Properties

Label 128.5.f.b
Level $128$
Weight $5$
Character orbit 128.f
Analytic conductor $13.231$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 128.f (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.2313552747\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - 4 x^{13} + 15 x^{12} - 34 x^{11} + 62 x^{10} - 312 x^{9} + 1432 x^{8} - 4960 x^{7} + 11456 x^{6} - 19968 x^{5} + 31744 x^{4} - 139264 x^{3} + 491520 x^{2} - 1048576 x + 2097152\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{42} \)
Twist minimal: no (minimal twist has level 16)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} -\beta_{3} q^{5} -\beta_{9} q^{7} + ( 19 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{12} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} -\beta_{3} q^{5} -\beta_{9} q^{7} + ( 19 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{12} ) q^{9} + ( -7 + 7 \beta_{1} - \beta_{7} - \beta_{8} ) q^{11} + ( 1 - \beta_{1} + \beta_{4} - \beta_{5} + \beta_{7} - \beta_{9} + \beta_{13} ) q^{13} + ( -24 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{8} + \beta_{10} - 2 \beta_{12} + \beta_{13} ) q^{15} + ( -2 + 2 \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + 3 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{17} + ( 49 + 49 \beta_{1} + 5 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{9} + 2 \beta_{10} + \beta_{11} + 2 \beta_{12} ) q^{19} + ( 13 + 13 \beta_{1} - 15 \beta_{2} - \beta_{3} - 3 \beta_{4} + 2 \beta_{6} - 3 \beta_{9} - \beta_{10} + 3 \beta_{11} + 2 \beta_{12} ) q^{21} + ( 76 - 4 \beta_{2} - 8 \beta_{3} + 4 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} + 8 \beta_{8} + 3 \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{13} ) q^{23} + ( 47 \beta_{1} + 13 \beta_{2} - \beta_{3} - 6 \beta_{4} + 13 \beta_{5} + 3 \beta_{7} - \beta_{8} + \beta_{10} + 3 \beta_{11} + 4 \beta_{12} + \beta_{13} ) q^{25} + ( 115 - 115 \beta_{1} - 2 \beta_{4} + 11 \beta_{5} + 6 \beta_{6} - 3 \beta_{7} + 17 \beta_{8} + 2 \beta_{9} - 6 \beta_{12} + 2 \beta_{13} ) q^{27} + ( -56 + 56 \beta_{1} - 6 \beta_{4} + 36 \beta_{5} - 2 \beta_{6} - 3 \beta_{8} + 6 \beta_{9} + 2 \beta_{12} + 4 \beta_{13} ) q^{29} + ( 20 \beta_{1} + 4 \beta_{2} + 16 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - 2 \beta_{7} + 16 \beta_{8} + 3 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} + 3 \beta_{13} ) q^{31} + ( 16 - 43 \beta_{2} - 4 \beta_{3} + 43 \beta_{5} + 3 \beta_{6} + 4 \beta_{8} - 13 \beta_{9} - 4 \beta_{10} + 4 \beta_{13} ) q^{33} + ( -102 - 102 \beta_{1} + 8 \beta_{2} - 34 \beta_{3} - 6 \beta_{4} - 2 \beta_{6} - 6 \beta_{9} + 6 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} ) q^{35} + ( 115 + 115 \beta_{1} + 59 \beta_{2} + 2 \beta_{3} + 13 \beta_{4} + 8 \beta_{6} + 13 \beta_{9} - 3 \beta_{10} + 5 \beta_{11} + 8 \beta_{12} ) q^{37} + ( 188 + 10 \beta_{2} + 26 \beta_{3} - 10 \beta_{5} + 4 \beta_{6} - 6 \beta_{7} - 26 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} + 6 \beta_{11} - 2 \beta_{13} ) q^{39} + ( 6 \beta_{1} - 67 \beta_{2} - 11 \beta_{3} + 12 \beta_{4} - 67 \beta_{5} + 5 \beta_{7} - 11 \beta_{8} + 3 \beta_{10} + 5 \beta_{11} + 2 \beta_{12} + 3 \beta_{13} ) q^{41} + ( -112 + 112 \beta_{1} + 8 \beta_{4} + 7 \beta_{5} + 4 \beta_{6} - 40 \beta_{8} - 8 \beta_{9} - 4 \beta_{12} + 4 \beta_{13} ) q^{43} + ( -115 + 115 \beta_{1} + 11 \beta_{4} - 129 \beta_{5} - 6 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} - 11 \beta_{9} + 6 \beta_{12} + \beta_{13} ) q^{45} + ( 400 \beta_{1} + 18 \beta_{2} - 34 \beta_{3} + 18 \beta_{5} - 4 \beta_{7} - 34 \beta_{8} - \beta_{10} - 4 \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{47} + ( 15 + 110 \beta_{2} - 16 \beta_{3} - 110 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} + 16 \beta_{8} + 6 \beta_{9} - 4 \beta_{11} ) q^{49} + ( 229 + 229 \beta_{1} + 5 \beta_{2} + 53 \beta_{3} - 12 \beta_{4} - 4 \beta_{6} - 12 \beta_{9} - 4 \beta_{10} - 3 \beta_{11} - 4 \beta_{12} ) q^{51} + ( 65 + 65 \beta_{1} - 187 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} - 2 \beta_{6} - 3 \beta_{9} + 3 \beta_{10} - 9 \beta_{11} - 2 \beta_{12} ) q^{53} + ( -844 + 26 \beta_{2} - 38 \beta_{3} - 26 \beta_{5} - 16 \beta_{6} + 6 \beta_{7} + 38 \beta_{8} - 13 \beta_{9} - 4 \beta_{10} - 6 \beta_{11} + 4 \beta_{13} ) q^{55} + ( -56 \beta_{1} + 165 \beta_{2} - 4 \beta_{3} - \beta_{4} + 165 \beta_{5} - 12 \beta_{7} - 4 \beta_{8} - 4 \beta_{10} - 12 \beta_{11} - 9 \beta_{12} - 4 \beta_{13} ) q^{57} + ( 194 - 194 \beta_{1} + 6 \beta_{4} - 39 \beta_{5} - 22 \beta_{6} + 6 \beta_{7} + 50 \beta_{8} - 6 \beta_{9} + 22 \beta_{12} - 10 \beta_{13} ) q^{59} + ( 303 - 303 \beta_{1} + 5 \beta_{4} + 261 \beta_{5} + 10 \beta_{6} + 7 \beta_{7} + 14 \beta_{8} - 5 \beta_{9} - 10 \beta_{12} - 13 \beta_{13} ) q^{61} + ( -1424 \beta_{1} - 14 \beta_{2} + 30 \beta_{3} - 17 \beta_{4} - 14 \beta_{5} + 12 \beta_{7} + 30 \beta_{8} - 9 \beta_{10} + 12 \beta_{11} - 14 \beta_{12} - 9 \beta_{13} ) q^{63} + ( -108 - 177 \beta_{2} + 31 \beta_{3} + 177 \beta_{5} - 10 \beta_{6} - 7 \beta_{7} - 31 \beta_{8} + 36 \beta_{9} + 17 \beta_{10} + 7 \beta_{11} - 17 \beta_{13} ) q^{65} + ( -573 - 573 \beta_{1} - 50 \beta_{2} - 5 \beta_{3} + 44 \beta_{4} + 16 \beta_{6} + 44 \beta_{9} - 24 \beta_{10} - 5 \beta_{11} + 16 \beta_{12} ) q^{67} + ( -713 - 713 \beta_{1} + 379 \beta_{2} + \beta_{3} - 49 \beta_{4} - 42 \beta_{6} - 49 \beta_{9} + 13 \beta_{10} - 15 \beta_{11} - 42 \beta_{12} ) q^{69} + ( 1484 - 84 \beta_{2} + 8 \beta_{3} + 84 \beta_{5} - 10 \beta_{6} + 30 \beta_{7} - 8 \beta_{8} + 3 \beta_{9} - 7 \beta_{10} - 30 \beta_{11} + 7 \beta_{13} ) q^{71} + ( -242 \beta_{1} - 243 \beta_{2} + 80 \beta_{3} - 41 \beta_{4} - 243 \beta_{5} - 20 \beta_{7} + 80 \beta_{8} - 16 \beta_{10} - 20 \beta_{11} - 17 \beta_{12} - 16 \beta_{13} ) q^{73} + ( -1268 + 1268 \beta_{1} - 58 \beta_{4} - 27 \beta_{5} - 14 \beta_{6} + 12 \beta_{7} - 24 \beta_{8} + 58 \beta_{9} + 14 \beta_{12} - 18 \beta_{13} ) q^{75} + ( 595 - 595 \beta_{1} - 51 \beta_{4} - 359 \beta_{5} + 38 \beta_{6} - 5 \beta_{7} + 3 \beta_{8} + 51 \beta_{9} - 38 \beta_{12} - 17 \beta_{13} ) q^{77} + ( 2172 \beta_{1} - 174 \beta_{2} + 2 \beta_{3} + 20 \beta_{4} - 174 \beta_{5} + 22 \beta_{7} + 2 \beta_{8} - 10 \beta_{10} + 22 \beta_{11} + 28 \beta_{12} - 10 \beta_{13} ) q^{79} + ( 39 + 325 \beta_{2} + 88 \beta_{3} - 325 \beta_{5} - 5 \beta_{6} - 88 \beta_{8} - 37 \beta_{9} + 16 \beta_{10} - 16 \beta_{13} ) q^{81} + ( 1236 + 1236 \beta_{1} - 3 \beta_{2} - 16 \beta_{3} + 30 \beta_{4} - 22 \beta_{6} + 30 \beta_{9} - 14 \beta_{10} - 4 \beta_{11} - 22 \beta_{12} ) q^{83} + ( -649 - 649 \beta_{1} - 377 \beta_{2} + 29 \beta_{3} + 37 \beta_{4} - 28 \beta_{6} + 37 \beta_{9} + \beta_{10} - 15 \beta_{11} - 28 \beta_{12} ) q^{85} + ( -3460 - 84 \beta_{2} + 64 \beta_{3} + 84 \beta_{5} - 42 \beta_{6} + 6 \beta_{7} - 64 \beta_{8} - 5 \beta_{9} + \beta_{10} - 6 \beta_{11} - \beta_{13} ) q^{87} + ( -534 \beta_{1} + 295 \beta_{2} + 44 \beta_{3} + 81 \beta_{4} + 295 \beta_{5} + 44 \beta_{8} + 4 \beta_{10} - 31 \beta_{12} + 4 \beta_{13} ) q^{89} + ( 2044 - 2044 \beta_{1} + 22 \beta_{4} - 30 \beta_{5} - 14 \beta_{6} + 28 \beta_{7} - 136 \beta_{8} - 22 \beta_{9} + 14 \beta_{12} + 14 \beta_{13} ) q^{91} + ( -582 + 582 \beta_{1} + 70 \beta_{4} + 310 \beta_{5} - 20 \beta_{6} - 30 \beta_{7} - 2 \beta_{8} - 70 \beta_{9} + 20 \beta_{12} - 14 \beta_{13} ) q^{93} + ( -3064 \beta_{1} + 20 \beta_{2} - 108 \beta_{3} - 9 \beta_{4} + 20 \beta_{5} - 28 \beta_{7} - 108 \beta_{8} - 16 \beta_{10} - 28 \beta_{11} + 16 \beta_{12} - 16 \beta_{13} ) q^{95} + ( 74 - 248 \beta_{2} - 105 \beta_{3} + 248 \beta_{5} - 15 \beta_{6} + 37 \beta_{7} + 105 \beta_{8} + 23 \beta_{9} - 7 \beta_{10} - 37 \beta_{11} + 7 \beta_{13} ) q^{97} + ( -3450 - 3450 \beta_{1} + 19 \beta_{2} + 106 \beta_{3} - 122 \beta_{4} - 34 \beta_{6} - 122 \beta_{9} - 2 \beta_{10} - 18 \beta_{11} - 34 \beta_{12} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q + 2q^{3} + 2q^{5} - 4q^{7} + O(q^{10}) \) \( 14q + 2q^{3} + 2q^{5} - 4q^{7} - 94q^{11} + 2q^{13} - 4q^{17} + 706q^{19} + 164q^{21} + 1148q^{23} + 1664q^{27} - 862q^{29} - 4q^{33} - 1340q^{35} + 1826q^{37} + 2684q^{39} - 1694q^{43} - 1410q^{45} + 682q^{49} + 3012q^{51} + 482q^{53} - 11780q^{55} + 2786q^{59} + 3778q^{61} - 2020q^{65} - 7998q^{67} - 9628q^{69} + 19964q^{71} - 17570q^{75} + 9508q^{77} + 1454q^{81} + 17282q^{83} - 9948q^{85} - 49284q^{87} + 28036q^{91} - 8896q^{93} - 4q^{97} - 49214q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} - 4 x^{13} + 15 x^{12} - 34 x^{11} + 62 x^{10} - 312 x^{9} + 1432 x^{8} - 4960 x^{7} + 11456 x^{6} - 19968 x^{5} + 31744 x^{4} - 139264 x^{3} + 491520 x^{2} - 1048576 x + 2097152\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-2875 \nu^{13} + 13444 \nu^{12} - 26581 \nu^{11} + 16062 \nu^{10} - 24954 \nu^{9} + 748984 \nu^{8} - 4619080 \nu^{7} + 10623840 \nu^{6} - 11499840 \nu^{5} + 5960704 \nu^{4} - 51930112 \nu^{3} + 429211648 \nu^{2} - 1316257792 \nu + 1279524864\)\()/ 678952960 \)
\(\beta_{2}\)\(=\)\((\)\(-11609 \nu^{13} + 81068 \nu^{12} - 216535 \nu^{11} + 92170 \nu^{10} - 445086 \nu^{9} + 4629992 \nu^{8} - 22287000 \nu^{7} + 63665440 \nu^{6} - 81223104 \nu^{5} + 58993664 \nu^{4} - 414198784 \nu^{3} + 2235252736 \nu^{2} - 5795840000 \nu + 7904428032\)\()/ 678952960 \)
\(\beta_{3}\)\(=\)\((\)\(23279 \nu^{13} + 167756 \nu^{12} - 482591 \nu^{11} + 959162 \nu^{10} - 197678 \nu^{9} + 877992 \nu^{8} - 35554520 \nu^{7} + 170800160 \nu^{6} - 398883776 \nu^{5} + 133770240 \nu^{4} - 180005888 \nu^{3} + 2196488192 \nu^{2} - 17134354432 \nu + 45595492352\)\()/ 678952960 \)
\(\beta_{4}\)\(=\)\((\)\(-3499 \nu^{13} + 6652 \nu^{12} + 13019 \nu^{11} - 10778 \nu^{10} + 122070 \nu^{9} + 1232776 \nu^{8} - 3340360 \nu^{7} - 34400 \nu^{6} + 24398016 \nu^{5} + 5933568 \nu^{4} - 170023936 \nu^{3} + 637231104 \nu^{2} - 494927872 \nu - 3419275264\)\()/84869120\)
\(\beta_{5}\)\(=\)\((\)\(-30299 \nu^{13} + 175428 \nu^{12} - 396085 \nu^{11} + 537790 \nu^{10} - 1159226 \nu^{9} + 9651512 \nu^{8} - 53625160 \nu^{7} + 150852960 \nu^{6} - 203316544 \nu^{5} + 134759424 \nu^{4} - 982549504 \nu^{3} + 4972855296 \nu^{2} - 16046653440 \nu + 20694433792\)\()/ 678952960 \)
\(\beta_{6}\)\(=\)\((\)\(187 \nu^{13} + 588 \nu^{12} - 3179 \nu^{11} + 8306 \nu^{10} - 18854 \nu^{9} + 936 \nu^{8} - 206648 \nu^{7} + 1252384 \nu^{6} - 3000512 \nu^{5} + 5232640 \nu^{4} - 7527424 \nu^{3} + 7045120 \nu^{2} - 93421568 \nu + 359268352\)\()/2424832\)
\(\beta_{7}\)\(=\)\((\)\(54457 \nu^{13} - 666028 \nu^{12} + 2372471 \nu^{11} - 2827082 \nu^{10} - 5318178 \nu^{9} - 5702120 \nu^{8} + 151698840 \nu^{7} - 675003680 \nu^{6} + 1063292352 \nu^{5} + 531149824 \nu^{4} - 2507641856 \nu^{3} - 7324450816 \nu^{2} + 52640448512 \nu - 123104133120\)\()/ 678952960 \)
\(\beta_{8}\)\(=\)\((\)\(-63941 \nu^{13} + 248956 \nu^{12} - 325931 \nu^{11} + 303682 \nu^{10} - 1485958 \nu^{9} + 16189512 \nu^{8} - 70629560 \nu^{7} + 157086880 \nu^{6} - 67802816 \nu^{5} + 199142400 \nu^{4} - 922754048 \nu^{3} + 7022559232 \nu^{2} - 16313188352 \nu + 4344512512\)\()/ 678952960 \)
\(\beta_{9}\)\(=\)\((\)\(-1373 \nu^{13} - 1028 \nu^{12} + 16237 \nu^{11} - 30894 \nu^{10} + 21354 \nu^{9} + 156488 \nu^{8} + 410632 \nu^{7} - 4973408 \nu^{6} + 13415232 \nu^{5} - 9518080 \nu^{4} - 7715840 \nu^{3} - 46219264 \nu^{2} + 423264256 \nu - 1804861440\)\()/9699328\)
\(\beta_{10}\)\(=\)\((\)\(-60737 \nu^{13} - 163572 \nu^{12} + 819089 \nu^{11} - 871398 \nu^{10} - 2291022 \nu^{9} + 15043880 \nu^{8} + 15561640 \nu^{7} - 288799200 \nu^{6} + 569406528 \nu^{5} + 16526336 \nu^{4} - 1198050304 \nu^{3} + 1135230976 \nu^{2} + 35844947968 \nu - 98412789760\)\()/ 339476480 \)
\(\beta_{11}\)\(=\)\((\)\(191167 \nu^{13} - 198708 \nu^{12} - 780399 \nu^{11} + 1949018 \nu^{10} + 2939442 \nu^{9} - 59111000 \nu^{8} + 130720680 \nu^{7} + 55001120 \nu^{6} - 991562688 \nu^{5} + 198482944 \nu^{4} + 4742575104 \nu^{3} - 12649414656 \nu^{2} + 6744604672 \nu + 139540561920\)\()/ 678952960 \)
\(\beta_{12}\)\(=\)\((\)\(-107577 \nu^{13} + 484524 \nu^{12} - 1270135 \nu^{11} + 2145610 \nu^{10} - 3926878 \nu^{9} + 32206056 \nu^{8} - 152847000 \nu^{7} + 393864480 \nu^{6} - 547344832 \nu^{5} + 682642432 \nu^{4} - 2131233792 \nu^{3} + 12085280768 \nu^{2} - 41333391360 \nu + 33873985536\)\()/ 339476480 \)
\(\beta_{13}\)\(=\)\((\)\(108593 \nu^{13} - 103372 \nu^{12} - 236161 \nu^{11} + 473862 \nu^{10} + 1410158 \nu^{9} - 32112040 \nu^{8} + 64026840 \nu^{7} - 34664480 \nu^{6} - 122103872 \nu^{5} - 344812544 \nu^{4} + 2863123456 \nu^{3} - 10052583424 \nu^{2} + 11285659648 \nu + 31120424960\)\()/ 339476480 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{12} + \beta_{10} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{2} - 9 \beta_{1} + 17\)\()/64\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{12} + 2 \beta_{11} + \beta_{10} - 4 \beta_{9} + 7 \beta_{8} + \beta_{7} - 3 \beta_{6} + 7 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} - 21 \beta_{1} - 63\)\()/64\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{12} - 2 \beta_{11} + \beta_{10} - 12 \beta_{9} + 31 \beta_{8} + \beta_{7} - 7 \beta_{6} - 65 \beta_{5} - 12 \beta_{4} + 10 \beta_{3} - 20 \beta_{2} + 167 \beta_{1} - 67\)\()/64\)
\(\nu^{4}\)\(=\)\((\)\(-6 \beta_{13} + 11 \beta_{12} + 12 \beta_{11} - 5 \beta_{10} + 12 \beta_{9} + 45 \beta_{8} + 3 \beta_{7} + 29 \beta_{6} - 255 \beta_{5} + 28 \beta_{4} - 4 \beta_{3} + 82 \beta_{2} + 967 \beta_{1} + 121\)\()/64\)
\(\nu^{5}\)\(=\)\((\)\(44 \beta_{13} + 15 \beta_{12} - 34 \beta_{11} - 23 \beta_{10} - 36 \beta_{9} + 107 \beta_{8} + 5 \beta_{7} - 3 \beta_{6} - 29 \beta_{5} + 108 \beta_{4} - 118 \beta_{3} - 4 \beta_{2} - 1909 \beta_{1} + 5561\)\()/64\)
\(\nu^{6}\)\(=\)\((\)\(-38 \beta_{13} - 189 \beta_{12} - 108 \beta_{11} - 5 \beta_{10} - 132 \beta_{9} + 445 \beta_{8} - 157 \beta_{7} + 133 \beta_{6} + 657 \beta_{5} - 68 \beta_{4} - 316 \beta_{3} - 422 \beta_{2} - 8849 \beta_{1} - 12447\)\()/64\)
\(\nu^{7}\)\(=\)\((\)\(-280 \beta_{13} + 7 \beta_{12} + 242 \beta_{11} - 291 \beta_{10} - 84 \beta_{9} + 1143 \beta_{8} + 89 \beta_{7} - 371 \beta_{6} + 1255 \beta_{5} + 156 \beta_{4} - 266 \beta_{3} + 1616 \beta_{2} - 44397 \beta_{1} - 10215\)\()/64\)
\(\nu^{8}\)\(=\)\((\)\(-858 \beta_{13} + 611 \beta_{12} - 1136 \beta_{11} - 153 \beta_{10} - 1156 \beta_{9} + 409 \beta_{8} - 185 \beta_{7} - 867 \beta_{6} - 8315 \beta_{5} - 1284 \beta_{4} + 3976 \beta_{3} + 5494 \beta_{2} - 59217 \beta_{1} + 66329\)\()/64\)
\(\nu^{9}\)\(=\)\((\)\(-2564 \beta_{13} - 153 \beta_{12} + 1158 \beta_{11} - 127 \beta_{10} + 3660 \beta_{9} - 6677 \beta_{8} - 3323 \beta_{7} + 3381 \beta_{6} - 23677 \beta_{5} + 8796 \beta_{4} + 4802 \beta_{3} + 4708 \beta_{2} + 108659 \beta_{1} + 89825\)\()/64\)
\(\nu^{10}\)\(=\)\((\)\(3434 \beta_{13} + 6691 \beta_{12} + 548 \beta_{11} + 2587 \beta_{10} - 4804 \beta_{9} + 3021 \beta_{8} - 1357 \beta_{7} - 5915 \beta_{6} + 53569 \beta_{5} + 21852 \beta_{4} + 7444 \beta_{3} - 121494 \beta_{2} - 580657 \beta_{1} + 628385\)\()/64\)
\(\nu^{11}\)\(=\)\((\)\(-15088 \beta_{13} - 31849 \beta_{12} - 5174 \beta_{11} + 8437 \beta_{10} + 2892 \beta_{9} + 90719 \beta_{8} - 9359 \beta_{7} + 15405 \beta_{6} + 84255 \beta_{5} - 32196 \beta_{4} + 79198 \beta_{3} - 343064 \beta_{2} - 15805 \beta_{1} - 1111399\)\()/64\)
\(\nu^{12}\)\(=\)\((\)\(9886 \beta_{13} + 1059 \beta_{12} + 65352 \beta_{11} - 12833 \beta_{10} + 144956 \beta_{9} + 270289 \beta_{8} + 38927 \beta_{7} - 10163 \beta_{6} - 170835 \beta_{5} + 47548 \beta_{4} + 313424 \beta_{3} + 396334 \beta_{2} - 2474769 \beta_{1} + 1768521\)\()/64\)
\(\nu^{13}\)\(=\)\((\)\(124644 \beta_{13} + 4807 \beta_{12} - 241810 \beta_{11} + 41769 \beta_{10} - 273012 \beta_{9} + 107123 \beta_{8} + 93757 \beta_{7} - 97819 \beta_{6} - 1479205 \beta_{5} + 169308 \beta_{4} + 1013274 \beta_{3} + 1212684 \beta_{2} + 85715 \beta_{1} - 514479\)\()/64\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(\beta_{1}\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
1.03712 2.63142i
2.24452 + 1.72109i
−2.15805 + 1.82834i
−2.40693 1.48549i
2.79265 0.448449i
0.336831 2.80830i
0.153862 + 2.82424i
1.03712 + 2.63142i
2.24452 1.72109i
−2.15805 1.82834i
−2.40693 + 1.48549i
2.79265 + 0.448449i
0.336831 + 2.80830i
0.153862 2.82424i
0 −11.5209 11.5209i 0 14.6016 + 14.6016i 0 −24.0210 0 184.461i 0
31.2 0 −5.54016 5.54016i 0 −21.7374 21.7374i 0 −6.62054 0 19.6133i 0
31.3 0 −3.91498 3.91498i 0 −4.72348 4.72348i 0 45.3712 0 50.3458i 0
31.4 0 0.0461995 + 0.0461995i 0 8.04297 + 8.04297i 0 −49.8797 0 80.9957i 0
31.5 0 4.63552 + 4.63552i 0 29.2002 + 29.2002i 0 59.6196 0 38.0239i 0
31.6 0 7.86839 + 7.86839i 0 −27.2309 27.2309i 0 50.3097 0 42.8233i 0
31.7 0 9.42589 + 9.42589i 0 2.84710 + 2.84710i 0 −76.7794 0 96.6949i 0
95.1 0 −11.5209 + 11.5209i 0 14.6016 14.6016i 0 −24.0210 0 184.461i 0
95.2 0 −5.54016 + 5.54016i 0 −21.7374 + 21.7374i 0 −6.62054 0 19.6133i 0
95.3 0 −3.91498 + 3.91498i 0 −4.72348 + 4.72348i 0 45.3712 0 50.3458i 0
95.4 0 0.0461995 0.0461995i 0 8.04297 8.04297i 0 −49.8797 0 80.9957i 0
95.5 0 4.63552 4.63552i 0 29.2002 29.2002i 0 59.6196 0 38.0239i 0
95.6 0 7.86839 7.86839i 0 −27.2309 + 27.2309i 0 50.3097 0 42.8233i 0
95.7 0 9.42589 9.42589i 0 2.84710 2.84710i 0 −76.7794 0 96.6949i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 95.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 128.5.f.b 14
4.b odd 2 1 128.5.f.a 14
8.b even 2 1 16.5.f.a 14
8.d odd 2 1 64.5.f.a 14
16.e even 4 1 64.5.f.a 14
16.e even 4 1 128.5.f.a 14
16.f odd 4 1 16.5.f.a 14
16.f odd 4 1 inner 128.5.f.b 14
24.f even 2 1 576.5.m.a 14
24.h odd 2 1 144.5.m.a 14
48.i odd 4 1 576.5.m.a 14
48.k even 4 1 144.5.m.a 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.5.f.a 14 8.b even 2 1
16.5.f.a 14 16.f odd 4 1
64.5.f.a 14 8.d odd 2 1
64.5.f.a 14 16.e even 4 1
128.5.f.a 14 4.b odd 2 1
128.5.f.a 14 16.e even 4 1
128.5.f.b 14 1.a even 1 1 trivial
128.5.f.b 14 16.f odd 4 1 inner
144.5.m.a 14 24.h odd 2 1
144.5.m.a 14 48.k even 4 1
576.5.m.a 14 24.f even 2 1
576.5.m.a 14 48.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{14} - \cdots\) acting on \(S_{5}^{\mathrm{new}}(128, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 2 T + 2 T^{2} - 610 T^{3} - 4613 T^{4} - 31732 T^{5} + 258740 T^{6} - 2114868 T^{7} + 70347753 T^{8} - 127844190 T^{9} + 3102089886 T^{10} - 22201539966 T^{11} + 75486949083 T^{12} - 2442289052376 T^{13} + 20677791784536 T^{14} - 197825413242456 T^{15} + 495269872933563 T^{16} - 11798808601071006 T^{17} + 133534797839563806 T^{18} - 445765127450480190 T^{19} + 19868283272269877193 T^{20} - 48381396305638460148 T^{21} + \)\(47\!\cdots\!40\)\( T^{22} - \)\(47\!\cdots\!72\)\( T^{23} - \)\(56\!\cdots\!13\)\( T^{24} - \)\(60\!\cdots\!10\)\( T^{25} + \)\(15\!\cdots\!22\)\( T^{26} - \)\(12\!\cdots\!82\)\( T^{27} + \)\(52\!\cdots\!21\)\( T^{28} \)
$5$ \( 1 - 2 T + 2 T^{2} - 3938 T^{3} - 94565 T^{4} + 9371916 T^{5} - 10800780 T^{6} + 6597908556 T^{7} - 151886931511 T^{8} - 563896366366 T^{9} + 18344031697054 T^{10} + 1149733079121218 T^{11} + 70375486014886875 T^{12} - 1339233603531363800 T^{13} + 14585682673141755608 T^{14} - \)\(83\!\cdots\!00\)\( T^{15} + \)\(27\!\cdots\!75\)\( T^{16} + \)\(28\!\cdots\!50\)\( T^{17} + \)\(27\!\cdots\!50\)\( T^{18} - \)\(53\!\cdots\!50\)\( T^{19} - \)\(90\!\cdots\!75\)\( T^{20} + \)\(24\!\cdots\!00\)\( T^{21} - \)\(25\!\cdots\!00\)\( T^{22} + \)\(13\!\cdots\!00\)\( T^{23} - \)\(86\!\cdots\!25\)\( T^{24} - \)\(22\!\cdots\!50\)\( T^{25} + \)\(71\!\cdots\!50\)\( T^{26} - \)\(44\!\cdots\!50\)\( T^{27} + \)\(13\!\cdots\!25\)\( T^{28} \)
$7$ \( ( 1 + 2 T + 8235 T^{2} + 64404 T^{3} + 38860249 T^{4} + 447351454 T^{5} + 125587217723 T^{6} + 1378109878936 T^{7} + 301534909752923 T^{8} + 2578892109370654 T^{9} + 537875867111373049 T^{10} + 2140333660404582804 T^{11} + \)\(65\!\cdots\!35\)\( T^{12} + \)\(38\!\cdots\!02\)\( T^{13} + \)\(45\!\cdots\!01\)\( T^{14} )^{2} \)
$11$ \( 1 + 94 T + 4418 T^{2} - 965570 T^{3} - 470088133 T^{4} - 3120961844 T^{5} + 2249641670708 T^{6} + 796542815073292 T^{7} + 97369777194709097 T^{8} - 5317394377195027646 T^{9} - \)\(63\!\cdots\!78\)\( T^{10} - \)\(14\!\cdots\!86\)\( T^{11} - \)\(23\!\cdots\!21\)\( T^{12} + \)\(22\!\cdots\!52\)\( T^{13} + \)\(18\!\cdots\!92\)\( T^{14} + \)\(33\!\cdots\!32\)\( T^{15} - \)\(50\!\cdots\!01\)\( T^{16} - \)\(45\!\cdots\!06\)\( T^{17} - \)\(29\!\cdots\!58\)\( T^{18} - \)\(35\!\cdots\!46\)\( T^{19} + \)\(95\!\cdots\!77\)\( T^{20} + \)\(11\!\cdots\!52\)\( T^{21} + \)\(47\!\cdots\!68\)\( T^{22} - \)\(96\!\cdots\!84\)\( T^{23} - \)\(21\!\cdots\!33\)\( T^{24} - \)\(63\!\cdots\!70\)\( T^{25} + \)\(42\!\cdots\!58\)\( T^{26} + \)\(13\!\cdots\!74\)\( T^{27} + \)\(20\!\cdots\!61\)\( T^{28} \)
$13$ \( 1 - 2 T + 2 T^{2} - 6883234 T^{3} + 1464853339 T^{4} - 65707775476 T^{5} + 23817940993652 T^{6} - 16199073445624116 T^{7} + 1142495439970904649 T^{8} - \)\(10\!\cdots\!70\)\( T^{9} + \)\(78\!\cdots\!46\)\( T^{10} - \)\(14\!\cdots\!90\)\( T^{11} + \)\(74\!\cdots\!35\)\( T^{12} - \)\(22\!\cdots\!32\)\( T^{13} + \)\(94\!\cdots\!60\)\( T^{14} - \)\(65\!\cdots\!52\)\( T^{15} + \)\(60\!\cdots\!35\)\( T^{16} - \)\(33\!\cdots\!90\)\( T^{17} + \)\(52\!\cdots\!86\)\( T^{18} - \)\(19\!\cdots\!70\)\( T^{19} + \)\(62\!\cdots\!89\)\( T^{20} - \)\(25\!\cdots\!36\)\( T^{21} + \)\(10\!\cdots\!12\)\( T^{22} - \)\(83\!\cdots\!16\)\( T^{23} + \)\(52\!\cdots\!39\)\( T^{24} - \)\(71\!\cdots\!74\)\( T^{25} + \)\(58\!\cdots\!42\)\( T^{26} - \)\(16\!\cdots\!62\)\( T^{27} + \)\(24\!\cdots\!41\)\( T^{28} \)
$17$ \( ( 1 + 2 T + 333755 T^{2} - 12776716 T^{3} + 56419031945 T^{4} - 2961382342882 T^{5} + 6454907058757691 T^{6} - 326099715157486120 T^{7} + \)\(53\!\cdots\!11\)\( T^{8} - \)\(20\!\cdots\!62\)\( T^{9} + \)\(32\!\cdots\!45\)\( T^{10} - \)\(62\!\cdots\!96\)\( T^{11} + \)\(13\!\cdots\!55\)\( T^{12} + \)\(67\!\cdots\!42\)\( T^{13} + \)\(28\!\cdots\!41\)\( T^{14} )^{2} \)
$19$ \( 1 - 706 T + 249218 T^{2} - 101381538 T^{3} + 32599242619 T^{4} - 3617133340788 T^{5} - 431513784802892 T^{6} + 960761925731564364 T^{7} - \)\(71\!\cdots\!35\)\( T^{8} + \)\(22\!\cdots\!14\)\( T^{9} - \)\(57\!\cdots\!66\)\( T^{10} + \)\(18\!\cdots\!22\)\( T^{11} - \)\(19\!\cdots\!61\)\( T^{12} - \)\(38\!\cdots\!56\)\( T^{13} + \)\(11\!\cdots\!48\)\( T^{14} - \)\(50\!\cdots\!76\)\( T^{15} - \)\(32\!\cdots\!01\)\( T^{16} + \)\(40\!\cdots\!42\)\( T^{17} - \)\(16\!\cdots\!46\)\( T^{18} + \)\(84\!\cdots\!14\)\( T^{19} - \)\(35\!\cdots\!35\)\( T^{20} + \)\(61\!\cdots\!24\)\( T^{21} - \)\(35\!\cdots\!12\)\( T^{22} - \)\(39\!\cdots\!28\)\( T^{23} + \)\(46\!\cdots\!19\)\( T^{24} - \)\(18\!\cdots\!98\)\( T^{25} + \)\(59\!\cdots\!38\)\( T^{26} - \)\(22\!\cdots\!66\)\( T^{27} + \)\(40\!\cdots\!81\)\( T^{28} \)
$23$ \( ( 1 - 574 T + 1024043 T^{2} - 635922028 T^{3} + 551773439769 T^{4} - 314763003369506 T^{5} + 213700110666561659 T^{6} - \)\(10\!\cdots\!08\)\( T^{7} + \)\(59\!\cdots\!19\)\( T^{8} - \)\(24\!\cdots\!86\)\( T^{9} + \)\(12\!\cdots\!49\)\( T^{10} - \)\(38\!\cdots\!08\)\( T^{11} + \)\(17\!\cdots\!43\)\( T^{12} - \)\(27\!\cdots\!34\)\( T^{13} + \)\(13\!\cdots\!81\)\( T^{14} )^{2} \)
$29$ \( 1 + 862 T + 371522 T^{2} + 1045006654 T^{3} + 1721779716827 T^{4} + 691721668187596 T^{5} + 502604487474844916 T^{6} + \)\(98\!\cdots\!28\)\( T^{7} + \)\(75\!\cdots\!49\)\( T^{8} + \)\(31\!\cdots\!74\)\( T^{9} + \)\(39\!\cdots\!98\)\( T^{10} + \)\(45\!\cdots\!94\)\( T^{11} + \)\(34\!\cdots\!87\)\( T^{12} + \)\(29\!\cdots\!64\)\( T^{13} + \)\(24\!\cdots\!96\)\( T^{14} + \)\(20\!\cdots\!84\)\( T^{15} + \)\(17\!\cdots\!07\)\( T^{16} + \)\(15\!\cdots\!54\)\( T^{17} + \)\(98\!\cdots\!58\)\( T^{18} + \)\(54\!\cdots\!74\)\( T^{19} + \)\(94\!\cdots\!69\)\( T^{20} + \)\(87\!\cdots\!08\)\( T^{21} + \)\(31\!\cdots\!56\)\( T^{22} + \)\(30\!\cdots\!16\)\( T^{23} + \)\(53\!\cdots\!27\)\( T^{24} + \)\(23\!\cdots\!74\)\( T^{25} + \)\(58\!\cdots\!42\)\( T^{26} + \)\(95\!\cdots\!42\)\( T^{27} + \)\(78\!\cdots\!21\)\( T^{28} \)
$31$ \( 1 - 6904334 T^{2} + 24182883262811 T^{4} - 57459081771770667372 T^{6} + \)\(10\!\cdots\!69\)\( T^{8} - \)\(14\!\cdots\!50\)\( T^{10} + \)\(17\!\cdots\!83\)\( T^{12} - \)\(17\!\cdots\!20\)\( T^{14} + \)\(15\!\cdots\!03\)\( T^{16} - \)\(10\!\cdots\!50\)\( T^{18} + \)\(64\!\cdots\!49\)\( T^{20} - \)\(30\!\cdots\!92\)\( T^{22} + \)\(10\!\cdots\!11\)\( T^{24} - \)\(26\!\cdots\!94\)\( T^{26} + \)\(32\!\cdots\!81\)\( T^{28} \)
$37$ \( 1 - 1826 T + 1667138 T^{2} - 4976934274 T^{3} + 7030206539163 T^{4} - 1716272691299380 T^{5} + 3798526848883495924 T^{6} - \)\(11\!\cdots\!60\)\( T^{7} - \)\(10\!\cdots\!63\)\( T^{8} + \)\(16\!\cdots\!22\)\( T^{9} + \)\(18\!\cdots\!50\)\( T^{10} + \)\(21\!\cdots\!18\)\( T^{11} - \)\(10\!\cdots\!97\)\( T^{12} + \)\(14\!\cdots\!60\)\( T^{13} - \)\(67\!\cdots\!04\)\( T^{14} + \)\(26\!\cdots\!60\)\( T^{15} - \)\(37\!\cdots\!37\)\( T^{16} + \)\(14\!\cdots\!58\)\( T^{17} + \)\(22\!\cdots\!50\)\( T^{18} + \)\(38\!\cdots\!22\)\( T^{19} - \)\(44\!\cdots\!43\)\( T^{20} - \)\(91\!\cdots\!60\)\( T^{21} + \)\(57\!\cdots\!44\)\( T^{22} - \)\(48\!\cdots\!80\)\( T^{23} + \)\(37\!\cdots\!63\)\( T^{24} - \)\(49\!\cdots\!14\)\( T^{25} + \)\(31\!\cdots\!98\)\( T^{26} - \)\(64\!\cdots\!06\)\( T^{27} + \)\(65\!\cdots\!41\)\( T^{28} \)
$41$ \( 1 - 24523982 T^{2} + 302442312166171 T^{4} - \)\(24\!\cdots\!76\)\( T^{6} + \)\(14\!\cdots\!01\)\( T^{8} - \)\(70\!\cdots\!70\)\( T^{10} + \)\(27\!\cdots\!51\)\( T^{12} - \)\(84\!\cdots\!88\)\( T^{14} + \)\(21\!\cdots\!71\)\( T^{16} - \)\(45\!\cdots\!70\)\( T^{18} + \)\(76\!\cdots\!61\)\( T^{20} - \)\(10\!\cdots\!56\)\( T^{22} + \)\(98\!\cdots\!71\)\( T^{24} - \)\(63\!\cdots\!22\)\( T^{26} + \)\(20\!\cdots\!41\)\( T^{28} \)
$43$ \( 1 + 1694 T + 1434818 T^{2} + 14278395262 T^{3} + 44454402050619 T^{4} + 13474016894363980 T^{5} + 60977294006539554100 T^{6} + \)\(38\!\cdots\!44\)\( T^{7} + \)\(23\!\cdots\!81\)\( T^{8} - \)\(78\!\cdots\!82\)\( T^{9} + \)\(12\!\cdots\!62\)\( T^{10} + \)\(18\!\cdots\!22\)\( T^{11} - \)\(91\!\cdots\!45\)\( T^{12} - \)\(10\!\cdots\!08\)\( T^{13} + \)\(16\!\cdots\!52\)\( T^{14} - \)\(36\!\cdots\!08\)\( T^{15} - \)\(10\!\cdots\!45\)\( T^{16} + \)\(73\!\cdots\!22\)\( T^{17} + \)\(16\!\cdots\!62\)\( T^{18} - \)\(36\!\cdots\!82\)\( T^{19} + \)\(37\!\cdots\!81\)\( T^{20} + \)\(21\!\cdots\!44\)\( T^{21} + \)\(11\!\cdots\!00\)\( T^{22} + \)\(85\!\cdots\!80\)\( T^{23} + \)\(96\!\cdots\!19\)\( T^{24} + \)\(10\!\cdots\!62\)\( T^{25} + \)\(36\!\cdots\!18\)\( T^{26} + \)\(14\!\cdots\!94\)\( T^{27} + \)\(29\!\cdots\!01\)\( T^{28} \)
$47$ \( 1 - 51887758 T^{2} + 1309844227745755 T^{4} - \)\(21\!\cdots\!56\)\( T^{6} + \)\(24\!\cdots\!09\)\( T^{8} - \)\(21\!\cdots\!30\)\( T^{10} + \)\(15\!\cdots\!39\)\( T^{12} - \)\(82\!\cdots\!04\)\( T^{14} + \)\(35\!\cdots\!79\)\( T^{16} - \)\(12\!\cdots\!30\)\( T^{18} + \)\(33\!\cdots\!29\)\( T^{20} - \)\(68\!\cdots\!96\)\( T^{22} + \)\(10\!\cdots\!55\)\( T^{24} - \)\(94\!\cdots\!38\)\( T^{26} + \)\(43\!\cdots\!21\)\( T^{28} \)
$53$ \( 1 - 482 T + 116162 T^{2} + 5558326078 T^{3} + 43583027341595 T^{4} - 155411473123116980 T^{5} + 85293132817975457012 T^{6} - \)\(11\!\cdots\!76\)\( T^{7} + \)\(35\!\cdots\!13\)\( T^{8} + \)\(92\!\cdots\!94\)\( T^{9} + \)\(12\!\cdots\!18\)\( T^{10} - \)\(13\!\cdots\!10\)\( T^{11} + \)\(28\!\cdots\!31\)\( T^{12} - \)\(18\!\cdots\!24\)\( T^{13} + \)\(12\!\cdots\!68\)\( T^{14} - \)\(14\!\cdots\!44\)\( T^{15} + \)\(17\!\cdots\!91\)\( T^{16} - \)\(67\!\cdots\!10\)\( T^{17} + \)\(47\!\cdots\!78\)\( T^{18} + \)\(28\!\cdots\!94\)\( T^{19} + \)\(84\!\cdots\!53\)\( T^{20} - \)\(21\!\cdots\!36\)\( T^{21} + \)\(12\!\cdots\!92\)\( T^{22} - \)\(18\!\cdots\!80\)\( T^{23} + \)\(40\!\cdots\!95\)\( T^{24} + \)\(41\!\cdots\!18\)\( T^{25} + \)\(67\!\cdots\!82\)\( T^{26} - \)\(22\!\cdots\!62\)\( T^{27} + \)\(36\!\cdots\!21\)\( T^{28} \)
$59$ \( 1 - 2786 T + 3880898 T^{2} - 95235375746 T^{3} + 282835430943931 T^{4} + 951817240129082700 T^{5} + \)\(78\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!24\)\( T^{7} - \)\(10\!\cdots\!59\)\( T^{8} + \)\(16\!\cdots\!78\)\( T^{9} + \)\(95\!\cdots\!50\)\( T^{10} + \)\(27\!\cdots\!94\)\( T^{11} + \)\(66\!\cdots\!23\)\( T^{12} - \)\(40\!\cdots\!76\)\( T^{13} + \)\(34\!\cdots\!76\)\( T^{14} - \)\(49\!\cdots\!36\)\( T^{15} + \)\(97\!\cdots\!83\)\( T^{16} + \)\(49\!\cdots\!14\)\( T^{17} + \)\(20\!\cdots\!50\)\( T^{18} + \)\(42\!\cdots\!78\)\( T^{19} - \)\(34\!\cdots\!99\)\( T^{20} + \)\(39\!\cdots\!04\)\( T^{21} + \)\(36\!\cdots\!20\)\( T^{22} + \)\(53\!\cdots\!00\)\( T^{23} + \)\(19\!\cdots\!31\)\( T^{24} - \)\(78\!\cdots\!06\)\( T^{25} + \)\(38\!\cdots\!58\)\( T^{26} - \)\(33\!\cdots\!66\)\( T^{27} + \)\(14\!\cdots\!41\)\( T^{28} \)
$61$ \( 1 - 3778 T + 7136642 T^{2} - 13584988130 T^{3} + 15885658886619 T^{4} + 322904339201283724 T^{5} - \)\(12\!\cdots\!20\)\( T^{6} + \)\(11\!\cdots\!80\)\( T^{7} - \)\(34\!\cdots\!59\)\( T^{8} + \)\(57\!\cdots\!82\)\( T^{9} - \)\(60\!\cdots\!50\)\( T^{10} + \)\(13\!\cdots\!70\)\( T^{11} + \)\(15\!\cdots\!55\)\( T^{12} - \)\(19\!\cdots\!60\)\( T^{13} + \)\(92\!\cdots\!08\)\( T^{14} - \)\(26\!\cdots\!60\)\( T^{15} + \)\(29\!\cdots\!55\)\( T^{16} + \)\(35\!\cdots\!70\)\( T^{17} - \)\(22\!\cdots\!50\)\( T^{18} + \)\(29\!\cdots\!82\)\( T^{19} - \)\(24\!\cdots\!19\)\( T^{20} + \)\(11\!\cdots\!80\)\( T^{21} - \)\(16\!\cdots\!20\)\( T^{22} + \)\(60\!\cdots\!64\)\( T^{23} + \)\(41\!\cdots\!19\)\( T^{24} - \)\(48\!\cdots\!30\)\( T^{25} + \)\(35\!\cdots\!02\)\( T^{26} - \)\(25\!\cdots\!38\)\( T^{27} + \)\(95\!\cdots\!61\)\( T^{28} \)
$67$ \( 1 + 7998 T + 31984002 T^{2} + 47670849246 T^{3} - 439076236005637 T^{4} - 648765759780793716 T^{5} + \)\(99\!\cdots\!64\)\( T^{6} + \)\(93\!\cdots\!52\)\( T^{7} + \)\(28\!\cdots\!21\)\( T^{8} - \)\(69\!\cdots\!70\)\( T^{9} - \)\(55\!\cdots\!14\)\( T^{10} - \)\(21\!\cdots\!74\)\( T^{11} - \)\(39\!\cdots\!49\)\( T^{12} - \)\(17\!\cdots\!60\)\( T^{13} - \)\(39\!\cdots\!76\)\( T^{14} - \)\(35\!\cdots\!60\)\( T^{15} - \)\(15\!\cdots\!09\)\( T^{16} - \)\(17\!\cdots\!14\)\( T^{17} - \)\(92\!\cdots\!34\)\( T^{18} - \)\(23\!\cdots\!70\)\( T^{19} + \)\(18\!\cdots\!41\)\( T^{20} + \)\(12\!\cdots\!32\)\( T^{21} + \)\(27\!\cdots\!04\)\( T^{22} - \)\(35\!\cdots\!96\)\( T^{23} - \)\(48\!\cdots\!37\)\( T^{24} + \)\(10\!\cdots\!66\)\( T^{25} + \)\(14\!\cdots\!82\)\( T^{26} + \)\(72\!\cdots\!78\)\( T^{27} + \)\(18\!\cdots\!81\)\( T^{28} \)
$71$ \( ( 1 - 9982 T + 145017323 T^{2} - 1165310044396 T^{3} + 10014081489420185 T^{4} - 63641985337531169890 T^{5} + \)\(40\!\cdots\!59\)\( T^{6} - \)\(20\!\cdots\!72\)\( T^{7} + \)\(10\!\cdots\!79\)\( T^{8} - \)\(41\!\cdots\!90\)\( T^{9} + \)\(16\!\cdots\!85\)\( T^{10} - \)\(48\!\cdots\!16\)\( T^{11} + \)\(15\!\cdots\!23\)\( T^{12} - \)\(26\!\cdots\!42\)\( T^{13} + \)\(68\!\cdots\!61\)\( T^{14} )^{2} \)
$73$ \( 1 - 168573838 T^{2} + 13353714116727067 T^{4} - \)\(70\!\cdots\!96\)\( T^{6} + \)\(30\!\cdots\!57\)\( T^{8} - \)\(11\!\cdots\!98\)\( T^{10} + \)\(39\!\cdots\!07\)\( T^{12} - \)\(11\!\cdots\!44\)\( T^{14} + \)\(31\!\cdots\!67\)\( T^{16} - \)\(74\!\cdots\!78\)\( T^{18} + \)\(15\!\cdots\!37\)\( T^{20} - \)\(29\!\cdots\!16\)\( T^{22} + \)\(45\!\cdots\!67\)\( T^{24} - \)\(46\!\cdots\!78\)\( T^{26} + \)\(22\!\cdots\!61\)\( T^{28} \)
$79$ \( 1 - 364033678 T^{2} + 65454647116587227 T^{4} - \)\(76\!\cdots\!56\)\( T^{6} + \)\(65\!\cdots\!81\)\( T^{8} - \)\(43\!\cdots\!10\)\( T^{10} + \)\(23\!\cdots\!67\)\( T^{12} - \)\(99\!\cdots\!88\)\( T^{14} + \)\(35\!\cdots\!87\)\( T^{16} - \)\(10\!\cdots\!10\)\( T^{18} + \)\(23\!\cdots\!61\)\( T^{20} - \)\(40\!\cdots\!96\)\( T^{22} + \)\(52\!\cdots\!27\)\( T^{24} - \)\(44\!\cdots\!58\)\( T^{26} + \)\(18\!\cdots\!21\)\( T^{28} \)
$83$ \( 1 - 17282 T + 149333762 T^{2} - 1088719641698 T^{3} + 16964332297412731 T^{4} - \)\(21\!\cdots\!96\)\( T^{5} + \)\(17\!\cdots\!52\)\( T^{6} - \)\(12\!\cdots\!12\)\( T^{7} + \)\(11\!\cdots\!61\)\( T^{8} - \)\(11\!\cdots\!62\)\( T^{9} + \)\(90\!\cdots\!74\)\( T^{10} - \)\(61\!\cdots\!82\)\( T^{11} + \)\(44\!\cdots\!67\)\( T^{12} - \)\(35\!\cdots\!48\)\( T^{13} + \)\(26\!\cdots\!76\)\( T^{14} - \)\(16\!\cdots\!08\)\( T^{15} + \)\(10\!\cdots\!47\)\( T^{16} - \)\(65\!\cdots\!02\)\( T^{17} + \)\(45\!\cdots\!94\)\( T^{18} - \)\(28\!\cdots\!62\)\( T^{19} + \)\(13\!\cdots\!81\)\( T^{20} - \)\(66\!\cdots\!92\)\( T^{21} + \)\(44\!\cdots\!72\)\( T^{22} - \)\(26\!\cdots\!76\)\( T^{23} + \)\(98\!\cdots\!31\)\( T^{24} - \)\(29\!\cdots\!58\)\( T^{25} + \)\(19\!\cdots\!42\)\( T^{26} - \)\(10\!\cdots\!02\)\( T^{27} + \)\(29\!\cdots\!81\)\( T^{28} \)
$89$ \( 1 - 548528910 T^{2} + 149200943223060123 T^{4} - \)\(26\!\cdots\!16\)\( T^{6} + \)\(35\!\cdots\!49\)\( T^{8} - \)\(37\!\cdots\!50\)\( T^{10} + \)\(31\!\cdots\!11\)\( T^{12} - \)\(21\!\cdots\!00\)\( T^{14} + \)\(12\!\cdots\!91\)\( T^{16} - \)\(57\!\cdots\!50\)\( T^{18} + \)\(21\!\cdots\!09\)\( T^{20} - \)\(64\!\cdots\!36\)\( T^{22} + \)\(14\!\cdots\!23\)\( T^{24} - \)\(20\!\cdots\!10\)\( T^{26} + \)\(14\!\cdots\!61\)\( T^{28} \)
$97$ \( ( 1 + 2 T + 387850619 T^{2} + 251760181236 T^{3} + 75114732161345545 T^{4} + 73269666487293981214 T^{5} + \)\(94\!\cdots\!67\)\( T^{6} + \)\(89\!\cdots\!44\)\( T^{7} + \)\(83\!\cdots\!27\)\( T^{8} + \)\(57\!\cdots\!54\)\( T^{9} + \)\(52\!\cdots\!45\)\( T^{10} + \)\(15\!\cdots\!56\)\( T^{11} + \)\(21\!\cdots\!19\)\( T^{12} + \)\(96\!\cdots\!62\)\( T^{13} + \)\(42\!\cdots\!61\)\( T^{14} )^{2} \)
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