# Properties

 Label 64.5.f.a Level $64$ Weight $5$ Character orbit 64.f Analytic conductor $6.616$ Analytic rank $0$ Dimension $14$ CM no Inner twists $2$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$64 = 2^{6}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 64.f (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.61567763737$$ 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 $$\left(\mathbb{Z}/64\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$63$$ $$\chi(n)$$ $$\beta_{1}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
15.1
 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
15.2 0 −5.54016 5.54016i 0 21.7374 + 21.7374i 0 6.62054 0 19.6133i 0
15.3 0 −3.91498 3.91498i 0 4.72348 + 4.72348i 0 −45.3712 0 50.3458i 0
15.4 0 0.0461995 + 0.0461995i 0 −8.04297 8.04297i 0 49.8797 0 80.9957i 0
15.5 0 4.63552 + 4.63552i 0 −29.2002 29.2002i 0 −59.6196 0 38.0239i 0
15.6 0 7.86839 + 7.86839i 0 27.2309 + 27.2309i 0 −50.3097 0 42.8233i 0
15.7 0 9.42589 + 9.42589i 0 −2.84710 2.84710i 0 76.7794 0 96.6949i 0
47.1 0 −11.5209 + 11.5209i 0 −14.6016 + 14.6016i 0 24.0210 0 184.461i 0
47.2 0 −5.54016 + 5.54016i 0 21.7374 21.7374i 0 6.62054 0 19.6133i 0
47.3 0 −3.91498 + 3.91498i 0 4.72348 4.72348i 0 −45.3712 0 50.3458i 0
47.4 0 0.0461995 0.0461995i 0 −8.04297 + 8.04297i 0 49.8797 0 80.9957i 0
47.5 0 4.63552 4.63552i 0 −29.2002 + 29.2002i 0 −59.6196 0 38.0239i 0
47.6 0 7.86839 7.86839i 0 27.2309 27.2309i 0 −50.3097 0 42.8233i 0
47.7 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 47.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 64.5.f.a 14
3.b odd 2 1 576.5.m.a 14
4.b odd 2 1 16.5.f.a 14
8.b even 2 1 128.5.f.a 14
8.d odd 2 1 128.5.f.b 14
12.b even 2 1 144.5.m.a 14
16.e even 4 1 16.5.f.a 14
16.e even 4 1 128.5.f.b 14
16.f odd 4 1 inner 64.5.f.a 14
16.f odd 4 1 128.5.f.a 14
48.i odd 4 1 144.5.m.a 14
48.k even 4 1 576.5.m.a 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.5.f.a 14 4.b odd 2 1
16.5.f.a 14 16.e even 4 1
64.5.f.a 14 1.a even 1 1 trivial
64.5.f.a 14 16.f odd 4 1 inner
128.5.f.a 14 8.b even 2 1
128.5.f.a 14 16.f odd 4 1
128.5.f.b 14 8.d odd 2 1
128.5.f.b 14 16.e even 4 1
144.5.m.a 14 12.b even 2 1
144.5.m.a 14 48.i odd 4 1
576.5.m.a 14 3.b odd 2 1
576.5.m.a 14 48.k even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(64, [\chi])$$.

## 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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