# Properties

 Label 384.3.l.b Level $384$ Weight $3$ Character orbit 384.l Analytic conductor $10.463$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 384.l (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.4632421514$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + 1408 x^{6} + 3584 x^{5} + 2560 x^{4} - 4096 x^{3} - 24576 x^{2} + 65536$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{24}$$ Twist minimal: no (minimal twist has level 48) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + \beta_{9} q^{5} + \beta_{5} q^{7} -3 \beta_{4} q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + \beta_{9} q^{5} + \beta_{5} q^{7} -3 \beta_{4} q^{9} + ( 2 + 2 \beta_{4} - \beta_{7} + \beta_{10} ) q^{11} + ( \beta_{1} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{13} ) q^{13} -\beta_{14} q^{15} + ( \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{17} + ( -2 - \beta_{1} + 2 \beta_{4} + \beta_{9} - \beta_{11} - 2 \beta_{12} - \beta_{15} ) q^{19} -\beta_{12} q^{21} + ( 8 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{23} + ( -4 \beta_{2} + 4 \beta_{3} - 5 \beta_{4} - 2 \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{25} -3 \beta_{3} q^{27} + ( -2 + \beta_{1} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{10} - 2 \beta_{13} - 3 \beta_{14} + \beta_{15} ) q^{29} + ( -8 \beta_{4} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{15} ) q^{31} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{33} + ( 6 - 2 \beta_{1} - 6 \beta_{4} + 2 \beta_{5} - 2 \beta_{8} + \beta_{9} + \beta_{11} - 2 \beta_{15} ) q^{35} + ( 6 - \beta_{1} - 6 \beta_{4} + \beta_{5} - 3 \beta_{6} - \beta_{8} - \beta_{9} + 2 \beta_{11} - \beta_{12} + 3 \beta_{14} - \beta_{15} ) q^{37} + ( -2 \beta_{1} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{39} + ( 6 \beta_{2} - 6 \beta_{3} + \beta_{7} - 3 \beta_{8} + \beta_{9} - 2 \beta_{12} - 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{41} + ( 10 + \beta_{1} + 10 \beta_{4} + 2 \beta_{5} - \beta_{7} + 2 \beta_{8} - \beta_{10} + 2 \beta_{13} + 2 \beta_{14} - 3 \beta_{15} ) q^{43} + 3 \beta_{7} q^{45} + ( 6 \beta_{2} - 6 \beta_{3} + 24 \beta_{4} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{12} - 2 \beta_{13} ) q^{47} + ( 7 - 8 \beta_{2} - 8 \beta_{3} - 4 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} + 4 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{49} + ( -6 - \beta_{1} + 6 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} - 5 \beta_{9} + \beta_{11} - 2 \beta_{12} + 2 \beta_{14} - \beta_{15} ) q^{51} + ( 10 + \beta_{1} - 8 \beta_{2} - 10 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{8} + 3 \beta_{9} + 2 \beta_{11} + 2 \beta_{12} - 3 \beta_{14} + \beta_{15} ) q^{53} + ( 16 + 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{5} + 4 \beta_{6} - 7 \beta_{7} + 7 \beta_{9} - \beta_{10} - \beta_{11} ) q^{55} + ( -2 \beta_{2} + 2 \beta_{3} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 3 \beta_{14} + \beta_{15} ) q^{57} + ( -8 - 2 \beta_{1} - 8 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{10} - 4 \beta_{14} + 2 \beta_{15} ) q^{59} + ( 2 + \beta_{1} - 16 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - 6 \beta_{10} + \beta_{13} - 2 \beta_{15} ) q^{61} -3 \beta_{8} q^{63} + ( -2 + \beta_{1} + 10 \beta_{2} + 10 \beta_{3} - 3 \beta_{5} + 3 \beta_{6} + \beta_{7} - \beta_{9} - 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{65} + ( 20 + 2 \beta_{1} - 4 \beta_{2} - 20 \beta_{4} - 2 \beta_{5} + 2 \beta_{8} - 4 \beta_{9} - 4 \beta_{11} + 2 \beta_{15} ) q^{67} + ( -6 + 2 \beta_{1} + 8 \beta_{2} + 6 \beta_{4} + 2 \beta_{5} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} + 2 \beta_{15} ) q^{69} + ( -32 - 2 \beta_{5} ) q^{71} + ( 8 \beta_{2} - 8 \beta_{3} + 6 \beta_{4} - 4 \beta_{7} - 4 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + 7 \beta_{14} - 5 \beta_{15} ) q^{73} + ( 12 - 3 \beta_{1} - 5 \beta_{3} + 12 \beta_{4} - 2 \beta_{6} + 6 \beta_{7} + 2 \beta_{13} + \beta_{15} ) q^{75} + ( -14 - \beta_{1} + 24 \beta_{3} - 14 \beta_{4} + 5 \beta_{5} + \beta_{6} + 5 \beta_{8} + 2 \beta_{10} + \beta_{14} + \beta_{15} ) q^{77} + ( -12 \beta_{2} + 12 \beta_{3} + 8 \beta_{4} + 6 \beta_{7} + 5 \beta_{8} + 6 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{79} -9 q^{81} + ( -10 + 2 \beta_{1} + 10 \beta_{4} + 4 \beta_{6} - \beta_{9} + 3 \beta_{11} + 4 \beta_{12} - 4 \beta_{14} + 2 \beta_{15} ) q^{83} + ( -10 - 4 \beta_{1} - 32 \beta_{2} + 10 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} + 4 \beta_{8} - 6 \beta_{9} - 2 \beta_{11} - 2 \beta_{12} + 2 \beta_{14} - 4 \beta_{15} ) q^{85} + ( -2 \beta_{2} - 2 \beta_{3} - \beta_{6} + 3 \beta_{7} - 3 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{87} + ( -12 \beta_{2} + 12 \beta_{3} + 10 \beta_{4} + 8 \beta_{8} + 2 \beta_{10} - 2 \beta_{11} - 4 \beta_{14} + 4 \beta_{15} ) q^{89} + ( -30 + 5 \beta_{1} + 12 \beta_{3} - 30 \beta_{4} + 8 \beta_{6} + 5 \beta_{7} - 3 \beta_{10} - 2 \beta_{13} + 6 \beta_{14} - 3 \beta_{15} ) q^{91} + ( -\beta_{1} - 8 \beta_{3} + 5 \beta_{5} - \beta_{6} + 2 \beta_{7} + 5 \beta_{8} + 4 \beta_{10} + \beta_{13} ) q^{93} + ( 10 \beta_{2} - 10 \beta_{3} - 40 \beta_{4} + 4 \beta_{7} + 4 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + 4 \beta_{15} ) q^{95} + ( -2 \beta_{1} + 12 \beta_{2} + 12 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} - 4 \beta_{10} - 4 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{97} + ( 6 - 6 \beta_{4} + 3 \beta_{9} + 3 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q + 32q^{11} - 32q^{19} + 128q^{23} - 32q^{29} + 96q^{35} + 96q^{37} + 160q^{43} + 112q^{49} - 96q^{51} + 160q^{53} + 256q^{55} - 128q^{59} + 32q^{61} - 32q^{65} + 320q^{67} - 96q^{69} - 512q^{71} + 192q^{75} - 224q^{77} - 144q^{81} - 160q^{83} - 160q^{85} - 480q^{91} + 96q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + 1408 x^{6} + 3584 x^{5} + 2560 x^{4} - 4096 x^{3} - 24576 x^{2} + 65536$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$14 \nu^{15} - 15 \nu^{14} - 80 \nu^{13} - 126 \nu^{12} + 80 \nu^{11} + 1258 \nu^{10} + 1392 \nu^{9} - 2184 \nu^{8} - 10752 \nu^{7} - 16752 \nu^{6} + 16960 \nu^{5} + 82304 \nu^{4} + 84480 \nu^{3} - 166400 \nu^{2} - 620544 \nu - 163840$$$$)/61440$$ $$\beta_{2}$$ $$=$$ $$($$$$-81 \nu^{15} - 268 \nu^{14} - 218 \nu^{13} + 588 \nu^{12} + 2310 \nu^{11} + 1616 \nu^{10} - 9208 \nu^{9} - 30752 \nu^{8} - 38416 \nu^{7} + 22336 \nu^{6} + 142976 \nu^{5} + 146432 \nu^{4} - 195072 \nu^{3} - 976896 \nu^{2} - 966656 \nu + 180224$$$$)/245760$$ $$\beta_{3}$$ $$=$$ $$($$$$131 \nu^{15} + 88 \nu^{14} - 1122 \nu^{13} - 2268 \nu^{12} - 610 \nu^{11} + 9944 \nu^{10} + 27688 \nu^{9} + 1472 \nu^{8} - 117584 \nu^{7} - 278656 \nu^{6} - 125056 \nu^{5} + 588288 \nu^{4} + 1316352 \nu^{3} + 741376 \nu^{2} - 4317184 \nu - 7716864$$$$)/245760$$ $$\beta_{4}$$ $$=$$ $$($$$$-347 \nu^{15} - 626 \nu^{14} + 1234 \nu^{13} + 4536 \nu^{12} + 5530 \nu^{11} - 11868 \nu^{10} - 59096 \nu^{9} - 66544 \nu^{8} + 88528 \nu^{7} + 450272 \nu^{6} + 454272 \nu^{5} - 499456 \nu^{4} - 2271744 \nu^{3} - 3177472 \nu^{2} + 3719168 \nu + 10231808$$$$)/368640$$ $$\beta_{5}$$ $$=$$ $$($$$$-47 \nu^{15} - 110 \nu^{14} + 90 \nu^{13} + 528 \nu^{12} + 610 \nu^{11} - 1684 \nu^{10} - 8376 \nu^{9} - 11728 \nu^{8} + 5136 \nu^{7} + 52256 \nu^{6} + 60800 \nu^{5} - 73472 \nu^{4} - 350720 \nu^{3} - 537600 \nu^{2} + 172032 \nu + 1228800$$$$)/40960$$ $$\beta_{6}$$ $$=$$ $$($$$$91 \nu^{15} + 1260 \nu^{14} + 3590 \nu^{13} + 2316 \nu^{12} - 9170 \nu^{11} - 28288 \nu^{10} - 4872 \nu^{9} + 162144 \nu^{8} + 452592 \nu^{7} + 476352 \nu^{6} - 428800 \nu^{5} - 1854464 \nu^{4} - 1497600 \nu^{3} + 4352000 \nu^{2} + 14905344 \nu + 14909440$$$$)/122880$$ $$\beta_{7}$$ $$=$$ $$($$$$-751 \nu^{15} - 448 \nu^{14} + 6626 \nu^{13} + 14076 \nu^{12} + 5306 \nu^{11} - 54888 \nu^{10} - 156376 \nu^{9} - 19520 \nu^{8} + 641360 \nu^{7} + 1598464 \nu^{6} + 829440 \nu^{5} - 3026432 \nu^{4} - 7073280 \nu^{3} - 4517888 \nu^{2} + 22466560 \nu + 41107456$$$$)/368640$$ $$\beta_{8}$$ $$=$$ $$($$$$1063 \nu^{15} + 2242 \nu^{14} - 2666 \nu^{13} - 14184 \nu^{12} - 21122 \nu^{11} + 26652 \nu^{10} + 189400 \nu^{9} + 278960 \nu^{8} - 106640 \nu^{7} - 1269472 \nu^{6} - 1748352 \nu^{5} + 775424 \nu^{4} + 6971904 \nu^{3} + 12336128 \nu^{2} - 4268032 \nu - 26214400$$$$)/368640$$ $$\beta_{9}$$ $$=$$ $$($$$$-545 \nu^{15} - 1574 \nu^{14} - 302 \nu^{13} + 5256 \nu^{12} + 12838 \nu^{11} - 1188 \nu^{10} - 82544 \nu^{9} - 190768 \nu^{8} - 127664 \nu^{7} + 372128 \nu^{6} + 897600 \nu^{5} + 303872 \nu^{4} - 2511360 \nu^{3} - 7066624 \nu^{2} - 3770368 \nu + 6053888$$$$)/184320$$ $$\beta_{10}$$ $$=$$ $$($$$$-134 \nu^{15} - 20 \nu^{14} + 1153 \nu^{13} + 2232 \nu^{12} + 622 \nu^{11} - 9756 \nu^{10} - 25118 \nu^{9} + 1448 \nu^{8} + 113704 \nu^{7} + 257408 \nu^{6} + 102288 \nu^{5} - 502912 \nu^{4} - 1184256 \nu^{3} - 538624 \nu^{2} + 3969536 \nu + 6232064$$$$)/46080$$ $$\beta_{11}$$ $$=$$ $$($$$$1417 \nu^{15} + 4300 \nu^{14} + 1186 \nu^{13} - 13356 \nu^{12} - 35366 \nu^{11} + 528 \nu^{10} + 219304 \nu^{9} + 508256 \nu^{8} + 401488 \nu^{7} - 933184 \nu^{6} - 2421504 \nu^{5} - 922624 \nu^{4} + 6455808 \nu^{3} + 18839552 \nu^{2} + 11743232 \nu - 13975552$$$$)/368640$$ $$\beta_{12}$$ $$=$$ $$($$$$411 \nu^{15} - 178 \nu^{14} - 4586 \nu^{13} - 7776 \nu^{12} + 198 \nu^{11} + 38228 \nu^{10} + 84104 \nu^{9} - 37808 \nu^{8} - 478864 \nu^{7} - 985376 \nu^{6} - 282112 \nu^{5} + 2124032 \nu^{4} + 4176384 \nu^{3} + 1102848 \nu^{2} - 16609280 \nu - 25722880$$$$)/122880$$ $$\beta_{13}$$ $$=$$ $$($$$$-1229 \nu^{15} + 844 \nu^{14} + 13750 \nu^{13} + 23436 \nu^{12} - 2786 \nu^{11} - 118848 \nu^{10} - 248264 \nu^{9} + 154016 \nu^{8} + 1503088 \nu^{7} + 2985920 \nu^{6} + 638208 \nu^{5} - 6613504 \nu^{4} - 12980736 \nu^{3} - 1914880 \nu^{2} + 52957184 \nu + 79364096$$$$)/368640$$ $$\beta_{14}$$ $$=$$ $$($$$$-151 \nu^{15} - 271 \nu^{14} + 512 \nu^{13} + 1878 \nu^{12} + 2402 \nu^{11} - 4854 \nu^{10} - 25204 \nu^{9} - 30296 \nu^{8} + 32192 \nu^{7} + 186640 \nu^{6} + 194400 \nu^{5} - 199808 \nu^{4} - 975360 \nu^{3} - 1435136 \nu^{2} + 1352704 \nu + 4022272$$$$)/30720$$ $$\beta_{15}$$ $$=$$ $$($$$$-4331 \nu^{15} - 7634 \nu^{14} + 15322 \nu^{13} + 56088 \nu^{12} + 66634 \nu^{11} - 143484 \nu^{10} - 729800 \nu^{9} - 826480 \nu^{8} + 1085200 \nu^{7} + 5587424 \nu^{6} + 5662464 \nu^{5} - 6193408 \nu^{4} - 28205568 \nu^{3} - 39095296 \nu^{2} + 45215744 \nu + 128368640$$$$)/368640$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{11} - \beta_{9} - \beta_{8} - \beta_{6} + 2 \beta_{4} - 2 \beta_{3} - \beta_{1}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{15} - 2 \beta_{14} - 2 \beta_{13} - \beta_{11} + \beta_{10} - \beta_{9} - 3 \beta_{7} + 2 \beta_{6} - 2 \beta_{4} - 4 \beta_{2} + 6$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{15} - 3 \beta_{14} - 4 \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + 3 \beta_{7} - 3 \beta_{6} - \beta_{5} + 10 \beta_{4} + 6 \beta_{3} - 6 \beta_{2} - 3 \beta_{1} + 6$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{15} - 3 \beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} + 2 \beta_{10} - 4 \beta_{9} - \beta_{8} - 5 \beta_{7} + \beta_{5} + 22 \beta_{4} + 4 \beta_{3} - 12 \beta_{2} + 3 \beta_{1} + 8$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-4 \beta_{14} - 4 \beta_{13} - 8 \beta_{12} - \beta_{11} - 7 \beta_{9} - 7 \beta_{8} - 4 \beta_{7} - \beta_{6} + 6 \beta_{5} - 14 \beta_{4} - 6 \beta_{3} + 40 \beta_{2} - \beta_{1} - 40$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$8 \beta_{15} + 2 \beta_{14} + 4 \beta_{13} + 2 \beta_{12} - 9 \beta_{11} - 13 \beta_{10} - 7 \beta_{9} + 8 \beta_{8} - 7 \beta_{7} + 8 \beta_{6} - 4 \beta_{5} - 2 \beta_{4} + 36 \beta_{3} - 48 \beta_{2} + 8 \beta_{1} - 102$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-11 \beta_{15} - 13 \beta_{14} - 4 \beta_{13} - 8 \beta_{12} + 3 \beta_{11} + 5 \beta_{10} + 57 \beta_{9} + 13 \beta_{8} - 19 \beta_{7} + 17 \beta_{6} - 11 \beta_{5} + 90 \beta_{4} - 42 \beta_{3} + 22 \beta_{2} + 21 \beta_{1} - 182$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$-8 \beta_{15} + 3 \beta_{14} + 21 \beta_{13} + 9 \beta_{12} - 26 \beta_{11} + 3 \beta_{10} - 35 \beta_{9} + 9 \beta_{8} + 24 \beta_{7} - 12 \beta_{6} + 9 \beta_{5} - 16 \beta_{4} + 24 \beta_{3} + 72 \beta_{2} + 17 \beta_{1} - 146$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-19 \beta_{15} + 71 \beta_{14} + 12 \beta_{13} + 16 \beta_{12} + 42 \beta_{11} - 45 \beta_{10} + 38 \beta_{9} + 12 \beta_{8} + 23 \beta_{7} + 38 \beta_{6} + 35 \beta_{5} + 20 \beta_{4} + 108 \beta_{3} + 270 \beta_{2} + 30 \beta_{1} - 138$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$50 \beta_{15} + 42 \beta_{14} + 38 \beta_{13} + 80 \beta_{12} - 5 \beta_{11} + 17 \beta_{10} + 107 \beta_{9} + 56 \beta_{8} + 53 \beta_{7} + 6 \beta_{6} - 128 \beta_{5} - 1018 \beta_{4} - 320 \beta_{3} - 116 \beta_{2} + 32 \beta_{1} - 786$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-215 \beta_{15} + 219 \beta_{14} + 48 \beta_{13} + 36 \beta_{12} - 143 \beta_{11} - 75 \beta_{10} + 175 \beta_{9} + 59 \beta_{8} - 47 \beta_{7} + 183 \beta_{6} - 199 \beta_{5} + 1006 \beta_{4} - 198 \beta_{3} + 446 \beta_{2} + 39 \beta_{1} + 1210$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$-38 \beta_{15} - 13 \beta_{14} + 165 \beta_{13} + 227 \beta_{12} + 175 \beta_{11} + 42 \beta_{10} + 152 \beta_{9} + 17 \beta_{8} + 541 \beta_{7} - 132 \beta_{6} + 71 \beta_{5} - 238 \beta_{4} + 516 \beta_{3} + 180 \beta_{2} - 123 \beta_{1} + 8$$ $$\nu^{13}$$ $$=$$ $$192 \beta_{15} + 300 \beta_{14} - 452 \beta_{13} + 16 \beta_{12} + 161 \beta_{11} + 304 \beta_{10} - 353 \beta_{9} - 9 \beta_{8} - 52 \beta_{7} + 33 \beta_{6} + 10 \beta_{5} - 1394 \beta_{4} - 250 \beta_{3} - 312 \beta_{2} + 73 \beta_{1} + 3224$$ $$\nu^{14}$$ $$=$$ $$-280 \beta_{15} + 390 \beta_{14} + 484 \beta_{13} - 426 \beta_{12} + 25 \beta_{11} - 371 \beta_{10} + 495 \beta_{9} - 888 \beta_{8} + 63 \beta_{7} - 416 \beta_{6} - 692 \beta_{5} - 4926 \beta_{4} - 500 \beta_{3} - 112 \beta_{2} - 1096 \beta_{1} + 7174$$ $$\nu^{15}$$ $$=$$ $$211 \beta_{15} - 779 \beta_{14} - 1244 \beta_{13} + 920 \beta_{12} + 565 \beta_{11} + 11 \beta_{10} - 977 \beta_{9} - 1245 \beta_{8} - 781 \beta_{7} + 591 \beta_{6} - 605 \beta_{5} + 14390 \beta_{4} - 2838 \beta_{3} - 10182 \beta_{2} - 149 \beta_{1} + 4502$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$133$$ $$257$$ $$\chi(n)$$ $$-1$$ $$-\beta_{4}$$ $$1$$

## 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
 1.80398 + 0.863518i −1.96679 − 0.362960i −0.455024 − 1.94755i 1.84258 − 0.777752i 0.125358 + 1.99607i 1.78012 − 0.911682i −1.25564 − 1.55672i −1.87459 + 0.697079i 1.80398 − 0.863518i −1.96679 + 0.362960i −0.455024 + 1.94755i 1.84258 + 0.777752i 0.125358 − 1.99607i 1.78012 + 0.911682i −1.25564 + 1.55672i −1.87459 − 0.697079i
0 −1.22474 1.22474i 0 −6.49473 6.49473i 0 −3.94273 0 3.00000i 0
31.2 0 −1.22474 1.22474i 0 −1.69930 1.69930i 0 5.74280 0 3.00000i 0
31.3 0 −1.22474 1.22474i 0 3.40572 + 3.40572i 0 −12.1303 0 3.00000i 0
31.4 0 −1.22474 1.22474i 0 4.78830 + 4.78830i 0 10.3302 0 3.00000i 0
31.5 0 1.22474 + 1.22474i 0 −3.32679 3.32679i 0 4.04088 0 3.00000i 0
31.6 0 1.22474 + 1.22474i 0 −1.00772 1.00772i 0 −10.0236 0 3.00000i 0
31.7 0 1.22474 + 1.22474i 0 −0.909023 0.909023i 0 0.654713 0 3.00000i 0
31.8 0 1.22474 + 1.22474i 0 5.24354 + 5.24354i 0 5.32796 0 3.00000i 0
223.1 0 −1.22474 + 1.22474i 0 −6.49473 + 6.49473i 0 −3.94273 0 3.00000i 0
223.2 0 −1.22474 + 1.22474i 0 −1.69930 + 1.69930i 0 5.74280 0 3.00000i 0
223.3 0 −1.22474 + 1.22474i 0 3.40572 3.40572i 0 −12.1303 0 3.00000i 0
223.4 0 −1.22474 + 1.22474i 0 4.78830 4.78830i 0 10.3302 0 3.00000i 0
223.5 0 1.22474 1.22474i 0 −3.32679 + 3.32679i 0 4.04088 0 3.00000i 0
223.6 0 1.22474 1.22474i 0 −1.00772 + 1.00772i 0 −10.0236 0 3.00000i 0
223.7 0 1.22474 1.22474i 0 −0.909023 + 0.909023i 0 0.654713 0 3.00000i 0
223.8 0 1.22474 1.22474i 0 5.24354 5.24354i 0 5.32796 0 3.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 223.8 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 384.3.l.b 16
3.b odd 2 1 1152.3.m.c 16
4.b odd 2 1 384.3.l.a 16
8.b even 2 1 192.3.l.a 16
8.d odd 2 1 48.3.l.a 16
12.b even 2 1 1152.3.m.f 16
16.e even 4 1 48.3.l.a 16
16.e even 4 1 384.3.l.a 16
16.f odd 4 1 192.3.l.a 16
16.f odd 4 1 inner 384.3.l.b 16
24.f even 2 1 144.3.m.c 16
24.h odd 2 1 576.3.m.c 16
48.i odd 4 1 144.3.m.c 16
48.i odd 4 1 1152.3.m.f 16
48.k even 4 1 576.3.m.c 16
48.k even 4 1 1152.3.m.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.3.l.a 16 8.d odd 2 1
48.3.l.a 16 16.e even 4 1
144.3.m.c 16 24.f even 2 1
144.3.m.c 16 48.i odd 4 1
192.3.l.a 16 8.b even 2 1
192.3.l.a 16 16.f odd 4 1
384.3.l.a 16 4.b odd 2 1
384.3.l.a 16 16.e even 4 1
384.3.l.b 16 1.a even 1 1 trivial
384.3.l.b 16 16.f odd 4 1 inner
576.3.m.c 16 24.h odd 2 1
576.3.m.c 16 48.k even 4 1
1152.3.m.c 16 3.b odd 2 1
1152.3.m.c 16 48.k even 4 1
1152.3.m.f 16 12.b even 2 1
1152.3.m.f 16 48.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} - 224 T_{7}^{6} + 448 T_{7}^{5} + 13704 T_{7}^{4} - 53248 T_{7}^{3} - 136576 T_{7}^{2} + 720640 T_{7} - 400880$$ acting on $$S_{3}^{\mathrm{new}}(384, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + 9 T^{4} )^{4}$$
$5$ $$1 - 32 T^{3} - 344 T^{4} - 5664 T^{5} + 512 T^{6} - 145600 T^{7} + 223452 T^{8} + 2255168 T^{9} + 20875776 T^{10} + 67282720 T^{11} + 753060504 T^{12} - 1881828576 T^{13} + 2740220928 T^{14} - 75471830656 T^{15} - 399298967994 T^{16} - 1886795766400 T^{17} + 1712638080000 T^{18} - 29403571500000 T^{19} + 294164259375000 T^{20} + 657057812500000 T^{21} + 5096625000000000 T^{22} + 13764453125000000 T^{23} + 34096069335937500 T^{24} - 555419921875000000 T^{25} + 48828125000000000 T^{26} - 13504028320312500000 T^{27} - 20503997802734375000 T^{28} - 47683715820312500000 T^{29} +$$$$23\!\cdots\!25$$$$T^{32}$$
$7$ $$( 1 + 168 T^{2} + 448 T^{3} + 15076 T^{4} + 56512 T^{5} + 1070392 T^{6} + 3649664 T^{7} + 60103046 T^{8} + 178833536 T^{9} + 2570011192 T^{10} + 6648580288 T^{11} + 86910139876 T^{12} + 126548911552 T^{13} + 2325336249768 T^{14} + 33232930569601 T^{16} )^{2}$$
$11$ $$1 - 32 T + 512 T^{2} - 8480 T^{3} + 137032 T^{4} - 1636576 T^{5} + 18165248 T^{6} - 219655136 T^{7} + 2263228700 T^{8} - 21867108000 T^{9} + 253620152832 T^{10} - 2916953293728 T^{11} + 33797606438392 T^{12} - 431768458252384 T^{13} + 5293166227138048 T^{14} - 61910274521995104 T^{15} + 703855220885889990 T^{16} - 7491143217161407584 T^{17} + 77497246731528160768 T^{18} -$$$$76\!\cdots\!24$$$$T^{19} +$$$$72\!\cdots\!52$$$$T^{20} -$$$$75\!\cdots\!28$$$$T^{21} +$$$$79\!\cdots\!72$$$$T^{22} -$$$$83\!\cdots\!00$$$$T^{23} +$$$$10\!\cdots\!00$$$$T^{24} -$$$$12\!\cdots\!16$$$$T^{25} +$$$$12\!\cdots\!48$$$$T^{26} -$$$$13\!\cdots\!96$$$$T^{27} +$$$$13\!\cdots\!12$$$$T^{28} -$$$$10\!\cdots\!80$$$$T^{29} +$$$$73\!\cdots\!72$$$$T^{30} -$$$$55\!\cdots\!32$$$$T^{31} +$$$$21\!\cdots\!21$$$$T^{32}$$
$13$ $$1 - 3200 T^{3} + 7608 T^{4} - 95360 T^{5} + 5120000 T^{6} - 68335872 T^{7} + 2004669468 T^{8} - 7270355200 T^{9} + 184268595200 T^{10} - 4889456013184 T^{11} + 5354592144136 T^{12} - 669839496880000 T^{13} + 7008632866619392 T^{14} - 70586941744778752 T^{15} + 2398056097119178950 T^{16} - 11929193154867609088 T^{17} +$$$$20\!\cdots\!12$$$$T^{18} -$$$$32\!\cdots\!00$$$$T^{19} +$$$$43\!\cdots\!56$$$$T^{20} -$$$$67\!\cdots\!16$$$$T^{21} +$$$$42\!\cdots\!00$$$$T^{22} -$$$$28\!\cdots\!00$$$$T^{23} +$$$$13\!\cdots\!88$$$$T^{24} -$$$$76\!\cdots\!88$$$$T^{25} +$$$$97\!\cdots\!00$$$$T^{26} -$$$$30\!\cdots\!40$$$$T^{27} +$$$$41\!\cdots\!88$$$$T^{28} -$$$$29\!\cdots\!00$$$$T^{29} +$$$$44\!\cdots\!81$$$$T^{32}$$
$17$ $$( 1 + 968 T^{2} - 2944 T^{3} + 516540 T^{4} - 3209600 T^{5} + 201700088 T^{6} - 1543904000 T^{7} + 63894476806 T^{8} - 446188256000 T^{9} + 16846193049848 T^{10} - 77471941462400 T^{11} + 3603257748574140 T^{12} - 5935086042921856 T^{13} + 563978325638408648 T^{14} + 48661191875666868481 T^{16} )^{2}$$
$19$ $$1 + 32 T + 512 T^{2} - 2656 T^{3} - 523448 T^{4} - 8424608 T^{5} + 1945088 T^{6} + 4454446304 T^{7} + 107916937244 T^{8} - 703649376 T^{9} - 30601835632128 T^{10} - 698985761087712 T^{11} + 998616856187896 T^{12} + 253693358084547040 T^{13} + 3161998119961945600 T^{14} - 30474951661580761248 T^{15} -$$$$19\!\cdots\!42$$$$T^{16} -$$$$11\!\cdots\!28$$$$T^{17} +$$$$41\!\cdots\!00$$$$T^{18} +$$$$11\!\cdots\!40$$$$T^{19} +$$$$16\!\cdots\!36$$$$T^{20} -$$$$42\!\cdots\!12$$$$T^{21} -$$$$67\!\cdots\!08$$$$T^{22} -$$$$56\!\cdots\!96$$$$T^{23} +$$$$31\!\cdots\!64$$$$T^{24} +$$$$46\!\cdots\!64$$$$T^{25} +$$$$73\!\cdots\!88$$$$T^{26} -$$$$11\!\cdots\!88$$$$T^{27} -$$$$25\!\cdots\!08$$$$T^{28} -$$$$46\!\cdots\!36$$$$T^{29} +$$$$32\!\cdots\!92$$$$T^{30} +$$$$73\!\cdots\!32$$$$T^{31} +$$$$83\!\cdots\!61$$$$T^{32}$$
$23$ $$( 1 - 64 T + 3496 T^{2} - 127936 T^{3} + 4410332 T^{4} - 130001728 T^{5} + 3673719192 T^{6} - 94049622208 T^{7} + 2261818535238 T^{8} - 49752250148032 T^{9} + 1028057252408472 T^{10} - 19244921376016192 T^{11} + 345377444336323292 T^{12} - 5299942138629398464 T^{13} + 76613527014343042216 T^{14} -$$$$74\!\cdots\!76$$$$T^{15} +$$$$61\!\cdots\!61$$$$T^{16} )^{2}$$
$29$ $$1 + 32 T + 512 T^{2} - 18368 T^{3} - 1552984 T^{4} - 20596992 T^{5} + 304715776 T^{6} + 25469097376 T^{7} + 491466517980 T^{8} - 9791032230816 T^{9} - 347423504794624 T^{10} - 2649303176415616 T^{11} + 694517140133881240 T^{12} + 20658732330776531008 T^{13} +$$$$19\!\cdots\!28$$$$T^{14} -$$$$11\!\cdots\!40$$$$T^{15} -$$$$82\!\cdots\!10$$$$T^{16} -$$$$96\!\cdots\!40$$$$T^{17} +$$$$13\!\cdots\!68$$$$T^{18} +$$$$12\!\cdots\!68$$$$T^{19} +$$$$34\!\cdots\!40$$$$T^{20} -$$$$11\!\cdots\!16$$$$T^{21} -$$$$12\!\cdots\!84$$$$T^{22} -$$$$29\!\cdots\!96$$$$T^{23} +$$$$12\!\cdots\!80$$$$T^{24} +$$$$53\!\cdots\!36$$$$T^{25} +$$$$53\!\cdots\!76$$$$T^{26} -$$$$30\!\cdots\!72$$$$T^{27} -$$$$19\!\cdots\!04$$$$T^{28} -$$$$19\!\cdots\!28$$$$T^{29} +$$$$45\!\cdots\!32$$$$T^{30} +$$$$23\!\cdots\!32$$$$T^{31} +$$$$62\!\cdots\!41$$$$T^{32}$$
$31$ $$1 - 7312 T^{2} + 29025544 T^{4} - 80335806576 T^{6} + 171125889681052 T^{8} - 295006946315669072 T^{10} +$$$$42\!\cdots\!64$$$$T^{12} -$$$$51\!\cdots\!00$$$$T^{14} +$$$$53\!\cdots\!38$$$$T^{16} -$$$$47\!\cdots\!00$$$$T^{18} +$$$$36\!\cdots\!24$$$$T^{20} -$$$$23\!\cdots\!92$$$$T^{22} +$$$$12\!\cdots\!12$$$$T^{24} -$$$$53\!\cdots\!76$$$$T^{26} +$$$$18\!\cdots\!24$$$$T^{28} -$$$$41\!\cdots\!92$$$$T^{30} +$$$$52\!\cdots\!61$$$$T^{32}$$
$37$ $$1 - 96 T + 4608 T^{2} - 145952 T^{3} + 4040888 T^{4} - 217733344 T^{5} + 12932982272 T^{6} - 602883756192 T^{7} + 21839639792924 T^{8} - 655265530977504 T^{9} + 21703692469355008 T^{10} - 815191556064282016 T^{11} + 35433653736114978312 T^{12} -$$$$14\!\cdots\!40$$$$T^{13} +$$$$50\!\cdots\!52$$$$T^{14} -$$$$14\!\cdots\!84$$$$T^{15} +$$$$43\!\cdots\!90$$$$T^{16} -$$$$20\!\cdots\!96$$$$T^{17} +$$$$95\!\cdots\!72$$$$T^{18} -$$$$37\!\cdots\!60$$$$T^{19} +$$$$12\!\cdots\!52$$$$T^{20} -$$$$39\!\cdots\!84$$$$T^{21} +$$$$14\!\cdots\!48$$$$T^{22} -$$$$59\!\cdots\!56$$$$T^{23} +$$$$26\!\cdots\!84$$$$T^{24} -$$$$10\!\cdots\!68$$$$T^{25} +$$$$29\!\cdots\!72$$$$T^{26} -$$$$68\!\cdots\!36$$$$T^{27} +$$$$17\!\cdots\!68$$$$T^{28} -$$$$86\!\cdots\!68$$$$T^{29} +$$$$37\!\cdots\!68$$$$T^{30} -$$$$10\!\cdots\!04$$$$T^{31} +$$$$15\!\cdots\!81$$$$T^{32}$$
$41$ $$1 - 13840 T^{2} + 102706104 T^{4} - 524939980080 T^{6} + 2044068651261084 T^{8} - 6376104819902485008 T^{10} +$$$$16\!\cdots\!68$$$$T^{12} -$$$$35\!\cdots\!72$$$$T^{14} +$$$$64\!\cdots\!06$$$$T^{16} -$$$$99\!\cdots\!92$$$$T^{18} +$$$$13\!\cdots\!28$$$$T^{20} -$$$$14\!\cdots\!48$$$$T^{22} +$$$$13\!\cdots\!44$$$$T^{24} -$$$$94\!\cdots\!80$$$$T^{26} +$$$$52\!\cdots\!44$$$$T^{28} -$$$$19\!\cdots\!40$$$$T^{30} +$$$$40\!\cdots\!81$$$$T^{32}$$
$43$ $$1 - 160 T + 12800 T^{2} - 978464 T^{3} + 71106632 T^{4} - 3813053664 T^{5} + 178619596288 T^{6} - 8719368905312 T^{7} + 336417491247900 T^{8} - 9339737479444512 T^{9} + 288453906337733120 T^{10} - 7137460469658328480 T^{11} -$$$$12\!\cdots\!76$$$$T^{12} +$$$$13\!\cdots\!84$$$$T^{13} -$$$$44\!\cdots\!20$$$$T^{14} +$$$$28\!\cdots\!76$$$$T^{15} -$$$$17\!\cdots\!30$$$$T^{16} +$$$$53\!\cdots\!24$$$$T^{17} -$$$$15\!\cdots\!20$$$$T^{18} +$$$$83\!\cdots\!16$$$$T^{19} -$$$$15\!\cdots\!76$$$$T^{20} -$$$$15\!\cdots\!20$$$$T^{21} +$$$$11\!\cdots\!20$$$$T^{22} -$$$$69\!\cdots\!88$$$$T^{23} +$$$$45\!\cdots\!00$$$$T^{24} -$$$$22\!\cdots\!88$$$$T^{25} +$$$$83\!\cdots\!88$$$$T^{26} -$$$$32\!\cdots\!36$$$$T^{27} +$$$$11\!\cdots\!32$$$$T^{28} -$$$$28\!\cdots\!36$$$$T^{29} +$$$$69\!\cdots\!00$$$$T^{30} -$$$$16\!\cdots\!40$$$$T^{31} +$$$$18\!\cdots\!01$$$$T^{32}$$
$47$ $$1 - 24144 T^{2} + 280869112 T^{4} - 2097883923184 T^{6} + 11327375509374492 T^{8} - 47271044690493269328 T^{10} +$$$$15\!\cdots\!16$$$$T^{12} -$$$$44\!\cdots\!04$$$$T^{14} +$$$$10\!\cdots\!58$$$$T^{16} -$$$$21\!\cdots\!24$$$$T^{18} +$$$$37\!\cdots\!76$$$$T^{20} -$$$$54\!\cdots\!48$$$$T^{22} +$$$$64\!\cdots\!32$$$$T^{24} -$$$$58\!\cdots\!84$$$$T^{26} +$$$$37\!\cdots\!72$$$$T^{28} -$$$$15\!\cdots\!84$$$$T^{30} +$$$$32\!\cdots\!41$$$$T^{32}$$
$53$ $$1 - 160 T + 12800 T^{2} - 602944 T^{3} + 3948712 T^{4} + 1481707264 T^{5} - 105845942272 T^{6} + 3791430241760 T^{7} + 34861972067036 T^{8} - 14471440004155872 T^{9} + 1098860393015073792 T^{10} - 54880211634179791488 T^{11} +$$$$13\!\cdots\!12$$$$T^{12} +$$$$17\!\cdots\!00$$$$T^{13} -$$$$18\!\cdots\!48$$$$T^{14} -$$$$16\!\cdots\!08$$$$T^{15} +$$$$51\!\cdots\!54$$$$T^{16} -$$$$46\!\cdots\!72$$$$T^{17} -$$$$14\!\cdots\!88$$$$T^{18} +$$$$38\!\cdots\!00$$$$T^{19} +$$$$84\!\cdots\!32$$$$T^{20} -$$$$95\!\cdots\!12$$$$T^{21} +$$$$53\!\cdots\!72$$$$T^{22} -$$$$19\!\cdots\!68$$$$T^{23} +$$$$13\!\cdots\!56$$$$T^{24} +$$$$41\!\cdots\!40$$$$T^{25} -$$$$32\!\cdots\!72$$$$T^{26} +$$$$12\!\cdots\!76$$$$T^{27} +$$$$95\!\cdots\!72$$$$T^{28} -$$$$40\!\cdots\!76$$$$T^{29} +$$$$24\!\cdots\!00$$$$T^{30} -$$$$85\!\cdots\!40$$$$T^{31} +$$$$15\!\cdots\!41$$$$T^{32}$$
$59$ $$1 + 128 T + 8192 T^{2} + 1121408 T^{3} + 136226184 T^{4} + 9279937408 T^{5} + 700645040128 T^{6} + 71627082366848 T^{7} + 5234572115355804 T^{8} + 316007889653226112 T^{9} + 25502997282495045632 T^{10} +$$$$19\!\cdots\!80$$$$T^{11} +$$$$10\!\cdots\!40$$$$T^{12} +$$$$69\!\cdots\!16$$$$T^{13} +$$$$51\!\cdots\!56$$$$T^{14} +$$$$29\!\cdots\!24$$$$T^{15} +$$$$15\!\cdots\!38$$$$T^{16} +$$$$10\!\cdots\!44$$$$T^{17} +$$$$62\!\cdots\!16$$$$T^{18} +$$$$29\!\cdots\!56$$$$T^{19} +$$$$16\!\cdots\!40$$$$T^{20} +$$$$98\!\cdots\!80$$$$T^{21} +$$$$45\!\cdots\!92$$$$T^{22} +$$$$19\!\cdots\!32$$$$T^{23} +$$$$11\!\cdots\!64$$$$T^{24} +$$$$53\!\cdots\!08$$$$T^{25} +$$$$18\!\cdots\!28$$$$T^{26} +$$$$84\!\cdots\!48$$$$T^{27} +$$$$43\!\cdots\!24$$$$T^{28} +$$$$12\!\cdots\!28$$$$T^{29} +$$$$31\!\cdots\!32$$$$T^{30} +$$$$17\!\cdots\!28$$$$T^{31} +$$$$46\!\cdots\!81$$$$T^{32}$$
$61$ $$1 - 32 T + 512 T^{2} + 38048 T^{3} - 60439624 T^{4} + 1520787552 T^{5} - 16996289024 T^{6} - 2981900088544 T^{7} + 2018049968078364 T^{8} - 40394929489472928 T^{9} + 275088896591278592 T^{10} +$$$$10\!\cdots\!56$$$$T^{11} -$$$$45\!\cdots\!20$$$$T^{12} +$$$$71\!\cdots\!16$$$$T^{13} -$$$$18\!\cdots\!28$$$$T^{14} -$$$$23\!\cdots\!64$$$$T^{15} +$$$$72\!\cdots\!42$$$$T^{16} -$$$$88\!\cdots\!44$$$$T^{17} -$$$$26\!\cdots\!48$$$$T^{18} +$$$$36\!\cdots\!76$$$$T^{19} -$$$$86\!\cdots\!20$$$$T^{20} +$$$$75\!\cdots\!56$$$$T^{21} +$$$$73\!\cdots\!32$$$$T^{22} -$$$$39\!\cdots\!48$$$$T^{23} +$$$$74\!\cdots\!04$$$$T^{24} -$$$$40\!\cdots\!64$$$$T^{25} -$$$$86\!\cdots\!24$$$$T^{26} +$$$$28\!\cdots\!92$$$$T^{27} -$$$$42\!\cdots\!84$$$$T^{28} +$$$$99\!\cdots\!28$$$$T^{29} +$$$$49\!\cdots\!72$$$$T^{30} -$$$$11\!\cdots\!32$$$$T^{31} +$$$$13\!\cdots\!21$$$$T^{32}$$
$67$ $$1 - 320 T + 51200 T^{2} - 6047552 T^{3} + 641735304 T^{4} - 64228593856 T^{5} + 5982745065472 T^{6} - 525110406070976 T^{7} + 43992629224199580 T^{8} - 3502096836597496384 T^{9} +$$$$26\!\cdots\!12$$$$T^{10} -$$$$19\!\cdots\!80$$$$T^{11} +$$$$14\!\cdots\!20$$$$T^{12} -$$$$10\!\cdots\!88$$$$T^{13} +$$$$71\!\cdots\!04$$$$T^{14} -$$$$48\!\cdots\!52$$$$T^{15} +$$$$32\!\cdots\!74$$$$T^{16} -$$$$21\!\cdots\!28$$$$T^{17} +$$$$14\!\cdots\!84$$$$T^{18} -$$$$93\!\cdots\!72$$$$T^{19} +$$$$58\!\cdots\!20$$$$T^{20} -$$$$36\!\cdots\!20$$$$T^{21} +$$$$21\!\cdots\!32$$$$T^{22} -$$$$12\!\cdots\!36$$$$T^{23} +$$$$72\!\cdots\!80$$$$T^{24} -$$$$38\!\cdots\!84$$$$T^{25} +$$$$19\!\cdots\!72$$$$T^{26} -$$$$95\!\cdots\!84$$$$T^{27} +$$$$42\!\cdots\!84$$$$T^{28} -$$$$18\!\cdots\!88$$$$T^{29} +$$$$69\!\cdots\!00$$$$T^{30} -$$$$19\!\cdots\!80$$$$T^{31} +$$$$27\!\cdots\!61$$$$T^{32}$$
$71$ $$( 1 + 256 T + 68104 T^{2} + 10692864 T^{3} + 1610923548 T^{4} + 179723087616 T^{5} + 18972832358712 T^{6} + 1588998739085056 T^{7} + 125568612540426694 T^{8} + 8010142643727767296 T^{9} +$$$$48\!\cdots\!72$$$$T^{10} +$$$$23\!\cdots\!36$$$$T^{11} +$$$$10\!\cdots\!28$$$$T^{12} +$$$$34\!\cdots\!64$$$$T^{13} +$$$$11\!\cdots\!64$$$$T^{14} +$$$$21\!\cdots\!36$$$$T^{15} +$$$$41\!\cdots\!21$$$$T^{16} )^{2}$$
$73$ $$1 - 42768 T^{2} + 946714744 T^{4} - 14391245893936 T^{6} + 167549428359087132 T^{8} -$$$$15\!\cdots\!24$$$$T^{10} +$$$$12\!\cdots\!76$$$$T^{12} -$$$$83\!\cdots\!96$$$$T^{14} +$$$$47\!\cdots\!22$$$$T^{16} -$$$$23\!\cdots\!36$$$$T^{18} +$$$$10\!\cdots\!56$$$$T^{20} -$$$$36\!\cdots\!04$$$$T^{22} +$$$$10\!\cdots\!52$$$$T^{24} -$$$$26\!\cdots\!36$$$$T^{26} +$$$$49\!\cdots\!04$$$$T^{28} -$$$$63\!\cdots\!08$$$$T^{30} +$$$$42\!\cdots\!21$$$$T^{32}$$
$79$ $$1 - 62928 T^{2} + 1905826568 T^{4} - 37296559235888 T^{6} + 534425714020543644 T^{8} -$$$$60\!\cdots\!08$$$$T^{10} +$$$$55\!\cdots\!96$$$$T^{12} -$$$$43\!\cdots\!36$$$$T^{14} +$$$$29\!\cdots\!62$$$$T^{16} -$$$$16\!\cdots\!16$$$$T^{18} +$$$$84\!\cdots\!56$$$$T^{20} -$$$$35\!\cdots\!28$$$$T^{22} +$$$$12\!\cdots\!24$$$$T^{24} -$$$$33\!\cdots\!88$$$$T^{26} +$$$$66\!\cdots\!08$$$$T^{28} -$$$$85\!\cdots\!08$$$$T^{30} +$$$$52\!\cdots\!41$$$$T^{32}$$
$83$ $$1 + 160 T + 12800 T^{2} + 895904 T^{3} + 107479624 T^{4} + 16432771168 T^{5} + 1654826188288 T^{6} + 174484645067104 T^{7} + 18280323695716892 T^{8} + 1483531366054758688 T^{9} +$$$$11\!\cdots\!96$$$$T^{10} +$$$$11\!\cdots\!36$$$$T^{11} +$$$$13\!\cdots\!00$$$$T^{12} +$$$$12\!\cdots\!44$$$$T^{13} +$$$$89\!\cdots\!76$$$$T^{14} +$$$$74\!\cdots\!76$$$$T^{15} +$$$$61\!\cdots\!66$$$$T^{16} +$$$$51\!\cdots\!64$$$$T^{17} +$$$$42\!\cdots\!96$$$$T^{18} +$$$$39\!\cdots\!36$$$$T^{19} +$$$$30\!\cdots\!00$$$$T^{20} +$$$$17\!\cdots\!64$$$$T^{21} +$$$$12\!\cdots\!56$$$$T^{22} +$$$$10\!\cdots\!52$$$$T^{23} +$$$$92\!\cdots\!52$$$$T^{24} +$$$$60\!\cdots\!36$$$$T^{25} +$$$$39\!\cdots\!88$$$$T^{26} +$$$$27\!\cdots\!52$$$$T^{27} +$$$$12\!\cdots\!04$$$$T^{28} +$$$$70\!\cdots\!76$$$$T^{29} +$$$$69\!\cdots\!00$$$$T^{30} +$$$$59\!\cdots\!40$$$$T^{31} +$$$$25\!\cdots\!61$$$$T^{32}$$
$89$ $$1 - 81008 T^{2} + 3201135736 T^{4} - 82544801381712 T^{6} + 1567286911309649436 T^{8} -$$$$23\!\cdots\!04$$$$T^{10} +$$$$28\!\cdots\!72$$$$T^{12} -$$$$29\!\cdots\!36$$$$T^{14} +$$$$25\!\cdots\!10$$$$T^{16} -$$$$18\!\cdots\!76$$$$T^{18} +$$$$11\!\cdots\!32$$$$T^{20} -$$$$57\!\cdots\!84$$$$T^{22} +$$$$24\!\cdots\!96$$$$T^{24} -$$$$80\!\cdots\!12$$$$T^{26} +$$$$19\!\cdots\!76$$$$T^{28} -$$$$31\!\cdots\!48$$$$T^{30} +$$$$24\!\cdots\!21$$$$T^{32}$$
$97$ $$( 1 + 38216 T^{2} + 116224 T^{3} + 770481564 T^{4} + 3485408768 T^{5} + 10857255215864 T^{6} + 49274039499776 T^{7} + 116292098553803590 T^{8} + 463619437653392384 T^{9} +$$$$96\!\cdots\!84$$$$T^{10} +$$$$29\!\cdots\!72$$$$T^{11} +$$$$60\!\cdots\!04$$$$T^{12} +$$$$85\!\cdots\!76$$$$T^{13} +$$$$26\!\cdots\!56$$$$T^{14} +$$$$61\!\cdots\!21$$$$T^{16} )^{2}$$