - 11 \beta_{19} ) q^{39} + ( -24773 - 627 \beta_{2} + 1463 \beta_{4} + 17 \beta_{5} - 68 \beta_{6} - 42 \beta_{7} + 51 \beta_{8} + 2 \beta_{9} - 29 \beta_{10} + 3 \beta_{14} ) q^{40} + ( -78649 + 722 \beta_{2} + 649 \beta_{4} - 14 \beta_{5} - 27 \beta_{6} - 30 \beta_{7} + 189 \beta_{8} - 14 \beta_{9} - 35 \beta_{10} + 2 \beta_{14} ) q^{41} + ( 1407 \beta_{1} + 49 \beta_{3} + 198 \beta_{11} - 242 \beta_{12} - 3 \beta_{13} + 20 \beta_{15} + 20 \beta_{16} - 64 \beta_{17} + 9 \beta_{18} - 2 \beta_{19} ) q^{42} + ( -4372 \beta_{1} + 56 \beta_{3} - 53 \beta_{11} - 411 \beta_{12} + 5 \beta_{13} + 28 \beta_{15} + 5 \beta_{16} - 26 \beta_{17} - 6 \beta_{18} - 16 \beta_{19} ) q^{43} + ( 34011 - 1482 \beta_{2} + 539 \beta_{4} + 62 \beta_{5} - 123 \beta_{6} + 69 \beta_{7} - 153 \beta_{8} + 17 \beta_{9} + 6 \beta_{10} + \beta_{14} ) q^{44} + ( 3759 \beta_{1} + 128 \beta_{3} + 84 \beta_{11} + 696 \beta_{12} - 9 \beta_{13} - 49 \beta_{15} + 16 \beta_{16} - 63 \beta_{17} - 3 \beta_{18} + 4 \beta_{19} ) q^{45} + ( -145283 + 901 \beta_{2} - 1913 \beta_{4} - 58 \beta_{5} - 125 \beta_{6} - 39 \beta_{7} + 49 \beta_{8} - 13 \beta_{9} + 4 \beta_{10} - 3 \beta_{14} ) q^{46} + ( -75522 + 1587 \beta_{2} - 1504 \beta_{4} - 58 \beta_{5} + 42 \beta_{6} + 114 \beta_{7} - 219 \beta_{8} - 16 \beta_{9} - 27 \beta_{10} + \beta_{14} ) q^{47} + ( 75434 - 2630 \beta_{2} - 1585 \beta_{4} + 48 \beta_{5} - 39 \beta_{6} + 83 \beta_{7} - 23 \beta_{8} + 27 \beta_{9} - 129 \beta_{10} - 2 \beta_{14} ) q^{48} + ( 163645 + 1292 \beta_{2} - 322 \beta_{4} + 42 \beta_{5} + 163 \beta_{6} - 116 \beta_{7} - 284 \beta_{8} - 28 \beta_{9} - 3 \beta_{10} - 4 \beta_{14} ) q^{49} + ( 9121 \beta_{1} - 335 \beta_{3} + 270 \beta_{11} - 35 \beta_{12} - 10 \beta_{13} - 49 \beta_{15} - 36 \beta_{16} + 52 \beta_{17} - 2 \beta_{18} + 4 \beta_{19} ) q^{50} + ( 8682 \beta_{1} - 298 \beta_{3} - 278 \beta_{11} + 157 \beta_{12} + 6 \beta_{13} + 6 \beta_{15} - 17 \beta_{16} - 52 \beta_{17} + 12 \beta_{18} - 6 \beta_{19} ) q^{51} + ( 6951 \beta_{1} - 183 \beta_{3} + 34 \beta_{11} - 908 \beta_{12} + 9 \beta_{13} + 22 \beta_{15} + 20 \beta_{16} + 52 \beta_{17} + 4 \beta_{18} + 18 \beta_{19} ) q^{52} + ( 149724 - 2646 \beta_{2} + 2528 \beta_{4} + 134 \beta_{5} - 165 \beta_{6} + 156 \beta_{7} + 288 \beta_{8} + 8 \beta_{9} + 159 \beta_{10} - 2 \beta_{14} ) q^{53} + ( -22180 \beta_{1} + 152 \beta_{3} + 622 \beta_{11} - 437 \beta_{12} - 20 \beta_{13} - 19 \beta_{15} - 44 \beta_{16} + 64 \beta_{17} + 15 \beta_{18} - 34 \beta_{19} ) q^{54} + ( -13295 \beta_{1} + 99 \beta_{3} + 447 \beta_{11} + 425 \beta_{12} + 21 \beta_{13} - 41 \beta_{15} + 11 \beta_{16} + 65 \beta_{17} + \beta_{18} + 21 \beta_{19} ) q^{55} + ( 10136 \beta_{1} - 474 \beta_{3} + 93 \beta_{11} - 416 \beta_{12} + 4 \beta_{13} + 70 \beta_{15} - 36 \beta_{16} + 62 \beta_{17} - 19 \beta_{18} + 26 \beta_{19} ) q^{56} + ( -6377 \beta_{1} + 365 \beta_{3} + 508 \beta_{11} + 1320 \beta_{12} - \beta_{13} + 21 \beta_{15} + 40 \beta_{16} - 128 \beta_{17} - 9 \beta_{18} + 29 \beta_{19} ) q^{57} + ( 19977 + 77 \beta_{2} - 2251 \beta_{4} - 154 \beta_{5} + 297 \beta_{6} + 95 \beta_{7} - 633 \beta_{8} + 21 \beta_{9} + 118 \beta_{10} + 13 \beta_{14} ) q^{58} + ( 35361 \beta_{1} - 456 \beta_{3} - 470 \beta_{11} + 35 \beta_{12} + 2 \beta_{13} + 55 \beta_{15} + 54 \beta_{16} + 76 \beta_{17} - 11 \beta_{18} + 4 \beta_{19} ) q^{59} + ( -11259 \beta_{1} + 365 \beta_{3} - 1075 \beta_{11} - 2511 \beta_{12} - 9 \beta_{13} + 77 \beta_{15} - 44 \beta_{16} - 12 \beta_{17} + 15 \beta_{18} - 2 \beta_{19} ) q^{60} + ( -21504 \beta_{1} + 461 \beta_{3} - 807 \beta_{11} - 170 \beta_{12} + 20 \beta_{13} - 20 \beta_{15} - 34 \beta_{16} + 38 \beta_{17} - 24 \beta_{18} + 17 \beta_{19} ) q^{61} + ( 185964 - 1746 \beta_{2} - 1013 \beta_{4} - 113 \beta_{5} + 192 \beta_{6} - 420 \beta_{7} - 585 \beta_{8} + 28 \beta_{9} + 36 \beta_{10} - 10 \beta_{14} ) q^{62} + ( -598717 - 257 \beta_{2} + 7852 \beta_{4} - 42 \beta_{5} - 490 \beta_{6} - 390 \beta_{7} + 1618 \beta_{8} + 24 \beta_{9} - 309 \beta_{10} + 9 \beta_{14} ) q^{63} + ( -554735 + 5381 \beta_{2} - 5754 \beta_{4} - 231 \beta_{5} - 175 \beta_{6} - 63 \beta_{7} + 1488 \beta_{8} - 11 \beta_{9} + 330 \beta_{10} - 11 \beta_{14} ) q^{64} + ( 211844 + 3314 \beta_{2} - 11613 \beta_{4} - 42 \beta_{5} + 630 \beta_{6} + 150 \beta_{7} - 636 \beta_{8} + 18 \beta_{9} - 281 \beta_{10} - 12 \beta_{14} ) q^{65} + ( -8921 \beta_{1} + 1190 \beta_{3} - 2031 \beta_{11} - 2416 \beta_{12} - 49 \beta_{13} + 14 \beta_{15} + 16 \beta_{16} + 116 \beta_{17} - 36 \beta_{19} ) q^{66} + ( 179709 - 372 \beta_{2} + 15914 \beta_{4} + 192 \beta_{5} - 2 \beta_{6} - 388 \beta_{7} - 125 \beta_{8} + 8 \beta_{9} + 267 \beta_{10} + 26 \beta_{14} ) q^{67} + ( -99865 \beta_{1} + 1490 \beta_{3} + 2104 \beta_{11} + 105 \beta_{12} - 33 \beta_{13} + 83 \beta_{15} - 60 \beta_{16} + 66 \beta_{17} + 78 \beta_{18} - 48 \beta_{19} ) q^{68} + ( 41473 \beta_{1} - 501 \beta_{3} - 2034 \beta_{11} + 1709 \beta_{12} - 37 \beta_{13} - 69 \beta_{15} - 66 \beta_{16} - 64 \beta_{17} + 69 \beta_{18} - 85 \beta_{19} ) q^{69} + ( 1080118 - 1719 \beta_{2} + 2367 \beta_{4} + 265 \beta_{5} + 948 \beta_{6} - 495 \beta_{7} + 370 \beta_{8} + 6 \beta_{9} + 305 \beta_{10} - 25 \beta_{14} ) q^{70} + ( -762648 + 345 \beta_{2} + 760 \beta_{4} - 254 \beta_{5} - 810 \beta_{6} + 654 \beta_{7} + 777 \beta_{8} + 16 \beta_{9} + 273 \beta_{10} - \beta_{14} ) q^{71} + ( 43668 \beta_{1} + 503 \beta_{3} - 712 \beta_{11} + 5083 \beta_{12} - 78 \beta_{13} - 69 \beta_{15} + 40 \beta_{16} + 2 \beta_{17} - 48 \beta_{18} + 42 \beta_{19} ) q^{72} + ( 553826 - 2921 \beta_{2} + 3700 \beta_{4} + 2 \beta_{5} + 1057 \beta_{6} + 670 \beta_{7} + 223 \beta_{8} + 8 \beta_{9} - 419 \beta_{10} + 15 \beta_{14} ) q^{73} + ( -99042 + 16617 \beta_{1} + 7683 \beta_{2} - 423 \beta_{3} + 11383 \beta_{4} - 134 \beta_{5} + 579 \beta_{6} - 342 \beta_{7} + 1128 \beta_{8} - 29 \beta_{9} + 282 \beta_{10} + 1797 \beta_{11} - 3238 \beta_{12} - 88 \beta_{13} + 11 \beta_{14} - 34 \beta_{15} - 36 \beta_{16} - 116 \beta_{17} + 55 \beta_{18} - 26 \beta_{19} ) q^{74} + ( 743633 + 5032 \beta_{2} - 8311 \beta_{4} + 176 \beta_{5} + 602 \beta_{6} + 596 \beta_{7} - 3873 \beta_{8} + 33 \beta_{10} - 14 \beta_{14} ) q^{75} + ( 40229 \beta_{1} - 940 \beta_{3} - 2966 \beta_{11} + 4443 \beta_{12} + 53 \beta_{13} + 13 \beta_{15} - 64 \beta_{16} + 130 \beta_{17} + 12 \beta_{18} + 104 \beta_{19} ) q^{76} + ( -121886 - 7694 \beta_{2} + 1838 \beta_{4} + 98 \beta_{5} - 1065 \beta_{6} - 744 \beta_{7} + 1080 \beta_{8} - 28 \beta_{9} - 235 \beta_{10} - 38 \beta_{14} ) q^{77} + ( -610096 - 3657 \beta_{2} - 9240 \beta_{4} + 26 \beta_{5} - 564 \beta_{6} - 337 \beta_{7} + 3797 \beta_{8} + 66 \beta_{9} - 543 \beta_{10} - 23 \beta_{14} ) q^{78} + ( 2438 \beta_{1} - 1279 \beta_{3} + 3737 \beta_{11} + 3082 \beta_{12} + 41 \beta_{13} - 34 \beta_{15} - 65 \beta_{16} + 77 \beta_{17} - 110 \beta_{18} + 23 \beta_{19} ) q^{79} + ( 88636 \beta_{1} - 426 \beta_{3} - 2163 \beta_{11} - 6149 \beta_{12} - 24 \beta_{13} - 35 \beta_{15} - 168 \beta_{17} + 69 \beta_{18} + 140 \beta_{19} ) q^{80} + ( -886847 + 11388 \beta_{2} - 8949 \beta_{4} + 102 \beta_{5} - 1306 \beta_{6} + 442 \beta_{7} - 1710 \beta_{8} - 78 \beta_{9} + 303 \beta_{10} - 10 \beta_{14} ) q^{81} + ( -159631 \beta_{1} + 2176 \beta_{3} + 318 \beta_{11} - 6003 \beta_{12} + 91 \beta_{13} + 217 \beta_{15} - 40 \beta_{16} + 116 \beta_{17} + 23 \beta_{18} - 38 \beta_{19} ) q^{82} + ( -647210 - 12995 \beta_{2} - 10434 \beta_{4} - 18 \beta_{5} - 1668 \beta_{6} - 90 \beta_{7} - 2253 \beta_{8} + 72 \beta_{9} - 163 \beta_{10} + 39 \beta_{14} ) q^{83} + ( 496519 + 738 \beta_{2} - 1856 \beta_{4} + 71 \beta_{5} + 353 \beta_{6} + 861 \beta_{7} - 3826 \beta_{8} + 81 \beta_{9} - 258 \beta_{10} - 15 \beta_{14} ) q^{84} + ( -135977 - 4613 \beta_{2} - 2880 \beta_{4} - 102 \beta_{5} - 154 \beta_{6} + 246 \beta_{7} + 1854 \beta_{8} - 188 \beta_{9} - 51 \beta_{10} + 25 \beta_{14} ) q^{85} + ( 764276 - 9301 \beta_{2} - 4859 \beta_{4} + 109 \beta_{5} + 726 \beta_{6} - 105 \beta_{7} - 4908 \beta_{8} + 184 \beta_{9} - 347 \beta_{10} + 53 \beta_{14} ) q^{86} + ( 40408 \beta_{1} - 2105 \beta_{3} + 6145 \beta_{11} - 13260 \beta_{12} + 110 \beta_{13} + 266 \beta_{15} + 200 \beta_{16} - 248 \beta_{17} + 18 \beta_{18} + 3 \beta_{19} ) q^{87} + ( 216358 \beta_{1} - 2746 \beta_{3} - 6792 \beta_{11} + 10779 \beta_{12} - 22 \beta_{13} - 133 \beta_{15} + 160 \beta_{16} + 130 \beta_{17} - 128 \beta_{18} + 50 \beta_{19} ) q^{88} + ( 14969 \beta_{1} - 1147 \beta_{3} + 2042 \beta_{11} - 9278 \beta_{12} - 111 \beta_{13} - 197 \beta_{15} - 36 \beta_{16} - 120 \beta_{17} - 39 \beta_{18} - 171 \beta_{19} ) q^{89} + ( -661691 - 15289 \beta_{2} + 16130 \beta_{4} - 133 \beta_{5} - 1095 \beta_{6} + 651 \beta_{7} + 5740 \beta_{8} - 63 \beta_{9} + 246 \beta_{10} + 27 \beta_{14} ) q^{90} + ( -78206 \beta_{1} - 686 \beta_{3} + 8396 \beta_{11} + 3411 \beta_{12} + 202 \beta_{13} + 74 \beta_{15} + 319 \beta_{16} - 142 \beta_{17} - 156 \beta_{18} + 226 \beta_{19} ) q^{91} + ( -166590 \beta_{1} + 2637 \beta_{3} + 6330 \beta_{11} - 5133 \beta_{12} - 108 \beta_{13} + 71 \beta_{15} - 120 \beta_{16} - 78 \beta_{17} - 18 \beta_{18} - 170 \beta_{19} ) q^{92} + ( 623 \beta_{1} - 1874 \beta_{3} + 4516 \beta_{11} + 6374 \beta_{12} - 23 \beta_{13} - 289 \beta_{15} + 200 \beta_{16} - 255 \beta_{17} + 117 \beta_{18} - 214 \beta_{19} ) q^{93} + ( -307114 \beta_{1} + 2673 \beta_{3} + 5409 \beta_{11} + 15181 \beta_{12} + 72 \beta_{13} - 97 \beta_{15} - 188 \beta_{16} + 352 \beta_{17} + 26 \beta_{18} - 240 \beta_{19} ) q^{94} + ( 2440402 - 11900 \beta_{2} - 56014 \beta_{4} + 296 \beta_{5} + 1512 \beta_{6} - 336 \beta_{7} - 3978 \beta_{8} - 184 \beta_{9} + 68 \beta_{10} - 68 \beta_{14} ) q^{95} + ( 198622 \beta_{1} - 5303 \beta_{3} - 2562 \beta_{11} + 9705 \beta_{12} + 116 \beta_{13} - 175 \beta_{15} - 388 \beta_{16} - 62 \beta_{17} + 42 \beta_{18} + 42 \beta_{19} ) q^{96} + ( 13839 \beta_{1} + 1190 \beta_{3} - 9749 \beta_{11} - 9133 \beta_{12} + 155 \beta_{13} - 259 \beta_{15} + 220 \beta_{16} - 120 \beta_{17} - 17 \beta_{18} - 46 \beta_{19} ) q^{97} + ( 482 \beta_{1} + 4611 \beta_{3} - 6868 \beta_{11} - 9890 \beta_{12} - 69 \beta_{13} + 16 \beta_{15} + 360 \beta_{16} - 360 \beta_{17} + 15 \beta_{18} + 54 \beta_{19} ) q^{98} + ( 2156834 - 12702 \beta_{2} + 45681 \beta_{4} - 112 \beta_{5} + 1080 \beta_{6} - 496 \beta_{7} - 6874 \beta_{8} - 312 \beta_{9} + 219 \beta_{10} + 28 \beta_{14} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 78q^{3} - 844q^{4} - 1746q^{7} + 12362q^{9} + O(q^{10}) \) \( 20q + 78q^{3} - 844q^{4} - 1746q^{7} + 12362q^{9} - 882q^{10} + 3498q^{11} - 30374q^{12} + 36116q^{16} + 113482q^{21} - 108112q^{25} + 49278q^{26} - 304110q^{27} - 41192q^{28} + 429776q^{30} + 305646q^{33} - 960356q^{34} + 484758q^{36} + 108732q^{37} + 1049916q^{38} - 496346q^{40} - 1577742q^{41} + 685266q^{44} - 2906298q^{46} - 1512786q^{47} + 1522958q^{48} + 3269246q^{49} + 2999358q^{53} + 405946q^{58} + 3728310q^{62} - 11995292q^{63} - 11109700q^{64} + 4251792q^{65} + 3562224q^{67} + 21605644q^{70} - 15259086q^{71} + 11088018q^{73} - 2036544q^{74} + 14882062q^{75} - 2419122q^{77} - 12178734q^{78} - 17764972q^{81} - 12873822q^{83} + 9944396q^{84} - 2698920q^{85} + 15345336q^{86} - 13219100q^{90} + 48981192q^{95} + 43111380q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} + 1702 x^{18} + 1194509 x^{16} + 450999516 x^{14} + 100204783492 x^{12} + 13461378480848 x^{10} + 1081011973644416 x^{8} + 49304995250225664 x^{6} + 1131877572418003968 x^{4} + 9402469145336696832 x^{2} + 130757963535876096\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 170 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 298 \nu \)
\(\beta_{4}\)\(=\)\((\)\(-2464505834717504057037311 \nu^{18} - 4062230331226013071114142142 \nu^{16} - 2730941476640764596857601696187 \nu^{14} - 972523355647324591990280529402392 \nu^{12} - 199177511378201200972500208639297308 \nu^{10} - 23775505782848904412782101038905822784 \nu^{8} - 1589913372117868473220608533833621991040 \nu^{6} - 52845024881973318142046045753052965016576 \nu^{4} - 623414686985079517664020650512923038010368 \nu^{2} - 211327012969517734318833939725568488570880\)\()/ \)\(18\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-72437390228236114481401559 \nu^{18} - 112799828149121085988898999406 \nu^{16} - 69752792845248289760031886510579 \nu^{14} - 21835209857675357350108039622694104 \nu^{12} - 3605155479800561631845347684396477308 \nu^{10} - 282472056520527883521692248269326017600 \nu^{8} - 4636145134099919875798984157928020959872 \nu^{6} + 533930292250735347926698609694380110253056 \nu^{4} + 18961059121622146897710082736570384660812800 \nu^{2} + 28822489008580498058808841307397225184690176\)\()/ \)\(18\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-101067482367253264859276263 \nu^{18} - 162305150863517882052894281742 \nu^{16} - 105337021018987775713575067257923 \nu^{14} - 35768085468339075101625706649539288 \nu^{12} - 6872514902533616673995941872889047996 \nu^{10} - 754953465509430144637985230470885469760 \nu^{8} - 45806240196839285119260586577289831441024 \nu^{6} - 1419291047654833611841014475136023845884928 \nu^{4} - 19150657176574798473130880036550685696650240 \nu^{2} - 55410934829190140382336869068041055632949248\)\()/ \)\(18\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(16619442297430347275982005 \nu^{18} + 27283761149645033201740833066 \nu^{16} + 18228281423432850953630707446217 \nu^{14} + 6425249942322883123269485587423112 \nu^{12} + 1292573216635505159764076557508921652 \nu^{10} + 149248035661097057248573721848857581248 \nu^{8} + 9356500178642552369820969763332920840064 \nu^{6} + 273082982693398246556987855258438923484160 \nu^{4} + 2418260399456169565892039579103286956082176 \nu^{2} - 31600021201546169727353558919804946808832\)\()/ \)\(15\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-148716068399547684394225237 \nu^{18} - 244445151347543262245234990442 \nu^{16} - 163385861984356710921400503459305 \nu^{14} - 57536667111050196571571778652234120 \nu^{12} - 11542877301549046622542392341582745012 \nu^{10} - 1327836725352669773138491119799966753472 \nu^{8} - 83253080249132096883033614276778078463872 \nu^{6} - 2481247060605210142141694366012318266555392 \nu^{4} - 24300764537987168728439389547680442091193344 \nu^{2} + 720553131452666542971219879063327302025216\)\()/ \)\(91\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-114067224462257589138622843 \nu^{18} - 185581231536150839772344817078 \nu^{16} - 122612795276310283962504197689895 \nu^{14} - 42663850445321252821035703207982200 \nu^{12} - 8471274478274206411133784188955697548 \nu^{10} - 970045393477506642686243143876443668288 \nu^{8} - 61352641342763749708877748348185159526528 \nu^{6} - 1885162297743743462970134876919354197173248 \nu^{4} - 16975494165485137648656619900241282500033536 \nu^{2} + 159697596327172862838930910118954532532322304\)\()/ \)\(61\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(165171011642634770730971683 \nu^{18} + 274106916426404600858491880838 \nu^{16} + 185471438741091425592115567015535 \nu^{14} + 66342044053553096868624394079108920 \nu^{12} + 13576996287333963807307576976633100588 \nu^{10} + 1602440530865773179690368432730607124288 \nu^{8} + 103980658135798114308712700948305142594688 \nu^{6} + 3266140417796693166430187885720753826505728 \nu^{4} + 36306439943578942599770935450690609371749376 \nu^{2} + 31159642359983761931280745581553362470240256\)\()/ \)\(61\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(2464505834717504057037311 \nu^{19} + 4062230331226013071114142142 \nu^{17} + 2730941476640764596857601696187 \nu^{15} + 972523355647324591990280529402392 \nu^{13} + 199177511378201200972500208639297308 \nu^{11} + 23775505782848904412782101038905822784 \nu^{9} + 1589913372117868473220608533833621991040 \nu^{7} + 52845024881973318142046045753052965016576 \nu^{5} + 623414686985079517664020650512923038010368 \nu^{3} + 358266031511265795676056328357071074426880 \nu\)\()/ \)\(18\!\cdots\!00\)\( \)
\(\beta_{12}\)\(=\)\((\)\(149205623889522174853968331 \nu^{19} + 238680118902384976575227678934 \nu^{17} + 153140903669826030528221896145591 \nu^{15} + 50523064565145096384152947691643256 \nu^{13} + 9041291190994924209985118746589434572 \nu^{11} + 820674021743754356805991733214979476800 \nu^{9} + 24248272179242725519693513439418885314688 \nu^{7} - 1258964413779277508063336113195278414019584 \nu^{5} - 87135115079495926056346188459433204109460480 \nu^{3} - 1038623212087779826978324643826625958792331264 \nu\)\()/ \)\(54\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(729822063149394118837889443 \nu^{19} + 1171805715531447039048952527558 \nu^{17} + 762382413637802330239947558182255 \nu^{15} + 261195117618999253722889564657776760 \nu^{13} + 51412366821125545376098083953189630508 \nu^{11} + 5993860648952919419107518331622864395328 \nu^{9} + 417884136696393193162770940159624519045248 \nu^{7} + 17391747657627371705565062673254299998901248 \nu^{5} + 394895243938276680982437690387624803456572416 \nu^{3} + 3063816581328457356584465379762302296771854336 \nu\)\()/ \)\(20\!\cdots\!00\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-4350166997645308103506370681 \nu^{18} - 7141291471039086937201056696498 \nu^{16} - 4765331788029016020751164009839005 \nu^{14} - 1674560577794386654585190655386659880 \nu^{12} - 335026570812640896424810073773872726468 \nu^{10} - 38397705895864666193484618182376914983360 \nu^{8} - 2393675572935401817000950052688222395878784 \nu^{6} - 70501807823924255345181808791712819805039616 \nu^{4} - 668945029458379111919865599051965164524835840 \nu^{2} + 1725306828150691736179408499614108811329536\)\()/ \)\(36\!\cdots\!00\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-62315448945520129718985456083 \nu^{19} - 103199799873420092732079738289638 \nu^{17} - 69730223279094568021147228042501855 \nu^{15} - 24956999561558916640599705983887327160 \nu^{13} - 5135960873104288190082978887869109585388 \nu^{11} - 616714561881838868137176281823778662922048 \nu^{9} - 41876507804550229669846423592833671610578048 \nu^{7} - 1481379049272752166688575976132712980942169088 \nu^{5} - 23293529227614741386861016426578115062412205056 \nu^{3} - 122789425747637104107053843906013718012351021056 \nu\)\()/ \)\(48\!\cdots\!00\)\( \)
\(\beta_{16}\)\(=\)\((\)\(92697198970605567323807656241 \nu^{19} + 150464369228458455738870660527394 \nu^{17} + 98682168353084453998552138092900661 \nu^{15} + 33724169481757103374160919133967037416 \nu^{13} + 6428309090677878028492878078689385111972 \nu^{11} + 670202260930524373751896232362062674960320 \nu^{9} + 33162699332908467965920029924017091569578368 \nu^{7} + 313775659666187653224044121183354600031620096 \nu^{5} - 24307664953523674676975011262836680502568094720 \nu^{3} - 437293503148256512793832612440447330452298072064 \nu\)\()/ \)\(48\!\cdots\!00\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-93494584868252911760372481719 \nu^{19} - 155815042287133465143144571744302 \nu^{17} - 106256640275374041915989412954038611 \nu^{15} - 38560071452454961890404077058819736536 \nu^{13} - 8107525082435521443330911289283372820220 \nu^{11} - 1007831820003429717817375217706094073846848 \nu^{9} - 72557751916385203650968031617891967282877056 \nu^{7} - 2849677431711812800735695257106267735991809024 \nu^{5} - 54653896471703559060129207815783472885427055616 \nu^{3} - 405774174175514817465375356122020925659481178112 \nu\)\()/ \)\(48\!\cdots\!00\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-28870075983610019882861554465 \nu^{19} - 50799496339937136862030236224514 \nu^{17} - 37210544776818150001556033919683813 \nu^{15} - 14834519115805829009828132356139423848 \nu^{13} - 3528284451962962502174943843969842371428 \nu^{11} - 515148362472635435878632433180400037482432 \nu^{9} - 45617299399335146062848171526066967437719936 \nu^{7} - 2314974914925616314625206494833109108446694400 \nu^{5} - 58814483967376471592621851792871466936295732224 \nu^{3} - 523462747630686718483457689547495033161979461632 \nu\)\()/ \)\(12\!\cdots\!00\)\( \)
\(\beta_{19}\)\(=\)\((\)\(-8456739437484389320049780135 \nu^{19} - 14222066674967869294729331257998 \nu^{17} - 9820756335196560683573500387702531 \nu^{15} - 3626707594284718663920869900355972056 \nu^{13} - 781615237036981798965987728921450649276 \nu^{11} - 100669860939577454640577105470166908480064 \nu^{9} - 7630268473601477358104475453663987598119552 \nu^{7} - 322477926047112602871112963969255043996881920 \nu^{5} - 6764258440109592244326335249383305383336537088 \nu^{3} - 51320688494251936745231981397930113856559448064 \nu\)\()/ \)\(25\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 170\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 298 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{10} + \beta_{9} - 2 \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - 6 \beta_{4} - 448 \beta_{2} + 50688\)
\(\nu^{5}\)\(=\)\(10 \beta_{19} - 5 \beta_{18} - 2 \beta_{17} - 8 \beta_{15} + 2 \beta_{13} + 250 \beta_{12} - 41 \beta_{11} - 581 \beta_{3} + 107960 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-11 \beta_{14} + 970 \beta_{10} - 651 \beta_{9} + 2768 \beta_{8} - 703 \beta_{7} + 465 \beta_{6} - 871 \beta_{5} - 1914 \beta_{4} + 193797 \beta_{2} - 18380527\)
\(\nu^{7}\)\(=\)\(-8050 \beta_{19} + 4363 \beta_{18} + 538 \beta_{17} - 660 \beta_{16} + 6402 \beta_{15} - 2224 \beta_{13} - 208888 \beta_{12} + 61423 \beta_{11} + 286937 \beta_{3} - 43171486 \beta_{1}\)
\(\nu^{8}\)\(=\)\(9567 \beta_{14} - 604362 \beta_{10} + 339039 \beta_{9} - 1930120 \beta_{8} + 417323 \beta_{7} - 230277 \beta_{6} + 556419 \beta_{5} + 6774826 \beta_{4} - 85047949 \beta_{2} + 7356193323\)
\(\nu^{9}\)\(=\)\(4809690 \beta_{19} - 2757911 \beta_{18} + 52510 \beta_{17} + 678180 \beta_{16} - 3642650 \beta_{15} + 1593872 \beta_{13} + 130255496 \beta_{12} - 51030763 \beta_{11} - 136140225 \beta_{3} + 18274909902 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-5774891 \beta_{14} + 327064038 \beta_{10} - 165205439 \beta_{9} + 1091155264 \beta_{8} - 230683451 \beta_{7} + 119017373 \beta_{6} - 315170019 \beta_{5} - 6306548266 \beta_{4} + 37965245697 \beta_{2} - 3116047703047\)
\(\nu^{11}\)\(=\)\(-2587179690 \beta_{19} + 1543681783 \beta_{18} - 147038606 \beta_{17} - 484360500 \beta_{16} + 1832478546 \beta_{15} - 961797400 \beta_{13} - 72399497408 \beta_{12} + 33518622891 \beta_{11} + 63844372521 \beta_{3} - 8013645796550 \beta_{1}\)
\(\nu^{12}\)\(=\)\(3038838435 \beta_{14} - 166734036758 \beta_{10} + 78569642423 \beta_{9} - 564579581136 \beta_{8} + 121539360243 \beta_{7} - 61461655013 \beta_{6} + 167858164043 \beta_{5} + 4306279184522 \beta_{4} - 17187322078969 \beta_{2} + 1367118555542847\)
\(\nu^{13}\)\(=\)\(1325256976250 \beta_{19} - 813119758175 \beta_{18} + 111130456894 \beta_{17} + 296377317780 \beta_{16} - 875227443314 \beta_{15} + 531829170296 \beta_{13} + 37913874693840 \beta_{12} - 19542265379763 \beta_{11} - 29862529158913 \beta_{3} + 3595118956976758 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-1503100340603 \beta_{14} + 82533398918870 \beta_{10} - 37034306669503 \beta_{9} + 280047778113504 \beta_{8} - 61935527232219 \beta_{7} + 31266223449565 \beta_{6} - 86193312276323 \beta_{5} - 2556428038705322 \beta_{4} + 7865537953198209 \beta_{2} - 613565538221477431\)
\(\nu^{15}\)\(=\)\(-660983474302250 \beta_{19} + 413570565364279 \beta_{18} - 65919848222606 \beta_{17} - 166574015615220 \beta_{16} + 409096393166226 \beta_{15} - 279511745712952 \beta_{13} - 19178116036789824 \beta_{12} + 10637856164447339 \beta_{11} + 13975696300149529 \beta_{3} - 1637644586730934662 \beta_{1}\)
\(\nu^{16}\)\(=\)\(722012791137891 \beta_{14} - 40202332506443974 \beta_{10} + 17408832726416935 \beta_{9} - 135974888525503376 \beta_{8} + 30851229410608867 \beta_{7} - 15648365918817109 \beta_{6} + 43227374131287835 \beta_{5} + 1405488490558677450 \beta_{4} - 3629185983177290041 \beta_{2} + 279572994164221596543\)
\(\nu^{17}\)\(=\)\(324393518025982810 \beta_{19} - 205789773848881583 \beta_{18} + 35387895383823454 \beta_{17} + 88833909796725780 \beta_{16} - 189788661212829074 \beta_{15} + 142279652116946072 \beta_{13} + 9495027656523550288 \beta_{12} - 5548134816166330307 \beta_{11} - 6550975475159496113 \beta_{3} + 753838420219553115446 \beta_{1}\)
\(\nu^{18}\)\(=\)\(-342132420648808331 \beta_{14} + 19397737494803189494 \beta_{10} - 8181925611967772175 \beta_{9} + 65284832893688088896 \beta_{8} - 15131882120342702827 \beta_{7} + 7724403906458336589 \beta_{6} - 21334515277928070355 \beta_{5} - 737482622129337657450 \beta_{4} + 1684923278142934179057 \beta_{2} - 128720745678166304922247\)
\(\nu^{19}\)\(=\)\(-157545503390487365770 \beta_{19} + 100928092393471761703 \beta_{18} - 18064007031742816814 \beta_{17} - 45767710250865026100 \beta_{16} + 87963899138175361554 \beta_{15} - 70908824283094853752 \beta_{13} - 4636402684353851147488 \beta_{12} + 2813303336205830229787 \beta_{11} + 3076107017597251199369 \beta_{3} - 349574716089700914349798 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/37\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
36.1
21.7545i
18.8806i
16.0010i
14.8861i
12.3896i
10.3219i
7.77325i
7.52533i
4.18617i
0.118026i
0.118026i
4.18617i
7.52533i
7.77325i
10.3219i
12.3896i
14.8861i
16.0010i
18.8806i
21.7545i
21.7545i 44.4956 −345.259 104.884i 967.981i −454.964 4726.36i −207.137 −2281.70
36.2 18.8806i −54.3989 −228.476 103.526i 1027.08i −202.536 1897.05i 772.237 1954.63
36.3 16.0010i 52.6587 −128.032 434.946i 842.592i 1479.04 0.510187i 585.941 6959.56
36.4 14.8861i −0.807131 −93.5969 387.405i 12.0151i 696.938 512.130i −2186.35 −5766.96
36.5 12.3896i 76.2286 −25.5018 93.6237i 944.440i −1237.82 1269.91i 3623.79 −1159.96
36.6 10.3219i 6.68770 21.4577 296.142i 69.0300i −1262.47 1542.69i −2142.27 3056.76
36.7 7.77325i −62.8536 67.5766 184.143i 488.577i 1071.60 1520.27i 1763.58 1431.39
36.8 7.52533i −79.9440 71.3694 468.997i 601.605i −1534.47 1500.32i 4204.05 −3529.36
36.9 4.18617i 63.9664 110.476 270.412i 267.774i 565.484 998.302i 1904.70 −1131.99
36.10 0.118026i −7.03331 127.986 225.549i 0.830113i 6.20125 30.2130i −2137.53 26.6207
36.11 0.118026i −7.03331 127.986 225.549i 0.830113i 6.20125 30.2130i −2137.53 26.6207
36.12 4.18617i 63.9664 110.476 270.412i 267.774i 565.484 998.302i 1904.70 −1131.99
36.13 7.52533i −79.9440 71.3694 468.997i 601.605i −1534.47 1500.32i 4204.05 −3529.36
36.14 7.77325i −62.8536 67.5766 184.143i 488.577i 1071.60 1520.27i 1763.58 1431.39
36.15 10.3219i 6.68770 21.4577 296.142i 69.0300i −1262.47 1542.69i −2142.27 3056.76
36.16 12.3896i 76.2286 −25.5018 93.6237i 944.440i −1237.82 1269.91i 3623.79 −1159.96
36.17 14.8861i −0.807131 −93.5969 387.405i 12.0151i 696.938 512.130i −2186.35 −5766.96
36.18 16.0010i 52.6587 −128.032 434.946i 842.592i 1479.04 0.510187i 585.941 6959.56
36.19 18.8806i −54.3989 −228.476 103.526i 1027.08i −202.536 1897.05i 772.237 1954.63
36.20 21.7545i 44.4956 −345.259 104.884i 967.981i −454.964 4726.36i −207.137 −2281.70
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 36.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.8.b.a 20
37.b even 2 1 inner 37.8.b.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.8.b.a 20 1.a even 1 1 trivial
37.8.b.a 20 37.b even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(37, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 130757963535876096 + 9402469145336696832 T^{2} + 1131877572418003968 T^{4} + 49304995250225664 T^{6} + 1081011973644416 T^{8} + 13461378480848 T^{10} + 100204783492 T^{12} + 450999516 T^{14} + 1194509 T^{16} + 1702 T^{18} + T^{20} \)
$3$ \( ( -118561742732844 - 143245528916031 T + 7106792645853 T^{2} + 3144726313731 T^{3} - 78378764805 T^{4} - 2274900624 T^{5} + 56195620 T^{6} + 530931 T^{7} - 13265 T^{8} - 39 T^{9} + T^{10} )^{2} \)
$5$ \( \)\(71\!\cdots\!00\)\( + \)\(27\!\cdots\!00\)\( T^{2} + \)\(42\!\cdots\!00\)\( T^{4} + \)\(33\!\cdots\!00\)\( T^{6} + \)\(14\!\cdots\!00\)\( T^{8} + \)\(37\!\cdots\!00\)\( T^{10} + \)\(56\!\cdots\!09\)\( T^{12} + 52782801048089202 T^{14} + 286976410091 T^{16} + 835306 T^{18} + T^{20} \)
$7$ \( ( -\)\(85\!\cdots\!36\)\( + \)\(13\!\cdots\!52\)\( T + \)\(63\!\cdots\!76\)\( T^{2} - \)\(11\!\cdots\!44\)\( T^{3} - 3991862887224512280 T^{4} + 3574736913497689 T^{5} + 6859107381585 T^{6} - 3317747194 T^{7} - 4553962 T^{8} + 873 T^{9} + T^{10} )^{2} \)
$11$ \( ( -\)\(11\!\cdots\!04\)\( - \)\(11\!\cdots\!81\)\( T + \)\(20\!\cdots\!93\)\( T^{2} + \)\(23\!\cdots\!21\)\( T^{3} - \)\(41\!\cdots\!69\)\( T^{4} - 2385450614623358160 T^{5} + 3010138410188052 T^{6} + 110661881733 T^{7} - 91710693 T^{8} - 1749 T^{9} + T^{10} )^{2} \)
$13$ \( \)\(38\!\cdots\!56\)\( + \)\(37\!\cdots\!40\)\( T^{2} + \)\(24\!\cdots\!40\)\( T^{4} + \)\(60\!\cdots\!20\)\( T^{6} + \)\(71\!\cdots\!20\)\( T^{8} + \)\(40\!\cdots\!68\)\( T^{10} + \)\(11\!\cdots\!85\)\( T^{12} + \)\(18\!\cdots\!10\)\( T^{14} + 154530555318142155 T^{16} + 632138730 T^{18} + T^{20} \)
$17$ \( \)\(26\!\cdots\!56\)\( + \)\(70\!\cdots\!24\)\( T^{2} + \)\(61\!\cdots\!76\)\( T^{4} + \)\(20\!\cdots\!24\)\( T^{6} + \)\(32\!\cdots\!84\)\( T^{8} + \)\(26\!\cdots\!64\)\( T^{10} + \)\(11\!\cdots\!24\)\( T^{12} + \)\(30\!\cdots\!08\)\( T^{14} + 4400558957572617776 T^{16} + 3321595492 T^{18} + T^{20} \)
$19$ \( \)\(50\!\cdots\!76\)\( + \)\(85\!\cdots\!68\)\( T^{2} + \)\(24\!\cdots\!56\)\( T^{4} + \)\(30\!\cdots\!72\)\( T^{6} + \)\(19\!\cdots\!08\)\( T^{8} + \)\(71\!\cdots\!32\)\( T^{10} + \)\(15\!\cdots\!84\)\( T^{12} + \)\(20\!\cdots\!40\)\( T^{14} + 15622122050561509632 T^{16} + 6260069400 T^{18} + T^{20} \)
$23$ \( \)\(33\!\cdots\!96\)\( + \)\(72\!\cdots\!44\)\( T^{2} + \)\(39\!\cdots\!76\)\( T^{4} + \)\(10\!\cdots\!04\)\( T^{6} + \)\(14\!\cdots\!36\)\( T^{8} + \)\(12\!\cdots\!92\)\( T^{10} + \)\(61\!\cdots\!89\)\( T^{12} + \)\(18\!\cdots\!02\)\( T^{14} + \)\(31\!\cdots\!75\)\( T^{16} + 28047457810 T^{18} + T^{20} \)
$29$ \( \)\(12\!\cdots\!96\)\( + \)\(41\!\cdots\!84\)\( T^{2} + \)\(46\!\cdots\!52\)\( T^{4} + \)\(20\!\cdots\!56\)\( T^{6} + \)\(42\!\cdots\!32\)\( T^{8} + \)\(45\!\cdots\!64\)\( T^{10} + \)\(27\!\cdots\!93\)\( T^{12} + \)\(10\!\cdots\!98\)\( T^{14} + \)\(20\!\cdots\!99\)\( T^{16} + 224854622446 T^{18} + T^{20} \)
$31$ \( \)\(14\!\cdots\!00\)\( + \)\(45\!\cdots\!40\)\( T^{2} + \)\(39\!\cdots\!16\)\( T^{4} + \)\(11\!\cdots\!48\)\( T^{6} + \)\(15\!\cdots\!16\)\( T^{8} + \)\(12\!\cdots\!80\)\( T^{10} + \)\(58\!\cdots\!53\)\( T^{12} + \)\(16\!\cdots\!22\)\( T^{14} + \)\(28\!\cdots\!55\)\( T^{16} + 263739369210 T^{18} + T^{20} \)
$37$ \( \)\(59\!\cdots\!49\)\( - \)\(68\!\cdots\!96\)\( T + \)\(25\!\cdots\!42\)\( T^{2} - \)\(35\!\cdots\!92\)\( T^{3} + \)\(53\!\cdots\!09\)\( T^{4} - \)\(95\!\cdots\!52\)\( T^{5} + \)\(73\!\cdots\!84\)\( T^{6} - \)\(17\!\cdots\!64\)\( T^{7} + \)\(76\!\cdots\!46\)\( T^{8} - \)\(23\!\cdots\!96\)\( T^{9} + \)\(76\!\cdots\!92\)\( T^{10} - \)\(25\!\cdots\!12\)\( T^{11} + \)\(84\!\cdots\!14\)\( T^{12} - \)\(20\!\cdots\!72\)\( T^{13} + \)\(90\!\cdots\!04\)\( T^{14} - \)\(12\!\cdots\!64\)\( T^{15} + \)\(73\!\cdots\!61\)\( T^{16} - 51550160046950796 T^{17} + 380826937262 T^{18} - 108732 T^{19} + T^{20} \)
$41$ \( ( \)\(23\!\cdots\!26\)\( + \)\(61\!\cdots\!89\)\( T - \)\(17\!\cdots\!55\)\( T^{2} - \)\(48\!\cdots\!81\)\( T^{3} - \)\(31\!\cdots\!41\)\( T^{4} + \)\(77\!\cdots\!30\)\( T^{5} + \)\(83\!\cdots\!08\)\( T^{6} - 435238491711196773 T^{7} - 513671240259 T^{8} + 788871 T^{9} + T^{10} )^{2} \)
$43$ \( \)\(79\!\cdots\!96\)\( + \)\(44\!\cdots\!24\)\( T^{2} + \)\(32\!\cdots\!76\)\( T^{4} + \)\(97\!\cdots\!76\)\( T^{6} + \)\(14\!\cdots\!52\)\( T^{8} + \)\(12\!\cdots\!44\)\( T^{10} + \)\(64\!\cdots\!24\)\( T^{12} + \)\(18\!\cdots\!00\)\( T^{14} + \)\(31\!\cdots\!84\)\( T^{16} + 2779292121648 T^{18} + T^{20} \)
$47$ \( ( \)\(74\!\cdots\!84\)\( - \)\(26\!\cdots\!28\)\( T + \)\(98\!\cdots\!20\)\( T^{2} + \)\(66\!\cdots\!28\)\( T^{3} - \)\(33\!\cdots\!36\)\( T^{4} + \)\(46\!\cdots\!29\)\( T^{5} + \)\(14\!\cdots\!37\)\( T^{6} - 777201452733370878 T^{7} - 2090433670110 T^{8} + 756393 T^{9} + T^{10} )^{2} \)
$53$ \( ( \)\(14\!\cdots\!64\)\( - \)\(17\!\cdots\!64\)\( T + \)\(38\!\cdots\!76\)\( T^{2} + \)\(18\!\cdots\!80\)\( T^{3} - \)\(12\!\cdots\!34\)\( T^{4} - \)\(23\!\cdots\!71\)\( T^{5} + \)\(16\!\cdots\!81\)\( T^{6} + 10195003023347821158 T^{7} - 7046784154368 T^{8} - 1499679 T^{9} + T^{10} )^{2} \)
$59$ \( \)\(12\!\cdots\!56\)\( + \)\(15\!\cdots\!72\)\( T^{2} + \)\(10\!\cdots\!08\)\( T^{4} + \)\(20\!\cdots\!72\)\( T^{6} + \)\(11\!\cdots\!56\)\( T^{8} + \)\(23\!\cdots\!72\)\( T^{10} + \)\(21\!\cdots\!92\)\( T^{12} + \)\(96\!\cdots\!32\)\( T^{14} + \)\(21\!\cdots\!00\)\( T^{16} + 23780290980220 T^{18} + T^{20} \)
$61$ \( \)\(27\!\cdots\!00\)\( + \)\(21\!\cdots\!60\)\( T^{2} + \)\(12\!\cdots\!56\)\( T^{4} + \)\(30\!\cdots\!52\)\( T^{6} + \)\(42\!\cdots\!56\)\( T^{8} + \)\(37\!\cdots\!28\)\( T^{10} + \)\(20\!\cdots\!45\)\( T^{12} + \)\(73\!\cdots\!22\)\( T^{14} + \)\(16\!\cdots\!31\)\( T^{16} + 19651838390478 T^{18} + T^{20} \)
$67$ \( ( \)\(25\!\cdots\!76\)\( - \)\(22\!\cdots\!52\)\( T + \)\(15\!\cdots\!20\)\( T^{2} + \)\(22\!\cdots\!92\)\( T^{3} - \)\(15\!\cdots\!76\)\( T^{4} - \)\(53\!\cdots\!40\)\( T^{5} + \)\(37\!\cdots\!01\)\( T^{6} + 50924299531511286848 T^{7} - 34005155980423 T^{8} - 1781112 T^{9} + T^{10} )^{2} \)
$71$ \( ( \)\(12\!\cdots\!76\)\( - \)\(77\!\cdots\!80\)\( T - \)\(23\!\cdots\!16\)\( T^{2} - \)\(89\!\cdots\!80\)\( T^{3} + \)\(62\!\cdots\!32\)\( T^{4} + \)\(33\!\cdots\!67\)\( T^{5} - \)\(23\!\cdots\!99\)\( T^{6} - \)\(32\!\cdots\!46\)\( T^{7} - 26624331913914 T^{8} + 7629543 T^{9} + T^{10} )^{2} \)
$73$ \( ( \)\(35\!\cdots\!06\)\( - \)\(58\!\cdots\!71\)\( T - \)\(13\!\cdots\!95\)\( T^{2} + \)\(41\!\cdots\!39\)\( T^{3} - \)\(42\!\cdots\!05\)\( T^{4} - \)\(62\!\cdots\!66\)\( T^{5} + \)\(96\!\cdots\!60\)\( T^{6} + \)\(32\!\cdots\!99\)\( T^{7} - 56050169782927 T^{8} - 5544009 T^{9} + T^{10} )^{2} \)
$79$ \( \)\(16\!\cdots\!36\)\( + \)\(94\!\cdots\!00\)\( T^{2} + \)\(13\!\cdots\!72\)\( T^{4} + \)\(12\!\cdots\!36\)\( T^{6} + \)\(42\!\cdots\!20\)\( T^{8} + \)\(71\!\cdots\!68\)\( T^{10} + \)\(66\!\cdots\!73\)\( T^{12} + \)\(35\!\cdots\!94\)\( T^{14} + \)\(10\!\cdots\!75\)\( T^{16} + 170634215807934 T^{18} + T^{20} \)
$83$ \( ( \)\(50\!\cdots\!16\)\( + \)\(14\!\cdots\!04\)\( T + \)\(40\!\cdots\!40\)\( T^{2} - \)\(39\!\cdots\!96\)\( T^{3} - \)\(95\!\cdots\!32\)\( T^{4} + \)\(32\!\cdots\!71\)\( T^{5} + \)\(64\!\cdots\!97\)\( T^{6} - \)\(82\!\cdots\!06\)\( T^{7} - 146045072089062 T^{8} + 6436911 T^{9} + T^{10} )^{2} \)
$89$ \( \)\(13\!\cdots\!56\)\( + \)\(69\!\cdots\!80\)\( T^{2} + \)\(11\!\cdots\!68\)\( T^{4} + \)\(79\!\cdots\!36\)\( T^{6} + \)\(23\!\cdots\!00\)\( T^{8} + \)\(35\!\cdots\!12\)\( T^{10} + \)\(27\!\cdots\!80\)\( T^{12} + \)\(11\!\cdots\!64\)\( T^{14} + \)\(25\!\cdots\!48\)\( T^{16} + 261084499656316 T^{18} + T^{20} \)
$97$ \( \)\(27\!\cdots\!96\)\( + \)\(10\!\cdots\!88\)\( T^{2} + \)\(17\!\cdots\!08\)\( T^{4} + \)\(15\!\cdots\!52\)\( T^{6} + \)\(75\!\cdots\!92\)\( T^{8} + \)\(21\!\cdots\!96\)\( T^{10} + \)\(38\!\cdots\!32\)\( T^{12} + \)\(40\!\cdots\!52\)\( T^{14} + \)\(24\!\cdots\!64\)\( T^{16} + 788101919779560 T^{18} + T^{20} \)
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