Properties

Label 74.7.d.b
Level $74$
Weight $7$
Character orbit 74.d
Analytic conductor $17.024$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 74.d (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.0240021879\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial: \(x^{20} + 10424 x^{18} + 44844916 x^{16} + 103219343022 x^{14} + 138101513095620 x^{12} + 109787124347520192 x^{10} + 51172837440825906441 x^{8} + 13199761586736849750156 x^{6} + 1629475196758519705300656 x^{4} + 65381450766316829487245376 x^{2} + 736627437450607950694158336\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2}\cdot 11^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -4 - 4 \beta_{6} ) q^{2} + ( -\beta_{1} + 3 \beta_{6} ) q^{3} + 32 \beta_{6} q^{4} + ( 3 - 3 \beta_{6} - \beta_{8} ) q^{5} + ( 12 + 4 \beta_{1} + 4 \beta_{2} - 12 \beta_{6} ) q^{6} + ( 5 + 3 \beta_{2} + \beta_{4} ) q^{7} + ( 128 - 128 \beta_{6} ) q^{8} + ( -323 - 3 \beta_{2} + \beta_{3} + \beta_{7} + \beta_{8} ) q^{9} +O(q^{10})\) \( q + ( -4 - 4 \beta_{6} ) q^{2} + ( -\beta_{1} + 3 \beta_{6} ) q^{3} + 32 \beta_{6} q^{4} + ( 3 - 3 \beta_{6} - \beta_{8} ) q^{5} + ( 12 + 4 \beta_{1} + 4 \beta_{2} - 12 \beta_{6} ) q^{6} + ( 5 + 3 \beta_{2} + \beta_{4} ) q^{7} + ( 128 - 128 \beta_{6} ) q^{8} + ( -323 - 3 \beta_{2} + \beta_{3} + \beta_{7} + \beta_{8} ) q^{9} + ( -24 + 4 \beta_{7} + 4 \beta_{8} ) q^{10} + ( -3 \beta_{1} + 111 \beta_{6} + 4 \beta_{7} - 4 \beta_{8} - \beta_{9} - \beta_{15} ) q^{11} + ( -96 - 32 \beta_{2} ) q^{12} + ( 77 + 5 \beta_{1} + 5 \beta_{2} - \beta_{4} - 77 \beta_{6} + \beta_{9} - \beta_{18} ) q^{13} + ( -20 + 12 \beta_{1} - 12 \beta_{2} - 4 \beta_{4} - 20 \beta_{6} - 4 \beta_{9} ) q^{14} + ( -112 - 26 \beta_{1} + 26 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 112 \beta_{6} - 4 \beta_{7} + \beta_{9} + \beta_{10} + \beta_{12} + \beta_{14} + \beta_{15} ) q^{15} -1024 q^{16} + ( 793 + 22 \beta_{1} + 22 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - \beta_{5} - 793 \beta_{6} + \beta_{8} + 3 \beta_{9} + \beta_{11} + 2 \beta_{12} - \beta_{15} + \beta_{17} + \beta_{18} ) q^{17} + ( 1292 - 12 \beta_{1} + 12 \beta_{2} - 4 \beta_{3} + 1292 \beta_{6} - 8 \beta_{7} - 4 \beta_{12} ) q^{18} + ( 187 + 33 \beta_{1} + 33 \beta_{2} + 4 \beta_{4} + \beta_{5} - 187 \beta_{6} - 11 \beta_{8} - 4 \beta_{9} - \beta_{11} - \beta_{13} + \beta_{15} + \beta_{17} ) q^{19} + ( 96 + 96 \beta_{6} - 32 \beta_{7} ) q^{20} + ( -23 \beta_{1} + 2674 \beta_{6} - 9 \beta_{7} + 9 \beta_{8} + 26 \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} + \beta_{17} + \beta_{18} - \beta_{19} ) q^{21} + ( 444 + 12 \beta_{1} + 12 \beta_{2} - 4 \beta_{4} + 4 \beta_{5} - 444 \beta_{6} + 32 \beta_{8} + 4 \beta_{9} + 4 \beta_{15} ) q^{22} + ( -308 - 74 \beta_{1} - 74 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + 308 \beta_{6} - 21 \beta_{8} + 2 \beta_{9} - \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{15} - 2 \beta_{17} - 3 \beta_{18} ) q^{23} + ( 384 - 128 \beta_{1} + 128 \beta_{2} + 384 \beta_{6} ) q^{24} + ( -16 \beta_{1} - 4753 \beta_{6} - 7 \beta_{7} + 7 \beta_{8} + 3 \beta_{9} + 4 \beta_{10} + 4 \beta_{11} - 6 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + 8 \beta_{15} + \beta_{16} + \beta_{17} - 4 \beta_{18} + 4 \beta_{19} ) q^{25} + ( -616 - 40 \beta_{2} + 8 \beta_{4} + 4 \beta_{18} + 4 \beta_{19} ) q^{26} + ( 351 \beta_{1} - 1661 \beta_{6} - 20 \beta_{7} + 20 \beta_{8} + 6 \beta_{9} - 3 \beta_{10} - 3 \beta_{11} + 10 \beta_{12} - 3 \beta_{13} - 3 \beta_{14} - 9 \beta_{15} + 3 \beta_{16} + 3 \beta_{17} ) q^{27} + ( -96 \beta_{1} + 160 \beta_{6} + 32 \beta_{9} ) q^{28} + ( -96 + 56 \beta_{1} - 56 \beta_{2} - 6 \beta_{3} - 28 \beta_{4} - 3 \beta_{5} - 96 \beta_{6} + 19 \beta_{7} - 28 \beta_{9} + 5 \beta_{10} - 6 \beta_{12} - 2 \beta_{14} + 3 \beta_{15} + 7 \beta_{16} + 2 \beta_{19} ) q^{29} + ( 208 \beta_{1} + 896 \beta_{6} + 16 \beta_{7} - 16 \beta_{8} - 8 \beta_{9} - 4 \beta_{10} - 4 \beta_{11} - 8 \beta_{12} - 4 \beta_{13} - 4 \beta_{14} - 8 \beta_{15} ) q^{30} + ( 5894 + 34 \beta_{1} - 34 \beta_{2} + 2 \beta_{3} + 18 \beta_{4} - 3 \beta_{5} + 5894 \beta_{6} - 4 \beta_{7} + 18 \beta_{9} - \beta_{10} + 2 \beta_{12} + 6 \beta_{14} + 3 \beta_{15} + 6 \beta_{16} + 4 \beta_{19} ) q^{31} + ( 4096 + 4096 \beta_{6} ) q^{32} + ( -4638 + 175 \beta_{2} + 7 \beta_{3} + 87 \beta_{4} - 12 \beta_{5} - 80 \beta_{7} - 80 \beta_{8} + 10 \beta_{10} - 10 \beta_{11} - 9 \beta_{13} + 9 \beta_{14} + 4 \beta_{16} - 4 \beta_{17} + 6 \beta_{18} + 6 \beta_{19} ) q^{33} + ( -6344 - 176 \beta_{2} + 16 \beta_{3} + 24 \beta_{4} + 8 \beta_{5} - 4 \beta_{7} - 4 \beta_{8} + 4 \beta_{10} - 4 \beta_{11} + 4 \beta_{16} - 4 \beta_{17} - 4 \beta_{18} - 4 \beta_{19} ) q^{34} + ( -5253 + 61 \beta_{1} + 61 \beta_{2} - 8 \beta_{3} - 25 \beta_{4} - 25 \beta_{5} + 5253 \beta_{6} - 21 \beta_{8} + 25 \beta_{9} - 15 \beta_{11} + 8 \beta_{12} - 11 \beta_{13} - 25 \beta_{15} - 4 \beta_{17} + \beta_{18} ) q^{35} + ( 96 \beta_{1} - 10336 \beta_{6} + 32 \beta_{7} - 32 \beta_{8} + 32 \beta_{12} ) q^{36} + ( -870 + 150 \beta_{1} - 74 \beta_{2} + \beta_{3} - 12 \beta_{4} + 4 \beta_{5} + 835 \beta_{6} + 9 \beta_{7} + 43 \beta_{8} + 47 \beta_{9} - 6 \beta_{10} + 5 \beta_{11} + 3 \beta_{12} - 10 \beta_{14} + \beta_{15} + 3 \beta_{16} + 11 \beta_{17} - 8 \beta_{18} + 3 \beta_{19} ) q^{37} + ( -1496 - 264 \beta_{2} - 32 \beta_{4} - 8 \beta_{5} + 44 \beta_{7} + 44 \beta_{8} - 4 \beta_{10} + 4 \beta_{11} + 4 \beta_{13} - 4 \beta_{14} + 4 \beta_{16} - 4 \beta_{17} ) q^{38} + ( 6019 - 214 \beta_{1} + 214 \beta_{2} - 11 \beta_{3} + 7 \beta_{4} + 13 \beta_{5} + 6019 \beta_{6} - 5 \beta_{7} + 7 \beta_{9} + 14 \beta_{10} - 11 \beta_{12} + 5 \beta_{14} - 13 \beta_{15} + 7 \beta_{16} + 4 \beta_{19} ) q^{39} + ( -768 \beta_{6} + 128 \beta_{7} - 128 \beta_{8} ) q^{40} + ( -73 \beta_{1} - 22714 \beta_{6} - 55 \beta_{7} + 55 \beta_{8} + 69 \beta_{9} + 5 \beta_{10} + 5 \beta_{11} + 37 \beta_{12} + \beta_{13} + \beta_{14} + 33 \beta_{15} + 3 \beta_{16} + 3 \beta_{17} - 15 \beta_{18} + 15 \beta_{19} ) q^{41} + ( 10696 + 92 \beta_{1} + 92 \beta_{2} + 4 \beta_{3} + 104 \beta_{4} - 4 \beta_{5} - 10696 \beta_{6} - 72 \beta_{8} - 104 \beta_{9} + 8 \beta_{11} - 4 \beta_{12} - 8 \beta_{13} - 4 \beta_{15} - 8 \beta_{17} - 8 \beta_{18} ) q^{42} + ( 3289 - 194 \beta_{1} - 194 \beta_{2} + 17 \beta_{3} - 36 \beta_{4} - 6 \beta_{5} - 3289 \beta_{6} + \beta_{8} + 36 \beta_{9} + 3 \beta_{11} - 17 \beta_{12} - 7 \beta_{13} - 6 \beta_{15} + 17 \beta_{17} + 10 \beta_{18} ) q^{43} + ( -3552 - 96 \beta_{2} + 32 \beta_{4} - 32 \beta_{5} - 128 \beta_{7} - 128 \beta_{8} ) q^{44} + ( -23472 + 548 \beta_{1} + 548 \beta_{2} + 50 \beta_{3} - 69 \beta_{4} + 29 \beta_{5} + 23472 \beta_{6} + 147 \beta_{8} + 69 \beta_{9} - 6 \beta_{11} - 50 \beta_{12} + 2 \beta_{13} + 29 \beta_{15} + 12 \beta_{17} + 7 \beta_{18} ) q^{45} + ( 2464 + 592 \beta_{2} + 8 \beta_{3} + 16 \beta_{4} - 8 \beta_{5} + 84 \beta_{7} + 84 \beta_{8} - 4 \beta_{10} + 4 \beta_{11} + 8 \beta_{13} - 8 \beta_{14} - 8 \beta_{16} + 8 \beta_{17} + 12 \beta_{18} + 12 \beta_{19} ) q^{46} + ( -8063 - 1579 \beta_{2} + 76 \beta_{3} - 67 \beta_{4} + 76 \beta_{5} + 58 \beta_{7} + 58 \beta_{8} - 6 \beta_{10} + 6 \beta_{11} - 6 \beta_{13} + 6 \beta_{14} + 12 \beta_{16} - 12 \beta_{17} - 33 \beta_{18} - 33 \beta_{19} ) q^{47} + ( 1024 \beta_{1} - 3072 \beta_{6} ) q^{48} + ( 27605 + 2883 \beta_{2} - 69 \beta_{3} + 74 \beta_{4} + 17 \beta_{5} + 253 \beta_{7} + 253 \beta_{8} - 16 \beta_{10} + 16 \beta_{11} - 6 \beta_{13} + 6 \beta_{14} - 11 \beta_{16} + 11 \beta_{17} - 11 \beta_{18} - 11 \beta_{19} ) q^{49} + ( -19012 + 64 \beta_{1} + 64 \beta_{2} - 24 \beta_{3} + 12 \beta_{4} - 32 \beta_{5} + 19012 \beta_{6} - 56 \beta_{8} - 12 \beta_{9} - 32 \beta_{11} + 24 \beta_{12} + 16 \beta_{13} - 32 \beta_{15} - 8 \beta_{17} + 32 \beta_{18} ) q^{50} + ( 27643 - 1788 \beta_{1} + 1788 \beta_{2} - 73 \beta_{3} - 45 \beta_{4} - 5 \beta_{5} + 27643 \beta_{6} + 167 \beta_{7} - 45 \beta_{9} + 9 \beta_{10} - 73 \beta_{12} + 9 \beta_{14} + 5 \beta_{15} - 30 \beta_{16} + 37 \beta_{19} ) q^{51} + ( 2464 - 160 \beta_{1} + 160 \beta_{2} - 32 \beta_{4} + 2464 \beta_{6} - 32 \beta_{9} - 32 \beta_{19} ) q^{52} + ( 28470 - 1691 \beta_{2} - 105 \beta_{3} + 126 \beta_{4} + 15 \beta_{5} - 289 \beta_{7} - 289 \beta_{8} + 6 \beta_{10} - 6 \beta_{11} + 4 \beta_{13} - 4 \beta_{14} - 6 \beta_{16} + 6 \beta_{17} + 9 \beta_{18} + 9 \beta_{19} ) q^{53} + ( -6644 - 1404 \beta_{1} - 1404 \beta_{2} + 40 \beta_{3} + 24 \beta_{4} + 36 \beta_{5} + 6644 \beta_{6} - 160 \beta_{8} - 24 \beta_{9} + 24 \beta_{11} - 40 \beta_{12} + 24 \beta_{13} + 36 \beta_{15} - 24 \beta_{17} ) q^{54} + ( -69657 + 931 \beta_{1} - 931 \beta_{2} + 83 \beta_{3} - 195 \beta_{4} - 10 \beta_{5} - 69657 \beta_{6} - 294 \beta_{7} - 195 \beta_{9} + 30 \beta_{10} + 83 \beta_{12} + 9 \beta_{14} + 10 \beta_{15} - 9 \beta_{16} - \beta_{19} ) q^{55} + ( 640 + 384 \beta_{1} + 384 \beta_{2} + 128 \beta_{4} - 640 \beta_{6} - 128 \beta_{9} ) q^{56} + ( 31491 - 139 \beta_{1} + 139 \beta_{2} - 64 \beta_{3} - 63 \beta_{4} - 24 \beta_{5} + 31491 \beta_{6} - 647 \beta_{7} - 63 \beta_{9} - 47 \beta_{10} - 64 \beta_{12} + 24 \beta_{15} + \beta_{16} - 39 \beta_{19} ) q^{57} + ( -448 \beta_{1} + 768 \beta_{6} - 76 \beta_{7} + 76 \beta_{8} + 224 \beta_{9} - 20 \beta_{10} - 20 \beta_{11} + 48 \beta_{12} + 8 \beta_{13} + 8 \beta_{14} - 24 \beta_{15} - 28 \beta_{16} - 28 \beta_{17} + 8 \beta_{18} - 8 \beta_{19} ) q^{58} + ( -18582 - 1669 \beta_{1} - 1669 \beta_{2} - 15 \beta_{3} - 82 \beta_{4} - 84 \beta_{5} + 18582 \beta_{6} + 428 \beta_{8} + 82 \beta_{9} + 28 \beta_{11} + 15 \beta_{12} + 28 \beta_{13} - 84 \beta_{15} + 9 \beta_{17} + 53 \beta_{18} ) q^{59} + ( 3584 - 832 \beta_{1} - 832 \beta_{2} - 32 \beta_{3} - 32 \beta_{4} + 32 \beta_{5} - 3584 \beta_{6} + 128 \beta_{8} + 32 \beta_{9} + 32 \beta_{11} + 32 \beta_{12} + 32 \beta_{13} + 32 \beta_{15} ) q^{60} + ( 41959 + 832 \beta_{1} - 832 \beta_{2} + 10 \beta_{3} - 66 \beta_{4} - 49 \beta_{5} + 41959 \beta_{6} - 620 \beta_{7} - 66 \beta_{9} + \beta_{10} + 10 \beta_{12} + 42 \beta_{14} + 49 \beta_{15} - 23 \beta_{16} - 57 \beta_{19} ) q^{61} + ( -272 \beta_{1} - 47152 \beta_{6} + 16 \beta_{7} - 16 \beta_{8} - 144 \beta_{9} + 4 \beta_{10} + 4 \beta_{11} - 16 \beta_{12} - 24 \beta_{13} - 24 \beta_{14} - 24 \beta_{15} - 24 \beta_{16} - 24 \beta_{17} + 16 \beta_{18} - 16 \beta_{19} ) q^{62} + ( -13604 - 2318 \beta_{2} - 172 \beta_{3} - 786 \beta_{4} + 88 \beta_{5} + 601 \beta_{7} + 601 \beta_{8} - 24 \beta_{10} + 24 \beta_{11} + 59 \beta_{13} - 59 \beta_{14} - 18 \beta_{16} + 18 \beta_{17} + 4 \beta_{18} + 4 \beta_{19} ) q^{63} -32768 \beta_{6} q^{64} + ( 474 \beta_{1} - 7352 \beta_{6} + 343 \beta_{7} - 343 \beta_{8} - 199 \beta_{9} + 8 \beta_{10} + 8 \beta_{11} + 56 \beta_{12} + 13 \beta_{13} + 13 \beta_{14} + 51 \beta_{15} - 5 \beta_{16} - 5 \beta_{17} + 5 \beta_{18} - 5 \beta_{19} ) q^{65} + ( 18552 + 700 \beta_{1} - 700 \beta_{2} - 28 \beta_{3} - 348 \beta_{4} + 48 \beta_{5} + 18552 \beta_{6} + 640 \beta_{7} - 348 \beta_{9} - 80 \beta_{10} - 28 \beta_{12} - 72 \beta_{14} - 48 \beta_{15} - 32 \beta_{16} - 48 \beta_{19} ) q^{66} + ( 1764 \beta_{1} - 42034 \beta_{6} + 345 \beta_{7} - 345 \beta_{8} - 86 \beta_{9} + 12 \beta_{10} + 12 \beta_{11} - 20 \beta_{12} + 29 \beta_{13} + 29 \beta_{14} + 31 \beta_{15} - 12 \beta_{16} - 12 \beta_{17} - 71 \beta_{18} + 71 \beta_{19} ) q^{67} + ( 25376 - 704 \beta_{1} + 704 \beta_{2} - 64 \beta_{3} - 96 \beta_{4} - 32 \beta_{5} + 25376 \beta_{6} + 32 \beta_{7} - 96 \beta_{9} - 32 \beta_{10} - 64 \beta_{12} + 32 \beta_{15} - 32 \beta_{16} + 32 \beta_{19} ) q^{68} + ( -81171 - 2224 \beta_{1} + 2224 \beta_{2} + 99 \beta_{3} + 302 \beta_{4} - 71 \beta_{5} - 81171 \beta_{6} - 846 \beta_{7} + 302 \beta_{9} + 63 \beta_{10} + 99 \beta_{12} + 34 \beta_{14} + 71 \beta_{15} - 5 \beta_{16} - 29 \beta_{19} ) q^{69} + ( 42024 - 488 \beta_{2} + 64 \beta_{3} + 200 \beta_{4} + 200 \beta_{5} + 84 \beta_{7} + 84 \beta_{8} - 60 \beta_{10} + 60 \beta_{11} + 44 \beta_{13} - 44 \beta_{14} - 16 \beta_{16} + 16 \beta_{17} - 4 \beta_{18} - 4 \beta_{19} ) q^{70} + ( -5077 + 2649 \beta_{2} + 192 \beta_{3} + 461 \beta_{4} - 96 \beta_{5} + 940 \beta_{7} + 940 \beta_{8} + 17 \beta_{10} - 17 \beta_{11} - 37 \beta_{13} + 37 \beta_{14} + 4 \beta_{16} - 4 \beta_{17} - 57 \beta_{18} - 57 \beta_{19} ) q^{71} + ( -41344 - 384 \beta_{1} - 384 \beta_{2} + 128 \beta_{3} + 41344 \beta_{6} + 256 \beta_{8} - 128 \beta_{12} ) q^{72} + ( -2039 \beta_{1} + 28526 \beta_{6} + 628 \beta_{7} - 628 \beta_{8} - 148 \beta_{9} + 87 \beta_{10} + 87 \beta_{11} - 77 \beta_{12} + 31 \beta_{13} + 31 \beta_{14} - 46 \beta_{15} + 25 \beta_{16} + 25 \beta_{17} + 56 \beta_{18} - 56 \beta_{19} ) q^{73} + ( 6820 - 896 \beta_{1} - 304 \beta_{2} + 8 \beta_{3} + 236 \beta_{4} - 20 \beta_{5} + 140 \beta_{6} - 208 \beta_{7} - 136 \beta_{8} - 140 \beta_{9} + 44 \beta_{10} + 4 \beta_{11} - 16 \beta_{12} + 40 \beta_{13} + 40 \beta_{14} + 12 \beta_{15} + 32 \beta_{16} - 56 \beta_{17} + 44 \beta_{18} + 20 \beta_{19} ) q^{74} + ( 1193 + 1355 \beta_{2} + 122 \beta_{3} + 1098 \beta_{4} + 147 \beta_{5} + 143 \beta_{7} + 143 \beta_{8} - 6 \beta_{10} + 6 \beta_{11} - 15 \beta_{13} + 15 \beta_{14} - 36 \beta_{16} + 36 \beta_{17} + 21 \beta_{18} + 21 \beta_{19} ) q^{75} + ( 5984 - 1056 \beta_{1} + 1056 \beta_{2} + 128 \beta_{4} + 32 \beta_{5} + 5984 \beta_{6} - 352 \beta_{7} + 128 \beta_{9} + 32 \beta_{10} + 32 \beta_{14} - 32 \beta_{15} - 32 \beta_{16} ) q^{76} + ( 8147 \beta_{1} - 61310 \beta_{6} + 1682 \beta_{7} - 1682 \beta_{8} + 338 \beta_{9} - 93 \beta_{10} - 93 \beta_{11} + 191 \beta_{12} - 60 \beta_{13} - 60 \beta_{14} - 281 \beta_{15} + 30 \beta_{16} + 30 \beta_{17} - 7 \beta_{18} + 7 \beta_{19} ) q^{77} + ( 1712 \beta_{1} - 48152 \beta_{6} + 20 \beta_{7} - 20 \beta_{8} - 56 \beta_{9} - 56 \beta_{10} - 56 \beta_{11} + 88 \beta_{12} - 20 \beta_{13} - 20 \beta_{14} + 104 \beta_{15} - 28 \beta_{16} - 28 \beta_{17} + 16 \beta_{18} - 16 \beta_{19} ) q^{78} + ( -126056 - 451 \beta_{1} - 451 \beta_{2} + 150 \beta_{3} + 159 \beta_{4} - 105 \beta_{5} + 126056 \beta_{6} + 90 \beta_{8} - 159 \beta_{9} - 80 \beta_{11} - 150 \beta_{12} + 45 \beta_{13} - 105 \beta_{15} - 40 \beta_{17} - 156 \beta_{18} ) q^{79} + ( -3072 + 3072 \beta_{6} + 1024 \beta_{8} ) q^{80} + ( 145451 + 4578 \beta_{2} - 244 \beta_{3} - 549 \beta_{4} - 123 \beta_{5} - 1618 \beta_{7} - 1618 \beta_{8} - 39 \beta_{10} + 39 \beta_{11} + 3 \beta_{13} - 3 \beta_{14} - 12 \beta_{16} + 12 \beta_{17} + 18 \beta_{18} + 18 \beta_{19} ) q^{81} + ( -90856 + 292 \beta_{1} + 292 \beta_{2} + 148 \beta_{3} + 276 \beta_{4} - 132 \beta_{5} + 90856 \beta_{6} - 440 \beta_{8} - 276 \beta_{9} - 40 \beta_{11} - 148 \beta_{12} - 8 \beta_{13} - 132 \beta_{15} - 24 \beta_{17} + 120 \beta_{18} ) q^{82} + ( 34941 + 3367 \beta_{2} + 492 \beta_{3} - 381 \beta_{4} + 32 \beta_{5} + 1076 \beta_{7} + 1076 \beta_{8} + 13 \beta_{10} - 13 \beta_{11} - 39 \beta_{13} + 39 \beta_{14} + 22 \beta_{16} - 22 \beta_{17} + 11 \beta_{18} + 11 \beta_{19} ) q^{83} + ( -85568 - 736 \beta_{2} - 32 \beta_{3} - 832 \beta_{4} + 32 \beta_{5} + 288 \beta_{7} + 288 \beta_{8} + 32 \beta_{10} - 32 \beta_{11} + 32 \beta_{13} - 32 \beta_{14} - 32 \beta_{16} + 32 \beta_{17} + 32 \beta_{18} + 32 \beta_{19} ) q^{84} + ( -2424 \beta_{1} + 1732 \beta_{6} + 1523 \beta_{7} - 1523 \beta_{8} - 328 \beta_{9} + \beta_{10} + \beta_{11} + 116 \beta_{12} + 7 \beta_{13} + 7 \beta_{14} - 58 \beta_{15} - 6 \beta_{16} - 6 \beta_{17} + 24 \beta_{18} - 24 \beta_{19} ) q^{85} + ( -26312 + 1552 \beta_{2} - 136 \beta_{3} + 288 \beta_{4} + 48 \beta_{5} - 4 \beta_{7} - 4 \beta_{8} + 12 \beta_{10} - 12 \beta_{11} + 28 \beta_{13} - 28 \beta_{14} + 68 \beta_{16} - 68 \beta_{17} - 40 \beta_{18} - 40 \beta_{19} ) q^{86} + ( 46335 - 4962 \beta_{1} - 4962 \beta_{2} - 15 \beta_{3} + 868 \beta_{4} + 112 \beta_{5} - 46335 \beta_{6} - 490 \beta_{8} - 868 \beta_{9} + 157 \beta_{11} + 15 \beta_{12} - 74 \beta_{13} + 112 \beta_{15} - 25 \beta_{17} + \beta_{18} ) q^{87} + ( 14208 - 384 \beta_{1} + 384 \beta_{2} - 128 \beta_{4} + 128 \beta_{5} + 14208 \beta_{6} + 1024 \beta_{7} - 128 \beta_{9} - 128 \beta_{15} ) q^{88} + ( -47408 + 1222 \beta_{1} - 1222 \beta_{2} + 97 \beta_{3} - 113 \beta_{4} + 16 \beta_{5} - 47408 \beta_{6} - 1083 \beta_{7} - 113 \beta_{9} + 5 \beta_{10} + 97 \beta_{12} - 124 \beta_{14} - 16 \beta_{15} + 49 \beta_{16} + 159 \beta_{19} ) q^{89} + ( 187776 - 4384 \beta_{2} - 400 \beta_{3} + 552 \beta_{4} - 232 \beta_{5} - 588 \beta_{7} - 588 \beta_{8} - 24 \beta_{10} + 24 \beta_{11} - 8 \beta_{13} + 8 \beta_{14} + 48 \beta_{16} - 48 \beta_{17} - 28 \beta_{18} - 28 \beta_{19} ) q^{90} + ( -67748 + 1305 \beta_{1} + 1305 \beta_{2} + 97 \beta_{3} + 430 \beta_{4} - 137 \beta_{5} + 67748 \beta_{6} - 1741 \beta_{8} - 430 \beta_{9} - 35 \beta_{11} - 97 \beta_{12} + 23 \beta_{13} - 137 \beta_{15} + 20 \beta_{17} + 59 \beta_{18} ) q^{91} + ( -9856 + 2368 \beta_{1} - 2368 \beta_{2} - 32 \beta_{3} - 64 \beta_{4} + 32 \beta_{5} - 9856 \beta_{6} - 672 \beta_{7} - 64 \beta_{9} + 32 \beta_{10} - 32 \beta_{12} + 64 \beta_{14} - 32 \beta_{15} + 64 \beta_{16} - 96 \beta_{19} ) q^{92} + ( 29959 - 7863 \beta_{1} - 7863 \beta_{2} - 245 \beta_{3} - 1932 \beta_{4} + 301 \beta_{5} - 29959 \beta_{6} + 3201 \beta_{8} + 1932 \beta_{9} + 38 \beta_{11} + 245 \beta_{12} + 68 \beta_{13} + 301 \beta_{15} - 58 \beta_{17} + 9 \beta_{18} ) q^{93} + ( 32252 - 6316 \beta_{1} + 6316 \beta_{2} - 304 \beta_{3} + 268 \beta_{4} - 304 \beta_{5} + 32252 \beta_{6} - 464 \beta_{7} + 268 \beta_{9} + 48 \beta_{10} - 304 \beta_{12} - 48 \beta_{14} + 304 \beta_{15} - 96 \beta_{16} + 264 \beta_{19} ) q^{94} + ( 12652 \beta_{1} - 187134 \beta_{6} + 35 \beta_{7} - 35 \beta_{8} + 192 \beta_{9} - 24 \beta_{10} - 24 \beta_{11} - 616 \beta_{12} - 156 \beta_{13} - 156 \beta_{14} + 82 \beta_{15} + 57 \beta_{16} + 57 \beta_{17} - 88 \beta_{18} + 88 \beta_{19} ) q^{95} + ( -12288 - 4096 \beta_{1} - 4096 \beta_{2} + 12288 \beta_{6} ) q^{96} + ( 9198 - 383 \beta_{1} - 383 \beta_{2} - 550 \beta_{3} + 355 \beta_{4} - 27 \beta_{5} - 9198 \beta_{6} + 1070 \beta_{8} - 355 \beta_{9} - 202 \beta_{11} + 550 \beta_{12} - 92 \beta_{13} - 27 \beta_{15} - 46 \beta_{17} - 92 \beta_{18} ) q^{97} + ( -110420 + 11532 \beta_{1} - 11532 \beta_{2} + 276 \beta_{3} - 296 \beta_{4} - 68 \beta_{5} - 110420 \beta_{6} - 2024 \beta_{7} - 296 \beta_{9} + 128 \beta_{10} + 276 \beta_{12} - 48 \beta_{14} + 68 \beta_{15} + 88 \beta_{16} + 88 \beta_{19} ) q^{98} + ( -1202 \beta_{1} + 179516 \beta_{6} - 4765 \beta_{7} + 4765 \beta_{8} + 2151 \beta_{9} - 111 \beta_{10} - 111 \beta_{11} - 44 \beta_{12} + 89 \beta_{13} + 89 \beta_{14} + 289 \beta_{15} + 78 \beta_{16} + 78 \beta_{17} + 211 \beta_{18} - 211 \beta_{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 80q^{2} + 60q^{5} + 256q^{6} + 104q^{7} + 2560q^{8} - 6472q^{9} + O(q^{10}) \) \( 20q - 80q^{2} + 60q^{5} + 256q^{6} + 104q^{7} + 2560q^{8} - 6472q^{9} - 480q^{10} - 2048q^{12} + 1560q^{13} - 416q^{14} - 2136q^{15} - 20480q^{16} + 16000q^{17} + 25888q^{18} + 3838q^{19} + 1920q^{20} + 8928q^{22} - 6478q^{23} + 8192q^{24} - 12480q^{26} - 1964q^{29} + 117662q^{31} + 81920q^{32} - 92624q^{33} - 128000q^{34} - 104456q^{35} - 17618q^{37} - 30704q^{38} + 121012q^{39} + 213472q^{42} + 65582q^{43} - 71424q^{44} - 466848q^{45} + 51824q^{46} - 168176q^{47} + 563124q^{49} - 379904q^{50} + 560888q^{51} + 49920q^{52} + 561604q^{53} - 139120q^{54} - 1395304q^{55} + 13312q^{56} + 631036q^{57} - 376510q^{59} + 68352q^{60} + 836700q^{61} - 275908q^{63} + 370496q^{66} + 512000q^{68} - 1616748q^{69} + 835648q^{70} - 94584q^{71} - 828416q^{72} + 133200q^{74} + 20340q^{75} + 122816q^{76} - 2525594q^{79} - 61440q^{80} + 2933572q^{81} - 1816480q^{82} + 715304q^{83} - 1707776q^{84} - 524656q^{86} + 900256q^{87} + 285696q^{88} - 952068q^{89} + 3734784q^{90} - 1351840q^{91} - 207296q^{92} + 580320q^{93} + 672704q^{94} - 262144q^{96} + 178132q^{97} - 2252496q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} + 10424 x^{18} + 44844916 x^{16} + 103219343022 x^{14} + 138101513095620 x^{12} + 109787124347520192 x^{10} + 51172837440825906441 x^{8} + 13199761586736849750156 x^{6} + 1629475196758519705300656 x^{4} + 65381450766316829487245376 x^{2} + 736627437450607950694158336\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(\)\(76\!\cdots\!91\)\( \nu^{18} + \)\(75\!\cdots\!88\)\( \nu^{16} + \)\(30\!\cdots\!88\)\( \nu^{14} + \)\(62\!\cdots\!54\)\( \nu^{12} + \)\(71\!\cdots\!96\)\( \nu^{10} + \)\(45\!\cdots\!76\)\( \nu^{8} + \)\(14\!\cdots\!55\)\( \nu^{6} + \)\(21\!\cdots\!36\)\( \nu^{4} + \)\(86\!\cdots\!60\)\( \nu^{2} + \)\(12\!\cdots\!56\)\(\)\()/ \)\(58\!\cdots\!60\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(14\!\cdots\!11\)\( \nu^{18} - \)\(12\!\cdots\!88\)\( \nu^{16} - \)\(43\!\cdots\!48\)\( \nu^{14} - \)\(75\!\cdots\!34\)\( \nu^{12} - \)\(64\!\cdots\!16\)\( \nu^{10} - \)\(21\!\cdots\!56\)\( \nu^{8} + \)\(27\!\cdots\!25\)\( \nu^{6} + \)\(34\!\cdots\!04\)\( \nu^{4} + \)\(65\!\cdots\!00\)\( \nu^{2} + \)\(49\!\cdots\!44\)\(\)\()/ \)\(28\!\cdots\!20\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(11\!\cdots\!83\)\( \nu^{18} + \)\(14\!\cdots\!44\)\( \nu^{16} + \)\(69\!\cdots\!84\)\( \nu^{14} + \)\(16\!\cdots\!02\)\( \nu^{12} + \)\(21\!\cdots\!08\)\( \nu^{10} + \)\(14\!\cdots\!68\)\( \nu^{8} + \)\(46\!\cdots\!15\)\( \nu^{6} + \)\(61\!\cdots\!88\)\( \nu^{4} + \)\(87\!\cdots\!20\)\( \nu^{2} - \)\(62\!\cdots\!72\)\(\)\()/ \)\(20\!\cdots\!80\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(11\!\cdots\!71\)\( \nu^{18} + \)\(10\!\cdots\!88\)\( \nu^{16} + \)\(43\!\cdots\!48\)\( \nu^{14} + \)\(89\!\cdots\!74\)\( \nu^{12} + \)\(10\!\cdots\!36\)\( \nu^{10} + \)\(69\!\cdots\!96\)\( \nu^{8} + \)\(25\!\cdots\!15\)\( \nu^{6} + \)\(46\!\cdots\!36\)\( \nu^{4} + \)\(35\!\cdots\!80\)\( \nu^{2} + \)\(70\!\cdots\!36\)\(\)\()/ \)\(10\!\cdots\!40\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(10\!\cdots\!93\)\( \nu^{19} - \)\(10\!\cdots\!04\)\( \nu^{17} - \)\(41\!\cdots\!84\)\( \nu^{15} - \)\(87\!\cdots\!42\)\( \nu^{13} - \)\(10\!\cdots\!28\)\( \nu^{11} - \)\(68\!\cdots\!88\)\( \nu^{9} - \)\(24\!\cdots\!05\)\( \nu^{7} - \)\(42\!\cdots\!68\)\( \nu^{5} - \)\(31\!\cdots\!20\)\( \nu^{3} - \)\(13\!\cdots\!88\)\( \nu\)\()/ \)\(36\!\cdots\!80\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(72\!\cdots\!19\)\( \nu^{19} + \)\(40\!\cdots\!30\)\( \nu^{18} - \)\(61\!\cdots\!24\)\( \nu^{17} + \)\(21\!\cdots\!44\)\( \nu^{16} - \)\(18\!\cdots\!12\)\( \nu^{15} + \)\(55\!\cdots\!80\)\( \nu^{14} - \)\(21\!\cdots\!22\)\( \nu^{13} - \)\(18\!\cdots\!80\)\( \nu^{12} + \)\(54\!\cdots\!76\)\( \nu^{11} - \)\(48\!\cdots\!20\)\( \nu^{10} + \)\(37\!\cdots\!52\)\( \nu^{9} - \)\(54\!\cdots\!64\)\( \nu^{8} + \)\(34\!\cdots\!93\)\( \nu^{7} - \)\(30\!\cdots\!18\)\( \nu^{6} + \)\(12\!\cdots\!20\)\( \nu^{5} - \)\(81\!\cdots\!36\)\( \nu^{4} + \)\(19\!\cdots\!36\)\( \nu^{3} - \)\(93\!\cdots\!64\)\( \nu^{2} + \)\(58\!\cdots\!88\)\( \nu - \)\(26\!\cdots\!56\)\(\)\()/ \)\(70\!\cdots\!24\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(72\!\cdots\!19\)\( \nu^{19} + \)\(40\!\cdots\!30\)\( \nu^{18} + \)\(61\!\cdots\!24\)\( \nu^{17} + \)\(21\!\cdots\!44\)\( \nu^{16} + \)\(18\!\cdots\!12\)\( \nu^{15} + \)\(55\!\cdots\!80\)\( \nu^{14} + \)\(21\!\cdots\!22\)\( \nu^{13} - \)\(18\!\cdots\!80\)\( \nu^{12} - \)\(54\!\cdots\!76\)\( \nu^{11} - \)\(48\!\cdots\!20\)\( \nu^{10} - \)\(37\!\cdots\!52\)\( \nu^{9} - \)\(54\!\cdots\!64\)\( \nu^{8} - \)\(34\!\cdots\!93\)\( \nu^{7} - \)\(30\!\cdots\!18\)\( \nu^{6} - \)\(12\!\cdots\!20\)\( \nu^{5} - \)\(81\!\cdots\!36\)\( \nu^{4} - \)\(19\!\cdots\!36\)\( \nu^{3} - \)\(93\!\cdots\!64\)\( \nu^{2} - \)\(58\!\cdots\!88\)\( \nu - \)\(26\!\cdots\!56\)\(\)\()/ \)\(70\!\cdots\!24\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(91\!\cdots\!79\)\( \nu^{19} + \)\(82\!\cdots\!72\)\( \nu^{17} + \)\(31\!\cdots\!92\)\( \nu^{15} + \)\(70\!\cdots\!46\)\( \nu^{13} + \)\(10\!\cdots\!24\)\( \nu^{11} + \)\(94\!\cdots\!84\)\( \nu^{9} + \)\(51\!\cdots\!75\)\( \nu^{7} + \)\(14\!\cdots\!04\)\( \nu^{5} + \)\(19\!\cdots\!40\)\( \nu^{3} + \)\(66\!\cdots\!04\)\( \nu\)\()/ \)\(52\!\cdots\!80\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(49\!\cdots\!21\)\( \nu^{19} - \)\(67\!\cdots\!32\)\( \nu^{18} + \)\(50\!\cdots\!28\)\( \nu^{17} - \)\(74\!\cdots\!76\)\( \nu^{16} + \)\(20\!\cdots\!28\)\( \nu^{15} - \)\(31\!\cdots\!76\)\( \nu^{14} + \)\(39\!\cdots\!94\)\( \nu^{13} - \)\(63\!\cdots\!88\)\( \nu^{12} + \)\(40\!\cdots\!96\)\( \nu^{11} - \)\(62\!\cdots\!72\)\( \nu^{10} + \)\(17\!\cdots\!96\)\( \nu^{9} - \)\(23\!\cdots\!92\)\( \nu^{8} + \)\(99\!\cdots\!05\)\( \nu^{7} + \)\(80\!\cdots\!80\)\( \nu^{6} - \)\(13\!\cdots\!44\)\( \nu^{5} + \)\(22\!\cdots\!48\)\( \nu^{4} - \)\(20\!\cdots\!60\)\( \nu^{3} + \)\(28\!\cdots\!80\)\( \nu^{2} + \)\(46\!\cdots\!56\)\( \nu + \)\(10\!\cdots\!88\)\(\)\()/ \)\(52\!\cdots\!80\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(49\!\cdots\!21\)\( \nu^{19} + \)\(67\!\cdots\!32\)\( \nu^{18} + \)\(50\!\cdots\!28\)\( \nu^{17} + \)\(74\!\cdots\!76\)\( \nu^{16} + \)\(20\!\cdots\!28\)\( \nu^{15} + \)\(31\!\cdots\!76\)\( \nu^{14} + \)\(39\!\cdots\!94\)\( \nu^{13} + \)\(63\!\cdots\!88\)\( \nu^{12} + \)\(40\!\cdots\!96\)\( \nu^{11} + \)\(62\!\cdots\!72\)\( \nu^{10} + \)\(17\!\cdots\!96\)\( \nu^{9} + \)\(23\!\cdots\!92\)\( \nu^{8} + \)\(99\!\cdots\!05\)\( \nu^{7} - \)\(80\!\cdots\!80\)\( \nu^{6} - \)\(13\!\cdots\!44\)\( \nu^{5} - \)\(22\!\cdots\!48\)\( \nu^{4} - \)\(20\!\cdots\!60\)\( \nu^{3} - \)\(28\!\cdots\!80\)\( \nu^{2} + \)\(46\!\cdots\!56\)\( \nu - \)\(10\!\cdots\!88\)\(\)\()/ \)\(52\!\cdots\!80\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(14\!\cdots\!37\)\( \nu^{19} - \)\(15\!\cdots\!36\)\( \nu^{17} - \)\(64\!\cdots\!76\)\( \nu^{15} - \)\(14\!\cdots\!98\)\( \nu^{13} - \)\(18\!\cdots\!92\)\( \nu^{11} - \)\(13\!\cdots\!52\)\( \nu^{9} - \)\(55\!\cdots\!65\)\( \nu^{7} - \)\(12\!\cdots\!12\)\( \nu^{5} - \)\(13\!\cdots\!40\)\( \nu^{3} - \)\(54\!\cdots\!72\)\( \nu\)\()/ \)\(10\!\cdots\!60\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(11\!\cdots\!81\)\( \nu^{19} + \)\(40\!\cdots\!06\)\( \nu^{18} - \)\(12\!\cdots\!88\)\( \nu^{17} + \)\(38\!\cdots\!48\)\( \nu^{16} - \)\(54\!\cdots\!68\)\( \nu^{15} + \)\(14\!\cdots\!08\)\( \nu^{14} - \)\(12\!\cdots\!54\)\( \nu^{13} + \)\(28\!\cdots\!84\)\( \nu^{12} - \)\(17\!\cdots\!56\)\( \nu^{11} + \)\(31\!\cdots\!16\)\( \nu^{10} - \)\(13\!\cdots\!76\)\( \nu^{9} + \)\(19\!\cdots\!16\)\( \nu^{8} - \)\(62\!\cdots\!05\)\( \nu^{7} + \)\(57\!\cdots\!50\)\( \nu^{6} - \)\(15\!\cdots\!96\)\( \nu^{5} + \)\(60\!\cdots\!56\)\( \nu^{4} - \)\(16\!\cdots\!20\)\( \nu^{3} - \)\(23\!\cdots\!00\)\( \nu^{2} - \)\(42\!\cdots\!76\)\( \nu - \)\(13\!\cdots\!64\)\(\)\()/ \)\(35\!\cdots\!20\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(11\!\cdots\!81\)\( \nu^{19} - \)\(40\!\cdots\!06\)\( \nu^{18} - \)\(12\!\cdots\!88\)\( \nu^{17} - \)\(38\!\cdots\!48\)\( \nu^{16} - \)\(54\!\cdots\!68\)\( \nu^{15} - \)\(14\!\cdots\!08\)\( \nu^{14} - \)\(12\!\cdots\!54\)\( \nu^{13} - \)\(28\!\cdots\!84\)\( \nu^{12} - \)\(17\!\cdots\!56\)\( \nu^{11} - \)\(31\!\cdots\!16\)\( \nu^{10} - \)\(13\!\cdots\!76\)\( \nu^{9} - \)\(19\!\cdots\!16\)\( \nu^{8} - \)\(62\!\cdots\!05\)\( \nu^{7} - \)\(57\!\cdots\!50\)\( \nu^{6} - \)\(15\!\cdots\!96\)\( \nu^{5} - \)\(60\!\cdots\!56\)\( \nu^{4} - \)\(16\!\cdots\!20\)\( \nu^{3} + \)\(23\!\cdots\!00\)\( \nu^{2} - \)\(42\!\cdots\!76\)\( \nu + \)\(13\!\cdots\!64\)\(\)\()/ \)\(35\!\cdots\!20\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(95\!\cdots\!11\)\( \nu^{19} + \)\(98\!\cdots\!68\)\( \nu^{17} + \)\(41\!\cdots\!68\)\( \nu^{15} + \)\(94\!\cdots\!14\)\( \nu^{13} + \)\(12\!\cdots\!16\)\( \nu^{11} + \)\(98\!\cdots\!56\)\( \nu^{9} + \)\(45\!\cdots\!15\)\( \nu^{7} + \)\(11\!\cdots\!16\)\( \nu^{5} + \)\(13\!\cdots\!20\)\( \nu^{3} + \)\(31\!\cdots\!56\)\( \nu\)\()/ \)\(26\!\cdots\!40\)\( \)
\(\beta_{16}\)\(=\)\((\)\(\)\(20\!\cdots\!04\)\( \nu^{19} - \)\(72\!\cdots\!49\)\( \nu^{18} + \)\(20\!\cdots\!72\)\( \nu^{17} - \)\(58\!\cdots\!92\)\( \nu^{16} + \)\(84\!\cdots\!92\)\( \nu^{15} - \)\(19\!\cdots\!92\)\( \nu^{14} + \)\(18\!\cdots\!56\)\( \nu^{13} - \)\(36\!\cdots\!66\)\( \nu^{12} + \)\(24\!\cdots\!04\)\( \nu^{11} - \)\(48\!\cdots\!44\)\( \nu^{10} + \)\(18\!\cdots\!64\)\( \nu^{9} - \)\(44\!\cdots\!64\)\( \nu^{8} + \)\(88\!\cdots\!00\)\( \nu^{7} - \)\(26\!\cdots\!05\)\( \nu^{6} + \)\(22\!\cdots\!84\)\( \nu^{5} - \)\(83\!\cdots\!44\)\( \nu^{4} + \)\(27\!\cdots\!60\)\( \nu^{3} - \)\(12\!\cdots\!80\)\( \nu^{2} + \)\(70\!\cdots\!04\)\( \nu - \)\(31\!\cdots\!24\)\(\)\()/ \)\(52\!\cdots\!80\)\( \)
\(\beta_{17}\)\(=\)\((\)\(\)\(20\!\cdots\!04\)\( \nu^{19} + \)\(72\!\cdots\!49\)\( \nu^{18} + \)\(20\!\cdots\!72\)\( \nu^{17} + \)\(58\!\cdots\!92\)\( \nu^{16} + \)\(84\!\cdots\!92\)\( \nu^{15} + \)\(19\!\cdots\!92\)\( \nu^{14} + \)\(18\!\cdots\!56\)\( \nu^{13} + \)\(36\!\cdots\!66\)\( \nu^{12} + \)\(24\!\cdots\!04\)\( \nu^{11} + \)\(48\!\cdots\!44\)\( \nu^{10} + \)\(18\!\cdots\!64\)\( \nu^{9} + \)\(44\!\cdots\!64\)\( \nu^{8} + \)\(88\!\cdots\!00\)\( \nu^{7} + \)\(26\!\cdots\!05\)\( \nu^{6} + \)\(22\!\cdots\!84\)\( \nu^{5} + \)\(83\!\cdots\!44\)\( \nu^{4} + \)\(27\!\cdots\!60\)\( \nu^{3} + \)\(12\!\cdots\!80\)\( \nu^{2} + \)\(70\!\cdots\!04\)\( \nu + \)\(31\!\cdots\!24\)\(\)\()/ \)\(52\!\cdots\!80\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-\)\(16\!\cdots\!97\)\( \nu^{19} + \)\(43\!\cdots\!14\)\( \nu^{18} - \)\(16\!\cdots\!76\)\( \nu^{17} + \)\(34\!\cdots\!12\)\( \nu^{16} - \)\(71\!\cdots\!56\)\( \nu^{15} + \)\(91\!\cdots\!92\)\( \nu^{14} - \)\(15\!\cdots\!58\)\( \nu^{13} + \)\(71\!\cdots\!76\)\( \nu^{12} - \)\(20\!\cdots\!92\)\( \nu^{11} - \)\(10\!\cdots\!76\)\( \nu^{10} - \)\(15\!\cdots\!32\)\( \nu^{9} - \)\(24\!\cdots\!16\)\( \nu^{8} - \)\(62\!\cdots\!65\)\( \nu^{7} - \)\(18\!\cdots\!90\)\( \nu^{6} - \)\(13\!\cdots\!72\)\( \nu^{5} - \)\(60\!\cdots\!76\)\( \nu^{4} - \)\(12\!\cdots\!80\)\( \nu^{3} - \)\(72\!\cdots\!40\)\( \nu^{2} - \)\(16\!\cdots\!52\)\( \nu - \)\(10\!\cdots\!36\)\(\)\()/ \)\(35\!\cdots\!20\)\( \)
\(\beta_{19}\)\(=\)\((\)\(\)\(16\!\cdots\!97\)\( \nu^{19} + \)\(43\!\cdots\!14\)\( \nu^{18} + \)\(16\!\cdots\!76\)\( \nu^{17} + \)\(34\!\cdots\!12\)\( \nu^{16} + \)\(71\!\cdots\!56\)\( \nu^{15} + \)\(91\!\cdots\!92\)\( \nu^{14} + \)\(15\!\cdots\!58\)\( \nu^{13} + \)\(71\!\cdots\!76\)\( \nu^{12} + \)\(20\!\cdots\!92\)\( \nu^{11} - \)\(10\!\cdots\!76\)\( \nu^{10} + \)\(15\!\cdots\!32\)\( \nu^{9} - \)\(24\!\cdots\!16\)\( \nu^{8} + \)\(62\!\cdots\!65\)\( \nu^{7} - \)\(18\!\cdots\!90\)\( \nu^{6} + \)\(13\!\cdots\!72\)\( \nu^{5} - \)\(60\!\cdots\!76\)\( \nu^{4} + \)\(12\!\cdots\!80\)\( \nu^{3} - \)\(72\!\cdots\!40\)\( \nu^{2} + \)\(16\!\cdots\!52\)\( \nu - \)\(10\!\cdots\!36\)\(\)\()/ \)\(35\!\cdots\!20\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} + \beta_{7} + \beta_{3} + 3 \beta_{2} - 1043\)
\(\nu^{3}\)\(=\)\(-3 \beta_{17} - 3 \beta_{16} + 9 \beta_{15} + 3 \beta_{14} + 3 \beta_{13} - \beta_{12} + 3 \beta_{11} + 3 \beta_{10} - 6 \beta_{9} - 29 \beta_{8} + 29 \beta_{7} - 3379 \beta_{6} - 1809 \beta_{1}\)
\(\nu^{4}\)\(=\)\(18 \beta_{19} + 18 \beta_{18} - 24 \beta_{17} + 24 \beta_{16} - 39 \beta_{14} + 39 \beta_{13} + 75 \beta_{11} - 75 \beta_{10} - 4099 \beta_{8} - 4099 \beta_{7} - 15 \beta_{5} - 477 \beta_{4} - 2365 \beta_{3} - 10515 \beta_{2} + 1898879\)
\(\nu^{5}\)\(=\)\(-1797 \beta_{19} + 1797 \beta_{18} + 8067 \beta_{17} + 8067 \beta_{16} - 23004 \beta_{15} - 9147 \beta_{14} - 9147 \beta_{13} - 8093 \beta_{12} - 10986 \beta_{11} - 10986 \beta_{10} + 5850 \beta_{9} + 102392 \beta_{8} - 102392 \beta_{7} + 11723761 \beta_{6} + 3805581 \beta_{1}\)
\(\nu^{6}\)\(=\)\(31029 \beta_{19} + 31029 \beta_{18} + 103632 \beta_{17} - 103632 \beta_{16} + 188199 \beta_{14} - 188199 \beta_{13} - 338142 \beta_{11} + 338142 \beta_{10} + 11186593 \beta_{8} + 11186593 \beta_{7} + 44643 \beta_{5} + 1727424 \beta_{4} + 5368057 \beta_{3} + 37157157 \beta_{2} - 4000451735\)
\(\nu^{7}\)\(=\)\(8112618 \beta_{19} - 8112618 \beta_{18} - 18955965 \beta_{17} - 18955965 \beta_{16} + 49055001 \beta_{15} + 24471573 \beta_{14} + 24471573 \beta_{13} + 44889725 \beta_{12} + 32401680 \beta_{11} + 32401680 \beta_{10} + 13625757 \beta_{9} - 319423280 \beta_{8} + 319423280 \beta_{7} - 40531196137 \beta_{6} - 8498959449 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-272663616 \beta_{19} - 272663616 \beta_{18} - 328628157 \beta_{17} + 328628157 \beta_{16} - 661501563 \beta_{14} + 661501563 \beta_{13} + 1150856904 \beta_{11} - 1150856904 \beta_{10} - 28750839679 \beta_{8} - 28750839679 \beta_{7} - 82512018 \beta_{5} - 4992940548 \beta_{4} - 12341515741 \beta_{3} - 122293466457 \beta_{2} + 8937896960033\)
\(\nu^{9}\)\(=\)\(-27164156934 \beta_{19} + 27164156934 \beta_{18} + 44201042085 \beta_{17} + 44201042085 \beta_{16} - 101912592981 \beta_{15} - 63835579071 \beta_{14} - 63835579071 \beta_{13} - 167967603047 \beta_{12} - 89423528991 \beta_{11} - 89423528991 \beta_{10} - 93272846256 \beta_{9} + 939653684939 \beta_{8} - 939653684939 \beta_{7} + 131664977919511 \beta_{6} + 19628801732133 \beta_{1}\)
\(\nu^{10}\)\(=\)\(1075174546878 \beta_{19} + 1075174546878 \beta_{18} + 963448531332 \beta_{17} - 963448531332 \beta_{16} + 2055488766627 \beta_{14} - 2055488766627 \beta_{13} - 3501428636451 \beta_{11} + 3501428636451 \beta_{10} + 73315042662037 \beta_{8} + 73315042662037 \beta_{7} + 55194702081 \beta_{5} + 13502332838619 \beta_{4} + 28936615308073 \beta_{3} + 378406152214887 \beta_{2} - 20648883753801563\)
\(\nu^{11}\)\(=\)\(80972935777833 \beta_{19} - 80972935777833 \beta_{18} - 104672239025007 \beta_{17} - 104672239025007 \beta_{16} + 214035333629814 \beta_{15} + 165393061801167 \beta_{14} + 165393061801167 \beta_{13} + 546860535250493 \beta_{12} + 240175499304408 \beta_{11} + 240175499304408 \beta_{10} + 352002255434658 \beta_{9} - 2666596457774456 \beta_{8} + 2666596457774456 \beta_{7} - 404440559501997817 \beta_{6} - 46468549919651745 \beta_{1}\)
\(\nu^{12}\)\(=\)\(-3463044732642381 \beta_{19} - 3463044732642381 \beta_{18} - 2741660647605186 \beta_{17} + 2741660647605186 \beta_{16} - 6000149260686627 \beta_{14} + 6000149260686627 \beta_{13} + 10064838840458646 \beta_{11} - 10064838840458646 \beta_{10} - 187408426350768577 \beta_{8} - 187408426350768577 \beta_{7} + 355104636108657 \beta_{5} - 35594502171769566 \beta_{4} - 69167004007536889 \beta_{3} - 1120388272842384549 \beta_{2} + 48900628334423979515\)
\(\nu^{13}\)\(=\)\(-228043230896180736 \beta_{19} + 228043230896180736 \beta_{18} + 252606665375692455 \beta_{17} + 252606665375692455 \beta_{16} - 459485570485847355 \beta_{15} - 427842346386223473 \beta_{14} - 427842346386223473 \beta_{13} - 1658231408948518217 \beta_{12} - 636731110704452694 \beta_{11} - 636731110704452694 \beta_{10} - 1122543766427417307 \beta_{9} + 7393826172178594058 \beta_{8} - 7393826172178594058 \beta_{7} + 1192376099025289713433 \beta_{6} + 112259912855554486041 \beta_{1}\)
\(\nu^{14}\)\(=\)\(10221261171504147336 \beta_{19} + 10221261171504147336 \beta_{18} + 7666935238577502483 \beta_{17} - 7666935238577502483 \beta_{16} + 16899079007535916695 \beta_{14} - 16899079007535916695 \beta_{13} - 28017893249502880584 \beta_{11} + 28017893249502880584 \beta_{10} + 481096910447771870701 \beta_{8} + 481096910447771870701 \beta_{7} - 2252822678908870968 \beta_{5} + 92827860065371914408 \beta_{4} + 168219121141192113169 \beta_{3} + 3215801426505238221621 \beta_{2} - 118181204461916003781593\)
\(\nu^{15}\)\(=\)\(622932879949996352664 \beta_{19} - 622932879949996352664 \beta_{18} - 620310066341045356227 \beta_{17} - 620310066341045356227 \beta_{16} + 1011632529979150116537 \beta_{15} + 1107323647367974960095 \beta_{14} + 1107323647367974960095 \beta_{13} + 4818378498475626298307 \beta_{12} + 1676916095183369436327 \beta_{11} + 1676916095183369436327 \beta_{10} + 3317156429441314608090 \beta_{9} - 20186967476892284716421 \beta_{8} + 20186967476892284716421 \beta_{7} - 3413441978600543899039807 \beta_{6} - 275830819599742583775993 \beta_{1}\)
\(\nu^{16}\)\(=\)\(-28831993899508385180946 \beta_{19} - 28831993899508385180946 \beta_{18} - 21163311044037852945684 \beta_{17} + 21163311044037852945684 \beta_{16} - 46557982882360402099491 \beta_{14} + 46557982882360402099491 \beta_{13} + 76497912258927965054895 \beta_{11} - 76497912258927965054895 \beta_{10} - 1240211058779295012938575 \beta_{8} - 1240211058779295012938575 \beta_{7} + 9118097182656424273053 \beta_{5} - 241047060345975298464333 \beta_{4} - 415295524686721683482341 \beta_{3} - 9027134142010457056757355 \beta_{2} + 290493790319475711004112735\)
\(\nu^{17}\)\(=\)\(-1672634201643342758698521 \beta_{19} + 1672634201643342758698521 \beta_{18} + 1545736553035640392239795 \beta_{17} + 1545736553035640392239795 \beta_{16} - 2284935353415953540493660 \beta_{15} - 2870133654976449692098827 \beta_{14} - 2870133654976449692098827 \beta_{13} - 13616534387379885866630669 \beta_{12} - 4401762594984982572070254 \beta_{11} - 4401762594984982572070254 \beta_{10} - 9405842514856774774135362 \beta_{9} + 54535892235001009237654136 \beta_{8} - 54535892235001009237654136 \beta_{7} + 9565704586283039470053350641 \beta_{6} + 687300694453071596617874661 \beta_{1}\)
\(\nu^{18}\)\(=\)\(79253224367788136109084729 \beta_{19} + 79253224367788136109084729 \beta_{18} + 57815512224729199151697456 \beta_{17} - 57815512224729199151697456 \beta_{16} + 126456785756676787935215871 \beta_{14} - 126456785756676787935215871 \beta_{13} - 206322365279700957859940550 \beta_{11} + 206322365279700957859940550 \beta_{10} + 3208935430882858023369570121 \beta_{8} + 3208935430882858023369570121 \beta_{7} - 31321515325755265313190801 \beta_{5} + 625218323893247649491233740 \beta_{4} + 1038288182009243653091275945 \beta_{3} + 24932530393656215772166346133 \beta_{2} - 724105322728911783079587166871\)
\(\nu^{19}\)\(=\)\(4446996595270712934030347958 \beta_{19} - 4446996595270712934030347958 \beta_{18} - 3897772669203956099706476289 \beta_{17} - 3897772669203956099706476289 \beta_{16} + 5288184179743499591353237005 \beta_{15} + 7452906955507467535561431909 \beta_{14} + 7452906955507467535561431909 \beta_{13} + 37750221781576082485337900621 \beta_{12} + 11536773617708464435400994576 \beta_{11} + 11536773617708464435400994576 \beta_{10} + 26020554696868481699686975197 \beta_{9} - 146244676183187106955296708392 \beta_{8} + 146244676183187106955296708392 \beta_{7} - 26390107620177497769030858076369 \beta_{6} - 1732253418802544925688700697033 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\) \(39\)
\(\chi(n)\) \(\beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
51.2623i
40.9203i
26.0545i
21.6219i
4.34837i
6.30032i
17.1892i
24.1581i
44.9177i
44.9453i
44.9453i
44.9177i
24.1581i
17.1892i
6.30032i
4.34837i
21.6219i
26.0545i
40.9203i
51.2623i
−4.00000 4.00000i 48.2623i 32.0000i 120.036 120.036i −193.049 + 193.049i −229.509 128.000 128.000i −1600.25 −960.285
31.2 −4.00000 4.00000i 37.9203i 32.0000i −19.0526 + 19.0526i −151.681 + 151.681i 96.0580 128.000 128.000i −708.946 152.420
31.3 −4.00000 4.00000i 23.0545i 32.0000i −19.9104 + 19.9104i −92.2182 + 92.2182i 259.314 128.000 128.000i 197.488 159.283
31.4 −4.00000 4.00000i 18.6219i 32.0000i −151.370 + 151.370i −74.4876 + 74.4876i −414.903 128.000 128.000i 382.225 1210.96
31.5 −4.00000 4.00000i 7.34837i 32.0000i 166.447 166.447i 29.3935 29.3935i −272.903 128.000 128.000i 675.001 −1331.57
31.6 −4.00000 4.00000i 9.30032i 32.0000i 9.75519 9.75519i 37.2013 37.2013i −106.968 128.000 128.000i 642.504 −78.0415
31.7 −4.00000 4.00000i 20.1892i 32.0000i 9.27541 9.27541i 80.7569 80.7569i 8.30790 128.000 128.000i 321.395 −74.2032
31.8 −4.00000 4.00000i 27.1581i 32.0000i −159.256 + 159.256i 108.632 108.632i 664.609 128.000 128.000i −8.56259 1274.05
31.9 −4.00000 4.00000i 47.9177i 32.0000i 99.4727 99.4727i 191.671 191.671i 584.317 128.000 128.000i −1567.10 −795.781
31.10 −4.00000 4.00000i 47.9453i 32.0000i −25.3958 + 25.3958i 191.781 191.781i −536.323 128.000 128.000i −1569.75 203.167
43.1 −4.00000 + 4.00000i 47.9453i 32.0000i −25.3958 25.3958i 191.781 + 191.781i −536.323 128.000 + 128.000i −1569.75 203.167
43.2 −4.00000 + 4.00000i 47.9177i 32.0000i 99.4727 + 99.4727i 191.671 + 191.671i 584.317 128.000 + 128.000i −1567.10 −795.781
43.3 −4.00000 + 4.00000i 27.1581i 32.0000i −159.256 159.256i 108.632 + 108.632i 664.609 128.000 + 128.000i −8.56259 1274.05
43.4 −4.00000 + 4.00000i 20.1892i 32.0000i 9.27541 + 9.27541i 80.7569 + 80.7569i 8.30790 128.000 + 128.000i 321.395 −74.2032
43.5 −4.00000 + 4.00000i 9.30032i 32.0000i 9.75519 + 9.75519i 37.2013 + 37.2013i −106.968 128.000 + 128.000i 642.504 −78.0415
43.6 −4.00000 + 4.00000i 7.34837i 32.0000i 166.447 + 166.447i 29.3935 + 29.3935i −272.903 128.000 + 128.000i 675.001 −1331.57
43.7 −4.00000 + 4.00000i 18.6219i 32.0000i −151.370 151.370i −74.4876 74.4876i −414.903 128.000 + 128.000i 382.225 1210.96
43.8 −4.00000 + 4.00000i 23.0545i 32.0000i −19.9104 19.9104i −92.2182 92.2182i 259.314 128.000 + 128.000i 197.488 159.283
43.9 −4.00000 + 4.00000i 37.9203i 32.0000i −19.0526 19.0526i −151.681 151.681i 96.0580 128.000 + 128.000i −708.946 152.420
43.10 −4.00000 + 4.00000i 48.2623i 32.0000i 120.036 + 120.036i −193.049 193.049i −229.509 128.000 + 128.000i −1600.25 −960.285
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.7.d.b 20
37.d odd 4 1 inner 74.7.d.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.7.d.b 20 1.a even 1 1 trivial
74.7.d.b 20 37.d odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(14\!\cdots\!08\)\( T_{3}^{12} + \)\(12\!\cdots\!54\)\( T_{3}^{10} + \)\(59\!\cdots\!65\)\( T_{3}^{8} + \)\(17\!\cdots\!50\)\( T_{3}^{6} + \)\(26\!\cdots\!13\)\( T_{3}^{4} + \)\(18\!\cdots\!88\)\( T_{3}^{2} + \)\(45\!\cdots\!64\)\( \)">\(T_{3}^{20} + \cdots\) acting on \(S_{7}^{\mathrm{new}}(74, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 32 + 8 T + T^{2} )^{10} \)
$3$ \( \)\(45\!\cdots\!64\)\( + \)\(18\!\cdots\!88\)\( T^{2} + \)\(26\!\cdots\!13\)\( T^{4} + \)\(17\!\cdots\!50\)\( T^{6} + 59375181221765926065 T^{8} + 120121546987515654 T^{10} + 146101349611908 T^{12} + 106976322102 T^{14} + 45811969 T^{16} + 10526 T^{18} + T^{20} \)
$5$ \( \)\(17\!\cdots\!00\)\( - \)\(14\!\cdots\!00\)\( T + \)\(56\!\cdots\!00\)\( T^{2} + \)\(44\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!00\)\( T^{4} - \)\(57\!\cdots\!00\)\( T^{5} + \)\(56\!\cdots\!00\)\( T^{6} + \)\(56\!\cdots\!00\)\( T^{7} + \)\(23\!\cdots\!00\)\( T^{8} + \)\(35\!\cdots\!00\)\( T^{9} + \)\(16\!\cdots\!00\)\( T^{10} - \)\(41\!\cdots\!00\)\( T^{11} + 4758852621863489025 T^{12} - 516798454310580 T^{13} + 9995025145608 T^{14} - 300785622024 T^{15} + 4501566271 T^{16} - 318456 T^{17} + 1800 T^{18} - 60 T^{19} + T^{20} \)
$7$ \( ( -\)\(11\!\cdots\!40\)\( + \)\(13\!\cdots\!72\)\( T + 74412597135075993708 T^{2} - 1660362747276316476 T^{3} - 8853805137498814 T^{4} + 24993484541190 T^{5} + 159846351707 T^{6} - 46133022 T^{7} - 727674 T^{8} - 52 T^{9} + T^{10} )^{2} \)
$11$ \( \)\(19\!\cdots\!00\)\( + \)\(48\!\cdots\!76\)\( T^{2} + \)\(10\!\cdots\!21\)\( T^{4} + \)\(40\!\cdots\!66\)\( T^{6} + \)\(70\!\cdots\!69\)\( T^{8} + \)\(66\!\cdots\!46\)\( T^{10} + \)\(36\!\cdots\!72\)\( T^{12} + \)\(11\!\cdots\!70\)\( T^{14} + 227561114269657 T^{16} + 23426422 T^{18} + T^{20} \)
$13$ \( \)\(16\!\cdots\!56\)\( - \)\(33\!\cdots\!32\)\( T + \)\(34\!\cdots\!52\)\( T^{2} - \)\(11\!\cdots\!12\)\( T^{3} + \)\(20\!\cdots\!08\)\( T^{4} - \)\(25\!\cdots\!68\)\( T^{5} + \)\(36\!\cdots\!72\)\( T^{6} - \)\(65\!\cdots\!96\)\( T^{7} + \)\(10\!\cdots\!84\)\( T^{8} - \)\(12\!\cdots\!24\)\( T^{9} + \)\(94\!\cdots\!48\)\( T^{10} - \)\(49\!\cdots\!68\)\( T^{11} + \)\(16\!\cdots\!05\)\( T^{12} - \)\(35\!\cdots\!72\)\( T^{13} + \)\(10\!\cdots\!28\)\( T^{14} - 588245058681844204 T^{15} + 224432625649231 T^{16} - 28941605424 T^{17} + 1216800 T^{18} - 1560 T^{19} + T^{20} \)
$17$ \( \)\(88\!\cdots\!64\)\( - \)\(31\!\cdots\!48\)\( T + \)\(55\!\cdots\!68\)\( T^{2} - \)\(16\!\cdots\!36\)\( T^{3} + \)\(45\!\cdots\!96\)\( T^{4} - \)\(11\!\cdots\!60\)\( T^{5} + \)\(13\!\cdots\!00\)\( T^{6} - \)\(91\!\cdots\!80\)\( T^{7} + \)\(40\!\cdots\!20\)\( T^{8} - \)\(12\!\cdots\!92\)\( T^{9} + \)\(29\!\cdots\!24\)\( T^{10} - \)\(71\!\cdots\!08\)\( T^{11} + \)\(21\!\cdots\!88\)\( T^{12} - \)\(57\!\cdots\!04\)\( T^{13} + \)\(11\!\cdots\!08\)\( T^{14} - \)\(13\!\cdots\!76\)\( T^{15} + 10595162920308356 T^{16} - 725609529096 T^{17} + 128000000 T^{18} - 16000 T^{19} + T^{20} \)
$19$ \( \)\(83\!\cdots\!00\)\( + \)\(73\!\cdots\!20\)\( T + \)\(32\!\cdots\!32\)\( T^{2} - \)\(73\!\cdots\!48\)\( T^{3} + \)\(11\!\cdots\!76\)\( T^{4} + \)\(15\!\cdots\!04\)\( T^{5} + \)\(14\!\cdots\!56\)\( T^{6} - \)\(84\!\cdots\!60\)\( T^{7} + \)\(44\!\cdots\!36\)\( T^{8} + \)\(37\!\cdots\!96\)\( T^{9} + \)\(32\!\cdots\!36\)\( T^{10} - \)\(15\!\cdots\!68\)\( T^{11} + \)\(45\!\cdots\!88\)\( T^{12} + \)\(17\!\cdots\!16\)\( T^{13} + \)\(16\!\cdots\!64\)\( T^{14} - 74368528795906304752 T^{15} + 15975104302679840 T^{16} - 6263225344 T^{17} + 7365122 T^{18} - 3838 T^{19} + T^{20} \)
$23$ \( \)\(90\!\cdots\!00\)\( - \)\(90\!\cdots\!80\)\( T + \)\(44\!\cdots\!28\)\( T^{2} - \)\(11\!\cdots\!08\)\( T^{3} + \)\(17\!\cdots\!64\)\( T^{4} - \)\(13\!\cdots\!92\)\( T^{5} + \)\(11\!\cdots\!96\)\( T^{6} - \)\(23\!\cdots\!04\)\( T^{7} + \)\(42\!\cdots\!00\)\( T^{8} - \)\(26\!\cdots\!76\)\( T^{9} + \)\(80\!\cdots\!94\)\( T^{10} - \)\(82\!\cdots\!14\)\( T^{11} + \)\(29\!\cdots\!97\)\( T^{12} - \)\(15\!\cdots\!64\)\( T^{13} + \)\(22\!\cdots\!06\)\( T^{14} + \)\(14\!\cdots\!34\)\( T^{15} + 85373302682543863 T^{16} - 2497688606240 T^{17} + 20982242 T^{18} + 6478 T^{19} + T^{20} \)
$29$ \( \)\(10\!\cdots\!36\)\( - \)\(32\!\cdots\!96\)\( T + \)\(47\!\cdots\!28\)\( T^{2} - \)\(38\!\cdots\!04\)\( T^{3} + \)\(21\!\cdots\!44\)\( T^{4} - \)\(13\!\cdots\!92\)\( T^{5} + \)\(15\!\cdots\!28\)\( T^{6} - \)\(12\!\cdots\!68\)\( T^{7} + \)\(60\!\cdots\!96\)\( T^{8} - \)\(12\!\cdots\!44\)\( T^{9} + \)\(23\!\cdots\!28\)\( T^{10} - \)\(19\!\cdots\!40\)\( T^{11} + \)\(12\!\cdots\!45\)\( T^{12} - \)\(15\!\cdots\!48\)\( T^{13} + \)\(59\!\cdots\!68\)\( T^{14} - \)\(41\!\cdots\!32\)\( T^{15} + 6486996540413129483 T^{16} - 37263697203960 T^{17} + 1928648 T^{18} + 1964 T^{19} + T^{20} \)
$31$ \( \)\(39\!\cdots\!24\)\( + \)\(58\!\cdots\!60\)\( T + \)\(43\!\cdots\!00\)\( T^{2} - \)\(27\!\cdots\!84\)\( T^{3} - \)\(46\!\cdots\!92\)\( T^{4} - \)\(54\!\cdots\!36\)\( T^{5} + \)\(19\!\cdots\!32\)\( T^{6} - \)\(24\!\cdots\!12\)\( T^{7} + \)\(17\!\cdots\!20\)\( T^{8} - \)\(80\!\cdots\!04\)\( T^{9} + \)\(24\!\cdots\!90\)\( T^{10} - \)\(49\!\cdots\!22\)\( T^{11} + \)\(10\!\cdots\!61\)\( T^{12} - \)\(31\!\cdots\!76\)\( T^{13} + \)\(93\!\cdots\!06\)\( T^{14} - \)\(16\!\cdots\!90\)\( T^{15} + 18629777650006423631 T^{16} - 237724191967904 T^{17} + 6922173122 T^{18} - 117662 T^{19} + T^{20} \)
$37$ \( \)\(12\!\cdots\!01\)\( + \)\(84\!\cdots\!02\)\( T + \)\(11\!\cdots\!78\)\( T^{2} + \)\(22\!\cdots\!02\)\( T^{3} + \)\(40\!\cdots\!93\)\( T^{4} + \)\(19\!\cdots\!08\)\( T^{5} + \)\(19\!\cdots\!52\)\( T^{6} + \)\(85\!\cdots\!04\)\( T^{7} + \)\(17\!\cdots\!62\)\( T^{8} + \)\(25\!\cdots\!56\)\( T^{9} + \)\(94\!\cdots\!88\)\( T^{10} + \)\(99\!\cdots\!84\)\( T^{11} + \)\(26\!\cdots\!02\)\( T^{12} + \)\(50\!\cdots\!76\)\( T^{13} + \)\(45\!\cdots\!32\)\( T^{14} + \)\(17\!\cdots\!92\)\( T^{15} + 14308014844443901473 T^{16} + 305502811122658 T^{17} + 5990594018 T^{18} + 17618 T^{19} + T^{20} \)
$41$ \( \)\(58\!\cdots\!00\)\( + \)\(16\!\cdots\!56\)\( T^{2} + \)\(17\!\cdots\!29\)\( T^{4} + \)\(88\!\cdots\!58\)\( T^{6} + \)\(22\!\cdots\!85\)\( T^{8} + \)\(28\!\cdots\!46\)\( T^{10} + \)\(17\!\cdots\!60\)\( T^{12} + \)\(49\!\cdots\!70\)\( T^{14} + \)\(67\!\cdots\!13\)\( T^{16} + 42621021158 T^{18} + T^{20} \)
$43$ \( \)\(58\!\cdots\!04\)\( - \)\(14\!\cdots\!92\)\( T + \)\(17\!\cdots\!08\)\( T^{2} - \)\(87\!\cdots\!76\)\( T^{3} + \)\(19\!\cdots\!48\)\( T^{4} + \)\(18\!\cdots\!08\)\( T^{5} + \)\(32\!\cdots\!92\)\( T^{6} - \)\(10\!\cdots\!72\)\( T^{7} + \)\(17\!\cdots\!20\)\( T^{8} + \)\(39\!\cdots\!24\)\( T^{9} + \)\(77\!\cdots\!80\)\( T^{10} - \)\(24\!\cdots\!04\)\( T^{11} + \)\(38\!\cdots\!36\)\( T^{12} + \)\(62\!\cdots\!84\)\( T^{13} + \)\(32\!\cdots\!48\)\( T^{14} - \)\(10\!\cdots\!76\)\( T^{15} + \)\(17\!\cdots\!16\)\( T^{16} + 56025395822096 T^{17} + 2150499362 T^{18} - 65582 T^{19} + T^{20} \)
$47$ \( ( \)\(75\!\cdots\!00\)\( - \)\(74\!\cdots\!80\)\( T + \)\(15\!\cdots\!44\)\( T^{2} + \)\(54\!\cdots\!08\)\( T^{3} - \)\(34\!\cdots\!16\)\( T^{4} + \)\(50\!\cdots\!14\)\( T^{5} + \)\(26\!\cdots\!87\)\( T^{6} - 4569427010726114 T^{7} - 84078588612 T^{8} + 84088 T^{9} + T^{10} )^{2} \)
$53$ \( ( -\)\(39\!\cdots\!00\)\( - \)\(12\!\cdots\!40\)\( T + \)\(45\!\cdots\!76\)\( T^{2} + \)\(12\!\cdots\!80\)\( T^{3} - \)\(10\!\cdots\!78\)\( T^{4} - \)\(26\!\cdots\!30\)\( T^{5} + \)\(10\!\cdots\!37\)\( T^{6} + 17909826829303988 T^{7} - 63398055440 T^{8} - 280802 T^{9} + T^{10} )^{2} \)
$59$ \( \)\(52\!\cdots\!36\)\( + \)\(25\!\cdots\!56\)\( T + \)\(64\!\cdots\!88\)\( T^{2} + \)\(42\!\cdots\!12\)\( T^{3} + \)\(21\!\cdots\!72\)\( T^{4} + \)\(62\!\cdots\!44\)\( T^{5} + \)\(21\!\cdots\!00\)\( T^{6} + \)\(97\!\cdots\!16\)\( T^{7} + \)\(24\!\cdots\!32\)\( T^{8} + \)\(16\!\cdots\!36\)\( T^{9} + \)\(38\!\cdots\!60\)\( T^{10} + \)\(18\!\cdots\!36\)\( T^{11} + \)\(46\!\cdots\!48\)\( T^{12} + \)\(79\!\cdots\!00\)\( T^{13} + \)\(15\!\cdots\!00\)\( T^{14} + \)\(77\!\cdots\!60\)\( T^{15} + \)\(19\!\cdots\!68\)\( T^{16} + 6129346997707360 T^{17} + 70879890050 T^{18} + 376510 T^{19} + T^{20} \)
$61$ \( \)\(22\!\cdots\!00\)\( + \)\(78\!\cdots\!80\)\( T + \)\(13\!\cdots\!68\)\( T^{2} + \)\(83\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!40\)\( T^{4} - \)\(11\!\cdots\!16\)\( T^{5} + \)\(31\!\cdots\!16\)\( T^{6} + \)\(16\!\cdots\!68\)\( T^{7} + \)\(30\!\cdots\!96\)\( T^{8} - \)\(66\!\cdots\!12\)\( T^{9} + \)\(44\!\cdots\!52\)\( T^{10} + \)\(38\!\cdots\!28\)\( T^{11} + \)\(29\!\cdots\!01\)\( T^{12} - \)\(57\!\cdots\!04\)\( T^{13} + \)\(34\!\cdots\!52\)\( T^{14} - \)\(69\!\cdots\!92\)\( T^{15} + \)\(11\!\cdots\!19\)\( T^{16} - 58880757445145752 T^{17} + 350033445000 T^{18} - 836700 T^{19} + T^{20} \)
$67$ \( \)\(12\!\cdots\!24\)\( + \)\(22\!\cdots\!08\)\( T^{2} + \)\(36\!\cdots\!24\)\( T^{4} + \)\(59\!\cdots\!32\)\( T^{6} + \)\(39\!\cdots\!96\)\( T^{8} + \)\(13\!\cdots\!48\)\( T^{10} + \)\(27\!\cdots\!45\)\( T^{12} + \)\(32\!\cdots\!10\)\( T^{14} + \)\(21\!\cdots\!15\)\( T^{16} + 730660905082 T^{18} + T^{20} \)
$71$ \( ( -\)\(23\!\cdots\!16\)\( - \)\(72\!\cdots\!12\)\( T + \)\(14\!\cdots\!72\)\( T^{2} + \)\(17\!\cdots\!84\)\( T^{3} - \)\(28\!\cdots\!00\)\( T^{4} + \)\(70\!\cdots\!10\)\( T^{5} + \)\(23\!\cdots\!63\)\( T^{6} - 21862733293959526 T^{7} - 843085918872 T^{8} + 47292 T^{9} + T^{10} )^{2} \)
$73$ \( \)\(35\!\cdots\!00\)\( + \)\(54\!\cdots\!00\)\( T^{2} + \)\(31\!\cdots\!49\)\( T^{4} + \)\(92\!\cdots\!74\)\( T^{6} + \)\(15\!\cdots\!33\)\( T^{8} + \)\(16\!\cdots\!62\)\( T^{10} + \)\(11\!\cdots\!28\)\( T^{12} + \)\(46\!\cdots\!34\)\( T^{14} + \)\(11\!\cdots\!41\)\( T^{16} + 1661457800902 T^{18} + T^{20} \)
$79$ \( \)\(32\!\cdots\!16\)\( + \)\(12\!\cdots\!96\)\( T + \)\(25\!\cdots\!88\)\( T^{2} + \)\(31\!\cdots\!36\)\( T^{3} + \)\(25\!\cdots\!80\)\( T^{4} + \)\(14\!\cdots\!80\)\( T^{5} + \)\(60\!\cdots\!68\)\( T^{6} + \)\(18\!\cdots\!28\)\( T^{7} + \)\(43\!\cdots\!08\)\( T^{8} + \)\(77\!\cdots\!32\)\( T^{9} + \)\(10\!\cdots\!46\)\( T^{10} + \)\(13\!\cdots\!06\)\( T^{11} + \)\(19\!\cdots\!17\)\( T^{12} + \)\(28\!\cdots\!32\)\( T^{13} + \)\(35\!\cdots\!70\)\( T^{14} + \)\(33\!\cdots\!86\)\( T^{15} + \)\(26\!\cdots\!23\)\( T^{16} + 2713260579018257600 T^{17} + 3189312526418 T^{18} + 2525594 T^{19} + T^{20} \)
$83$ \( ( -\)\(59\!\cdots\!80\)\( - \)\(47\!\cdots\!88\)\( T + \)\(36\!\cdots\!20\)\( T^{2} + \)\(20\!\cdots\!52\)\( T^{3} - \)\(67\!\cdots\!76\)\( T^{4} - \)\(19\!\cdots\!42\)\( T^{5} + \)\(50\!\cdots\!03\)\( T^{6} + 624025600090983958 T^{7} - 1446700409576 T^{8} - 357652 T^{9} + T^{10} )^{2} \)
$89$ \( \)\(40\!\cdots\!00\)\( + \)\(14\!\cdots\!40\)\( T + \)\(27\!\cdots\!68\)\( T^{2} - \)\(41\!\cdots\!44\)\( T^{3} + \)\(21\!\cdots\!16\)\( T^{4} - \)\(15\!\cdots\!92\)\( T^{5} + \)\(54\!\cdots\!52\)\( T^{6} + \)\(20\!\cdots\!40\)\( T^{7} + \)\(29\!\cdots\!00\)\( T^{8} - \)\(15\!\cdots\!76\)\( T^{9} + \)\(47\!\cdots\!84\)\( T^{10} + \)\(18\!\cdots\!28\)\( T^{11} + \)\(69\!\cdots\!12\)\( T^{12} - \)\(14\!\cdots\!00\)\( T^{13} + \)\(26\!\cdots\!76\)\( T^{14} + \)\(58\!\cdots\!64\)\( T^{15} + \)\(73\!\cdots\!24\)\( T^{16} - 273007290537482568 T^{17} + 453216738312 T^{18} + 952068 T^{19} + T^{20} \)
$97$ \( \)\(58\!\cdots\!96\)\( - \)\(58\!\cdots\!32\)\( T + \)\(29\!\cdots\!72\)\( T^{2} - \)\(75\!\cdots\!68\)\( T^{3} + \)\(11\!\cdots\!36\)\( T^{4} - \)\(94\!\cdots\!08\)\( T^{5} + \)\(45\!\cdots\!56\)\( T^{6} - \)\(11\!\cdots\!64\)\( T^{7} + \)\(16\!\cdots\!40\)\( T^{8} - \)\(17\!\cdots\!92\)\( T^{9} + \)\(99\!\cdots\!92\)\( T^{10} - \)\(22\!\cdots\!80\)\( T^{11} + \)\(25\!\cdots\!40\)\( T^{12} - \)\(76\!\cdots\!88\)\( T^{13} + \)\(22\!\cdots\!48\)\( T^{14} - \)\(53\!\cdots\!36\)\( T^{15} + \)\(93\!\cdots\!92\)\( T^{16} - 1687816495481933160 T^{17} + 15865504712 T^{18} - 178132 T^{19} + T^{20} \)
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