# Properties

 Label 252.2.bm.a Level 252 Weight 2 Character orbit 252.bm Analytic conductor 2.012 Analytic rank 0 Dimension 16 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 252.bm (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.01223013094$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$3^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{5} - \beta_{10} ) q^{3} + ( \beta_{7} - \beta_{11} ) q^{5} + ( 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{7} - \beta_{9} + \beta_{11} + \beta_{12} + \beta_{14} - 2 \beta_{15} ) q^{7} + ( -\beta_{2} + \beta_{3} + \beta_{4} - \beta_{14} + \beta_{15} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{5} - \beta_{10} ) q^{3} + ( \beta_{7} - \beta_{11} ) q^{5} + ( 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{7} - \beta_{9} + \beta_{11} + \beta_{12} + \beta_{14} - 2 \beta_{15} ) q^{7} + ( -\beta_{2} + \beta_{3} + \beta_{4} - \beta_{14} + \beta_{15} ) q^{9} + ( -1 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{12} + \beta_{14} - \beta_{15} ) q^{11} + ( \beta_{4} - \beta_{5} - \beta_{7} - \beta_{10} + \beta_{13} + \beta_{14} ) q^{13} + ( -1 + \beta_{1} - 2 \beta_{6} - \beta_{7} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{15} + ( -1 - \beta_{1} + \beta_{6} - \beta_{9} - \beta_{11} ) q^{17} + ( 1 - \beta_{1} + \beta_{2} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} - 2 \beta_{13} ) q^{19} + ( 1 + 2 \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} - \beta_{14} ) q^{21} + ( -2 - \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{9} - 2 \beta_{10} - \beta_{11} - 2 \beta_{12} + \beta_{13} + 2 \beta_{15} ) q^{23} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{12} + \beta_{14} - \beta_{15} ) q^{25} + ( \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{27} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{11} - \beta_{12} + 2 \beta_{13} ) q^{29} + ( -\beta_{2} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{15} ) q^{31} + ( \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{33} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{6} - \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{35} + ( 1 - \beta_{1} + 2 \beta_{6} + 2 \beta_{10} - \beta_{11} - \beta_{13} ) q^{37} + ( 1 - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{13} - \beta_{14} ) q^{39} + ( \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - 3 \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} + 3 \beta_{15} ) q^{41} + ( 1 + \beta_{1} + \beta_{2} + \beta_{5} - 3 \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} - 2 \beta_{15} ) q^{43} + ( 1 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{7} + 3 \beta_{9} - 2 \beta_{12} - 2 \beta_{14} + 3 \beta_{15} ) q^{45} + ( -1 - 3 \beta_{1} + \beta_{5} + 2 \beta_{6} - \beta_{9} + \beta_{10} + \beta_{11} + 3 \beta_{12} - \beta_{13} - 2 \beta_{15} ) q^{47} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - \beta_{7} + \beta_{8} + 3 \beta_{10} + \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{15} ) q^{49} + ( 2 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{7} - \beta_{9} + \beta_{10} - 2 \beta_{11} - \beta_{14} ) q^{51} + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{10} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{53} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{7} + \beta_{9} + \beta_{10} - 2 \beta_{12} + \beta_{13} + \beta_{15} ) q^{55} + ( 2 \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{12} + \beta_{13} - \beta_{14} + 3 \beta_{15} ) q^{57} + ( 1 + \beta_{1} - \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + \beta_{8} - 2 \beta_{9} + 4 \beta_{10} + \beta_{12} - 2 \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{59} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} + \beta_{12} - 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{61} + ( -4 - 2 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - 3 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} - 3 \beta_{10} + 2 \beta_{11} + 2 \beta_{13} + \beta_{14} ) q^{63} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{14} + 3 \beta_{15} ) q^{65} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{4} - 4 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} - 3 \beta_{10} - \beta_{12} + 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{67} + ( \beta_{1} - 5 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} - 3 \beta_{12} - 2 \beta_{14} + 5 \beta_{15} ) q^{69} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{9} - \beta_{11} - 2 \beta_{12} + 2 \beta_{13} + \beta_{15} ) q^{71} + ( -2 + 2 \beta_{2} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{10} + \beta_{11} + \beta_{12} + 2 \beta_{13} + 2 \beta_{14} ) q^{73} + ( -4 - 5 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} - \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{75} + ( -2 - 2 \beta_{1} + \beta_{2} - \beta_{4} + 5 \beta_{5} + \beta_{8} + \beta_{10} + 2 \beta_{11} - 2 \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{77} + ( \beta_{6} + \beta_{10} + \beta_{11} + \beta_{13} ) q^{79} + ( 3 + \beta_{1} + 5 \beta_{2} - \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{9} + \beta_{10} + 3 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} + \beta_{14} - 6 \beta_{15} ) q^{81} + ( -1 + 3 \beta_{2} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} - 3 \beta_{8} - \beta_{9} - 2 \beta_{10} + 2 \beta_{13} + 3 \beta_{14} - 2 \beta_{15} ) q^{83} + ( 1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{11} - \beta_{12} + 2 \beta_{14} - 4 \beta_{15} ) q^{85} + ( -1 - 6 \beta_{1} + 3 \beta_{2} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} + 2 \beta_{12} + 2 \beta_{14} - 3 \beta_{15} ) q^{87} + ( -1 + 3 \beta_{1} - 3 \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 3 \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} + 3 \beta_{15} ) q^{89} + ( \beta_{1} - 2 \beta_{2} + \beta_{4} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} - 2 \beta_{9} + 2 \beta_{11} - \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{91} + ( -4 + \beta_{1} + 4 \beta_{2} - 3 \beta_{3} - \beta_{4} - 2 \beta_{7} - 3 \beta_{9} + 3 \beta_{11} + 2 \beta_{12} + 3 \beta_{14} - 4 \beta_{15} ) q^{93} + ( 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} - \beta_{13} - \beta_{14} + 4 \beta_{15} ) q^{95} + ( \beta_{1} - 5 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + \beta_{6} + 3 \beta_{7} - 2 \beta_{8} - 6 \beta_{11} - \beta_{12} + \beta_{13} - 2 \beta_{14} + 4 \beta_{15} ) q^{97} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} + \beta_{10} - 3 \beta_{11} - 2 \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - q^{7} + O(q^{10})$$ $$16q - q^{7} + 3q^{13} - 3q^{15} - 9q^{17} + 16q^{25} - 9q^{27} + 6q^{29} + 6q^{31} - 27q^{33} + 15q^{35} + q^{37} - 3q^{39} + 6q^{41} - 2q^{43} - 15q^{45} - 18q^{47} + 13q^{49} + 15q^{51} + 15q^{57} - 15q^{59} + 3q^{61} - 9q^{63} - 39q^{65} - 7q^{67} - 21q^{69} - 15q^{75} - 45q^{77} - q^{79} + 6q^{85} - 3q^{87} - 21q^{89} + 9q^{91} - 69q^{93} + 6q^{95} + 3q^{97} - 9q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} - 156 x^{7} + 558 x^{6} - 837 x^{5} + 1782 x^{4} - 4131 x^{3} + 3645 x^{2} - 4374 x + 6561$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-1307 \nu^{15} + 5068 \nu^{14} + 824 \nu^{13} + 49267 \nu^{12} + 2716 \nu^{11} + 77018 \nu^{10} - 113602 \nu^{9} - 7210 \nu^{8} - 181946 \nu^{7} + 84090 \nu^{6} - 1174032 \nu^{5} + 900801 \nu^{4} - 2054484 \nu^{3} + 11094408 \nu^{2} + 4573017 \nu + 19166868$$$$)/621108$$ $$\beta_{2}$$ $$=$$ $$($$$$3695 \nu^{15} - 20725 \nu^{14} + 51544 \nu^{13} - 99223 \nu^{12} + 215537 \nu^{11} - 360098 \nu^{10} + 187876 \nu^{9} - 298928 \nu^{8} + 711356 \nu^{7} - 844320 \nu^{6} + 2586978 \nu^{5} - 9488205 \nu^{4} + 18766647 \nu^{3} - 20620980 \nu^{2} + 33241671 \nu - 46300977$$$$)/1242216$$ $$\beta_{3}$$ $$=$$ $$($$$$-6677 \nu^{15} + 21577 \nu^{14} - 16612 \nu^{13} + 91129 \nu^{12} - 145673 \nu^{11} + 11630 \nu^{10} - 101824 \nu^{9} + 277628 \nu^{8} - 214640 \nu^{7} + 202764 \nu^{6} - 3624714 \nu^{5} + 7713063 \nu^{4} - 3514995 \nu^{3} + 15997176 \nu^{2} - 16161201 \nu - 5872095$$$$)/1242216$$ $$\beta_{4}$$ $$=$$ $$($$$$-818 \nu^{15} - 2453 \nu^{14} - 9016 \nu^{13} - 10490 \nu^{12} - 12167 \nu^{11} + 10856 \nu^{10} + 25058 \nu^{9} + 37994 \nu^{8} + 39934 \nu^{7} + 148242 \nu^{6} + 64386 \nu^{5} - 91530 \nu^{4} - 1898397 \nu^{3} - 3112344 \nu^{2} - 4586868 \nu - 2858409$$$$)/103518$$ $$\beta_{5}$$ $$=$$ $$($$$$4659 \nu^{15} - 5459 \nu^{14} + 29980 \nu^{13} - 59671 \nu^{12} + 62083 \nu^{11} - 182738 \nu^{10} + 119048 \nu^{9} - 112444 \nu^{8} + 265832 \nu^{7} - 648764 \nu^{6} + 2006262 \nu^{5} - 2722977 \nu^{4} + 10424025 \nu^{3} - 11026368 \nu^{2} + 9412119 \nu - 25313067$$$$)/414072$$ $$\beta_{6}$$ $$=$$ $$($$$$-7279 \nu^{15} + 11960 \nu^{14} - 9158 \nu^{13} + 168785 \nu^{12} + 36326 \nu^{11} + 302098 \nu^{10} - 390434 \nu^{9} - 30734 \nu^{8} - 575494 \nu^{7} + 625362 \nu^{6} - 4453344 \nu^{5} + 1286793 \nu^{4} - 9909540 \nu^{3} + 36051966 \nu^{2} + 18445887 \nu + 70277058$$$$)/621108$$ $$\beta_{7}$$ $$=$$ $$($$$$-13862 \nu^{15} + 1333 \nu^{14} - 45760 \nu^{13} + 164782 \nu^{12} + 15775 \nu^{11} + 343040 \nu^{10} - 361210 \nu^{9} + 100070 \nu^{8} - 284726 \nu^{7} + 1370658 \nu^{6} - 5698206 \nu^{5} + 63018 \nu^{4} - 18020475 \nu^{3} + 29851092 \nu^{2} + 7676370 \nu + 61443765$$$$)/621108$$ $$\beta_{8}$$ $$=$$ $$($$$$27959 \nu^{15} + 45263 \nu^{14} + 129088 \nu^{13} - 58111 \nu^{12} - 95755 \nu^{11} - 442826 \nu^{10} + 157996 \nu^{9} - 326936 \nu^{8} - 426532 \nu^{7} - 3257256 \nu^{6} + 5542434 \nu^{5} + 10347075 \nu^{4} + 36196875 \nu^{3} + 5383908 \nu^{2} + 7598367 \nu - 47062053$$$$)/1242216$$ $$\beta_{9}$$ $$=$$ $$($$$$-7241 \nu^{15} + 4357 \nu^{14} - 32353 \nu^{13} + 38470 \nu^{12} - 78698 \nu^{11} + 72614 \nu^{10} - 29722 \nu^{9} + 229040 \nu^{8} - 92132 \nu^{7} + 637584 \nu^{6} - 2307114 \nu^{5} + 2838753 \nu^{4} - 8018433 \nu^{3} + 3449385 \nu^{2} - 14505642 \nu + 1180980$$$$)/310554$$ $$\beta_{10}$$ $$=$$ $$($$$$5228 \nu^{15} + 1525 \nu^{14} + 20418 \nu^{13} - 46974 \nu^{12} - 5255 \nu^{11} - 122904 \nu^{10} + 100970 \nu^{9} - 49402 \nu^{8} + 85518 \nu^{7} - 572506 \nu^{6} + 1811682 \nu^{5} + 307476 \nu^{4} + 7123329 \nu^{3} - 7628742 \nu^{2} - 1004562 \nu - 19887849$$$$)/207036$$ $$\beta_{11}$$ $$=$$ $$($$$$16952 \nu^{15} - 9175 \nu^{14} + 73804 \nu^{13} - 123904 \nu^{12} + 128807 \nu^{11} - 247964 \nu^{10} + 166462 \nu^{9} - 398066 \nu^{8} + 282422 \nu^{7} - 1677798 \nu^{6} + 5939262 \nu^{5} - 5124384 \nu^{4} + 20396205 \nu^{3} - 16524972 \nu^{2} + 21030192 \nu - 23648031$$$$)/621108$$ $$\beta_{12}$$ $$=$$ $$($$$$11445 \nu^{15} - 10925 \nu^{14} + 48460 \nu^{13} - 170497 \nu^{12} + 54037 \nu^{11} - 403262 \nu^{10} + 367448 \nu^{9} - 147196 \nu^{8} + 581144 \nu^{7} - 1306628 \nu^{6} + 5344746 \nu^{5} - 3588687 \nu^{4} + 19817487 \nu^{3} - 33430320 \nu^{2} + 2317977 \nu - 69748533$$$$)/414072$$ $$\beta_{13}$$ $$=$$ $$($$$$19793 \nu^{15} + 1430 \nu^{14} + 108688 \nu^{13} - 147133 \nu^{12} + 121322 \nu^{11} - 529214 \nu^{10} + 286834 \nu^{9} - 407870 \nu^{8} + 442766 \nu^{7} - 2453682 \nu^{6} + 6290388 \nu^{5} - 3336147 \nu^{4} + 34358094 \nu^{3} - 21512304 \nu^{2} + 26451765 \nu - 66410442$$$$)/621108$$ $$\beta_{14}$$ $$=$$ $$($$$$59447 \nu^{15} + 8969 \nu^{14} + 284176 \nu^{13} - 415519 \nu^{12} + 220139 \nu^{11} - 1262618 \nu^{10} + 758536 \nu^{9} - 1010876 \nu^{8} + 904208 \nu^{7} - 6809388 \nu^{6} + 18944982 \nu^{5} - 4839237 \nu^{4} + 86642541 \nu^{3} - 56752164 \nu^{2} + 46993527 \nu - 158830875$$$$)/1242216$$ $$\beta_{15}$$ $$=$$ $$($$$$24909 \nu^{15} - 6205 \nu^{14} + 135476 \nu^{13} - 180977 \nu^{12} + 218117 \nu^{11} - 566494 \nu^{10} + 299296 \nu^{9} - 605132 \nu^{8} + 530200 \nu^{7} - 2915308 \nu^{6} + 8256786 \nu^{5} - 7163775 \nu^{4} + 40104855 \nu^{3} - 24898752 \nu^{2} + 41926977 \nu - 60703101$$$$)/414072$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{15} - 2 \beta_{14} + \beta_{4} + 2 \beta_{3} - 2 \beta_{2}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{14} - \beta_{13} - 2 \beta_{12} + 2 \beta_{10} - \beta_{9} - \beta_{7} - \beta_{6} + 3 \beta_{5} + \beta_{3} + \beta_{2} + 2 \beta_{1}$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{15} + 2 \beta_{14} + 2 \beta_{13} - \beta_{12} - 3 \beta_{10} - \beta_{9} - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} - 5 \beta_{1} + 4$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$6 \beta_{15} - \beta_{13} + 2 \beta_{12} - 12 \beta_{10} + 2 \beta_{9} + 3 \beta_{8} - 2 \beta_{7} - \beta_{6} - 7 \beta_{5} + 5 \beta_{4} + 6 \beta_{3} - 2 \beta_{2} + \beta_{1} + 7$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-13 \beta_{15} + 5 \beta_{14} - 7 \beta_{13} + 9 \beta_{12} - 3 \beta_{11} + 10 \beta_{10} - 16 \beta_{9} - 9 \beta_{8} - 9 \beta_{7} + 8 \beta_{6} + 4 \beta_{5} - 10 \beta_{4} + 10 \beta_{3} + 8 \beta_{2} + 12 \beta_{1} - 4$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$\beta_{15} + 7 \beta_{14} + 24 \beta_{13} - 25 \beta_{12} - 12 \beta_{11} - 25 \beta_{10} - 3 \beta_{9} - 18 \beta_{8} - 23 \beta_{7} - 30 \beta_{6} + \beta_{5} - 6 \beta_{4} - \beta_{3} - 13 \beta_{2} + 31 \beta_{1} - 13$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$35 \beta_{15} + 4 \beta_{14} - 29 \beta_{13} - 8 \beta_{12} - 27 \beta_{11} + 3 \beta_{10} - 2 \beta_{9} - 3 \beta_{8} + 20 \beta_{7} + \beta_{6} + 4 \beta_{5} + 5 \beta_{4} + 17 \beta_{3} - 33 \beta_{2} - 82 \beta_{1} + 20$$$$)/3$$ $$\nu^{8}$$ $$=$$ $$($$$$73 \beta_{15} - 51 \beta_{14} - 30 \beta_{13} + 4 \beta_{12} - 63 \beta_{11} + 52 \beta_{10} - 24 \beta_{9} - 6 \beta_{8} + 23 \beta_{7} + 21 \beta_{6} + 65 \beta_{5} + 61 \beta_{4} + 39 \beta_{3} - 94 \beta_{2} + 59 \beta_{1} - 26$$$$)/3$$ $$\nu^{9}$$ $$=$$ $$($$$$-132 \beta_{15} + 4 \beta_{14} - 100 \beta_{13} + 15 \beta_{12} + 48 \beta_{11} + 61 \beta_{10} - 49 \beta_{9} + 90 \beta_{8} + 3 \beta_{7} - 37 \beta_{6} + 235 \beta_{5} - 110 \beta_{4} - 97 \beta_{3} + 85 \beta_{2} + 72 \beta_{1} - 16$$$$)/3$$ $$\nu^{10}$$ $$=$$ $$($$$$79 \beta_{15} + 41 \beta_{14} - 17 \beta_{13} - 71 \beta_{12} + 15 \beta_{11} - 174 \beta_{10} + 82 \beta_{9} + 63 \beta_{8} + 17 \beta_{7} + 19 \beta_{6} + 49 \beta_{5} + 36 \beta_{4} + 10 \beta_{3} - 113 \beta_{2} - 472 \beta_{1} - 463$$$$)/3$$ $$\nu^{11}$$ $$=$$ $$($$$$-5 \beta_{15} + 275 \beta_{14} - 241 \beta_{13} + 185 \beta_{12} - 225 \beta_{11} - 63 \beta_{10} - 28 \beta_{9} + 120 \beta_{8} + 118 \beta_{7} + 197 \beta_{6} - 142 \beta_{5} + 121 \beta_{4} - 152 \beta_{3} + 177 \beta_{2} - 113 \beta_{1} - 80$$$$)/3$$ $$\nu^{12}$$ $$=$$ $$($$$$-995 \beta_{15} + 438 \beta_{14} - 26 \beta_{13} + 764 \beta_{12} + 51 \beta_{11} + 139 \beta_{10} - 452 \beta_{9} - 39 \beta_{8} + 4 \beta_{7} + 331 \beta_{6} + 174 \beta_{5} - 547 \beta_{4} - 387 \beta_{3} + 397 \beta_{2} + 727 \beta_{1} - 198$$$$)/3$$ $$\nu^{13}$$ $$=$$ $$($$$$376 \beta_{15} - 4 \beta_{14} + 276 \beta_{13} - 699 \beta_{12} + 408 \beta_{11} - 924 \beta_{10} + 825 \beta_{9} - 129 \beta_{8} - 189 \beta_{7} - 1467 \beta_{6} - 393 \beta_{5} - 625 \beta_{4} - 476 \beta_{3} - 295 \beta_{2} + 558 \beta_{1} - 1935$$$$)/3$$ $$\nu^{14}$$ $$=$$ $$($$$$453 \beta_{15} + 679 \beta_{14} - 1199 \beta_{13} - 40 \beta_{12} - 1635 \beta_{11} + 2518 \beta_{10} - 287 \beta_{9} - 732 \beta_{8} + 1228 \beta_{7} + 1309 \beta_{6} - 150 \beta_{5} - 336 \beta_{4} - 781 \beta_{3} - 253 \beta_{2} - 4148 \beta_{1} - 1881$$$$)/3$$ $$\nu^{15}$$ $$=$$ $$($$$$1349 \beta_{15} - 1166 \beta_{14} + 1447 \beta_{13} + 472 \beta_{12} - 1887 \beta_{11} + 861 \beta_{10} + 454 \beta_{9} + 1194 \beta_{8} + 2750 \beta_{7} - 320 \beta_{6} + 949 \beta_{5} + 2555 \beta_{4} - 2791 \beta_{3} - 2679 \beta_{2} + 3770 \beta_{1} + 611$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$73$$ $$127$$ $$\chi(n)$$ $$1 + \beta_{1}$$ $$-\beta_{1}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
173.1
 1.68042 − 0.419752i −0.213160 + 1.71888i 1.08696 − 1.34852i −1.61108 − 0.635951i −0.811340 − 1.53027i −0.544978 − 1.64408i −0.268067 + 1.71118i 1.68124 + 0.416458i 1.68042 + 0.419752i −0.213160 − 1.71888i 1.08696 + 1.34852i −1.61108 + 0.635951i −0.811340 + 1.53027i −0.544978 + 1.64408i −0.268067 − 1.71118i 1.68124 − 0.416458i
0 −1.60579 + 0.649187i 0 −2.96988 0 2.38485 1.14563i 0 2.15711 2.08491i 0
173.2 0 −1.44376 0.956855i 0 2.86804 0 −1.83240 1.90848i 0 1.16886 + 2.76293i 0
173.3 0 −1.31579 + 1.12637i 0 0.0764245 0 −2.39886 + 1.11601i 0 0.462593 2.96412i 0
173.4 0 0.127742 + 1.72733i 0 2.18300 0 2.64473 0.0736382i 0 −2.96736 + 0.441307i 0
173.5 0 0.478402 1.66467i 0 2.74332 0 1.70417 + 2.02381i 0 −2.54226 1.59276i 0
173.6 0 0.615921 1.61884i 0 −3.91482 0 −2.51757 + 0.813537i 0 −2.24128 1.99416i 0
173.7 0 1.42826 + 0.979841i 0 −1.68574 0 −0.0236360 + 2.64565i 0 1.07983 + 2.79892i 0
173.8 0 1.71501 0.242362i 0 0.699656 0 −0.461278 2.60523i 0 2.88252 0.831308i 0
185.1 0 −1.60579 0.649187i 0 −2.96988 0 2.38485 + 1.14563i 0 2.15711 + 2.08491i 0
185.2 0 −1.44376 + 0.956855i 0 2.86804 0 −1.83240 + 1.90848i 0 1.16886 2.76293i 0
185.3 0 −1.31579 1.12637i 0 0.0764245 0 −2.39886 1.11601i 0 0.462593 + 2.96412i 0
185.4 0 0.127742 1.72733i 0 2.18300 0 2.64473 + 0.0736382i 0 −2.96736 0.441307i 0
185.5 0 0.478402 + 1.66467i 0 2.74332 0 1.70417 2.02381i 0 −2.54226 + 1.59276i 0
185.6 0 0.615921 + 1.61884i 0 −3.91482 0 −2.51757 0.813537i 0 −2.24128 + 1.99416i 0
185.7 0 1.42826 0.979841i 0 −1.68574 0 −0.0236360 2.64565i 0 1.07983 2.79892i 0
185.8 0 1.71501 + 0.242362i 0 0.699656 0 −0.461278 + 2.60523i 0 2.88252 + 0.831308i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 185.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.s even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.2.bm.a yes 16
3.b odd 2 1 756.2.bm.a 16
4.b odd 2 1 1008.2.df.d 16
7.b odd 2 1 1764.2.bm.a 16
7.c even 3 1 1764.2.w.b 16
7.c even 3 1 1764.2.x.b 16
7.d odd 6 1 252.2.w.a 16
7.d odd 6 1 1764.2.x.a 16
9.c even 3 1 756.2.w.a 16
9.c even 3 1 2268.2.t.a 16
9.d odd 6 1 252.2.w.a 16
9.d odd 6 1 2268.2.t.b 16
12.b even 2 1 3024.2.df.d 16
21.c even 2 1 5292.2.bm.a 16
21.g even 6 1 756.2.w.a 16
21.g even 6 1 5292.2.x.a 16
21.h odd 6 1 5292.2.w.b 16
21.h odd 6 1 5292.2.x.b 16
28.f even 6 1 1008.2.ca.d 16
36.f odd 6 1 3024.2.ca.d 16
36.h even 6 1 1008.2.ca.d 16
63.g even 3 1 5292.2.bm.a 16
63.h even 3 1 5292.2.x.a 16
63.i even 6 1 1764.2.x.b 16
63.i even 6 1 2268.2.t.a 16
63.j odd 6 1 1764.2.x.a 16
63.k odd 6 1 756.2.bm.a 16
63.l odd 6 1 5292.2.w.b 16
63.n odd 6 1 1764.2.bm.a 16
63.o even 6 1 1764.2.w.b 16
63.s even 6 1 inner 252.2.bm.a yes 16
63.t odd 6 1 2268.2.t.b 16
63.t odd 6 1 5292.2.x.b 16
84.j odd 6 1 3024.2.ca.d 16
252.n even 6 1 3024.2.df.d 16
252.bn odd 6 1 1008.2.df.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.w.a 16 7.d odd 6 1
252.2.w.a 16 9.d odd 6 1
252.2.bm.a yes 16 1.a even 1 1 trivial
252.2.bm.a yes 16 63.s even 6 1 inner
756.2.w.a 16 9.c even 3 1
756.2.w.a 16 21.g even 6 1
756.2.bm.a 16 3.b odd 2 1
756.2.bm.a 16 63.k odd 6 1
1008.2.ca.d 16 28.f even 6 1
1008.2.ca.d 16 36.h even 6 1
1008.2.df.d 16 4.b odd 2 1
1008.2.df.d 16 252.bn odd 6 1
1764.2.w.b 16 7.c even 3 1
1764.2.w.b 16 63.o even 6 1
1764.2.x.a 16 7.d odd 6 1
1764.2.x.a 16 63.j odd 6 1
1764.2.x.b 16 7.c even 3 1
1764.2.x.b 16 63.i even 6 1
1764.2.bm.a 16 7.b odd 2 1
1764.2.bm.a 16 63.n odd 6 1
2268.2.t.a 16 9.c even 3 1
2268.2.t.a 16 63.i even 6 1
2268.2.t.b 16 9.d odd 6 1
2268.2.t.b 16 63.t odd 6 1
3024.2.ca.d 16 36.f odd 6 1
3024.2.ca.d 16 84.j odd 6 1
3024.2.df.d 16 12.b even 2 1
3024.2.df.d 16 252.n even 6 1
5292.2.w.b 16 21.h odd 6 1
5292.2.w.b 16 63.l odd 6 1
5292.2.x.a 16 21.g even 6 1
5292.2.x.a 16 63.h even 3 1
5292.2.x.b 16 21.h odd 6 1
5292.2.x.b 16 63.t odd 6 1
5292.2.bm.a 16 21.c even 2 1
5292.2.bm.a 16 63.g even 3 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(252, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 + 3 T^{3} - 27 T^{5} + 27 T^{6} + 9 T^{7} - 72 T^{8} + 27 T^{9} + 243 T^{10} - 729 T^{11} + 729 T^{13} + 6561 T^{16}$$
$5$ $$( 1 + 16 T^{2} + 12 T^{3} + 151 T^{4} + 165 T^{5} + 1096 T^{6} + 1236 T^{7} + 6142 T^{8} + 6180 T^{9} + 27400 T^{10} + 20625 T^{11} + 94375 T^{12} + 37500 T^{13} + 250000 T^{14} + 390625 T^{16} )^{2}$$
$7$ $$1 + T - 6 T^{2} - 16 T^{3} - 10 T^{4} + 39 T^{5} + 385 T^{6} - 239 T^{7} - 2880 T^{8} - 1673 T^{9} + 18865 T^{10} + 13377 T^{11} - 24010 T^{12} - 268912 T^{13} - 705894 T^{14} + 823543 T^{15} + 5764801 T^{16}$$
$11$ $$1 - 86 T^{2} + 3801 T^{4} - 115279 T^{6} + 2684372 T^{8} - 50825121 T^{10} + 808796131 T^{12} - 11028714095 T^{14} + 130183029216 T^{16} - 1334474405495 T^{18} + 11841584153971 T^{20} - 90039802183881 T^{22} + 575418978107732 T^{24} - 2990040370578679 T^{26} + 11929166259916521 T^{28} - 32658485688158726 T^{30} + 45949729863572161 T^{32}$$
$13$ $$1 - 3 T + 53 T^{2} - 150 T^{3} + 1317 T^{4} - 4452 T^{5} + 23056 T^{6} - 90522 T^{7} + 327950 T^{8} - 1218621 T^{9} + 4082037 T^{10} - 12551466 T^{11} + 42977824 T^{12} - 136810599 T^{13} + 317458436 T^{14} - 1828010007 T^{15} + 2359677564 T^{16} - 23764130091 T^{17} + 53650475684 T^{18} - 300572886003 T^{19} + 1227489631264 T^{20} - 4660271465538 T^{21} + 19703212929933 T^{22} - 76466660535057 T^{23} + 267518889951950 T^{24} - 959940492242706 T^{25} + 3178465388070544 T^{26} - 7978698074252724 T^{27} + 30683578106307477 T^{28} - 45431265988837950 T^{29} + 208680948442062317 T^{30} - 153557679042272271 T^{31} + 665416609183179841 T^{32}$$
$17$ $$1 + 9 T - 31 T^{2} - 498 T^{3} + 6 T^{4} + 11613 T^{5} + 1180 T^{6} - 197118 T^{7} + 356723 T^{8} + 3738756 T^{9} - 13114257 T^{10} - 60977685 T^{11} + 321880693 T^{12} + 548795961 T^{13} - 7936687522 T^{14} - 1772823582 T^{15} + 160937528238 T^{16} - 30138000894 T^{17} - 2293702693858 T^{18} + 2696234556393 T^{19} + 26883797360053 T^{20} - 86579592891045 T^{21} - 316546283221233 T^{22} + 1534156175710788 T^{23} + 2488413121625843 T^{24} - 23375805039335646 T^{25} + 2378872802529820 T^{26} + 397999531820542029 T^{27} + 3495733423378566 T^{28} - 4932479860387156626 T^{29} - 5219712623341428799 T^{30} + 25761807463588342137 T^{31} + 48661191875666868481 T^{32}$$
$19$ $$1 + 59 T^{2} + 1482 T^{4} + 279 T^{5} + 21358 T^{6} - 23166 T^{7} + 284075 T^{8} - 1924515 T^{9} + 8259087 T^{10} - 60795468 T^{11} + 210571285 T^{12} - 1125692802 T^{13} + 3051652292 T^{14} - 14070928500 T^{15} + 40349090466 T^{16} - 267347641500 T^{17} + 1101646477412 T^{18} - 7721126928918 T^{19} + 27441860432485 T^{20} - 150535597519332 T^{21} + 388556024170647 T^{22} - 1720269569781585 T^{23} + 4824605670872075 T^{24} - 7475383206748314 T^{25} + 130947313134113758 T^{26} + 32500782232603101 T^{27} + 3280132710056050602 T^{28} + 47141394461190163139 T^{30} +$$$$28\!\cdots\!81$$$$T^{32}$$
$23$ $$1 - 125 T^{2} + 7680 T^{4} - 295747 T^{6} + 8228447 T^{8} - 202400565 T^{10} + 5673034564 T^{12} - 171801092192 T^{14} + 4463691201090 T^{16} - 90882777769568 T^{18} + 1587547665424324 T^{20} - 29962547573877285 T^{22} + 644377791902488607 T^{24} - 12251766411903050803 T^{26} +$$$$16\!\cdots\!80$$$$T^{28} -$$$$14\!\cdots\!25$$$$T^{30} +$$$$61\!\cdots\!61$$$$T^{32}$$
$29$ $$1 - 6 T + 106 T^{2} - 564 T^{3} + 5445 T^{4} - 23700 T^{5} + 153455 T^{6} - 499140 T^{7} + 2268674 T^{8} - 3839766 T^{9} + 25367247 T^{10} - 108756381 T^{11} + 2283543946 T^{12} - 15452150286 T^{13} + 152754700246 T^{14} - 806878620189 T^{15} + 5726182499856 T^{16} - 23399479985481 T^{17} + 128466702906886 T^{18} - 376862493325254 T^{19} + 1615107245670826 T^{20} - 2230718335391769 T^{21} + 15089030105167287 T^{22} - 66235488555503694 T^{23} + 1134896030677883714 T^{24} - 7241096842395252660 T^{25} + 64559628486082344455 T^{26} -$$$$28\!\cdots\!00$$$$T^{27} +$$$$19\!\cdots\!45$$$$T^{28} -$$$$57\!\cdots\!96$$$$T^{29} +$$$$31\!\cdots\!86$$$$T^{30} -$$$$51\!\cdots\!94$$$$T^{31} +$$$$25\!\cdots\!21$$$$T^{32}$$
$31$ $$1 - 6 T + 128 T^{2} - 696 T^{3} + 7467 T^{4} - 41745 T^{5} + 298711 T^{6} - 1778985 T^{7} + 10180130 T^{8} - 58737741 T^{9} + 320070333 T^{10} - 1687070526 T^{11} + 8941505953 T^{12} - 50372743224 T^{13} + 219778671713 T^{14} - 1637243989362 T^{15} + 5894911299924 T^{16} - 50754563670222 T^{17} + 211207303516193 T^{18} - 1500654393386184 T^{19} + 8257668519220513 T^{20} - 48299396836503426 T^{21} + 284063598716395773 T^{22} - 1616028801884863251 T^{23} + 8682541636984247330 T^{24} - 47035691229501298935 T^{25} +$$$$24\!\cdots\!11$$$$T^{26} -$$$$10\!\cdots\!95$$$$T^{27} +$$$$58\!\cdots\!87$$$$T^{28} -$$$$16\!\cdots\!36$$$$T^{29} +$$$$96\!\cdots\!88$$$$T^{30} -$$$$14\!\cdots\!06$$$$T^{31} +$$$$72\!\cdots\!81$$$$T^{32}$$
$37$ $$1 - T - 188 T^{2} + 55 T^{3} + 18431 T^{4} + 6958 T^{5} - 1220598 T^{6} - 1466820 T^{7} + 61023285 T^{8} + 128763363 T^{9} - 2426865150 T^{10} - 7087005807 T^{11} + 80710069782 T^{12} + 247085609469 T^{13} - 2444398029834 T^{14} - 3801878390997 T^{15} + 81128049964254 T^{16} - 140669500466889 T^{17} - 3346380902842746 T^{18} + 12515627376433257 T^{19} + 151263665092702902 T^{20} - 491441025939358299 T^{21} - 6226672006436746350 T^{22} + 12223747755547878279 T^{23} +$$$$21\!\cdots\!85$$$$T^{24} -$$$$19\!\cdots\!40$$$$T^{25} -$$$$58\!\cdots\!02$$$$T^{26} +$$$$12\!\cdots\!54$$$$T^{27} +$$$$12\!\cdots\!11$$$$T^{28} +$$$$13\!\cdots\!35$$$$T^{29} -$$$$16\!\cdots\!32$$$$T^{30} -$$$$33\!\cdots\!93$$$$T^{31} +$$$$12\!\cdots\!41$$$$T^{32}$$
$41$ $$1 - 6 T - 142 T^{2} + 1146 T^{3} + 7575 T^{4} - 89148 T^{5} - 186515 T^{6} + 3858942 T^{7} + 5165546 T^{8} - 137269398 T^{9} - 383379603 T^{10} + 6275799867 T^{11} + 14048576428 T^{12} - 262470296220 T^{13} + 63020850350 T^{14} + 4818424458177 T^{15} - 18114488698896 T^{16} + 197555402785257 T^{17} + 105938049438350 T^{18} - 18089715285778620 T^{19} + 39697919375761708 T^{20} + 727090330826925267 T^{21} - 1821093078123196323 T^{22} - 26733801933571993638 T^{23} + 41246498577585065066 T^{24} +$$$$12\!\cdots\!62$$$$T^{25} -$$$$25\!\cdots\!15$$$$T^{26} -$$$$49\!\cdots\!68$$$$T^{27} +$$$$17\!\cdots\!75$$$$T^{28} +$$$$10\!\cdots\!66$$$$T^{29} -$$$$53\!\cdots\!62$$$$T^{30} -$$$$93\!\cdots\!06$$$$T^{31} +$$$$63\!\cdots\!41$$$$T^{32}$$
$43$ $$1 + 2 T - 137 T^{2} - 620 T^{3} + 8660 T^{4} + 62377 T^{5} - 131688 T^{6} - 3324402 T^{7} - 16902063 T^{8} + 47132751 T^{9} + 1398120027 T^{10} + 5063156292 T^{11} - 36164979717 T^{12} - 388120390392 T^{13} - 815732360646 T^{14} + 8046953157708 T^{15} + 93229603031718 T^{16} + 346018985781444 T^{17} - 1508289134834454 T^{18} - 30858287878896744 T^{19} - 123640868821459317 T^{20} + 744326723152573356 T^{21} + 8838024276744682323 T^{22} + 12811558914472065357 T^{23} -$$$$19\!\cdots\!63$$$$T^{24} -$$$$16\!\cdots\!86$$$$T^{25} -$$$$28\!\cdots\!12$$$$T^{26} +$$$$57\!\cdots\!39$$$$T^{27} +$$$$34\!\cdots\!60$$$$T^{28} -$$$$10\!\cdots\!60$$$$T^{29} -$$$$10\!\cdots\!13$$$$T^{30} +$$$$63\!\cdots\!14$$$$T^{31} +$$$$13\!\cdots\!01$$$$T^{32}$$
$47$ $$1 + 18 T + 62 T^{2} - 192 T^{3} + 3579 T^{4} + 45093 T^{5} + 305398 T^{6} + 2890521 T^{7} + 680339 T^{8} - 61965303 T^{9} + 840301383 T^{10} + 4943829948 T^{11} + 3397740844 T^{12} + 111292595154 T^{13} - 711051013816 T^{14} - 459408198168 T^{15} + 101830826927016 T^{16} - 21592185313896 T^{17} - 1570711689519544 T^{18} + 11554731106673742 T^{19} + 16579891439390764 T^{20} + 1133842714030869636 T^{21} + 9057789548613500007 T^{22} - 31393055166295295289 T^{23} + 16199746956175816979 T^{24} +$$$$32\!\cdots\!07$$$$T^{25} +$$$$16\!\cdots\!02$$$$T^{26} +$$$$11\!\cdots\!79$$$$T^{27} +$$$$41\!\cdots\!39$$$$T^{28} -$$$$10\!\cdots\!84$$$$T^{29} +$$$$15\!\cdots\!78$$$$T^{30} +$$$$21\!\cdots\!74$$$$T^{31} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$1 + 271 T^{2} + 39798 T^{4} - 8019 T^{5} + 3882470 T^{6} - 3112830 T^{7} + 272618249 T^{8} - 573793713 T^{9} + 13751201223 T^{10} - 69938411256 T^{11} + 461723008303 T^{12} - 6218775316710 T^{13} + 7202514917530 T^{14} - 423085276358208 T^{15} - 24546838659498 T^{16} - 22423519646985024 T^{17} + 20231864403341770 T^{18} - 925832612825834670 T^{19} + 3643216624277663743 T^{20} - 29247928374839669208 T^{21} +$$$$30\!\cdots\!67$$$$T^{22} -$$$$67\!\cdots\!81$$$$T^{23} +$$$$16\!\cdots\!89$$$$T^{24} -$$$$10\!\cdots\!90$$$$T^{25} +$$$$67\!\cdots\!30$$$$T^{26} -$$$$74\!\cdots\!43$$$$T^{27} +$$$$19\!\cdots\!18$$$$T^{28} +$$$$37\!\cdots\!99$$$$T^{30} +$$$$38\!\cdots\!21$$$$T^{32}$$
$59$ $$1 + 15 T - 79 T^{2} - 1428 T^{3} + 12489 T^{4} + 104847 T^{5} - 1041029 T^{6} + 349611 T^{7} + 90472109 T^{8} - 405874929 T^{9} - 2700289995 T^{10} + 51789934650 T^{11} - 105592258130 T^{12} - 2611128991413 T^{13} + 30211727298980 T^{14} + 82730812140738 T^{15} - 2120569425031284 T^{16} + 4881117916303542 T^{17} + 105167022727749380 T^{18} - 536271061127410527 T^{19} - 1279499510566394930 T^{20} + 37025882724907060350 T^{21} -$$$$11\!\cdots\!95$$$$T^{22} -$$$$10\!\cdots\!51$$$$T^{23} +$$$$13\!\cdots\!89$$$$T^{24} +$$$$30\!\cdots\!29$$$$T^{25} -$$$$53\!\cdots\!29$$$$T^{26} +$$$$31\!\cdots\!73$$$$T^{27} +$$$$22\!\cdots\!09$$$$T^{28} -$$$$14\!\cdots\!12$$$$T^{29} -$$$$48\!\cdots\!19$$$$T^{30} +$$$$54\!\cdots\!85$$$$T^{31} +$$$$21\!\cdots\!41$$$$T^{32}$$
$61$ $$1 - 3 T + 377 T^{2} - 1122 T^{3} + 74481 T^{4} - 239253 T^{5} + 10537021 T^{6} - 36111681 T^{7} + 1196449751 T^{8} - 4163017557 T^{9} + 114788648541 T^{10} - 392960339076 T^{11} + 9549397216930 T^{12} - 31727661059931 T^{13} + 697503406538990 T^{14} - 2222103948983802 T^{15} + 45117478421428590 T^{16} - 135548340888011922 T^{17} + 2595410175731581790 T^{18} - 7201576235044198311 T^{19} +$$$$13\!\cdots\!30$$$$T^{20} -$$$$33\!\cdots\!76$$$$T^{21} +$$$$59\!\cdots\!01$$$$T^{22} -$$$$13\!\cdots\!97$$$$T^{23} +$$$$22\!\cdots\!31$$$$T^{24} -$$$$42\!\cdots\!21$$$$T^{25} +$$$$75\!\cdots\!21$$$$T^{26} -$$$$10\!\cdots\!33$$$$T^{27} +$$$$19\!\cdots\!01$$$$T^{28} -$$$$18\!\cdots\!82$$$$T^{29} +$$$$37\!\cdots\!57$$$$T^{30} -$$$$18\!\cdots\!03$$$$T^{31} +$$$$36\!\cdots\!61$$$$T^{32}$$
$67$ $$1 + 7 T - 266 T^{2} - 955 T^{3} + 40565 T^{4} + 13739 T^{5} - 4302801 T^{6} + 7895916 T^{7} + 333320832 T^{8} - 1176952818 T^{9} - 19419549429 T^{10} + 109762768995 T^{11} + 793012008891 T^{12} - 6957819527658 T^{13} - 12193565486865 T^{14} + 198908265810519 T^{15} - 461225363558112 T^{16} + 13326853809304773 T^{17} - 54736915470536985 T^{18} - 2092654674597003054 T^{19} + 15980080945615616811 T^{20} +$$$$14\!\cdots\!65$$$$T^{21} -$$$$17\!\cdots\!01$$$$T^{22} -$$$$71\!\cdots\!14$$$$T^{23} +$$$$13\!\cdots\!12$$$$T^{24} +$$$$21\!\cdots\!52$$$$T^{25} -$$$$78\!\cdots\!49$$$$T^{26} +$$$$16\!\cdots\!37$$$$T^{27} +$$$$33\!\cdots\!65$$$$T^{28} -$$$$52\!\cdots\!85$$$$T^{29} -$$$$97\!\cdots\!14$$$$T^{30} +$$$$17\!\cdots\!01$$$$T^{31} +$$$$16\!\cdots\!81$$$$T^{32}$$
$71$ $$1 - 389 T^{2} + 95718 T^{4} - 16982206 T^{6} + 2417461733 T^{8} - 285171142929 T^{10} + 28739565041956 T^{12} - 2501948752776044 T^{14} + 190114711967546784 T^{16} - 12612323662744037804 T^{18} +$$$$73\!\cdots\!36$$$$T^{20} -$$$$36\!\cdots\!09$$$$T^{22} +$$$$15\!\cdots\!13$$$$T^{24} -$$$$55\!\cdots\!06$$$$T^{26} +$$$$15\!\cdots\!38$$$$T^{28} -$$$$32\!\cdots\!09$$$$T^{30} +$$$$41\!\cdots\!21$$$$T^{32}$$
$73$ $$1 + 278 T^{2} + 39699 T^{4} - 160362 T^{5} + 4049527 T^{6} - 43185312 T^{7} + 316493258 T^{8} - 5978310966 T^{9} + 30516141699 T^{10} - 573490190745 T^{11} + 3892519182646 T^{12} - 40210747978752 T^{13} + 436575836052170 T^{14} - 2488085380264887 T^{15} + 37348173312201540 T^{16} - 181630232759336751 T^{17} + 2326512630322013930 T^{18} - 15642664546450166784 T^{19} +$$$$11\!\cdots\!86$$$$T^{20} -$$$$11\!\cdots\!85$$$$T^{21} +$$$$46\!\cdots\!11$$$$T^{22} -$$$$66\!\cdots\!02$$$$T^{23} +$$$$25\!\cdots\!98$$$$T^{24} -$$$$25\!\cdots\!56$$$$T^{25} +$$$$17\!\cdots\!23$$$$T^{26} -$$$$50\!\cdots\!74$$$$T^{27} +$$$$90\!\cdots\!79$$$$T^{28} +$$$$33\!\cdots\!02$$$$T^{30} +$$$$65\!\cdots\!61$$$$T^{32}$$
$79$ $$1 + T - 488 T^{2} + 305 T^{3} + 130067 T^{4} - 217447 T^{5} - 23515767 T^{6} + 58439790 T^{7} + 3200266602 T^{8} - 9493225932 T^{9} - 346875510927 T^{10} + 1047490944351 T^{11} + 31673904979161 T^{12} - 75618180243888 T^{13} - 2595852946891617 T^{14} + 2451851231878899 T^{15} + 205361771983476300 T^{16} + 193696247318433021 T^{17} - 16200718241550581697 T^{18} - 37282711969266295632 T^{19} +$$$$12\!\cdots\!41$$$$T^{20} +$$$$32\!\cdots\!49$$$$T^{21} -$$$$84\!\cdots\!67$$$$T^{22} -$$$$18\!\cdots\!88$$$$T^{23} +$$$$48\!\cdots\!22$$$$T^{24} +$$$$70\!\cdots\!10$$$$T^{25} -$$$$22\!\cdots\!67$$$$T^{26} -$$$$16\!\cdots\!13$$$$T^{27} +$$$$76\!\cdots\!47$$$$T^{28} +$$$$14\!\cdots\!95$$$$T^{29} -$$$$17\!\cdots\!28$$$$T^{30} +$$$$29\!\cdots\!99$$$$T^{31} +$$$$23\!\cdots\!21$$$$T^{32}$$
$83$ $$1 - 334 T^{2} + 1272 T^{3} + 54816 T^{4} - 395796 T^{5} - 5397056 T^{6} + 60133572 T^{7} + 310945691 T^{8} - 6455120304 T^{9} - 3389068248 T^{10} + 563598757704 T^{11} - 2368219763012 T^{12} - 39202056035964 T^{13} + 437295746267318 T^{14} + 1352773893773628 T^{15} - 45259898774669136 T^{16} + 112280233183211124 T^{17} + 3012530396035553702 T^{18} - 22415226014635747668 T^{19} -$$$$11\!\cdots\!52$$$$T^{20} +$$$$22\!\cdots\!72$$$$T^{21} -$$$$11\!\cdots\!12$$$$T^{22} -$$$$17\!\cdots\!08$$$$T^{23} +$$$$70\!\cdots\!31$$$$T^{24} +$$$$11\!\cdots\!16$$$$T^{25} -$$$$83\!\cdots\!44$$$$T^{26} -$$$$50\!\cdots\!32$$$$T^{27} +$$$$58\!\cdots\!76$$$$T^{28} +$$$$11\!\cdots\!36$$$$T^{29} -$$$$24\!\cdots\!86$$$$T^{30} +$$$$50\!\cdots\!81$$$$T^{32}$$
$89$ $$1 + 21 T - 253 T^{2} - 6084 T^{3} + 64134 T^{4} + 1086543 T^{5} - 14444834 T^{6} - 154075794 T^{7} + 2457441293 T^{8} + 16916386392 T^{9} - 343956627405 T^{10} - 1437006555873 T^{11} + 41653185355711 T^{12} + 92739461592729 T^{13} - 4399434577615834 T^{14} - 3097874885577318 T^{15} + 411982470004053510 T^{16} - 275710864816381302 T^{17} - 34847921289295021114 T^{18} + 65378445499564570401 T^{19} +$$$$26\!\cdots\!51$$$$T^{20} -$$$$80\!\cdots\!77$$$$T^{21} -$$$$17\!\cdots\!05$$$$T^{22} +$$$$74\!\cdots\!68$$$$T^{23} +$$$$96\!\cdots\!33$$$$T^{24} -$$$$53\!\cdots\!46$$$$T^{25} -$$$$45\!\cdots\!34$$$$T^{26} +$$$$30\!\cdots\!27$$$$T^{27} +$$$$15\!\cdots\!14$$$$T^{28} -$$$$13\!\cdots\!96$$$$T^{29} -$$$$49\!\cdots\!73$$$$T^{30} +$$$$36\!\cdots\!29$$$$T^{31} +$$$$15\!\cdots\!61$$$$T^{32}$$
$97$ $$1 - 3 T + 392 T^{2} - 1167 T^{3} + 75243 T^{4} - 248802 T^{5} + 9281977 T^{6} - 31870980 T^{7} + 787182476 T^{8} - 2086019346 T^{9} + 40966656945 T^{10} + 99548858823 T^{11} - 1131878124686 T^{12} + 48785193558327 T^{13} - 618766130152816 T^{14} + 7413615555033420 T^{15} - 81803373889459032 T^{16} + 719120708838241740 T^{17} - 5821970518607845744 T^{18} + 44524928960458978071 T^{19} -$$$$10\!\cdots\!66$$$$T^{20} +$$$$85\!\cdots\!11$$$$T^{21} +$$$$34\!\cdots\!05$$$$T^{22} -$$$$16\!\cdots\!98$$$$T^{23} +$$$$61\!\cdots\!36$$$$T^{24} -$$$$24\!\cdots\!60$$$$T^{25} +$$$$68\!\cdots\!73$$$$T^{26} -$$$$17\!\cdots\!06$$$$T^{27} +$$$$52\!\cdots\!63$$$$T^{28} -$$$$78\!\cdots\!59$$$$T^{29} +$$$$25\!\cdots\!48$$$$T^{30} -$$$$18\!\cdots\!79$$$$T^{31} +$$$$61\!\cdots\!21$$$$T^{32}$$