Properties

Label 91.10.a.c
Level $91$
Weight $10$
Character orbit 91.a
Self dual yes
Analytic conductor $46.868$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(46.8682610909\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - x^{13} - 4752 x^{12} + 9346 x^{11} + 8576824 x^{10} - 26923636 x^{9} - 7450416552 x^{8} + 31594524240 x^{7} + 3232668379296 x^{6} - 15784728192704 x^{5} - 654706244016512 x^{4} + 3031992312058112 x^{3} + 50459693673905664 x^{2} - 177878730644221952 x - 247584423217725440\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + ( 2 - \beta_{1} ) q^{2} + ( 12 - \beta_{4} ) q^{3} + ( 171 - 6 \beta_{1} + \beta_{2} ) q^{4} + ( 213 - 13 \beta_{1} + \beta_{2} + \beta_{6} ) q^{5} + ( 35 - 25 \beta_{1} + \beta_{4} - \beta_{9} ) q^{6} + 2401 q^{7} + ( 3102 - 139 \beta_{1} + 5 \beta_{2} - \beta_{3} - 4 \beta_{4} + 3 \beta_{6} - \beta_{9} ) q^{8} + ( 10292 + 120 \beta_{1} - 4 \beta_{2} - 15 \beta_{4} + 3 \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{11} ) q^{9} +O(q^{10})\) \( q + ( 2 - \beta_{1} ) q^{2} + ( 12 - \beta_{4} ) q^{3} + ( 171 - 6 \beta_{1} + \beta_{2} ) q^{4} + ( 213 - 13 \beta_{1} + \beta_{2} + \beta_{6} ) q^{5} + ( 35 - 25 \beta_{1} + \beta_{4} - \beta_{9} ) q^{6} + 2401 q^{7} + ( 3102 - 139 \beta_{1} + 5 \beta_{2} - \beta_{3} - 4 \beta_{4} + 3 \beta_{6} - \beta_{9} ) q^{8} + ( 10292 + 120 \beta_{1} - 4 \beta_{2} - 15 \beta_{4} + 3 \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{11} ) q^{9} + ( 9066 - 561 \beta_{1} + 36 \beta_{2} - \beta_{3} + 30 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} ) q^{10} + ( 5871 - 74 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} - 41 \beta_{4} + \beta_{5} + 12 \beta_{6} + \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{11} + ( 11297 - 336 \beta_{1} + 67 \beta_{2} - 89 \beta_{4} - 3 \beta_{5} + 9 \beta_{6} - 4 \beta_{7} - \beta_{8} - 9 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + 2 \beta_{13} ) q^{12} -28561 q^{13} + ( 4802 - 2401 \beta_{1} ) q^{14} + ( 11776 + 601 \beta_{1} + 128 \beta_{2} - 5 \beta_{3} - 339 \beta_{4} - 6 \beta_{5} + 36 \beta_{6} + 4 \beta_{7} + 10 \beta_{8} - 11 \beta_{9} - 14 \beta_{10} + 3 \beta_{11} - 2 \beta_{12} - 6 \beta_{13} ) q^{15} + ( 12267 - 2688 \beta_{1} + 183 \beta_{2} - 6 \beta_{3} - 480 \beta_{4} + 4 \beta_{5} + 64 \beta_{6} + 5 \beta_{7} - 7 \beta_{8} - 6 \beta_{9} - 10 \beta_{10} - 5 \beta_{12} + 9 \beta_{13} ) q^{16} + ( -3294 + 2062 \beta_{1} + 93 \beta_{2} + 24 \beta_{3} - 111 \beta_{4} + 6 \beta_{5} + 33 \beta_{6} - 16 \beta_{7} - 9 \beta_{8} + 18 \beta_{9} + 14 \beta_{10} + 4 \beta_{11} + 6 \beta_{12} - 6 \beta_{13} ) q^{17} + ( -58226 - 8227 \beta_{1} + 117 \beta_{2} + 2 \beta_{3} - 715 \beta_{4} + 5 \beta_{5} - 94 \beta_{6} - 14 \beta_{7} + 11 \beta_{8} + 19 \beta_{9} + 14 \beta_{10} - 12 \beta_{11} + \beta_{12} + 3 \beta_{13} ) q^{18} + ( 12060 + 4599 \beta_{1} + 243 \beta_{2} + 20 \beta_{3} - 299 \beta_{4} - 28 \beta_{5} + 92 \beta_{6} + 16 \beta_{7} - 2 \beta_{8} - 10 \beta_{9} + 9 \beta_{10} - 9 \beta_{11} + 7 \beta_{12} - 6 \beta_{13} ) q^{19} + ( 280332 - 20598 \beta_{1} + 840 \beta_{2} - 16 \beta_{3} - 967 \beta_{4} + 15 \beta_{5} + 7 \beta_{6} + 4 \beta_{7} - 31 \beta_{8} + 29 \beta_{9} - 3 \beta_{10} - \beta_{11} + 3 \beta_{12} + 14 \beta_{13} ) q^{20} + ( 28812 - 2401 \beta_{4} ) q^{21} + ( 65960 - 2792 \beta_{1} + 246 \beta_{2} + 43 \beta_{3} - 644 \beta_{4} + 17 \beta_{5} - 90 \beta_{6} - 38 \beta_{7} - 23 \beta_{8} + 51 \beta_{9} + 61 \beta_{10} - 17 \beta_{11} + \beta_{12} - 8 \beta_{13} ) q^{22} + ( 137782 + 3760 \beta_{1} + 526 \beta_{2} + 47 \beta_{3} - 527 \beta_{4} + 27 \beta_{5} - 19 \beta_{6} + 56 \beta_{7} + 41 \beta_{8} + 142 \beta_{9} + 20 \beta_{10} - 10 \beta_{11} + 10 \beta_{12} + 5 \beta_{13} ) q^{23} + ( 217867 - 31608 \beta_{1} - 329 \beta_{2} - 142 \beta_{3} - 5544 \beta_{4} - 36 \beta_{5} - 64 \beta_{6} + 19 \beta_{7} + 51 \beta_{8} - 62 \beta_{9} - 78 \beta_{10} + 28 \beta_{11} - 43 \beta_{12} - 5 \beta_{13} ) q^{24} + ( 589269 - 13909 \beta_{1} - 220 \beta_{2} - 66 \beta_{3} - 2670 \beta_{4} + 18 \beta_{5} + 101 \beta_{6} + 16 \beta_{7} - 25 \beta_{8} + 86 \beta_{9} - 57 \beta_{10} + 85 \beta_{11} - 15 \beta_{12} - 26 \beta_{13} ) q^{25} + ( -57122 + 28561 \beta_{1} ) q^{26} + ( 300929 - 29668 \beta_{1} - 2501 \beta_{2} - 60 \beta_{3} - 9041 \beta_{4} + 124 \beta_{5} - 466 \beta_{6} + 116 \beta_{7} + 191 \beta_{8} + 206 \beta_{9} - 85 \beta_{10} + 29 \beta_{11} - 83 \beta_{12} - 10 \beta_{13} ) q^{27} + ( 410571 - 14406 \beta_{1} + 2401 \beta_{2} ) q^{28} + ( -267565 - 34216 \beta_{1} - 3895 \beta_{2} + 47 \beta_{3} - 3738 \beta_{4} - 149 \beta_{5} - 551 \beta_{6} - 204 \beta_{7} - 160 \beta_{8} - 32 \beta_{9} + 153 \beta_{10} + \beta_{11} + 89 \beta_{12} - 17 \beta_{13} ) q^{29} + ( -417119 - 68595 \beta_{1} - 973 \beta_{2} - 124 \beta_{3} - 3214 \beta_{4} - 269 \beta_{5} - 986 \beta_{6} - 52 \beta_{7} - 133 \beta_{8} - 536 \beta_{9} - 40 \beta_{10} - 46 \beta_{11} + 81 \beta_{12} + 117 \beta_{13} ) q^{30} + ( -310546 - 57485 \beta_{1} - 2107 \beta_{2} + 196 \beta_{3} + 1485 \beta_{4} + 38 \beta_{5} - 1550 \beta_{6} + 81 \beta_{7} + 83 \beta_{8} + 296 \beta_{9} + 32 \beta_{10} - 165 \beta_{11} + 248 \beta_{12} + 20 \beta_{13} ) q^{31} + ( 225617 - 33044 \beta_{1} + 2659 \beta_{2} - 172 \beta_{3} + 1262 \beta_{4} - 56 \beta_{5} - 246 \beta_{6} + 25 \beta_{7} - 269 \beta_{8} - 570 \beta_{9} - 200 \beta_{10} + 16 \beta_{11} - 189 \beta_{12} + 33 \beta_{13} ) q^{32} + ( 1258394 - 27001 \beta_{1} - 5252 \beta_{2} - 33 \beta_{3} - 17167 \beta_{4} + \beta_{5} + 428 \beta_{6} + 136 \beta_{7} + 380 \beta_{8} - 154 \beta_{9} - 253 \beta_{10} + 145 \beta_{11} - 185 \beta_{12} + 11 \beta_{13} ) q^{33} + ( -1434206 - 24630 \beta_{1} - 8677 \beta_{2} + 409 \beta_{3} + 9135 \beta_{4} + 366 \beta_{5} - 224 \beta_{6} - 440 \beta_{7} - 176 \beta_{8} + 300 \beta_{9} + 647 \beta_{10} - 89 \beta_{11} + 158 \beta_{12} - 149 \beta_{13} ) q^{34} + ( 511413 - 31213 \beta_{1} + 2401 \beta_{2} + 2401 \beta_{6} ) q^{35} + ( 155282 - 110968 \beta_{1} + 4482 \beta_{2} + 302 \beta_{3} + 10400 \beta_{4} + 14 \beta_{5} - 776 \beta_{6} - 479 \beta_{7} + 45 \beta_{8} - 250 \beta_{9} + 276 \beta_{10} - 220 \beta_{11} + 145 \beta_{12} - 161 \beta_{13} ) q^{36} + ( 2094282 - 104811 \beta_{1} - 7357 \beta_{2} + 557 \beta_{3} + 791 \beta_{4} + 134 \beta_{5} + 865 \beta_{6} - 196 \beta_{7} - 627 \beta_{8} - 177 \beta_{9} + 152 \beta_{10} + 271 \beta_{11} + 48 \beta_{12} + 296 \beta_{13} ) q^{37} + ( -3150778 - 85587 \beta_{1} - 7466 \beta_{2} - 335 \beta_{3} - 2556 \beta_{4} - 87 \beta_{5} - 414 \beta_{6} + 824 \beta_{7} + 627 \beta_{8} - 1057 \beta_{9} - 489 \beta_{10} - 451 \beta_{11} - 385 \beta_{12} - 174 \beta_{13} ) q^{38} + ( -342732 + 28561 \beta_{4} ) q^{39} + ( 9671719 - 466081 \beta_{1} + 12344 \beta_{2} - 1109 \beta_{3} - 1244 \beta_{4} - 156 \beta_{5} + 3673 \beta_{6} + 125 \beta_{7} - 99 \beta_{8} - 805 \beta_{9} - 946 \beta_{10} + 612 \beta_{11} + 139 \beta_{12} + 29 \beta_{13} ) q^{40} + ( 3449948 - 54890 \beta_{1} - 4173 \beta_{2} + 658 \beta_{3} + 36140 \beta_{4} + 236 \beta_{5} + 391 \beta_{6} + 904 \beta_{7} - 337 \beta_{8} + 830 \beta_{9} + 576 \beta_{10} - 334 \beta_{11} - 272 \beta_{12} + 24 \beta_{13} ) q^{41} + ( 84035 - 60025 \beta_{1} + 2401 \beta_{4} - 2401 \beta_{9} ) q^{42} + ( 3773515 - 260341 \beta_{1} + 13634 \beta_{2} + 354 \beta_{3} + 20357 \beta_{4} + 408 \beta_{5} + 3876 \beta_{6} + 880 \beta_{7} + 289 \beta_{8} + 2190 \beta_{9} - 22 \beta_{10} - 972 \beta_{11} - 378 \beta_{12} + 78 \beta_{13} ) q^{43} + ( -1060027 - 163012 \beta_{1} - 6109 \beta_{2} + 260 \beta_{3} + 36077 \beta_{4} + 395 \beta_{5} - 613 \beta_{6} - 802 \beta_{7} + 275 \beta_{8} + 537 \beta_{9} + 593 \beta_{10} + 491 \beta_{11} - 131 \beta_{12} + 132 \beta_{13} ) q^{44} + ( 7069325 + 153082 \beta_{1} + 3292 \beta_{2} - 361 \beta_{3} + 27255 \beta_{4} - 1245 \beta_{5} + 11624 \beta_{6} - 1992 \beta_{7} - 57 \beta_{8} + 140 \beta_{9} - 243 \beta_{10} + 1821 \beta_{11} + 1097 \beta_{12} - 551 \beta_{13} ) q^{45} + ( -2461476 - 345583 \beta_{1} - 20946 \beta_{2} - 426 \beta_{3} + 77980 \beta_{4} + 492 \beta_{5} - 2908 \beta_{6} + 12 \beta_{7} + 1176 \beta_{8} + 1816 \beta_{9} + 1438 \beta_{10} - 262 \beta_{11} - 100 \beta_{12} - 832 \beta_{13} ) q^{46} + ( 11489548 - 182594 \beta_{1} + 6076 \beta_{2} - 1779 \beta_{3} - 16984 \beta_{4} - 661 \beta_{5} + 4287 \beta_{6} + 607 \beta_{7} + 476 \beta_{8} + 1504 \beta_{9} - 997 \beta_{10} + 484 \beta_{11} - 737 \beta_{12} + 813 \beta_{13} ) q^{47} + ( 16249643 - 118654 \beta_{1} + 32979 \beta_{2} - 458 \beta_{3} + 36770 \beta_{4} + 240 \beta_{5} - 3808 \beta_{6} + 3191 \beta_{7} + 85 \beta_{8} - 1236 \beta_{9} - 1712 \beta_{10} - 328 \beta_{11} - 331 \beta_{12} + 303 \beta_{13} ) q^{48} + 5764801 q^{49} + ( 10683950 - 559126 \beta_{1} + 45313 \beta_{2} - 134 \beta_{3} + 86359 \beta_{4} + 863 \beta_{5} - 1510 \beta_{6} - 1040 \beta_{7} - 3085 \beta_{8} + 3559 \beta_{9} + 46 \beta_{10} - 708 \beta_{11} + 461 \beta_{12} + 1005 \beta_{13} ) q^{50} + ( 4037113 + 545783 \beta_{1} - 7205 \beta_{2} + 1295 \beta_{3} + 7748 \beta_{4} - 2211 \beta_{5} + 1918 \beta_{6} - 720 \beta_{7} + 2581 \beta_{8} - 872 \beta_{9} - 710 \beta_{10} + 698 \beta_{11} + 656 \beta_{12} - 457 \beta_{13} ) q^{51} + ( -4883931 + 171366 \beta_{1} - 28561 \beta_{2} ) q^{52} + ( 5799456 - 546977 \beta_{1} + 28596 \beta_{2} - 1972 \beta_{3} + 8065 \beta_{4} + 119 \beta_{5} - 10431 \beta_{6} - 2620 \beta_{7} - 1469 \beta_{8} + 2335 \beta_{9} + 350 \beta_{10} - 1515 \beta_{11} - 144 \beta_{12} + 983 \beta_{13} ) q^{53} + ( 21434257 + 530873 \beta_{1} + 15023 \beta_{2} + 3623 \beta_{3} + 137876 \beta_{4} - 1132 \beta_{5} - 10676 \beta_{6} - 2824 \beta_{7} - 494 \beta_{8} + 1593 \beta_{9} + 3849 \beta_{10} + 813 \beta_{11} + 2536 \beta_{12} + 343 \beta_{13} ) q^{54} + ( 23487804 + 233310 \beta_{1} - 4463 \beta_{2} + 1130 \beta_{3} - 18570 \beta_{4} - 633 \beta_{5} + 8255 \beta_{6} - 970 \beta_{7} - 457 \beta_{8} + 4947 \beta_{9} + 1765 \beta_{10} - 10 \beta_{11} + 1441 \beta_{12} - 2059 \beta_{13} ) q^{55} + ( 7447902 - 333739 \beta_{1} + 12005 \beta_{2} - 2401 \beta_{3} - 9604 \beta_{4} + 7203 \beta_{6} - 2401 \beta_{9} ) q^{56} + ( 10815826 + 51993 \beta_{1} + 42332 \beta_{2} - 143 \beta_{3} - 66342 \beta_{4} + 2891 \beta_{5} + 3692 \beta_{6} - 3264 \beta_{7} - 1788 \beta_{8} + 7742 \beta_{9} + 465 \beta_{10} - 1105 \beta_{11} - 2807 \beta_{12} + 173 \beta_{13} ) q^{57} + ( 23808429 + 1803024 \beta_{1} + 27225 \beta_{2} + 5669 \beta_{3} - 27088 \beta_{4} + 1836 \beta_{5} - 8584 \beta_{6} + 3380 \beta_{7} + 2910 \beta_{8} - 5839 \beta_{9} - 3529 \beta_{10} - 89 \beta_{11} - 936 \beta_{12} - 573 \beta_{13} ) q^{58} + ( 31548674 + 526419 \beta_{1} + 26397 \beta_{2} + 3281 \beta_{3} + 48876 \beta_{4} + 4774 \beta_{5} - 10537 \beta_{6} + 3808 \beta_{7} - 4623 \beta_{8} + 3491 \beta_{9} - 640 \beta_{10} - 3249 \beta_{11} - 80 \beta_{12} - 148 \beta_{13} ) q^{59} + ( 40116648 + 36766 \beta_{1} + 25428 \beta_{2} - 4178 \beta_{3} - 158979 \beta_{4} - 935 \beta_{5} - 22893 \beta_{6} + 6015 \beta_{7} + 860 \beta_{8} - 19269 \beta_{9} - 5899 \beta_{10} + 219 \beta_{11} - 3178 \beta_{12} + 3367 \beta_{13} ) q^{60} + ( 19328020 - 53201 \beta_{1} - 26477 \beta_{2} - 6897 \beta_{3} - 51815 \beta_{4} + 215 \beta_{5} + 13075 \beta_{6} + 1083 \beta_{7} - 1192 \beta_{8} + 9232 \beta_{9} + 544 \beta_{10} - 1359 \beta_{11} - 770 \beta_{12} - 1447 \beta_{13} ) q^{61} + ( 38739510 + 871695 \beta_{1} - 43809 \beta_{2} + 529 \beta_{3} + 52013 \beta_{4} - 1996 \beta_{5} - 48512 \beta_{6} + 66 \beta_{7} + 6680 \beta_{8} + 2388 \beta_{9} + 1295 \beta_{10} + 3423 \beta_{11} - 750 \beta_{12} - 3723 \beta_{13} ) q^{62} + ( 24711092 + 288120 \beta_{1} - 9604 \beta_{2} - 36015 \beta_{4} + 7203 \beta_{6} + 2401 \beta_{7} + 2401 \beta_{8} - 4802 \beta_{9} + 2401 \beta_{11} ) q^{63} + ( 16025055 - 306400 \beta_{1} + 193 \beta_{2} - 5676 \beta_{3} - 69576 \beta_{4} - 6246 \beta_{5} + 17288 \beta_{6} - 1191 \beta_{7} - 4763 \beta_{8} - 6864 \beta_{9} - 4210 \beta_{10} - 290 \beta_{11} + 2293 \beta_{12} - 711 \beta_{13} ) q^{64} + ( -6083493 + 371293 \beta_{1} - 28561 \beta_{2} - 28561 \beta_{6} ) q^{65} + ( 22594518 + 810438 \beta_{1} - 47319 \beta_{2} + 4718 \beta_{3} + 30707 \beta_{4} - 4961 \beta_{5} - 36338 \beta_{6} - 3644 \beta_{7} + 1171 \beta_{8} - 15107 \beta_{9} + 4006 \beta_{10} - 288 \beta_{11} + 2965 \beta_{12} + 827 \beta_{13} ) q^{66} + ( 6656912 + 544931 \beta_{1} - 102223 \beta_{2} - 1739 \beta_{3} - 231811 \beta_{4} - 2859 \beta_{5} - 401 \beta_{6} - 5013 \beta_{7} - 1710 \beta_{8} - 19084 \beta_{9} + 5480 \beta_{10} - 2099 \beta_{11} - 86 \beta_{12} + 449 \beta_{13} ) q^{67} + ( 17931872 + 4307126 \beta_{1} - 48528 \beta_{2} + 15738 \beta_{3} + 108831 \beta_{4} + 9167 \beta_{5} - 14143 \beta_{6} - 3777 \beta_{7} + 2618 \beta_{8} + 28585 \beta_{9} + 11775 \beta_{10} - 1067 \beta_{11} + 2692 \beta_{12} + 255 \beta_{13} ) q^{68} + ( 20766504 + 3872195 \beta_{1} - 94396 \beta_{2} + 5535 \beta_{3} - 526568 \beta_{4} + 4787 \beta_{5} - 30774 \beta_{6} + 4447 \beta_{7} + 2027 \beta_{8} - 12120 \beta_{9} - 2626 \beta_{10} + 2175 \beta_{11} - 2596 \beta_{12} + 3863 \beta_{13} ) q^{69} + ( 21767466 - 1346961 \beta_{1} + 86436 \beta_{2} - 2401 \beta_{3} + 72030 \beta_{4} + 2401 \beta_{5} + 4802 \beta_{6} - 2401 \beta_{8} - 2401 \beta_{9} + 2401 \beta_{10} - 2401 \beta_{11} - 2401 \beta_{12} ) q^{70} + ( 6070134 + 2719487 \beta_{1} + 18242 \beta_{2} - 3643 \beta_{3} - 645571 \beta_{4} - 2786 \beta_{5} + 20722 \beta_{6} + 2994 \beta_{7} + 3944 \beta_{8} + 3667 \beta_{9} - 6344 \beta_{10} - 3871 \beta_{11} - 4408 \beta_{12} + 1072 \beta_{13} ) q^{71} + ( 103956641 + 1519511 \beta_{1} - 97864 \beta_{2} + 8965 \beta_{3} + 26196 \beta_{4} - 3282 \beta_{5} + 53877 \beta_{6} + 945 \beta_{7} - 2075 \beta_{8} - 14647 \beta_{9} + 1738 \beta_{10} + 10142 \beta_{11} + 4849 \beta_{12} - 3219 \beta_{13} ) q^{72} + ( 26150757 + 2296610 \beta_{1} - 1519 \beta_{2} - 5257 \beta_{3} - 142840 \beta_{4} + 4180 \beta_{5} + 69375 \beta_{6} - 10970 \beta_{7} - 1238 \beta_{8} - 3065 \beta_{9} + 7552 \beta_{10} + 3701 \beta_{11} + 7128 \beta_{12} + 4240 \beta_{13} ) q^{73} + ( 77919170 + 948250 \beta_{1} - 17096 \beta_{2} - 3109 \beta_{3} - 21884 \beta_{4} + 7877 \beta_{5} - 9386 \beta_{6} - 7116 \beta_{7} - 6021 \beta_{8} - 2917 \beta_{9} + 2465 \beta_{10} - 1957 \beta_{11} - 4381 \beta_{12} - 4310 \beta_{13} ) q^{74} + ( 82534443 + 3352576 \beta_{1} - 224777 \beta_{2} - 19320 \beta_{3} - 831995 \beta_{4} - 7348 \beta_{5} + 102522 \beta_{6} + 15776 \beta_{7} + 10461 \beta_{8} - 2422 \beta_{9} - 18273 \beta_{10} + 10721 \beta_{11} - 3743 \beta_{12} - 1344 \beta_{13} ) q^{75} + ( 48081986 + 3601114 \beta_{1} + 31906 \beta_{2} - 4748 \beta_{3} - 458077 \beta_{4} - 6007 \beta_{5} - 43739 \beta_{6} - 1962 \beta_{7} + 2709 \beta_{8} + 4843 \beta_{9} - 13113 \beta_{10} + 4445 \beta_{11} - 221 \beta_{12} + 8832 \beta_{13} ) q^{76} + ( 14096271 - 177674 \beta_{1} - 12005 \beta_{2} + 4802 \beta_{3} - 98441 \beta_{4} + 2401 \beta_{5} + 28812 \beta_{6} + 2401 \beta_{7} - 2401 \beta_{9} + 2401 \beta_{10} - 2401 \beta_{11} + 2401 \beta_{12} - 2401 \beta_{13} ) q^{77} + ( -999635 + 714025 \beta_{1} - 28561 \beta_{4} + 28561 \beta_{9} ) q^{78} + ( 31118599 + 3466397 \beta_{1} - 99243 \beta_{2} + 4566 \beta_{3} - 644628 \beta_{4} - 5935 \beta_{5} + 147723 \beta_{6} + 3846 \beta_{7} - 2208 \beta_{8} - 12103 \beta_{9} - 7769 \beta_{10} - 10586 \beta_{11} + 2703 \beta_{12} - 635 \beta_{13} ) q^{79} + ( 189159878 - 6313666 \beta_{1} + 505762 \beta_{2} - 25560 \beta_{3} + 325470 \beta_{4} - 10484 \beta_{5} - 29920 \beta_{6} - 4696 \beta_{7} - 6954 \beta_{8} - 25566 \beta_{9} - 15038 \beta_{10} - 168 \beta_{11} - 8228 \beta_{12} + 4460 \beta_{13} ) q^{80} + ( 45973076 + 4154727 \beta_{1} - 480843 \beta_{2} + 10613 \beta_{3} - 547889 \beta_{4} - 1562 \beta_{5} + 61552 \beta_{6} + 4318 \beta_{7} - 9761 \beta_{8} - 43359 \beta_{9} + 4823 \beta_{10} + 3514 \beta_{11} + 17025 \beta_{12} - 1206 \beta_{13} ) q^{81} + ( 45245223 - 934407 \beta_{1} + 49545 \beta_{2} + 3551 \beta_{3} + 453164 \beta_{4} + 8950 \beta_{5} + 89232 \beta_{6} + 10272 \beta_{7} + 2692 \beta_{8} + 78285 \beta_{9} + 2745 \beta_{10} - 12023 \beta_{11} - 4594 \beta_{12} - 5755 \beta_{13} ) q^{82} + ( 72526210 - 1748146 \beta_{1} + 256156 \beta_{2} - 4921 \beta_{3} - 140545 \beta_{4} - 8890 \beta_{5} - 92231 \beta_{6} - 21256 \beta_{7} - 12947 \beta_{8} + 39619 \beta_{9} + 6079 \beta_{10} - 8626 \beta_{11} + 4305 \beta_{12} - 1450 \beta_{13} ) q^{83} + ( 27124097 - 806736 \beta_{1} + 160867 \beta_{2} - 213689 \beta_{4} - 7203 \beta_{5} + 21609 \beta_{6} - 9604 \beta_{7} - 2401 \beta_{8} - 21609 \beta_{9} - 2401 \beta_{10} + 2401 \beta_{11} + 2401 \beta_{12} + 4802 \beta_{13} ) q^{84} + ( 78667008 + 3566285 \beta_{1} - 155684 \beta_{2} + 20425 \beta_{3} - 140626 \beta_{4} + 12637 \beta_{5} + 45412 \beta_{6} + 20462 \beta_{7} + 18712 \beta_{8} + 78108 \beta_{9} + 18717 \beta_{10} - 5525 \beta_{11} - 899 \beta_{12} - 9287 \beta_{13} ) q^{85} + ( 179873552 - 9618307 \beta_{1} + 120975 \beta_{2} - 16044 \beta_{3} + 1146497 \beta_{4} - 2637 \beta_{5} + 92298 \beta_{6} + 13448 \beta_{7} + 2171 \beta_{8} + 25763 \beta_{9} + 9692 \beta_{10} - 4510 \beta_{11} - 6263 \beta_{12} - 8077 \beta_{13} ) q^{86} + ( 74053024 - 1744562 \beta_{1} - 25059 \beta_{2} - 5364 \beta_{3} + 709598 \beta_{4} + 12963 \beta_{5} - 142641 \beta_{6} - 8172 \beta_{7} + 1773 \beta_{8} - 16919 \beta_{9} + 19779 \beta_{10} - 12900 \beta_{11} - 13437 \beta_{12} - 9777 \beta_{13} ) q^{87} + ( 75963339 + 4931760 \beta_{1} - 139445 \beta_{2} + 3918 \beta_{3} + 631664 \beta_{4} + 6896 \beta_{5} + 40016 \beta_{6} + 1535 \beta_{7} + 18447 \beta_{8} + 23422 \beta_{9} - 3822 \beta_{10} + 11816 \beta_{11} + 9309 \beta_{12} + 167 \beta_{13} ) q^{88} + ( 76272467 + 3986352 \beta_{1} - 439692 \beta_{2} - 3071 \beta_{3} - 379632 \beta_{4} - 13496 \beta_{5} - 29838 \beta_{6} - 6264 \beta_{7} + 2195 \beta_{8} + 52757 \beta_{9} - 6490 \beta_{10} + 9507 \beta_{11} - 6294 \beta_{12} - 428 \beta_{13} ) q^{89} + ( -90802365 - 5789556 \beta_{1} + 79412 \beta_{2} + 12364 \beta_{3} + 833435 \beta_{4} + 26420 \beta_{5} - 281620 \beta_{6} - 11004 \beta_{7} - 11124 \beta_{8} + 15919 \beta_{9} + 9832 \beta_{10} - 13164 \beta_{11} - 6396 \beta_{12} + 5178 \beta_{13} ) q^{90} -68574961 q^{91} + ( 163349607 + 9501342 \beta_{1} + 413461 \beta_{2} + 60784 \beta_{3} + 1121274 \beta_{4} + 8406 \beta_{5} + 29934 \beta_{6} - 30198 \beta_{7} - 1760 \beta_{8} + 130094 \beta_{9} + 26126 \beta_{10} + 1066 \beta_{11} + 15380 \beta_{12} - 434 \beta_{13} ) q^{92} + ( -67033981 + 2275335 \beta_{1} - 162923 \beta_{2} + 28709 \beta_{3} + 401746 \beta_{4} + 45178 \beta_{5} - 174590 \beta_{6} - 642 \beta_{7} + 813 \beta_{8} - 96215 \beta_{9} + 12717 \beta_{10} - 2888 \beta_{11} - 27037 \beta_{12} + 18724 \beta_{13} ) q^{93} + ( 145110789 - 14107856 \beta_{1} + 565159 \beta_{2} - 49958 \beta_{3} + 1460304 \beta_{4} - 3913 \beta_{5} + 38034 \beta_{6} + 43162 \beta_{7} + 12737 \beta_{8} - 55162 \beta_{9} - 40630 \beta_{10} - 5136 \beta_{11} - 22005 \beta_{12} + 4475 \beta_{13} ) q^{94} + ( 178648945 + 6428351 \beta_{1} - 413976 \beta_{2} + 37434 \beta_{3} - 714026 \beta_{4} - 5269 \beta_{5} - 113050 \beta_{6} + 1476 \beta_{7} - 32899 \beta_{8} - 101509 \beta_{9} - 20605 \beta_{10} + 12282 \beta_{11} + 14151 \beta_{12} - 5667 \beta_{13} ) q^{95} + ( -7858293 - 15869984 \beta_{1} + 525617 \beta_{2} - 25704 \beta_{3} + 2021064 \beta_{4} - 17410 \beta_{5} + 124472 \beta_{6} + 4029 \beta_{7} - 42423 \beta_{8} + 66204 \beta_{9} - 1722 \beta_{10} - 18878 \beta_{11} + 9365 \beta_{12} + 9525 \beta_{13} ) q^{96} + ( 203322341 - 10466288 \beta_{1} + 670105 \beta_{2} - 17097 \beta_{3} + 130907 \beta_{4} - 52929 \beta_{5} - 284651 \beta_{6} - 48530 \beta_{7} - 13628 \beta_{8} - 45452 \beta_{9} + 6675 \beta_{10} + 38149 \beta_{11} + 28191 \beta_{12} + 14141 \beta_{13} ) q^{97} + ( 11529602 - 5764801 \beta_{1} ) q^{98} + ( 361745986 + 4384245 \beta_{1} + 5828 \beta_{2} - 3367 \beta_{3} - 1167665 \beta_{4} - 12950 \beta_{5} + 110242 \beta_{6} + 1172 \beta_{7} + 15446 \beta_{8} - 65201 \beta_{9} + 6332 \beta_{10} + 48191 \beta_{11} + 22104 \beta_{12} - 7344 \beta_{13} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 27 q^{2} + 163 q^{3} + 2389 q^{4} + 2964 q^{5} + 471 q^{6} + 33614 q^{7} + 43263 q^{8} + 144129 q^{9} + O(q^{10}) \) \( 14 q + 27 q^{2} + 163 q^{3} + 2389 q^{4} + 2964 q^{5} + 471 q^{6} + 33614 q^{7} + 43263 q^{8} + 144129 q^{9} + 126524 q^{10} + 81825 q^{11} + 157399 q^{12} - 399854 q^{13} + 64827 q^{14} + 163856 q^{15} + 166361 q^{16} - 44922 q^{17} - 826396 q^{18} + 171756 q^{19} + 3899724 q^{20} + 391363 q^{21} + 917579 q^{22} + 1930479 q^{23} + 2992373 q^{24} + 8222344 q^{25} - 771147 q^{26} + 4139125 q^{27} + 5735989 q^{28} - 3799608 q^{29} - 5918004 q^{30} - 4392203 q^{31} + 3135663 q^{32} + 17499977 q^{33} - 20071132 q^{34} + 7116564 q^{35} + 2121398 q^{36} + 29198909 q^{37} - 44208366 q^{38} - 4655443 q^{39} + 134932928 q^{40} + 48410973 q^{41} + 1130871 q^{42} + 52650242 q^{43} - 14827353 q^{44} + 99215088 q^{45} - 34410455 q^{46} + 160580841 q^{47} + 227620515 q^{48} + 80707214 q^{49} + 149462949 q^{50} + 57114360 q^{51} - 68232229 q^{52} + 80753796 q^{53} + 301368833 q^{54} + 328919412 q^{55} + 103874463 q^{56} + 151101102 q^{57} + 335044204 q^{58} + 442445502 q^{59} + 561078360 q^{60} + 270199089 q^{61} + 543824517 q^{62} + 346053729 q^{63} + 223643137 q^{64} - 84654804 q^{65} + 317483345 q^{66} + 92500909 q^{67} + 255771204 q^{68} + 292017029 q^{69} + 303784124 q^{70} + 84383796 q^{71} + 1456696818 q^{72} + 367274315 q^{73} + 1091659407 q^{74} + 1154152501 q^{75} + 674789222 q^{76} + 196461825 q^{77} - 13452231 q^{78} + 434861545 q^{79} + 2644363752 q^{80} + 644207518 q^{81} + 634104331 q^{82} + 1013603934 q^{83} + 377914999 q^{84} + 1103701048 q^{85} + 2514069096 q^{86} + 1039292304 q^{87} + 1071310221 q^{88} + 1069739706 q^{89} - 1271572324 q^{90} - 960049454 q^{91} + 2301673917 q^{92} - 933838861 q^{93} + 2025486277 q^{94} + 2504029998 q^{95} - 116199027 q^{96} + 2839636281 q^{97} + 155649627 q^{98} + 5063037274 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} - x^{13} - 4752 x^{12} + 9346 x^{11} + 8576824 x^{10} - 26923636 x^{9} - 7450416552 x^{8} + 31594524240 x^{7} + 3232668379296 x^{6} - 15784728192704 x^{5} - 654706244016512 x^{4} + 3031992312058112 x^{3} + 50459693673905664 x^{2} - 177878730644221952 x - 247584423217725440\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 2 \nu - 679 \)
\(\beta_{3}\)\(=\)\((\)\(\)\(64\!\cdots\!57\)\( \nu^{13} - \)\(66\!\cdots\!99\)\( \nu^{12} - \)\(10\!\cdots\!34\)\( \nu^{11} + \)\(29\!\cdots\!74\)\( \nu^{10} + \)\(29\!\cdots\!52\)\( \nu^{9} - \)\(47\!\cdots\!52\)\( \nu^{8} - \)\(22\!\cdots\!96\)\( \nu^{7} + \)\(35\!\cdots\!00\)\( \nu^{6} - \)\(24\!\cdots\!16\)\( \nu^{5} - \)\(11\!\cdots\!32\)\( \nu^{4} + \)\(70\!\cdots\!40\)\( \nu^{3} + \)\(12\!\cdots\!28\)\( \nu^{2} - \)\(15\!\cdots\!40\)\( \nu + \)\(10\!\cdots\!40\)\(\)\()/ \)\(45\!\cdots\!20\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(13\!\cdots\!69\)\( \nu^{13} - \)\(22\!\cdots\!05\)\( \nu^{12} + \)\(59\!\cdots\!10\)\( \nu^{11} + \)\(92\!\cdots\!26\)\( \nu^{10} - \)\(97\!\cdots\!08\)\( \nu^{9} - \)\(13\!\cdots\!80\)\( \nu^{8} + \)\(74\!\cdots\!92\)\( \nu^{7} + \)\(80\!\cdots\!24\)\( \nu^{6} - \)\(26\!\cdots\!44\)\( \nu^{5} - \)\(21\!\cdots\!68\)\( \nu^{4} + \)\(41\!\cdots\!40\)\( \nu^{3} + \)\(20\!\cdots\!40\)\( \nu^{2} - \)\(18\!\cdots\!24\)\( \nu - \)\(57\!\cdots\!92\)\(\)\()/ \)\(91\!\cdots\!64\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(39\!\cdots\!31\)\( \nu^{13} + \)\(44\!\cdots\!97\)\( \nu^{12} + \)\(19\!\cdots\!62\)\( \nu^{11} - \)\(20\!\cdots\!42\)\( \nu^{10} - \)\(32\!\cdots\!16\)\( \nu^{9} + \)\(32\!\cdots\!96\)\( \nu^{8} + \)\(21\!\cdots\!08\)\( \nu^{7} - \)\(23\!\cdots\!00\)\( \nu^{6} - \)\(17\!\cdots\!52\)\( \nu^{5} + \)\(74\!\cdots\!96\)\( \nu^{4} - \)\(27\!\cdots\!20\)\( \nu^{3} - \)\(74\!\cdots\!84\)\( \nu^{2} + \)\(54\!\cdots\!80\)\( \nu + \)\(27\!\cdots\!00\)\(\)\()/ \)\(91\!\cdots\!40\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(20\!\cdots\!17\)\( \nu^{13} + \)\(19\!\cdots\!56\)\( \nu^{12} - \)\(90\!\cdots\!69\)\( \nu^{11} - \)\(77\!\cdots\!46\)\( \nu^{10} + \)\(14\!\cdots\!22\)\( \nu^{9} + \)\(10\!\cdots\!48\)\( \nu^{8} - \)\(10\!\cdots\!36\)\( \nu^{7} - \)\(55\!\cdots\!60\)\( \nu^{6} + \)\(32\!\cdots\!24\)\( \nu^{5} + \)\(12\!\cdots\!48\)\( \nu^{4} - \)\(27\!\cdots\!20\)\( \nu^{3} - \)\(14\!\cdots\!92\)\( \nu^{2} - \)\(12\!\cdots\!20\)\( \nu + \)\(47\!\cdots\!00\)\(\)\()/ \)\(38\!\cdots\!60\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(67\!\cdots\!27\)\( \nu^{13} - \)\(86\!\cdots\!23\)\( \nu^{12} + \)\(18\!\cdots\!98\)\( \nu^{11} + \)\(36\!\cdots\!50\)\( \nu^{10} - \)\(47\!\cdots\!16\)\( \nu^{9} - \)\(55\!\cdots\!60\)\( \nu^{8} - \)\(22\!\cdots\!92\)\( \nu^{7} + \)\(37\!\cdots\!36\)\( \nu^{6} + \)\(20\!\cdots\!16\)\( \nu^{5} - \)\(11\!\cdots\!52\)\( \nu^{4} - \)\(59\!\cdots\!68\)\( \nu^{3} + \)\(11\!\cdots\!32\)\( \nu^{2} + \)\(57\!\cdots\!52\)\( \nu - \)\(12\!\cdots\!00\)\(\)\()/ \)\(91\!\cdots\!40\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(38\!\cdots\!01\)\( \nu^{13} + \)\(13\!\cdots\!77\)\( \nu^{12} - \)\(15\!\cdots\!50\)\( \nu^{11} - \)\(54\!\cdots\!86\)\( \nu^{10} + \)\(21\!\cdots\!64\)\( \nu^{9} + \)\(78\!\cdots\!08\)\( \nu^{8} - \)\(10\!\cdots\!32\)\( \nu^{7} - \)\(50\!\cdots\!72\)\( \nu^{6} + \)\(10\!\cdots\!72\)\( \nu^{5} + \)\(14\!\cdots\!12\)\( \nu^{4} + \)\(57\!\cdots\!36\)\( \nu^{3} - \)\(17\!\cdots\!56\)\( \nu^{2} - \)\(75\!\cdots\!04\)\( \nu + \)\(29\!\cdots\!80\)\(\)\()/ \)\(30\!\cdots\!80\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(20\!\cdots\!67\)\( \nu^{13} + \)\(36\!\cdots\!63\)\( \nu^{12} - \)\(87\!\cdots\!70\)\( \nu^{11} - \)\(15\!\cdots\!70\)\( \nu^{10} + \)\(13\!\cdots\!40\)\( \nu^{9} + \)\(22\!\cdots\!56\)\( \nu^{8} - \)\(10\!\cdots\!08\)\( \nu^{7} - \)\(14\!\cdots\!08\)\( \nu^{6} + \)\(34\!\cdots\!12\)\( \nu^{5} + \)\(40\!\cdots\!84\)\( \nu^{4} - \)\(49\!\cdots\!48\)\( \nu^{3} - \)\(43\!\cdots\!72\)\( \nu^{2} + \)\(18\!\cdots\!76\)\( \nu + \)\(31\!\cdots\!88\)\(\)\()/ \)\(91\!\cdots\!64\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(22\!\cdots\!11\)\( \nu^{13} - \)\(10\!\cdots\!75\)\( \nu^{12} + \)\(91\!\cdots\!86\)\( \nu^{11} + \)\(44\!\cdots\!62\)\( \nu^{10} - \)\(13\!\cdots\!40\)\( \nu^{9} - \)\(67\!\cdots\!56\)\( \nu^{8} + \)\(84\!\cdots\!40\)\( \nu^{7} + \)\(46\!\cdots\!96\)\( \nu^{6} - \)\(24\!\cdots\!32\)\( \nu^{5} - \)\(13\!\cdots\!28\)\( \nu^{4} + \)\(31\!\cdots\!72\)\( \nu^{3} + \)\(13\!\cdots\!16\)\( \nu^{2} - \)\(57\!\cdots\!08\)\( \nu - \)\(45\!\cdots\!40\)\(\)\()/ \)\(91\!\cdots\!40\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(91\!\cdots\!79\)\( \nu^{13} + \)\(22\!\cdots\!59\)\( \nu^{12} - \)\(41\!\cdots\!18\)\( \nu^{11} - \)\(95\!\cdots\!42\)\( \nu^{10} + \)\(69\!\cdots\!92\)\( \nu^{9} + \)\(15\!\cdots\!92\)\( \nu^{8} - \)\(56\!\cdots\!32\)\( \nu^{7} - \)\(10\!\cdots\!80\)\( \nu^{6} + \)\(22\!\cdots\!12\)\( \nu^{5} + \)\(35\!\cdots\!32\)\( \nu^{4} - \)\(40\!\cdots\!24\)\( \nu^{3} - \)\(42\!\cdots\!12\)\( \nu^{2} + \)\(21\!\cdots\!84\)\( \nu + \)\(31\!\cdots\!36\)\(\)\()/ \)\(18\!\cdots\!28\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(11\!\cdots\!09\)\( \nu^{13} - \)\(16\!\cdots\!47\)\( \nu^{12} + \)\(49\!\cdots\!48\)\( \nu^{11} + \)\(63\!\cdots\!22\)\( \nu^{10} - \)\(79\!\cdots\!84\)\( \nu^{9} - \)\(88\!\cdots\!56\)\( \nu^{8} + \)\(57\!\cdots\!52\)\( \nu^{7} + \)\(51\!\cdots\!00\)\( \nu^{6} - \)\(18\!\cdots\!08\)\( \nu^{5} - \)\(13\!\cdots\!36\)\( \nu^{4} + \)\(20\!\cdots\!00\)\( \nu^{3} + \)\(16\!\cdots\!44\)\( \nu^{2} + \)\(12\!\cdots\!60\)\( \nu - \)\(42\!\cdots\!80\)\(\)\()/ \)\(11\!\cdots\!80\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(77\!\cdots\!61\)\( \nu^{13} - \)\(13\!\cdots\!45\)\( \nu^{12} + \)\(34\!\cdots\!26\)\( \nu^{11} + \)\(54\!\cdots\!42\)\( \nu^{10} - \)\(55\!\cdots\!80\)\( \nu^{9} - \)\(79\!\cdots\!56\)\( \nu^{8} + \)\(41\!\cdots\!00\)\( \nu^{7} + \)\(49\!\cdots\!36\)\( \nu^{6} - \)\(14\!\cdots\!52\)\( \nu^{5} - \)\(13\!\cdots\!68\)\( \nu^{4} + \)\(21\!\cdots\!72\)\( \nu^{3} + \)\(12\!\cdots\!36\)\( \nu^{2} - \)\(76\!\cdots\!88\)\( \nu + \)\(92\!\cdots\!60\)\(\)\()/ \)\(45\!\cdots\!20\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 2 \beta_{1} + 679\)
\(\nu^{3}\)\(=\)\(\beta_{9} - 3 \beta_{6} + 4 \beta_{4} + \beta_{3} + \beta_{2} + 1139 \beta_{1} - 1068\)
\(\nu^{4}\)\(=\)\(9 \beta_{13} - 5 \beta_{12} - 10 \beta_{10} + 2 \beta_{9} - 7 \beta_{8} + 5 \beta_{7} + 40 \beta_{6} + 4 \beta_{5} - 448 \beta_{4} + 2 \beta_{3} + 1703 \beta_{2} - 2712 \beta_{1} + 774355\)
\(\nu^{5}\)\(=\)\(57 \beta_{13} + 139 \beta_{12} - 16 \beta_{11} + 100 \beta_{10} + 2598 \beta_{9} + 199 \beta_{8} + 25 \beta_{7} - 5378 \beta_{6} + 96 \beta_{5} + 2290 \beta_{4} + 2200 \beta_{3} + 4171 \beta_{2} + 1555516 \beta_{1} - 1359331\)
\(\nu^{6}\)\(=\)\(22473 \beta_{13} - 8539 \beta_{12} - 482 \beta_{11} - 28010 \beta_{10} + 8992 \beta_{9} - 19875 \beta_{8} + 11609 \beta_{7} + 113712 \beta_{6} + 4906 \beta_{5} - 1243376 \beta_{4} + 5404 \beta_{3} + 2775761 \beta_{2} - 2331632 \beta_{1} + 1058849655\)
\(\nu^{7}\)\(=\)\(166839 \beta_{13} + 450971 \beta_{12} - 29406 \beta_{11} + 389990 \beta_{10} + 5351348 \beta_{9} + 463823 \beta_{8} - 62865 \beta_{7} - 8126240 \beta_{6} + 266430 \beta_{5} - 3501380 \beta_{4} + 4143764 \beta_{3} + 10889863 \beta_{2} + 2324624612 \beta_{1} - 806964759\)
\(\nu^{8}\)\(=\)\(45212397 \beta_{13} - 12253027 \beta_{12} - 675838 \beta_{11} - 59179538 \beta_{10} + 24047884 \beta_{9} - 41169943 \beta_{8} + 21878741 \beta_{7} + 233781560 \beta_{6} + 1882670 \beta_{5} - 2651444072 \beta_{4} + 10906232 \beta_{3} + 4567368629 \beta_{2} - 321174376 \beta_{1} + 1584160438431\)
\(\nu^{9}\)\(=\)\(373693479 \beta_{13} + 1029825399 \beta_{12} - 43167866 \beta_{11} + 981276858 \beta_{10} + 10194694092 \beta_{9} + 804596651 \beta_{8} - 327648225 \beta_{7} - 12129012144 \beta_{6} + 565614890 \beta_{5} - 15803701976 \beta_{4} + 7453251472 \beta_{3} + 24194685119 \beta_{2} + 3656441800656 \beta_{1} + 1051406974421\)
\(\nu^{10}\)\(=\)\(84497221453 \beta_{13} - 17004072115 \beta_{12} - 496041614 \beta_{11} - 112706667682 \beta_{10} + 53523605148 \beta_{9} - 76892259527 \beta_{8} + 38543884469 \beta_{7} + 429418987928 \beta_{6} - 6836886114 \beta_{5} - 5151892940680 \beta_{4} + 21473009160 \beta_{3} + 7612613144301 \beta_{2} + 4717927256296 \beta_{1} + 2494205825161607\)
\(\nu^{11}\)\(=\)\(761884384903 \beta_{13} + 2053769756647 \beta_{12} - 64990791498 \beta_{11} + 2057998443082 \beta_{10} + 18702580757268 \beta_{9} + 1272899982587 \beta_{8} - 869562659361 \beta_{7} - 18438553252488 \beta_{6} + 1090226734042 \beta_{5} - 39404566045192 \beta_{4} + 13132832163880 \beta_{3} + 49896357551943 \beta_{2} + 5935247579580120 \beta_{1} + 5313443647356709\)
\(\nu^{12}\)\(=\)\(152576903255957 \beta_{13} - 23489568476763 \beta_{12} + 195383404914 \beta_{11} - 204409231580338 \beta_{10} + 109580433247180 \beta_{9} - 137405816296127 \beta_{8} + 66015861992477 \beta_{7} + 752689572901496 \beta_{6} - 24685867788674 \beta_{5} - 9565006605932808 \beta_{4} + 42978904431032 \beta_{3} + 12818188739223237 \beta_{2} + 15660947162435016 \beta_{1} + 4052171210194371167\)
\(\nu^{13}\)\(=\)\(1484922819394479 \beta_{13} + 3835560510341407 \beta_{12} - 105595703369098 \beta_{11} + 3929311777879146 \beta_{10} + 33567784526722052 \beta_{9} + 1950252710875859 \beta_{8} - 1862184869414265 \beta_{7} - 28651060073133720 \beta_{6} + 2002753882676506 \beta_{5} - 83287147892104504 \beta_{4} + 22895079576540008 \beta_{3} + 98714011252899519 \beta_{2} + 9830041532425062008 \beta_{1} + 14182845921622361837\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
41.8127
33.1611
26.5220
21.0054
19.1573
15.6702
4.59170
−1.08476
−12.8857
−14.8710
−25.1291
−31.6981
−34.2541
−40.9975
−39.8127 138.193 1073.05 553.584 −5501.81 2401.00 −22336.8 −585.818 −22039.7
1.2 −31.1611 −183.696 459.017 −1660.96 5724.18 2401.00 1651.02 14061.3 51757.3
1.3 −24.5220 116.541 89.3306 −1444.79 −2857.83 2401.00 10364.7 −6101.17 35429.2
1.4 −19.0054 −267.441 −150.796 1106.90 5082.81 2401.00 12596.7 51841.8 −21037.1
1.5 −17.1573 244.102 −217.626 2666.11 −4188.13 2401.00 12518.4 39902.6 −45743.4
1.6 −13.6702 −25.3649 −325.126 344.296 346.743 2401.00 11443.7 −19039.6 −4706.60
1.7 −2.59170 254.265 −505.283 −1903.58 −658.979 2401.00 2636.49 44967.7 4933.51
1.8 3.08476 −166.247 −502.484 1085.59 −512.832 2401.00 −3129.44 7955.00 3348.80
1.9 14.8857 54.8703 −290.415 1171.15 816.786 2401.00 −11944.5 −16672.2 17433.4
1.10 16.8710 −26.4361 −227.369 −2366.09 −446.004 2401.00 −12473.9 −18984.1 −39918.3
1.11 27.1291 −167.275 223.990 −1059.06 −4538.03 2401.00 −7813.47 8297.99 −28731.4
1.12 33.6981 231.670 623.565 224.065 7806.84 2401.00 3759.53 33987.9 7550.59
1.13 36.2541 −166.815 802.361 2128.44 −6047.74 2401.00 10526.8 8144.30 77164.9
1.14 42.9975 126.635 1336.79 2118.33 5444.99 2401.00 35463.9 −3646.62 91082.8
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(13\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.10.a.c 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.10.a.c 14 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(28\!\cdots\!92\)\( T_{2}^{6} - \)\(22\!\cdots\!00\)\( T_{2}^{5} - \)\(61\!\cdots\!80\)\( T_{2}^{4} + \)\(23\!\cdots\!96\)\( T_{2}^{3} + \)\(52\!\cdots\!40\)\( T_{2}^{2} - \)\(38\!\cdots\!16\)\( T_{2} - \)\(38\!\cdots\!08\)\( \)">\(T_{2}^{14} - \cdots\) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(91))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -388018559954321408 - 38757753051119616 T + 52459614902435840 T^{2} + 2315491181469696 T^{3} - 618084913262080 T^{4} - 22338126537600 T^{5} + 2851128216192 T^{6} + 83326151472 T^{7} - 6416361792 T^{8} - 138358068 T^{9} + 7541636 T^{10} + 102102 T^{11} - 4414 T^{12} - 27 T^{13} + T^{14} \)
$3$ \( -\)\(24\!\cdots\!20\)\( - \)\(12\!\cdots\!24\)\( T + \)\(14\!\cdots\!80\)\( T^{2} + \)\(87\!\cdots\!48\)\( T^{3} - \)\(77\!\cdots\!60\)\( T^{4} - \)\(11\!\cdots\!72\)\( T^{5} + 9923145536387783808 T^{6} + 70069440473321544 T^{7} - 544886479526616 T^{8} - 2079077196156 T^{9} + 14832154624 T^{10} + 29964431 T^{11} - 196561 T^{12} - 163 T^{13} + T^{14} \)
$5$ \( -\)\(82\!\cdots\!00\)\( + \)\(82\!\cdots\!00\)\( T - \)\(25\!\cdots\!00\)\( T^{2} + \)\(18\!\cdots\!50\)\( T^{3} + \)\(44\!\cdots\!85\)\( T^{4} - \)\(65\!\cdots\!88\)\( T^{5} - \)\(17\!\cdots\!07\)\( T^{6} + \)\(60\!\cdots\!96\)\( T^{7} - 64084684021470397214 T^{8} - 240012706574222784 T^{9} + 58247350908802 T^{10} + 43824749658 T^{11} - 13390399 T^{12} - 2964 T^{13} + T^{14} \)
$7$ \( ( -2401 + T )^{14} \)
$11$ \( -\)\(15\!\cdots\!16\)\( - \)\(82\!\cdots\!20\)\( T - \)\(32\!\cdots\!96\)\( T^{2} + \)\(64\!\cdots\!80\)\( T^{3} + \)\(22\!\cdots\!12\)\( T^{4} - \)\(20\!\cdots\!88\)\( T^{5} - \)\(38\!\cdots\!52\)\( T^{6} + \)\(32\!\cdots\!72\)\( T^{7} + \)\(15\!\cdots\!72\)\( T^{8} - \)\(25\!\cdots\!32\)\( T^{9} + 12054937327049929588 T^{10} + 820098778692165 T^{11} - 7913513657 T^{12} - 81825 T^{13} + T^{14} \)
$13$ \( ( 28561 + T )^{14} \)
$17$ \( -\)\(57\!\cdots\!12\)\( - \)\(52\!\cdots\!84\)\( T + \)\(56\!\cdots\!28\)\( T^{2} + \)\(61\!\cdots\!40\)\( T^{3} - \)\(61\!\cdots\!08\)\( T^{4} - \)\(28\!\cdots\!44\)\( T^{5} + \)\(22\!\cdots\!60\)\( T^{6} + \)\(43\!\cdots\!68\)\( T^{7} - \)\(36\!\cdots\!08\)\( T^{8} - \)\(23\!\cdots\!12\)\( T^{9} + \)\(26\!\cdots\!92\)\( T^{10} + 30191112750073560 T^{11} - 853542087124 T^{12} + 44922 T^{13} + T^{14} \)
$19$ \( -\)\(42\!\cdots\!28\)\( + \)\(26\!\cdots\!72\)\( T + \)\(82\!\cdots\!08\)\( T^{2} - \)\(25\!\cdots\!26\)\( T^{3} - \)\(32\!\cdots\!63\)\( T^{4} + \)\(14\!\cdots\!60\)\( T^{5} + \)\(31\!\cdots\!37\)\( T^{6} + \)\(99\!\cdots\!16\)\( T^{7} - \)\(13\!\cdots\!86\)\( T^{8} - \)\(17\!\cdots\!52\)\( T^{9} + \)\(26\!\cdots\!06\)\( T^{10} + 322867042467008742 T^{11} - 2658222650731 T^{12} - 171756 T^{13} + T^{14} \)
$23$ \( \)\(10\!\cdots\!00\)\( + \)\(20\!\cdots\!25\)\( T + \)\(59\!\cdots\!61\)\( T^{2} - \)\(66\!\cdots\!86\)\( T^{3} - \)\(19\!\cdots\!02\)\( T^{4} + \)\(12\!\cdots\!75\)\( T^{5} + \)\(70\!\cdots\!67\)\( T^{6} + \)\(18\!\cdots\!36\)\( T^{7} - \)\(75\!\cdots\!08\)\( T^{8} - \)\(31\!\cdots\!69\)\( T^{9} + \)\(37\!\cdots\!19\)\( T^{10} + 13871178886717584870 T^{11} - 9390056446138 T^{12} - 1930479 T^{13} + T^{14} \)
$29$ \( \)\(52\!\cdots\!08\)\( + \)\(15\!\cdots\!24\)\( T - \)\(40\!\cdots\!40\)\( T^{2} - \)\(12\!\cdots\!66\)\( T^{3} - \)\(27\!\cdots\!55\)\( T^{4} + \)\(62\!\cdots\!64\)\( T^{5} + \)\(21\!\cdots\!61\)\( T^{6} - \)\(11\!\cdots\!20\)\( T^{7} - \)\(42\!\cdots\!02\)\( T^{8} + \)\(90\!\cdots\!00\)\( T^{9} + \)\(32\!\cdots\!06\)\( T^{10} - \)\(31\!\cdots\!94\)\( T^{11} - 98068268646871 T^{12} + 3799608 T^{13} + T^{14} \)
$31$ \( \)\(17\!\cdots\!92\)\( - \)\(50\!\cdots\!29\)\( T + \)\(28\!\cdots\!73\)\( T^{2} + \)\(70\!\cdots\!82\)\( T^{3} - \)\(69\!\cdots\!70\)\( T^{4} + \)\(71\!\cdots\!69\)\( T^{5} + \)\(49\!\cdots\!23\)\( T^{6} - \)\(24\!\cdots\!48\)\( T^{7} - \)\(17\!\cdots\!80\)\( T^{8} + \)\(82\!\cdots\!05\)\( T^{9} + \)\(30\!\cdots\!39\)\( T^{10} - \)\(10\!\cdots\!98\)\( T^{11} - 280596677221674 T^{12} + 4392203 T^{13} + T^{14} \)
$37$ \( \)\(21\!\cdots\!00\)\( + \)\(63\!\cdots\!00\)\( T - \)\(10\!\cdots\!60\)\( T^{2} + \)\(20\!\cdots\!56\)\( T^{3} + \)\(30\!\cdots\!84\)\( T^{4} - \)\(81\!\cdots\!28\)\( T^{5} - \)\(27\!\cdots\!92\)\( T^{6} + \)\(11\!\cdots\!04\)\( T^{7} + \)\(21\!\cdots\!72\)\( T^{8} - \)\(77\!\cdots\!96\)\( T^{9} + \)\(13\!\cdots\!62\)\( T^{10} + \)\(24\!\cdots\!65\)\( T^{11} - 672005573127447 T^{12} - 29198909 T^{13} + T^{14} \)
$41$ \( \)\(13\!\cdots\!88\)\( - \)\(62\!\cdots\!56\)\( T - \)\(13\!\cdots\!64\)\( T^{2} + \)\(80\!\cdots\!08\)\( T^{3} + \)\(64\!\cdots\!12\)\( T^{4} - \)\(21\!\cdots\!84\)\( T^{5} + \)\(97\!\cdots\!80\)\( T^{6} + \)\(16\!\cdots\!20\)\( T^{7} - \)\(13\!\cdots\!44\)\( T^{8} - \)\(55\!\cdots\!44\)\( T^{9} + \)\(68\!\cdots\!82\)\( T^{10} + \)\(84\!\cdots\!37\)\( T^{11} - 1401448986744707 T^{12} - 48410973 T^{13} + T^{14} \)
$43$ \( -\)\(65\!\cdots\!00\)\( - \)\(91\!\cdots\!00\)\( T - \)\(12\!\cdots\!00\)\( T^{2} + \)\(36\!\cdots\!00\)\( T^{3} + \)\(28\!\cdots\!25\)\( T^{4} - \)\(50\!\cdots\!34\)\( T^{5} - \)\(17\!\cdots\!79\)\( T^{6} + \)\(31\!\cdots\!08\)\( T^{7} + \)\(30\!\cdots\!10\)\( T^{8} - \)\(89\!\cdots\!52\)\( T^{9} + \)\(45\!\cdots\!30\)\( T^{10} + \)\(11\!\cdots\!60\)\( T^{11} - 1629419639151555 T^{12} - 52650242 T^{13} + T^{14} \)
$47$ \( \)\(78\!\cdots\!84\)\( + \)\(24\!\cdots\!39\)\( T - \)\(30\!\cdots\!11\)\( T^{2} - \)\(94\!\cdots\!90\)\( T^{3} + \)\(76\!\cdots\!74\)\( T^{4} + \)\(76\!\cdots\!29\)\( T^{5} - \)\(60\!\cdots\!81\)\( T^{6} - \)\(14\!\cdots\!56\)\( T^{7} + \)\(16\!\cdots\!60\)\( T^{8} - \)\(25\!\cdots\!47\)\( T^{9} - \)\(18\!\cdots\!81\)\( T^{10} + \)\(26\!\cdots\!10\)\( T^{11} + 5741999010187402 T^{12} - 160580841 T^{13} + T^{14} \)
$53$ \( \)\(51\!\cdots\!16\)\( + \)\(12\!\cdots\!28\)\( T - \)\(35\!\cdots\!88\)\( T^{2} + \)\(72\!\cdots\!14\)\( T^{3} + \)\(16\!\cdots\!49\)\( T^{4} - \)\(66\!\cdots\!84\)\( T^{5} - \)\(21\!\cdots\!31\)\( T^{6} + \)\(12\!\cdots\!12\)\( T^{7} + \)\(29\!\cdots\!82\)\( T^{8} - \)\(77\!\cdots\!36\)\( T^{9} + \)\(57\!\cdots\!94\)\( T^{10} + \)\(14\!\cdots\!26\)\( T^{11} - 15519460885977751 T^{12} - 80753796 T^{13} + T^{14} \)
$59$ \( -\)\(44\!\cdots\!08\)\( + \)\(39\!\cdots\!40\)\( T - \)\(42\!\cdots\!68\)\( T^{2} - \)\(18\!\cdots\!24\)\( T^{3} + \)\(32\!\cdots\!56\)\( T^{4} + \)\(23\!\cdots\!76\)\( T^{5} - \)\(73\!\cdots\!76\)\( T^{6} + \)\(63\!\cdots\!76\)\( T^{7} + \)\(70\!\cdots\!56\)\( T^{8} - \)\(33\!\cdots\!44\)\( T^{9} - \)\(26\!\cdots\!84\)\( T^{10} + \)\(22\!\cdots\!20\)\( T^{11} + 8219810705996748 T^{12} - 442445502 T^{13} + T^{14} \)
$61$ \( \)\(28\!\cdots\!36\)\( - \)\(19\!\cdots\!72\)\( T - \)\(55\!\cdots\!08\)\( T^{2} + \)\(16\!\cdots\!12\)\( T^{3} + \)\(25\!\cdots\!08\)\( T^{4} - \)\(22\!\cdots\!16\)\( T^{5} - \)\(14\!\cdots\!72\)\( T^{6} + \)\(16\!\cdots\!04\)\( T^{7} + \)\(23\!\cdots\!76\)\( T^{8} - \)\(37\!\cdots\!80\)\( T^{9} - \)\(47\!\cdots\!66\)\( T^{10} + \)\(18\!\cdots\!73\)\( T^{11} - 49833520489012207 T^{12} - 270199089 T^{13} + T^{14} \)
$67$ \( -\)\(38\!\cdots\!32\)\( - \)\(63\!\cdots\!04\)\( T + \)\(72\!\cdots\!28\)\( T^{2} + \)\(59\!\cdots\!20\)\( T^{3} - \)\(73\!\cdots\!24\)\( T^{4} - \)\(14\!\cdots\!28\)\( T^{5} + \)\(18\!\cdots\!64\)\( T^{6} + \)\(13\!\cdots\!72\)\( T^{7} - \)\(17\!\cdots\!64\)\( T^{8} - \)\(57\!\cdots\!72\)\( T^{9} + \)\(78\!\cdots\!44\)\( T^{10} + \)\(12\!\cdots\!57\)\( T^{11} - 148766925122995473 T^{12} - 92500909 T^{13} + T^{14} \)
$71$ \( -\)\(17\!\cdots\!44\)\( + \)\(24\!\cdots\!96\)\( T - \)\(50\!\cdots\!28\)\( T^{2} - \)\(38\!\cdots\!16\)\( T^{3} + \)\(10\!\cdots\!80\)\( T^{4} - \)\(17\!\cdots\!72\)\( T^{5} - \)\(20\!\cdots\!12\)\( T^{6} + \)\(19\!\cdots\!04\)\( T^{7} - \)\(11\!\cdots\!68\)\( T^{8} - \)\(63\!\cdots\!68\)\( T^{9} + \)\(22\!\cdots\!16\)\( T^{10} + \)\(49\!\cdots\!48\)\( T^{11} - 278843502265534524 T^{12} - 84383796 T^{13} + T^{14} \)
$73$ \( -\)\(19\!\cdots\!22\)\( + \)\(75\!\cdots\!77\)\( T - \)\(78\!\cdots\!27\)\( T^{2} - \)\(43\!\cdots\!30\)\( T^{3} + \)\(12\!\cdots\!88\)\( T^{4} - \)\(42\!\cdots\!29\)\( T^{5} - \)\(39\!\cdots\!05\)\( T^{6} + \)\(28\!\cdots\!64\)\( T^{7} + \)\(17\!\cdots\!66\)\( T^{8} - \)\(41\!\cdots\!93\)\( T^{9} + \)\(63\!\cdots\!83\)\( T^{10} + \)\(21\!\cdots\!98\)\( T^{11} - 479710240282112964 T^{12} - 367274315 T^{13} + T^{14} \)
$79$ \( -\)\(16\!\cdots\!12\)\( + \)\(44\!\cdots\!51\)\( T + \)\(72\!\cdots\!05\)\( T^{2} - \)\(37\!\cdots\!86\)\( T^{3} - \)\(69\!\cdots\!74\)\( T^{4} + \)\(41\!\cdots\!41\)\( T^{5} + \)\(24\!\cdots\!23\)\( T^{6} + \)\(23\!\cdots\!92\)\( T^{7} - \)\(36\!\cdots\!52\)\( T^{8} - \)\(51\!\cdots\!31\)\( T^{9} + \)\(25\!\cdots\!83\)\( T^{10} + \)\(28\!\cdots\!54\)\( T^{11} - 843529147273705582 T^{12} - 434861545 T^{13} + T^{14} \)
$83$ \( \)\(34\!\cdots\!92\)\( + \)\(70\!\cdots\!88\)\( T - \)\(18\!\cdots\!32\)\( T^{2} + \)\(44\!\cdots\!16\)\( T^{3} + \)\(13\!\cdots\!77\)\( T^{4} - \)\(39\!\cdots\!58\)\( T^{5} - \)\(30\!\cdots\!75\)\( T^{6} + \)\(13\!\cdots\!08\)\( T^{7} + \)\(22\!\cdots\!22\)\( T^{8} - \)\(15\!\cdots\!36\)\( T^{9} + \)\(77\!\cdots\!86\)\( T^{10} + \)\(68\!\cdots\!72\)\( T^{11} - 482916424143170595 T^{12} - 1013603934 T^{13} + T^{14} \)
$89$ \( \)\(68\!\cdots\!36\)\( - \)\(13\!\cdots\!32\)\( T + \)\(10\!\cdots\!36\)\( T^{2} + \)\(80\!\cdots\!04\)\( T^{3} - \)\(43\!\cdots\!91\)\( T^{4} - \)\(14\!\cdots\!94\)\( T^{5} + \)\(68\!\cdots\!45\)\( T^{6} + \)\(12\!\cdots\!28\)\( T^{7} - \)\(67\!\cdots\!50\)\( T^{8} - \)\(56\!\cdots\!28\)\( T^{9} + \)\(37\!\cdots\!14\)\( T^{10} + \)\(12\!\cdots\!00\)\( T^{11} - 997085078287531555 T^{12} - 1069739706 T^{13} + T^{14} \)
$97$ \( -\)\(70\!\cdots\!42\)\( - \)\(38\!\cdots\!53\)\( T + \)\(15\!\cdots\!13\)\( T^{2} + \)\(22\!\cdots\!38\)\( T^{3} + \)\(77\!\cdots\!92\)\( T^{4} - \)\(55\!\cdots\!71\)\( T^{5} - \)\(16\!\cdots\!77\)\( T^{6} + \)\(71\!\cdots\!68\)\( T^{7} - \)\(44\!\cdots\!02\)\( T^{8} - \)\(50\!\cdots\!99\)\( T^{9} + \)\(86\!\cdots\!91\)\( T^{10} + \)\(18\!\cdots\!50\)\( T^{11} - 5078304648302817364 T^{12} - 2839636281 T^{13} + T^{14} \)
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