Properties

Label 74.5.d.b
Level $74$
Weight $5$
Character orbit 74.d
Analytic conductor $7.649$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 74.d (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.64937726820\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
Defining polynomial: \(x^{14} + 727 x^{12} + 207381 x^{10} + 29788577 x^{8} + 2302194203 x^{6} + 92916575085 x^{4} + 1658451585508 x^{2} + 6531254919424\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 + 2 \beta_{5} ) q^{2} + ( \beta_{1} - \beta_{5} ) q^{3} + 8 \beta_{5} q^{4} + ( 3 - 3 \beta_{5} - \beta_{7} ) q^{5} + ( 2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{5} ) q^{6} + ( -3 + 2 \beta_{2} - \beta_{4} ) q^{7} + ( -16 + 16 \beta_{5} ) q^{8} + ( -24 + \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( 2 + 2 \beta_{5} ) q^{2} + ( \beta_{1} - \beta_{5} ) q^{3} + 8 \beta_{5} q^{4} + ( 3 - 3 \beta_{5} - \beta_{7} ) q^{5} + ( 2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{5} ) q^{6} + ( -3 + 2 \beta_{2} - \beta_{4} ) q^{7} + ( -16 + 16 \beta_{5} ) q^{8} + ( -24 + \beta_{2} + \beta_{3} ) q^{9} + ( 12 - 2 \beta_{7} + 2 \beta_{8} ) q^{10} + ( -\beta_{1} + 11 \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} ) q^{11} + ( 8 - 8 \beta_{2} ) q^{12} + ( -8 + 3 \beta_{1} - 3 \beta_{2} + 8 \beta_{5} - \beta_{7} - \beta_{12} ) q^{13} + ( -6 + 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{4} - 6 \beta_{5} + 2 \beta_{11} ) q^{14} + ( -27 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 27 \beta_{5} + 2 \beta_{6} - 3 \beta_{8} - \beta_{10} - \beta_{13} ) q^{15} -64 q^{16} + ( 35 + 4 \beta_{1} - 4 \beta_{2} - \beta_{3} - \beta_{4} - 35 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} - \beta_{11} ) q^{17} + ( -48 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 48 \beta_{5} + 2 \beta_{6} ) q^{18} + ( -29 + 4 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 29 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} - 5 \beta_{9} - 3 \beta_{11} ) q^{19} + ( 24 + 24 \beta_{5} + 8 \beta_{8} ) q^{20} + ( -12 \beta_{1} + 207 \beta_{5} - 6 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 3 \beta_{11} - \beta_{12} - \beta_{13} ) q^{21} + ( -22 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 22 \beta_{5} + 2 \beta_{6} + 4 \beta_{9} ) q^{22} + ( 19 - 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} - 19 \beta_{5} - 3 \beta_{6} - 10 \beta_{7} + \beta_{9} + 4 \beta_{11} + \beta_{12} ) q^{23} + ( 16 - 16 \beta_{1} - 16 \beta_{2} + 16 \beta_{5} ) q^{24} + ( 17 \beta_{1} - 271 \beta_{5} - 12 \beta_{7} - 12 \beta_{8} + 3 \beta_{11} + 3 \beta_{12} + 3 \beta_{13} ) q^{25} + ( -32 - 12 \beta_{2} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{12} + 2 \beta_{13} ) q^{26} + ( 10 \beta_{1} + 57 \beta_{5} + \beta_{6} + 12 \beta_{7} + 12 \beta_{8} - 9 \beta_{11} ) q^{27} + ( 16 \beta_{1} - 24 \beta_{5} + 8 \beta_{11} ) q^{28} + ( 127 + 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 127 \beta_{5} - 3 \beta_{6} + 8 \beta_{8} - 3 \beta_{10} - 3 \beta_{11} + 5 \beta_{13} ) q^{29} + ( 8 \beta_{1} - 108 \beta_{5} + 8 \beta_{6} - 6 \beta_{7} - 6 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{12} - 2 \beta_{13} ) q^{30} + ( 275 + 4 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} - \beta_{4} + 275 \beta_{5} - 6 \beta_{6} + 4 \beta_{8} - 7 \beta_{10} + \beta_{11} - 4 \beta_{13} ) q^{31} + ( -128 - 128 \beta_{5} ) q^{32} + ( 153 + 18 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 8 \beta_{9} + 8 \beta_{10} + \beta_{12} - \beta_{13} ) q^{33} + ( 140 - 16 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} ) q^{34} + ( -94 - 36 \beta_{1} + 36 \beta_{2} + 11 \beta_{3} + 3 \beta_{4} + 94 \beta_{5} - 11 \beta_{6} + 15 \beta_{7} + 7 \beta_{9} + 3 \beta_{11} + 4 \beta_{12} ) q^{35} + ( 8 \beta_{1} - 192 \beta_{5} + 8 \beta_{6} ) q^{36} + ( 189 + 42 \beta_{1} + 22 \beta_{2} - 6 \beta_{3} + 11 \beta_{4} + 154 \beta_{5} + 7 \beta_{6} + 8 \beta_{7} - 19 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} - \beta_{12} - 2 \beta_{13} ) q^{37} + ( -116 - 16 \beta_{2} - 8 \beta_{3} - 12 \beta_{4} + 6 \beta_{7} - 6 \beta_{8} - 10 \beta_{9} + 10 \beta_{10} ) q^{38} + ( -343 - 21 \beta_{1} - 21 \beta_{2} + 8 \beta_{3} + 12 \beta_{4} - 343 \beta_{5} + 8 \beta_{6} - 39 \beta_{8} + 2 \beta_{10} - 12 \beta_{11} - 4 \beta_{13} ) q^{39} + ( 96 \beta_{5} + 16 \beta_{7} + 16 \beta_{8} ) q^{40} + ( -89 \beta_{1} - 369 \beta_{5} - 6 \beta_{6} - 17 \beta_{7} - 17 \beta_{8} - 10 \beta_{9} - 10 \beta_{10} - 22 \beta_{11} - \beta_{12} - \beta_{13} ) q^{41} + ( -414 - 24 \beta_{1} + 24 \beta_{2} + 12 \beta_{3} + 6 \beta_{4} + 414 \beta_{5} - 12 \beta_{6} + 12 \beta_{7} + 8 \beta_{9} + 6 \beta_{11} - 4 \beta_{12} ) q^{42} + ( 240 + 12 \beta_{1} - 12 \beta_{2} + 10 \beta_{3} - 6 \beta_{4} - 240 \beta_{5} - 10 \beta_{6} + 29 \beta_{7} - 7 \beta_{9} - 6 \beta_{11} ) q^{43} + ( -88 + 8 \beta_{2} - 8 \beta_{3} + 8 \beta_{9} - 8 \beta_{10} ) q^{44} + ( -123 - 115 \beta_{1} + 115 \beta_{2} + 7 \beta_{3} - 9 \beta_{4} + 123 \beta_{5} - 7 \beta_{6} + 24 \beta_{7} + 8 \beta_{9} - 9 \beta_{11} + 5 \beta_{12} ) q^{45} + ( 76 + 12 \beta_{2} + 12 \beta_{3} + 16 \beta_{4} - 20 \beta_{7} + 20 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{12} - 2 \beta_{13} ) q^{46} + ( -501 + 14 \beta_{2} + 8 \beta_{3} + 9 \beta_{4} + 23 \beta_{7} - 23 \beta_{8} - 10 \beta_{9} + 10 \beta_{10} - 3 \beta_{12} + 3 \beta_{13} ) q^{47} + ( -64 \beta_{1} + 64 \beta_{5} ) q^{48} + ( 920 - 38 \beta_{2} - 10 \beta_{3} + 23 \beta_{4} - 12 \beta_{7} + 12 \beta_{8} - 5 \beta_{9} + 5 \beta_{10} - 4 \beta_{12} + 4 \beta_{13} ) q^{49} + ( 542 + 34 \beta_{1} - 34 \beta_{2} + 6 \beta_{4} - 542 \beta_{5} - 48 \beta_{7} + 6 \beta_{11} + 12 \beta_{12} ) q^{50} + ( -465 + 63 \beta_{1} + 63 \beta_{2} - 9 \beta_{4} - 465 \beta_{5} - 9 \beta_{8} + 11 \beta_{10} + 9 \beta_{11} - 4 \beta_{13} ) q^{51} + ( -64 - 24 \beta_{1} - 24 \beta_{2} - 64 \beta_{5} + 8 \beta_{8} + 8 \beta_{13} ) q^{52} + ( 905 + 26 \beta_{2} - 42 \beta_{3} + \beta_{4} - 5 \beta_{7} + 5 \beta_{8} + 9 \beta_{9} - 9 \beta_{10} - \beta_{12} + \beta_{13} ) q^{53} + ( -114 + 20 \beta_{1} - 20 \beta_{2} - 2 \beta_{3} - 18 \beta_{4} + 114 \beta_{5} + 2 \beta_{6} + 48 \beta_{7} - 18 \beta_{11} ) q^{54} + ( 394 - 18 \beta_{1} - 18 \beta_{2} - 11 \beta_{3} + 394 \beta_{5} - 11 \beta_{6} + 27 \beta_{8} + 4 \beta_{10} + \beta_{13} ) q^{55} + ( 48 + 32 \beta_{1} - 32 \beta_{2} + 16 \beta_{4} - 48 \beta_{5} + 16 \beta_{11} ) q^{56} + ( -467 + 86 \beta_{1} + 86 \beta_{2} - 18 \beta_{3} - 42 \beta_{4} - 467 \beta_{5} - 18 \beta_{6} - 81 \beta_{8} - 25 \beta_{10} + 42 \beta_{11} + 2 \beta_{13} ) q^{57} + ( 8 \beta_{1} + 508 \beta_{5} - 12 \beta_{6} + 16 \beta_{7} + 16 \beta_{8} - 6 \beta_{9} - 6 \beta_{10} - 12 \beta_{11} + 10 \beta_{12} + 10 \beta_{13} ) q^{58} + ( -544 - 122 \beta_{1} + 122 \beta_{2} - 14 \beta_{3} + 9 \beta_{4} + 544 \beta_{5} + 14 \beta_{6} - 12 \beta_{7} + 10 \beta_{9} + 9 \beta_{11} + 4 \beta_{12} ) q^{59} + ( 216 + 16 \beta_{1} - 16 \beta_{2} - 16 \beta_{3} - 216 \beta_{5} + 16 \beta_{6} - 24 \beta_{7} - 8 \beta_{9} - 8 \beta_{12} ) q^{60} + ( -457 - 41 \beta_{1} - 41 \beta_{2} + 13 \beta_{3} + 40 \beta_{4} - 457 \beta_{5} + 13 \beta_{6} - 16 \beta_{8} - 15 \beta_{10} - 40 \beta_{11} + 2 \beta_{13} ) q^{61} + ( 16 \beta_{1} + 1100 \beta_{5} - 24 \beta_{6} + 8 \beta_{7} + 8 \beta_{8} - 14 \beta_{9} - 14 \beta_{10} + 4 \beta_{11} - 8 \beta_{12} - 8 \beta_{13} ) q^{62} + ( 1494 - 354 \beta_{2} + 6 \beta_{3} + 18 \beta_{4} - 60 \beta_{7} + 60 \beta_{8} - 16 \beta_{9} + 16 \beta_{10} - \beta_{12} + \beta_{13} ) q^{63} -512 \beta_{5} q^{64} + ( 360 \beta_{1} - 748 \beta_{5} + 29 \beta_{6} + 116 \beta_{7} + 116 \beta_{8} - 9 \beta_{9} - 9 \beta_{10} + 12 \beta_{11} + 3 \beta_{12} + 3 \beta_{13} ) q^{65} + ( 306 + 36 \beta_{1} + 36 \beta_{2} - 6 \beta_{3} - 6 \beta_{4} + 306 \beta_{5} - 6 \beta_{6} + 32 \beta_{10} + 6 \beta_{11} - 4 \beta_{13} ) q^{66} + ( -111 \beta_{1} + 618 \beta_{5} + 3 \beta_{6} - 49 \beta_{7} - 49 \beta_{8} + 17 \beta_{9} + 17 \beta_{10} + 25 \beta_{11} - 4 \beta_{12} - 4 \beta_{13} ) q^{67} + ( 280 - 32 \beta_{1} - 32 \beta_{2} - 8 \beta_{3} - 8 \beta_{4} + 280 \beta_{5} - 8 \beta_{6} + 8 \beta_{8} - 8 \beta_{10} + 8 \beta_{11} ) q^{68} + ( 138 - 93 \beta_{1} - 93 \beta_{2} + 36 \beta_{3} + 30 \beta_{4} + 138 \beta_{5} + 36 \beta_{6} + 66 \beta_{8} - 21 \beta_{10} - 30 \beta_{11} ) q^{69} + ( -376 + 144 \beta_{2} + 44 \beta_{3} + 12 \beta_{4} + 30 \beta_{7} - 30 \beta_{8} + 14 \beta_{9} - 14 \beta_{10} + 8 \beta_{12} - 8 \beta_{13} ) q^{70} + ( 429 - 386 \beta_{2} + 12 \beta_{3} - 35 \beta_{4} + 20 \beta_{7} - 20 \beta_{8} + 3 \beta_{9} - 3 \beta_{10} - 4 \beta_{12} + 4 \beta_{13} ) q^{71} + ( 384 + 16 \beta_{1} - 16 \beta_{2} - 16 \beta_{3} - 384 \beta_{5} + 16 \beta_{6} ) q^{72} + ( 41 \beta_{1} + 1151 \beta_{5} - 57 \beta_{6} + 17 \beta_{7} + 17 \beta_{8} + 11 \beta_{9} + 11 \beta_{10} + 78 \beta_{11} ) q^{73} + ( 70 + 128 \beta_{1} - 40 \beta_{2} - 26 \beta_{3} + 28 \beta_{4} + 686 \beta_{5} + 2 \beta_{6} - 22 \beta_{7} - 54 \beta_{8} + 10 \beta_{9} + 2 \beta_{10} - 16 \beta_{11} - 6 \beta_{12} - 2 \beta_{13} ) q^{74} + ( -2513 + 346 \beta_{2} + 59 \beta_{3} - 63 \beta_{4} - 63 \beta_{7} + 63 \beta_{8} + 27 \beta_{9} - 27 \beta_{10} ) q^{75} + ( -232 - 32 \beta_{1} - 32 \beta_{2} - 16 \beta_{3} - 24 \beta_{4} - 232 \beta_{5} - 16 \beta_{6} - 24 \beta_{8} + 40 \beta_{10} + 24 \beta_{11} ) q^{76} + ( 126 \beta_{1} - 1945 \beta_{5} + 70 \beta_{6} - 32 \beta_{7} - 32 \beta_{8} + 38 \beta_{9} + 38 \beta_{10} + 41 \beta_{11} - \beta_{12} - \beta_{13} ) q^{77} + ( -84 \beta_{1} - 1372 \beta_{5} + 32 \beta_{6} - 78 \beta_{7} - 78 \beta_{8} + 4 \beta_{9} + 4 \beta_{10} - 48 \beta_{11} - 8 \beta_{12} - 8 \beta_{13} ) q^{78} + ( -2001 - 196 \beta_{1} + 196 \beta_{2} - 24 \beta_{3} - 76 \beta_{4} + 2001 \beta_{5} + 24 \beta_{6} + 21 \beta_{7} - 6 \beta_{9} - 76 \beta_{11} - 5 \beta_{12} ) q^{79} + ( -192 + 192 \beta_{5} + 64 \beta_{7} ) q^{80} + ( -2397 + 14 \beta_{2} + 5 \beta_{3} - 36 \beta_{4} - 51 \beta_{7} + 51 \beta_{8} - 30 \beta_{9} + 30 \beta_{10} - 3 \beta_{12} + 3 \beta_{13} ) q^{81} + ( 738 - 178 \beta_{1} + 178 \beta_{2} + 12 \beta_{3} - 44 \beta_{4} - 738 \beta_{5} - 12 \beta_{6} - 68 \beta_{7} - 40 \beta_{9} - 44 \beta_{11} - 4 \beta_{12} ) q^{82} + ( -1441 + 106 \beta_{2} + 40 \beta_{3} + 23 \beta_{4} - 136 \beta_{7} + 136 \beta_{8} - 13 \beta_{9} + 13 \beta_{10} ) q^{83} + ( -1656 + 96 \beta_{2} + 48 \beta_{3} + 24 \beta_{4} + 24 \beta_{7} - 24 \beta_{8} + 16 \beta_{9} - 16 \beta_{10} - 8 \beta_{12} + 8 \beta_{13} ) q^{84} + ( 66 \beta_{1} - 896 \beta_{5} - 2 \beta_{6} - 36 \beta_{7} - 36 \beta_{8} - \beta_{9} - \beta_{10} + 18 \beta_{11} - \beta_{12} - \beta_{13} ) q^{85} + ( 960 - 48 \beta_{2} + 40 \beta_{3} - 24 \beta_{4} + 58 \beta_{7} - 58 \beta_{8} - 14 \beta_{9} + 14 \beta_{10} ) q^{86} + ( 132 + 200 \beta_{1} - 200 \beta_{2} - 35 \beta_{3} - 51 \beta_{4} - 132 \beta_{5} + 35 \beta_{6} - 258 \beta_{7} + 19 \beta_{9} - 51 \beta_{11} - 20 \beta_{12} ) q^{87} + ( -176 + 16 \beta_{1} + 16 \beta_{2} - 16 \beta_{3} - 176 \beta_{5} - 16 \beta_{6} - 32 \beta_{10} ) q^{88} + ( 1107 - 359 \beta_{1} - 359 \beta_{2} - 13 \beta_{3} - 58 \beta_{4} + 1107 \beta_{5} - 13 \beta_{6} - 37 \beta_{8} - 13 \beta_{10} + 58 \beta_{11} - 24 \beta_{13} ) q^{89} + ( -492 + 460 \beta_{2} + 28 \beta_{3} - 36 \beta_{4} + 48 \beta_{7} - 48 \beta_{8} + 16 \beta_{9} - 16 \beta_{10} + 10 \beta_{12} - 10 \beta_{13} ) q^{90} + ( -578 + 323 \beta_{1} - 323 \beta_{2} + 26 \beta_{3} - 42 \beta_{4} + 578 \beta_{5} - 26 \beta_{6} + 181 \beta_{7} + 7 \beta_{9} - 42 \beta_{11} + 4 \beta_{12} ) q^{91} + ( 152 + 24 \beta_{1} + 24 \beta_{2} + 24 \beta_{3} + 32 \beta_{4} + 152 \beta_{5} + 24 \beta_{6} + 80 \beta_{8} - 8 \beta_{10} - 32 \beta_{11} - 8 \beta_{13} ) q^{92} + ( -209 + 529 \beta_{1} - 529 \beta_{2} + 24 \beta_{3} + 84 \beta_{4} + 209 \beta_{5} - 24 \beta_{6} + 78 \beta_{7} + 44 \beta_{9} + 84 \beta_{11} - \beta_{12} ) q^{93} + ( -1002 + 28 \beta_{1} + 28 \beta_{2} + 16 \beta_{3} + 18 \beta_{4} - 1002 \beta_{5} + 16 \beta_{6} - 92 \beta_{8} + 40 \beta_{10} - 18 \beta_{11} + 12 \beta_{13} ) q^{94} + ( 870 \beta_{1} + 910 \beta_{5} + 64 \beta_{6} + 46 \beta_{7} + 46 \beta_{8} - 28 \beta_{9} - 28 \beta_{10} + 90 \beta_{11} - 25 \beta_{12} - 25 \beta_{13} ) q^{95} + ( -128 - 128 \beta_{1} + 128 \beta_{2} + 128 \beta_{5} ) q^{96} + ( 1722 - 111 \beta_{1} + 111 \beta_{2} - 33 \beta_{3} - 16 \beta_{4} - 1722 \beta_{5} + 33 \beta_{6} - 92 \beta_{7} - 20 \beta_{9} - 16 \beta_{11} + 22 \beta_{12} ) q^{97} + ( 1840 - 76 \beta_{1} - 76 \beta_{2} - 20 \beta_{3} + 46 \beta_{4} + 1840 \beta_{5} - 20 \beta_{6} + 48 \beta_{8} + 20 \beta_{10} - 46 \beta_{11} + 16 \beta_{13} ) q^{98} + ( 354 \beta_{1} + 2214 \beta_{5} + 39 \beta_{6} - 87 \beta_{7} - 87 \beta_{8} + 20 \beta_{9} + 20 \beta_{10} + 108 \beta_{11} + 8 \beta_{12} + 8 \beta_{13} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q + 28q^{2} + 36q^{5} + 40q^{6} - 48q^{7} - 224q^{8} - 346q^{9} + O(q^{10}) \) \( 14q + 28q^{2} + 36q^{5} + 40q^{6} - 48q^{7} - 224q^{8} - 346q^{9} + 144q^{10} + 160q^{12} - 104q^{13} - 96q^{14} - 378q^{15} - 896q^{16} + 516q^{17} - 692q^{18} - 328q^{19} + 288q^{20} - 320q^{22} + 154q^{23} + 320q^{24} - 416q^{26} + 1686q^{29} + 3834q^{31} - 1792q^{32} + 2104q^{33} + 2064q^{34} - 1502q^{35} + 2640q^{37} - 1312q^{38} - 4526q^{39} - 5984q^{42} + 3616q^{43} - 1280q^{44} - 2238q^{45} + 616q^{46} - 6892q^{47} + 12854q^{49} + 7516q^{50} - 6742q^{51} - 832q^{52} + 12572q^{53} - 1072q^{54} + 5510q^{55} + 768q^{56} - 6302q^{57} - 8422q^{59} + 3024q^{60} - 6386q^{61} + 22244q^{63} + 4208q^{66} + 4128q^{68} + 1728q^{69} - 6008q^{70} + 8680q^{71} + 5536q^{72} + 1316q^{74} - 37980q^{75} - 2624q^{76} - 28520q^{79} - 2304q^{80} - 33962q^{81} + 9136q^{82} - 22688q^{83} - 23936q^{84} + 14464q^{86} + 1828q^{87} - 2560q^{88} + 18344q^{89} - 8952q^{90} - 4918q^{91} + 1232q^{92} + 24q^{93} - 13784q^{94} - 2560q^{96} + 23246q^{97} + 25708q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} + 727 x^{12} + 207381 x^{10} + 29788577 x^{8} + 2302194203 x^{6} + 92916575085 x^{4} + 1658451585508 x^{2} + 6531254919424\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(1200481 \nu^{12} + 766025276 \nu^{10} + 180814035481 \nu^{8} + 19656581442877 \nu^{6} + 1009461419465036 \nu^{4} + 21193213130990143 \nu^{2} + 86861278035132816\)\()/ 146604462483438 \)
\(\beta_{3}\)\(=\)\((\)\(1200481 \nu^{12} + 766025276 \nu^{10} + 180814035481 \nu^{8} + 19656581442877 \nu^{6} + 1009461419465036 \nu^{4} + 21339817593473581 \nu^{2} + 102108142133410368\)\()/ 146604462483438 \)
\(\beta_{4}\)\(=\)\((\)\(-18530585749 \nu^{12} - 12189038115456 \nu^{10} - 2996164678179132 \nu^{8} - 342866615492081801 \nu^{6} - 18588349221343017966 \nu^{4} - 409784965701915956508 \nu^{2} - 1745247274455234458404\)\()/ 659573476712987562 \)
\(\beta_{5}\)\(=\)\((\)\(33988179063 \nu^{13} + 21641418519809 \nu^{11} + 5090823854109571 \nu^{9} + 550365353983584359 \nu^{7} + 28012400263341938525 \nu^{5} + 578253285559395562603 \nu^{3} + 2205695795183558683628 \nu\)\()/ \)\(37\!\cdots\!16\)\( \)
\(\beta_{6}\)\(=\)\((\)\(58347870445 \nu^{13} + 36628602238213 \nu^{11} + 8418943212877049 \nu^{9} + 875376033533767509 \nu^{7} + 41684715129661590481 \nu^{5} + 747036004724817171917 \nu^{3} + 972696005909128487552 \nu\)\()/ 46833381958184202852 \)
\(\beta_{7}\)\(=\)\((\)\(-22445404523491577 \nu^{13} - 155243406075960656 \nu^{12} - 14471788650326385711 \nu^{11} - 100048466622082126344 \nu^{10} - 3464991525145285954053 \nu^{9} - 23929914713528647385640 \nu^{8} - 383925799030011746313577 \nu^{7} - 2645613399579714312614080 \nu^{6} - 20158568820480814425826899 \nu^{5} - 138309601140173362252447032 \nu^{4} - 432811414972773528929623917 \nu^{3} - 2948726950221537034451940168 \nu^{2} - 1781081962896386418390714788 \nu - 12157309359089096049365592992\)\()/ \)\(50\!\cdots\!52\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-22445404523491577 \nu^{13} + 155243406075960656 \nu^{12} - 14471788650326385711 \nu^{11} + 100048466622082126344 \nu^{10} - 3464991525145285954053 \nu^{9} + 23929914713528647385640 \nu^{8} - 383925799030011746313577 \nu^{7} + 2645613399579714312614080 \nu^{6} - 20158568820480814425826899 \nu^{5} + 138309601140173362252447032 \nu^{4} - 432811414972773528929623917 \nu^{3} + 2948726950221537034451940168 \nu^{2} - 1781081962896386418390714788 \nu + 12157309359089096049365592992\)\()/ \)\(50\!\cdots\!52\)\( \)
\(\beta_{9}\)\(=\)\((\)\(55606431993876725 \nu^{13} + 480815119945316600 \nu^{12} + 36066439831193159139 \nu^{11} + 311738695429199186688 \nu^{10} + 8704143374303469485985 \nu^{9} + 75194027247580052536224 \nu^{8} + 973929512862299534484085 \nu^{7} + 8407739673470737264177576 \nu^{6} + 51611979782069343804979287 \nu^{5} + 445245477535151967270304464 \nu^{4} + 1112649075626524591992886185 \nu^{3} + 9605035138212124877533882176 \nu^{2} + 4473234457393093588383983588 \nu + 39484136002131238010170637216\)\()/ \)\(50\!\cdots\!52\)\( \)
\(\beta_{10}\)\(=\)\((\)\(55606431993876725 \nu^{13} - 480815119945316600 \nu^{12} + 36066439831193159139 \nu^{11} - 311738695429199186688 \nu^{10} + 8704143374303469485985 \nu^{9} - 75194027247580052536224 \nu^{8} + 973929512862299534484085 \nu^{7} - 8407739673470737264177576 \nu^{6} + 51611979782069343804979287 \nu^{5} - 445245477535151967270304464 \nu^{4} + 1112649075626524591992886185 \nu^{3} - 9605035138212124877533882176 \nu^{2} + 4473234457393093588383983588 \nu - 39484136002131238010170637216\)\()/ \)\(50\!\cdots\!52\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-30922802471753989 \nu^{13} - 19942208056019730243 \nu^{11} - 4776802289589477812553 \nu^{9} - 529688844306014376116693 \nu^{7} - 27852316273638046900350039 \nu^{5} - 600195741886697743076279697 \nu^{3} - 2549605984538831135283948484 \nu\)\()/ \)\(25\!\cdots\!76\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-54789470857345017 \nu^{13} - 685919125159600360 \nu^{12} - 35423856886786876679 \nu^{11} - 442607820683305359496 \nu^{10} - 8511682600749441864253 \nu^{9} - 106087314758820635007440 \nu^{8} - 946976952524500235869865 \nu^{7} - 11769881957402815861004952 \nu^{6} - 49892182157368074960248603 \nu^{5} - 618728232100748351210616424 \nu^{4} - 1073913902298288245492874373 \nu^{3} - 13297418331660156424157006048 \nu^{2} - 4525955917204621793527238612 \nu - 55230926641064437288257315104\)\()/ \)\(16\!\cdots\!84\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-54789470857345017 \nu^{13} + 685919125159600360 \nu^{12} - 35423856886786876679 \nu^{11} + 442607820683305359496 \nu^{10} - 8511682600749441864253 \nu^{9} + 106087314758820635007440 \nu^{8} - 946976952524500235869865 \nu^{7} + 11769881957402815861004952 \nu^{6} - 49892182157368074960248603 \nu^{5} + 618728232100748351210616424 \nu^{4} - 1073913902298288245492874373 \nu^{3} + 13297418331660156424157006048 \nu^{2} - 4525955917204621793527238612 \nu + 55230926641064437288257315104\)\()/ \)\(16\!\cdots\!84\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - \beta_{2} - 104\)
\(\nu^{3}\)\(=\)\(-9 \beta_{11} + 12 \beta_{8} + 12 \beta_{7} + 4 \beta_{6} - 94 \beta_{5} - 152 \beta_{1}\)
\(\nu^{4}\)\(=\)\(3 \beta_{13} - 3 \beta_{12} + 30 \beta_{10} - 30 \beta_{9} + 3 \beta_{8} - 3 \beta_{7} - 72 \beta_{4} - 248 \beta_{3} + 377 \beta_{2} + 16308\)
\(\nu^{5}\)\(=\)\(-51 \beta_{13} - 51 \beta_{12} + 2628 \beta_{11} - 60 \beta_{10} - 60 \beta_{9} - 3234 \beta_{8} - 3234 \beta_{7} - 1379 \beta_{6} + 34766 \beta_{5} + 28155 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-648 \beta_{13} + 648 \beta_{12} - 8757 \beta_{10} + 8757 \beta_{9} - 360 \beta_{8} + 360 \beta_{7} + 23787 \beta_{4} + 56046 \beta_{3} - 110694 \beta_{2} - 3063460\)
\(\nu^{7}\)\(=\)\(16686 \beta_{13} + 16686 \beta_{12} - 636552 \beta_{11} + 12141 \beta_{10} + 12141 \beta_{9} + 730323 \beta_{8} + 730323 \beta_{7} + 368652 \beta_{6} - 10399512 \beta_{5} - 5735587 \beta_{1}\)
\(\nu^{8}\)\(=\)\(115002 \beta_{13} - 115002 \beta_{12} + 2038356 \beta_{10} - 2038356 \beta_{9} - 307134 \beta_{8} + 307134 \beta_{7} - 6191694 \beta_{4} - 12409159 \beta_{3} + 29375347 \beta_{2} + 626407172\)
\(\nu^{9}\)\(=\)\(-4076208 \beta_{13} - 4076208 \beta_{12} + 145919295 \beta_{11} - 1232964 \beta_{10} - 1232964 \beta_{9} - 158320902 \beta_{8} - 158320902 \beta_{7} - 91908124 \beta_{6} + 2819629006 \beta_{5} + 1217832374 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-19321059 \beta_{13} + 19321059 \beta_{12} - 446561616 \beta_{10} + 446561616 \beta_{9} + 186958743 \beta_{8} - 186958743 \beta_{7} + 1501281504 \beta_{4} + 2734974782 \beta_{3} - 7436146997 \beta_{2} - 132932680284\)
\(\nu^{11}\)\(=\)\(906403263 \beta_{13} + 906403263 \beta_{12} - 32835563334 \beta_{11} - 121553616 \beta_{10} - 121553616 \beta_{9} + 33996232902 \beta_{8} + 33996232902 \beta_{7} + 22278105977 \beta_{6} - 726963844178 \beta_{5} - 263349429657 \beta_{1}\)
\(\nu^{12}\)\(=\)\(3095029710 \beta_{13} - 3095029710 \beta_{12} + 96097274469 \beta_{10} - 96097274469 \beta_{9} - 69666241842 \beta_{8} + 69666241842 \beta_{7} - 354328207137 \beta_{4} - 602949352110 \beta_{3} + 1833798293952 \beta_{2} + 28687394074840\)
\(\nu^{13}\)\(=\)\(-194754750246 \beta_{13} - 194754750246 \beta_{12} + 7346159931312 \beta_{11} + 114926823963 \beta_{10} + 114926823963 \beta_{9} - 7297663483413 \beta_{8} - 7297663483413 \beta_{7} - 5320053967938 \beta_{6} + 181906675763964 \beta_{5} + 57465660764815 \beta_{1}\)

Character values

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

\(n\) \(39\)
\(\chi(n)\) \(\beta_{5}\)

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
14.0933i
9.07204i
8.86388i
2.31020i
6.93683i
9.34569i
15.0569i
15.0569i
9.34569i
6.93683i
2.31020i
8.86388i
9.07204i
14.0933i
2.00000 + 2.00000i 15.0933i 8.00000i 30.6150 30.6150i 30.1866 30.1866i −89.3132 −16.0000 + 16.0000i −146.808 122.460
31.2 2.00000 + 2.00000i 10.0720i 8.00000i −28.1665 + 28.1665i 20.1441 20.1441i −38.8317 −16.0000 + 16.0000i −20.4460 −112.666
31.3 2.00000 + 2.00000i 9.86388i 8.00000i 0.419185 0.419185i 19.7278 19.7278i 41.6300 −16.0000 + 16.0000i −16.2962 1.67674
31.4 2.00000 + 2.00000i 3.31020i 8.00000i −1.68780 + 1.68780i 6.62039 6.62039i 49.9045 −16.0000 + 16.0000i 70.0426 −6.75118
31.5 2.00000 + 2.00000i 5.93683i 8.00000i 34.8311 34.8311i −11.8737 + 11.8737i 33.1642 −16.0000 + 16.0000i 45.7541 139.324
31.6 2.00000 + 2.00000i 8.34569i 8.00000i −6.01044 + 6.01044i −16.6914 + 16.6914i −74.3653 −16.0000 + 16.0000i 11.3495 −24.0418
31.7 2.00000 + 2.00000i 14.0569i 8.00000i −12.0005 + 12.0005i −28.1138 + 28.1138i 53.8114 −16.0000 + 16.0000i −116.596 −48.0021
43.1 2.00000 2.00000i 14.0569i 8.00000i −12.0005 12.0005i −28.1138 28.1138i 53.8114 −16.0000 16.0000i −116.596 −48.0021
43.2 2.00000 2.00000i 8.34569i 8.00000i −6.01044 6.01044i −16.6914 16.6914i −74.3653 −16.0000 16.0000i 11.3495 −24.0418
43.3 2.00000 2.00000i 5.93683i 8.00000i 34.8311 + 34.8311i −11.8737 11.8737i 33.1642 −16.0000 16.0000i 45.7541 139.324
43.4 2.00000 2.00000i 3.31020i 8.00000i −1.68780 1.68780i 6.62039 + 6.62039i 49.9045 −16.0000 16.0000i 70.0426 −6.75118
43.5 2.00000 2.00000i 9.86388i 8.00000i 0.419185 + 0.419185i 19.7278 + 19.7278i 41.6300 −16.0000 16.0000i −16.2962 1.67674
43.6 2.00000 2.00000i 10.0720i 8.00000i −28.1665 28.1665i 20.1441 + 20.1441i −38.8317 −16.0000 16.0000i −20.4460 −112.666
43.7 2.00000 2.00000i 15.0933i 8.00000i 30.6150 + 30.6150i 30.1866 + 30.1866i −89.3132 −16.0000 16.0000i −146.808 122.460
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.7
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 74.5.d.b 14
37.d odd 4 1 inner 74.5.d.b 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.5.d.b 14 1.a even 1 1 trivial
74.5.d.b 14 37.d odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} + 740 T_{3}^{12} + 215344 T_{3}^{10} + 31575042 T_{3}^{8} + 2486540160 T_{3}^{6} + 102603150036 T_{3}^{4} + \)\(19\!\cdots\!29\)\( T_{3}^{2} + \)\(11\!\cdots\!00\)\( \) acting on \(S_{5}^{\mathrm{new}}(74, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 8 - 4 T + T^{2} )^{7} \)
$3$ \( 11951402126400 + 1954984086729 T^{2} + 102603150036 T^{4} + 2486540160 T^{6} + 31575042 T^{8} + 215344 T^{10} + 740 T^{12} + T^{14} \)
$5$ \( 300707794649088 - 471887822075904 T + 370256642138916 T^{2} + 476059947160464 T^{3} + 247053334680864 T^{4} + 43642696444128 T^{5} + 4153339436949 T^{6} + 156309619956 T^{7} + 1878387624 T^{8} - 80041344 T^{9} + 3834139 T^{10} + 54432 T^{11} + 648 T^{12} - 36 T^{13} + T^{14} \)
$7$ \( ( 956231061528 - 40544142492 T - 671608836 T^{2} + 41671754 T^{3} - 11998 T^{4} - 11329 T^{5} + 24 T^{6} + T^{7} )^{2} \)
$11$ \( \)\(15\!\cdots\!00\)\( + \)\(84\!\cdots\!61\)\( T^{2} + \)\(26\!\cdots\!36\)\( T^{4} + 245404805208044704 T^{6} + 57947254086242 T^{8} + 4780355216 T^{10} + 124196 T^{12} + T^{14} \)
$13$ \( \)\(78\!\cdots\!32\)\( - \)\(15\!\cdots\!88\)\( T + \)\(15\!\cdots\!96\)\( T^{2} - \)\(51\!\cdots\!60\)\( T^{3} + \)\(47\!\cdots\!28\)\( T^{4} - \)\(84\!\cdots\!96\)\( T^{5} + 9081330393476223205 T^{6} - 29420552362752148 T^{7} + 31767866394688 T^{8} - 116215787500 T^{9} + 5760038475 T^{10} - 12247824 T^{11} + 5408 T^{12} + 104 T^{13} + T^{14} \)
$17$ \( \)\(35\!\cdots\!92\)\( - \)\(37\!\cdots\!44\)\( T + \)\(20\!\cdots\!04\)\( T^{2} - \)\(51\!\cdots\!44\)\( T^{3} + \)\(72\!\cdots\!80\)\( T^{4} - 39831699061504858752 T^{5} + 462152889612923008 T^{6} - 9523156833109024 T^{7} + 128806883339840 T^{8} - 664733820080 T^{9} + 1695089184 T^{10} - 4789168 T^{11} + 133128 T^{12} - 516 T^{13} + T^{14} \)
$19$ \( \)\(29\!\cdots\!88\)\( + \)\(16\!\cdots\!72\)\( T + \)\(45\!\cdots\!84\)\( T^{2} - \)\(57\!\cdots\!56\)\( T^{3} + \)\(36\!\cdots\!32\)\( T^{4} + \)\(23\!\cdots\!72\)\( T^{5} + \)\(19\!\cdots\!28\)\( T^{6} - 15336670201889682496 T^{7} + 9704215607924256 T^{8} + 61541754860864 T^{9} + 269329986752 T^{10} - 89513336 T^{11} + 53792 T^{12} + 328 T^{13} + T^{14} \)
$23$ \( \)\(46\!\cdots\!72\)\( + \)\(39\!\cdots\!16\)\( T + \)\(16\!\cdots\!24\)\( T^{2} - \)\(18\!\cdots\!92\)\( T^{3} + \)\(10\!\cdots\!66\)\( T^{4} + \)\(14\!\cdots\!06\)\( T^{5} + \)\(90\!\cdots\!77\)\( T^{6} + 6354199382892462624 T^{7} + 3776254544221590 T^{8} + 47112242127762 T^{9} + 483735782755 T^{10} + 183126616 T^{11} + 11858 T^{12} - 154 T^{13} + T^{14} \)
$29$ \( \)\(21\!\cdots\!12\)\( + \)\(66\!\cdots\!44\)\( T + \)\(10\!\cdots\!64\)\( T^{2} - \)\(96\!\cdots\!28\)\( T^{3} + \)\(42\!\cdots\!90\)\( T^{4} - \)\(43\!\cdots\!50\)\( T^{5} + \)\(26\!\cdots\!61\)\( T^{6} - \)\(31\!\cdots\!28\)\( T^{7} + 11045564830990431806 T^{8} - 11983361341836734 T^{9} + 6657359672547 T^{10} - 779371024 T^{11} + 1421298 T^{12} - 1686 T^{13} + T^{14} \)
$31$ \( \)\(32\!\cdots\!68\)\( - \)\(20\!\cdots\!92\)\( T + \)\(65\!\cdots\!24\)\( T^{2} - \)\(95\!\cdots\!84\)\( T^{3} + \)\(10\!\cdots\!98\)\( T^{4} - \)\(17\!\cdots\!30\)\( T^{5} + \)\(39\!\cdots\!77\)\( T^{6} - \)\(56\!\cdots\!48\)\( T^{7} + 51199139268637628126 T^{8} - 28142904800994510 T^{9} + 12386645547199 T^{10} - 8287139096 T^{11} + 7349778 T^{12} - 3834 T^{13} + T^{14} \)
$37$ \( \)\(81\!\cdots\!21\)\( - \)\(11\!\cdots\!40\)\( T + \)\(93\!\cdots\!35\)\( T^{2} + \)\(11\!\cdots\!04\)\( T^{3} + \)\(72\!\cdots\!89\)\( T^{4} - \)\(80\!\cdots\!96\)\( T^{5} + \)\(23\!\cdots\!03\)\( T^{6} + \)\(13\!\cdots\!28\)\( T^{7} + 12632469232882347423 T^{8} - 22952596914211376 T^{9} + 10998650866769 T^{10} + 965104544 T^{11} + 405935 T^{12} - 2640 T^{13} + T^{14} \)
$41$ \( \)\(68\!\cdots\!00\)\( + \)\(43\!\cdots\!61\)\( T^{2} + \)\(16\!\cdots\!92\)\( T^{4} + \)\(18\!\cdots\!96\)\( T^{6} + \)\(91\!\cdots\!98\)\( T^{8} + 221507314509904 T^{10} + 24812448 T^{12} + T^{14} \)
$43$ \( \)\(89\!\cdots\!52\)\( - \)\(28\!\cdots\!68\)\( T + \)\(45\!\cdots\!56\)\( T^{2} - \)\(48\!\cdots\!12\)\( T^{3} - \)\(13\!\cdots\!20\)\( T^{4} + \)\(19\!\cdots\!56\)\( T^{5} + \)\(22\!\cdots\!92\)\( T^{6} + \)\(51\!\cdots\!12\)\( T^{7} + 56151638992570525216 T^{8} - 29382368068018784 T^{9} + 9997767704240 T^{10} + 5525881272 T^{11} + 6537728 T^{12} - 3616 T^{13} + T^{14} \)
$47$ \( ( \)\(14\!\cdots\!12\)\( + 11555448102801912276 T + 27198296465816592 T^{2} + 10923957705060 T^{3} - 22048272426 T^{4} - 6405629 T^{5} + 3446 T^{6} + T^{7} )^{2} \)
$53$ \( ( \)\(29\!\cdots\!72\)\( - 33680603023796491044 T - 255746267192775828 T^{2} - 57897320171762 T^{3} + 124885834346 T^{4} - 14557243 T^{5} - 6286 T^{6} + T^{7} )^{2} \)
$59$ \( \)\(15\!\cdots\!08\)\( - \)\(25\!\cdots\!92\)\( T + \)\(20\!\cdots\!04\)\( T^{2} + \)\(28\!\cdots\!80\)\( T^{3} + \)\(17\!\cdots\!92\)\( T^{4} - \)\(15\!\cdots\!24\)\( T^{5} + \)\(27\!\cdots\!80\)\( T^{6} + \)\(47\!\cdots\!08\)\( T^{7} + \)\(34\!\cdots\!36\)\( T^{8} + 948545137835192496 T^{9} + 143678070372424 T^{10} + 33451522192 T^{11} + 35465042 T^{12} + 8422 T^{13} + T^{14} \)
$61$ \( \)\(31\!\cdots\!32\)\( + \)\(47\!\cdots\!72\)\( T + \)\(34\!\cdots\!56\)\( T^{2} + \)\(50\!\cdots\!80\)\( T^{3} + \)\(38\!\cdots\!86\)\( T^{4} + \)\(12\!\cdots\!22\)\( T^{5} + \)\(13\!\cdots\!49\)\( T^{6} + \)\(16\!\cdots\!36\)\( T^{7} + \)\(11\!\cdots\!82\)\( T^{8} + 3717520305710100146 T^{9} + 608995280373879 T^{10} + 25049902856 T^{11} + 20390498 T^{12} + 6386 T^{13} + T^{14} \)
$67$ \( \)\(83\!\cdots\!00\)\( + \)\(30\!\cdots\!64\)\( T^{2} + \)\(27\!\cdots\!24\)\( T^{4} + \)\(10\!\cdots\!21\)\( T^{6} + \)\(19\!\cdots\!38\)\( T^{8} + 1741663748981011 T^{10} + 70890294 T^{12} + T^{14} \)
$71$ \( ( \)\(10\!\cdots\!56\)\( - \)\(92\!\cdots\!52\)\( T - 2601574867820465352 T^{2} + 1575403186537828 T^{3} + 199741110946 T^{4} - 72917211 T^{5} - 4340 T^{6} + T^{7} )^{2} \)
$73$ \( \)\(48\!\cdots\!00\)\( + \)\(38\!\cdots\!21\)\( T^{2} + \)\(46\!\cdots\!32\)\( T^{4} + \)\(17\!\cdots\!24\)\( T^{6} + \)\(23\!\cdots\!22\)\( T^{8} + 10974486380822472 T^{10} + 186617128 T^{12} + T^{14} \)
$79$ \( \)\(13\!\cdots\!32\)\( - \)\(38\!\cdots\!12\)\( T + \)\(52\!\cdots\!96\)\( T^{2} - \)\(22\!\cdots\!72\)\( T^{3} - \)\(23\!\cdots\!32\)\( T^{4} + \)\(77\!\cdots\!12\)\( T^{5} + \)\(61\!\cdots\!85\)\( T^{6} + \)\(14\!\cdots\!88\)\( T^{7} + \)\(19\!\cdots\!92\)\( T^{8} + \)\(15\!\cdots\!64\)\( T^{9} + 17543959774309483 T^{10} + 3066441988632 T^{11} + 406695200 T^{12} + 28520 T^{13} + T^{14} \)
$83$ \( ( \)\(62\!\cdots\!68\)\( - \)\(31\!\cdots\!72\)\( T + 22150411316622907944 T^{2} + 1643185835400412 T^{3} - 1211711672918 T^{4} - 98902891 T^{5} + 11344 T^{6} + T^{7} )^{2} \)
$89$ \( \)\(15\!\cdots\!48\)\( - \)\(15\!\cdots\!24\)\( T + \)\(73\!\cdots\!56\)\( T^{2} - \)\(58\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!36\)\( T^{4} - \)\(21\!\cdots\!60\)\( T^{5} + \)\(12\!\cdots\!88\)\( T^{6} - \)\(41\!\cdots\!76\)\( T^{7} + \)\(82\!\cdots\!36\)\( T^{8} - \)\(84\!\cdots\!76\)\( T^{9} + 47060591586064140 T^{10} - 1080296741632 T^{11} + 168251168 T^{12} - 18344 T^{13} + T^{14} \)
$97$ \( \)\(28\!\cdots\!48\)\( - \)\(31\!\cdots\!88\)\( T + \)\(17\!\cdots\!64\)\( T^{2} + \)\(69\!\cdots\!08\)\( T^{3} + \)\(70\!\cdots\!32\)\( T^{4} + \)\(44\!\cdots\!72\)\( T^{5} - \)\(92\!\cdots\!24\)\( T^{6} - \)\(22\!\cdots\!88\)\( T^{7} + \)\(75\!\cdots\!88\)\( T^{8} - 27490488997971138904 T^{9} + 1763067895821212 T^{10} - 1085446093000 T^{11} + 270188258 T^{12} - 23246 T^{13} + T^{14} \)
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