# Properties

 Label 378.2.t.a Level 378 Weight 2 Character orbit 378.t Analytic conductor 3.018 Analytic rank 0 Dimension 16 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 378.t (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.01834519640$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{4}$$ Twist minimal: no (minimal twist has level 126) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 1 - \beta_{8} ) q^{4} + ( -\beta_{12} - \beta_{14} ) q^{5} + ( \beta_{10} - \beta_{14} ) q^{7} + ( -\beta_{1} - \beta_{7} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 1 - \beta_{8} ) q^{4} + ( -\beta_{12} - \beta_{14} ) q^{5} + ( \beta_{10} - \beta_{14} ) q^{7} + ( -\beta_{1} - \beta_{7} ) q^{8} + ( \beta_{11} + \beta_{15} ) q^{10} + ( 1 + \beta_{2} - \beta_{4} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} - \beta_{13} - \beta_{15} ) q^{11} + ( -1 - 2 \beta_{1} - \beta_{2} - \beta_{7} - \beta_{9} - \beta_{11} - \beta_{13} ) q^{13} + ( 1 + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{13} ) q^{14} -\beta_{8} q^{16} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} - 3 \beta_{8} - \beta_{9} + \beta_{13} + \beta_{15} ) q^{17} + ( -\beta_{1} + \beta_{4} - 2 \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{19} + ( -\beta_{5} - \beta_{10} - \beta_{12} ) q^{20} + ( -1 + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{8} + \beta_{10} - \beta_{14} ) q^{22} + ( 1 - 2 \beta_{2} + 2 \beta_{3} - \beta_{6} - \beta_{8} - \beta_{9} + \beta_{11} + \beta_{13} + \beta_{15} ) q^{23} + ( 2 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + 3 \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{13} - 2 \beta_{15} ) q^{25} + ( 1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{7} + \beta_{10} + \beta_{12} - \beta_{14} ) q^{26} -\beta_{14} q^{28} + ( -1 - \beta_{4} - \beta_{6} + \beta_{8} - \beta_{9} + \beta_{13} ) q^{29} + ( 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} ) q^{31} -\beta_{7} q^{32} + ( -\beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - 3 \beta_{7} - \beta_{8} + \beta_{14} ) q^{34} + ( 1 + 3 \beta_{1} + 2 \beta_{7} + 4 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{14} - 2 \beta_{15} ) q^{35} + ( -3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{37} + ( -1 - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{38} + ( -\beta_{2} + \beta_{3} - \beta_{9} + \beta_{11} + \beta_{13} + \beta_{15} ) q^{40} + ( -1 - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} - \beta_{12} + \beta_{14} ) q^{41} + ( -\beta_{2} - \beta_{3} + \beta_{4} - \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{43} + ( 2 \beta_{2} - \beta_{3} - \beta_{6} + \beta_{9} - \beta_{13} ) q^{44} + ( -1 - \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{10} + \beta_{14} ) q^{46} + ( -3 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{8} - 2 \beta_{9} + \beta_{11} + \beta_{12} + 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{47} + ( 1 + 4 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - 3 \beta_{9} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{49} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{14} ) q^{50} + ( 1 - \beta_{1} + \beta_{3} - 2 \beta_{7} - \beta_{13} - \beta_{15} ) q^{52} + ( -2 - \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{53} + ( -1 + 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - \beta_{6} + 6 \beta_{7} + 3 \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{13} - \beta_{15} ) q^{55} + ( 1 + \beta_{1} + \beta_{3} + \beta_{6} - \beta_{8} + \beta_{11} ) q^{56} + ( -1 + \beta_{2} - \beta_{3} - \beta_{6} + \beta_{8} + \beta_{12} + \beta_{14} ) q^{58} + ( -4 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + 4 \beta_{8} + 2 \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{12} - 2 \beta_{15} ) q^{59} + ( -6 + \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + 5 \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{61} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{62} - q^{64} + ( -4 + 3 \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{6} + 4 \beta_{8} + 3 \beta_{10} + 2 \beta_{11} + 3 \beta_{12} + 2 \beta_{15} ) q^{65} + ( 2 + \beta_{2} + \beta_{4} + 3 \beta_{6} - 3 \beta_{8} - \beta_{10} - \beta_{12} + \beta_{14} ) q^{67} + ( -2 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} ) q^{68} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} + 5 \beta_{7} + 3 \beta_{8} + 2 \beta_{11} + 3 \beta_{12} + \beta_{13} ) q^{70} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} - 2 \beta_{8} + 3 \beta_{9} + \beta_{10} - 3 \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{71} + ( 1 + 5 \beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{73} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - 3 \beta_{8} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{74} + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{76} + ( 4 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + \beta_{7} - 2 \beta_{11} - \beta_{12} - 3 \beta_{13} - \beta_{14} ) q^{77} + ( 2 + 2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} - 2 \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{79} + ( -\beta_{5} - \beta_{10} + \beta_{14} ) q^{80} + ( -\beta_{1} + \beta_{4} - \beta_{6} - 2 \beta_{7} - \beta_{9} - \beta_{11} + \beta_{15} ) q^{82} + ( -2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} - 2 \beta_{11} - \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{83} + ( -3 \beta_{1} + \beta_{3} - \beta_{4} - 4 \beta_{5} - 6 \beta_{7} - 2 \beta_{8} - 3 \beta_{10} - \beta_{12} + 4 \beta_{14} ) q^{85} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{8} + 2 \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{86} + ( -1 - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{8} - \beta_{12} - \beta_{14} ) q^{88} + ( -3 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - \beta_{5} + 3 \beta_{6} - 3 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{12} - 2 \beta_{13} ) q^{89} + ( -1 - 5 \beta_{1} + 2 \beta_{2} + 3 \beta_{4} - 3 \beta_{5} + 4 \beta_{6} - 7 \beta_{7} - \beta_{10} - \beta_{11} - \beta_{12} - 2 \beta_{13} + 2 \beta_{14} ) q^{91} + ( 1 - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{8} - \beta_{9} + \beta_{13} ) q^{92} + ( 2 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + 2 \beta_{14} - \beta_{15} ) q^{94} + ( 4 + 2 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{6} + 7 \beta_{7} + 5 \beta_{8} - 3 \beta_{10} + 2 \beta_{11} + 3 \beta_{12} + 2 \beta_{13} + 3 \beta_{14} - 2 \beta_{15} ) q^{95} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{5} + \beta_{6} - 4 \beta_{7} + 2 \beta_{9} + \beta_{11} - \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{97} + ( -3 + \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} + 4 \beta_{8} + \beta_{11} + 2 \beta_{12} + \beta_{14} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 8q^{4} + 2q^{7} + O(q^{10})$$ $$16q + 8q^{4} + 2q^{7} - 6q^{13} + 6q^{14} - 8q^{16} - 18q^{17} + 16q^{25} + 12q^{26} - 2q^{28} - 6q^{29} + 6q^{31} + 30q^{35} - 2q^{37} - 6q^{41} - 2q^{43} - 12q^{44} + 6q^{46} + 18q^{47} + 10q^{49} + 12q^{50} - 36q^{53} - 12q^{58} - 30q^{59} - 60q^{61} + 36q^{62} - 16q^{64} - 42q^{65} + 14q^{67} - 36q^{68} + 18q^{77} - 16q^{79} - 12q^{85} - 24q^{89} - 12q^{91} - 6q^{92} + 66q^{95} - 6q^{97} - 24q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 8 x^{15} + 23 x^{14} - 8 x^{13} - 131 x^{12} + 380 x^{11} - 289 x^{10} - 880 x^{9} + 2785 x^{8} - 2640 x^{7} - 2601 x^{6} + 10260 x^{5} - 10611 x^{4} - 1944 x^{3} + 16767 x^{2} - 17496 x + 6561$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$50 \nu^{15} + 1352 \nu^{14} - 6827 \nu^{13} + 7676 \nu^{12} + 27422 \nu^{11} - 107246 \nu^{10} + 107467 \nu^{9} + 206194 \nu^{8} - 757363 \nu^{7} + 724572 \nu^{6} + 756198 \nu^{5} - 2730942 \nu^{4} + 2372247 \nu^{3} + 1318518 \nu^{2} - 4444713 \nu + 2825604$$$$)/142155$$ $$\beta_{2}$$ $$=$$ $$($$$$169 \nu^{15} - 866 \nu^{14} + 1319 \nu^{13} + 2308 \nu^{12} - 13199 \nu^{11} + 19055 \nu^{10} + 12131 \nu^{9} - 90010 \nu^{8} + 128722 \nu^{7} + 13734 \nu^{6} - 313347 \nu^{5} + 432351 \nu^{4} - 55431 \nu^{3} - 524799 \nu^{2} + 629856 \nu - 244944$$$$)/47385$$ $$\beta_{3}$$ $$=$$ $$($$$$154 \nu^{15} - 1325 \nu^{14} + 3608 \nu^{13} - 224 \nu^{12} - 22478 \nu^{11} + 55022 \nu^{10} - 23518 \nu^{9} - 159688 \nu^{8} + 382978 \nu^{7} - 226785 \nu^{6} - 566733 \nu^{5} + 1355805 \nu^{4} - 871830 \nu^{3} - 949725 \nu^{2} + 2135727 \nu - 1285227$$$$)/47385$$ $$\beta_{4}$$ $$=$$ $$($$$$1445 \nu^{15} - 9836 \nu^{14} + 21081 \nu^{13} + 15627 \nu^{12} - 172766 \nu^{11} + 334353 \nu^{10} + 13019 \nu^{9} - 1283242 \nu^{8} + 2419149 \nu^{7} - 693301 \nu^{6} - 4779069 \nu^{5} + 8827731 \nu^{4} - 3909951 \nu^{3} - 8165529 \nu^{2} + 13935564 \nu - 7117227$$$$)/47385$$ $$\beta_{5}$$ $$=$$ $$($$$$2858 \nu^{15} - 19265 \nu^{14} + 40866 \nu^{13} + 31392 \nu^{12} - 338066 \nu^{11} + 649239 \nu^{10} + 37829 \nu^{9} - 2521996 \nu^{8} + 4717386 \nu^{7} - 1293175 \nu^{6} - 9430341 \nu^{5} + 17293545 \nu^{4} - 7519770 \nu^{3} - 16197165 \nu^{2} + 27464589 \nu - 14041269$$$$)/47385$$ $$\beta_{6}$$ $$=$$ $$($$$$-11192 \nu^{15} + 70123 \nu^{14} - 136087 \nu^{13} - 145859 \nu^{12} + 1215277 \nu^{11} - 2154880 \nu^{10} - 494788 \nu^{9} + 9027170 \nu^{8} - 15633236 \nu^{7} + 2544783 \nu^{6} + 33710346 \nu^{5} - 57047193 \nu^{4} + 20576268 \nu^{3} + 57644217 \nu^{2} - 89475273 \nu + 42998607$$$$)/142155$$ $$\beta_{7}$$ $$=$$ $$($$$$16898 \nu^{15} - 108472 \nu^{14} + 217033 \nu^{13} + 208526 \nu^{12} - 1883968 \nu^{11} + 3436840 \nu^{10} + 565132 \nu^{9} - 13978340 \nu^{8} + 24892859 \nu^{7} - 5100702 \nu^{6} - 52089894 \nu^{5} + 90692622 \nu^{4} - 35165097 \nu^{3} - 88915158 \nu^{2} + 142069707 \nu - 69445998$$$$)/142155$$ $$\beta_{8}$$ $$=$$ $$($$$$-4120 \nu^{15} + 25571 \nu^{14} - 48788 \nu^{13} - 55006 \nu^{12} + 441224 \nu^{11} - 771188 \nu^{10} - 200900 \nu^{9} + 3269857 \nu^{8} - 5585140 \nu^{7} + 796053 \nu^{6} + 12187116 \nu^{5} - 20322468 \nu^{4} + 7028856 \nu^{3} + 20790351 \nu^{2} - 31653180 \nu + 14963454$$$$)/28431$$ $$\beta_{9}$$ $$=$$ $$($$$$8586 \nu^{15} - 53254 \nu^{14} + 101261 \nu^{13} + 115567 \nu^{12} - 919066 \nu^{11} + 1601225 \nu^{10} + 428149 \nu^{9} - 6806990 \nu^{8} + 11591968 \nu^{7} - 1602394 \nu^{6} - 25345758 \nu^{5} + 42144849 \nu^{4} - 14468679 \nu^{3} - 43148781 \nu^{2} + 65494089 \nu - 30865131$$$$)/47385$$ $$\beta_{10}$$ $$=$$ $$($$$$-26068 \nu^{15} + 165707 \nu^{14} - 325403 \nu^{13} - 334861 \nu^{12} + 2873363 \nu^{11} - 5145350 \nu^{10} - 1063457 \nu^{9} + 21324130 \nu^{8} - 37270684 \nu^{7} + 6581337 \nu^{6} + 79507809 \nu^{5} - 135722682 \nu^{4} + 50026572 \nu^{3} + 135776493 \nu^{2} - 212064642 \nu + 102032298$$$$)/142155$$ $$\beta_{11}$$ $$=$$ $$($$$$-31130 \nu^{15} + 194263 \nu^{14} - 374983 \nu^{13} - 405716 \nu^{12} + 3354958 \nu^{11} - 5936959 \nu^{10} - 1366837 \nu^{9} + 24841391 \nu^{8} - 43003982 \nu^{7} + 7063563 \nu^{6} + 92428227 \nu^{5} - 156501153 \nu^{4} + 56669868 \nu^{3} + 157282722 \nu^{2} - 244082322 \nu + 117273501$$$$)/142155$$ $$\beta_{12}$$ $$=$$ $$($$$$33311 \nu^{15} - 207064 \nu^{14} + 396466 \nu^{13} + 441842 \nu^{12} - 3574291 \nu^{11} + 6268915 \nu^{10} + 1580329 \nu^{9} - 26474600 \nu^{8} + 45384563 \nu^{7} - 6735609 \nu^{6} - 98586783 \nu^{5} + 165029724 \nu^{4} - 57791394 \nu^{3} - 167924421 \nu^{2} + 256798269 \nu - 121811526$$$$)/142155$$ $$\beta_{13}$$ $$=$$ $$($$$$5368 \nu^{15} - 32959 \nu^{14} + 62025 \nu^{13} + 72930 \nu^{12} - 567820 \nu^{11} + 981351 \nu^{10} + 280045 \nu^{9} - 4204439 \nu^{8} + 7108452 \nu^{7} - 906329 \nu^{6} - 15657912 \nu^{5} + 25851399 \nu^{4} - 8698959 \nu^{3} - 26672436 \nu^{2} + 40175190 \nu - 18826182$$$$)/15795$$ $$\beta_{14}$$ $$=$$ $$($$$$-62668 \nu^{15} + 397070 \nu^{14} - 778856 \nu^{13} - 802747 \nu^{12} + 6878696 \nu^{11} - 12322064 \nu^{10} - 2527469 \nu^{9} + 51016561 \nu^{8} - 89230126 \nu^{7} + 15905550 \nu^{6} + 190088901 \nu^{5} - 324842130 \nu^{4} + 120144465 \nu^{3} + 324340605 \nu^{2} - 507326409 \nu + 244346949$$$$)/142155$$ $$\beta_{15}$$ $$=$$ $$($$$$5105 \nu^{15} - 31804 \nu^{14} + 61039 \nu^{13} + 67583 \nu^{12} - 549514 \nu^{11} + 965497 \nu^{10} + 239686 \nu^{9} - 4072523 \nu^{8} + 6992981 \nu^{7} - 1054329 \nu^{6} - 15172956 \nu^{5} + 25447554 \nu^{4} - 8949609 \nu^{3} - 25866621 \nu^{2} + 39638646 \nu - 18816948$$$$)/10935$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{15} + 2 \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{6} - \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_{1} + 2$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{14} - 2 \beta_{13} - \beta_{11} + \beta_{10} + 3 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - \beta_{6} - 2 \beta_{4} - \beta_{3} - 2 \beta_{1} + 3$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{13} + \beta_{12} + 2 \beta_{11} + 3 \beta_{10} + 6 \beta_{9} + 2 \beta_{8} + 5 \beta_{7} + 2 \beta_{5} - 3 \beta_{4} - 5 \beta_{3} + 8 \beta_{2} + \beta_{1} - 4$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$5 \beta_{15} - 3 \beta_{13} - \beta_{12} + 3 \beta_{11} + 6 \beta_{10} + 2 \beta_{9} - 3 \beta_{8} + \beta_{7} + 7 \beta_{6} + 4 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + 11 \beta_{1} + 8$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-10 \beta_{14} + 3 \beta_{13} + 3 \beta_{12} + 9 \beta_{11} + 5 \beta_{10} + 6 \beta_{9} + 30 \beta_{8} - 3 \beta_{7} - 2 \beta_{6} + 6 \beta_{5} - 7 \beta_{4} + \beta_{3} + 16 \beta_{2} + 21 \beta_{1} - 1$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$-7 \beta_{15} - 2 \beta_{14} + 7 \beta_{13} - 2 \beta_{12} + 11 \beta_{11} - 2 \beta_{10} + 8 \beta_{9} + 15 \beta_{8} + 30 \beta_{7} - 3 \beta_{6} - 4 \beta_{5} - 12 \beta_{4} + 32 \beta_{3} - 21 \beta_{2} + 51 \beta_{1} + 20$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$-36 \beta_{15} + 3 \beta_{14} + 6 \beta_{13} + 37 \beta_{12} + 24 \beta_{11} - 21 \beta_{10} + 33 \beta_{9} + 49 \beta_{8} + 93 \beta_{7} - \beta_{5} - 39 \beta_{4} - 17 \beta_{3} + 46 \beta_{2} + 90 \beta_{1} - 3$$$$)/3$$ $$\nu^{8}$$ $$=$$ $$($$$$-42 \beta_{15} + 68 \beta_{14} + 10 \beta_{13} + 66 \beta_{12} + 50 \beta_{11} - 25 \beta_{10} + 42 \beta_{9} - 71 \beta_{8} + 179 \beta_{7} + 61 \beta_{6} + 18 \beta_{5} + 50 \beta_{4} + 31 \beta_{3} - 16 \beta_{2} + 163 \beta_{1} + 94$$$$)/3$$ $$\nu^{9}$$ $$=$$ $$($$$$-44 \beta_{15} + 32 \beta_{14} + 216 \beta_{12} + 72 \beta_{11} + 14 \beta_{10} + 40 \beta_{9} + 50 \beta_{8} - 19 \beta_{7} + 66 \beta_{6} + 126 \beta_{5} + 66 \beta_{4} - 64 \beta_{3} + 110 \beta_{2} + 67 \beta_{1} + 168$$$$)/3$$ $$\nu^{10}$$ $$=$$ $$($$$$-58 \beta_{15} - 18 \beta_{14} + 90 \beta_{13} + 172 \beta_{12} + 126 \beta_{11} + 162 \beta_{10} + 62 \beta_{9} + 14 \beta_{8} - 161 \beta_{7} - 8 \beta_{6} + 167 \beta_{5} + 158 \beta_{4} + 153 \beta_{3} - 144 \beta_{2} - 61 \beta_{1} + 188$$$$)/3$$ $$\nu^{11}$$ $$=$$ $$($$$$-126 \beta_{15} - 351 \beta_{14} - 100 \beta_{13} + 231 \beta_{12} + 13 \beta_{11} + 333 \beta_{10} + 48 \beta_{9} + 80 \beta_{8} - 482 \beta_{7} - 81 \beta_{6} + 141 \beta_{5} - 219 \beta_{4} - 138 \beta_{3} + 86 \beta_{2} - 46 \beta_{1} + 217$$$$)/3$$ $$\nu^{12}$$ $$=$$ $$($$$$-209 \beta_{15} - 418 \beta_{14} - 251 \beta_{13} - 170 \beta_{12} + 68 \beta_{11} + 416 \beta_{10} + 301 \beta_{9} - 320 \beta_{8} + 325 \beta_{7} - 36 \beta_{6} - 298 \beta_{5} + 153 \beta_{4} - 357 \beta_{3} + 376 \beta_{2} + 1031 \beta_{1} - 97$$$$)/3$$ $$\nu^{13}$$ $$=$$ $$($$$$385 \beta_{15} - 609 \beta_{14} - 1440 \beta_{13} + 402 \beta_{12} - 186 \beta_{11} + 414 \beta_{10} - 155 \beta_{9} - 478 \beta_{8} - 220 \beta_{7} + 98 \beta_{6} - 597 \beta_{5} + 277 \beta_{4} - 1199 \beta_{3} + 1540 \beta_{2} + 2320 \beta_{1} + 1243$$$$)/3$$ $$\nu^{14}$$ $$=$$ $$($$$$1281 \beta_{15} - 511 \beta_{14} - 835 \beta_{13} + 960 \beta_{12} + 1042 \beta_{11} + 812 \beta_{10} - 27 \beta_{9} + 3809 \beta_{8} + 541 \beta_{7} - 1481 \beta_{6} - 882 \beta_{5} + 1178 \beta_{4} - 1454 \beta_{3} + 2715 \beta_{2} + 2486 \beta_{1} - 1641$$$$)/3$$ $$\nu^{15}$$ $$=$$ $$($$$$2826 \beta_{15} - 1179 \beta_{14} - 1012 \beta_{13} + 2828 \beta_{12} + 1522 \beta_{11} + 2838 \beta_{10} - 2730 \beta_{9} + 6361 \beta_{8} - 11 \beta_{7} - 2559 \beta_{6} - 431 \beta_{5} - 2124 \beta_{4} + 809 \beta_{3} + 757 \beta_{2} + 1787 \beta_{1} + 208$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$325$$ $$\chi(n)$$ $$\beta_{8}$$ $$1 - \beta_{8}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 1.73109 + 0.0577511i −1.68301 − 0.409224i 0.320287 − 1.70218i 0.765614 + 1.55365i 1.27866 + 1.16834i −1.70672 + 0.295146i 1.58110 − 0.707199i 1.71298 − 0.256290i 1.73109 − 0.0577511i −1.68301 + 0.409224i 0.320287 + 1.70218i 0.765614 − 1.55365i 1.27866 − 1.16834i −1.70672 − 0.295146i 1.58110 + 0.707199i 1.71298 + 0.256290i
−0.866025 0.500000i 0 0.500000 + 0.866025i −2.28190 0 1.21977 + 2.34780i 1.00000i 0 1.97618 + 1.14095i
17.2 −0.866025 0.500000i 0 0.500000 + 0.866025i −1.42985 0 −2.43739 1.02913i 1.00000i 0 1.23829 + 0.714925i
17.3 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.0676069 0 −2.64192 0.142361i 1.00000i 0 −0.0585493 0.0338034i
17.4 −0.866025 0.500000i 0 0.500000 + 0.866025i 3.64414 0 2.62749 0.310282i 1.00000i 0 −3.15592 1.82207i
17.5 0.866025 + 0.500000i 0 0.500000 + 0.866025i −3.55225 0 −1.49384 + 2.18368i 1.00000i 0 −3.07634 1.77612i
17.6 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.967324 0 2.40137 1.11060i 1.00000i 0 −0.837727 0.483662i
17.7 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.900258 0 1.05755 + 2.42520i 1.00000i 0 0.779646 + 0.450129i
17.8 0.866025 + 0.500000i 0 0.500000 + 0.866025i 3.61932 0 0.266972 2.63225i 1.00000i 0 3.13442 + 1.80966i
89.1 −0.866025 + 0.500000i 0 0.500000 0.866025i −2.28190 0 1.21977 2.34780i 1.00000i 0 1.97618 1.14095i
89.2 −0.866025 + 0.500000i 0 0.500000 0.866025i −1.42985 0 −2.43739 + 1.02913i 1.00000i 0 1.23829 0.714925i
89.3 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.0676069 0 −2.64192 + 0.142361i 1.00000i 0 −0.0585493 + 0.0338034i
89.4 −0.866025 + 0.500000i 0 0.500000 0.866025i 3.64414 0 2.62749 + 0.310282i 1.00000i 0 −3.15592 + 1.82207i
89.5 0.866025 0.500000i 0 0.500000 0.866025i −3.55225 0 −1.49384 2.18368i 1.00000i 0 −3.07634 + 1.77612i
89.6 0.866025 0.500000i 0 0.500000 0.866025i −0.967324 0 2.40137 + 1.11060i 1.00000i 0 −0.837727 + 0.483662i
89.7 0.866025 0.500000i 0 0.500000 0.866025i 0.900258 0 1.05755 2.42520i 1.00000i 0 0.779646 0.450129i
89.8 0.866025 0.500000i 0 0.500000 0.866025i 3.61932 0 0.266972 + 2.63225i 1.00000i 0 3.13442 1.80966i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 89.8 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 378.2.t.a 16
3.b odd 2 1 126.2.t.a yes 16
4.b odd 2 1 3024.2.df.c 16
7.b odd 2 1 2646.2.t.b 16
7.c even 3 1 2646.2.l.a 16
7.c even 3 1 2646.2.m.a 16
7.d odd 6 1 378.2.l.a 16
7.d odd 6 1 2646.2.m.b 16
9.c even 3 1 126.2.l.a 16
9.c even 3 1 1134.2.k.a 16
9.d odd 6 1 378.2.l.a 16
9.d odd 6 1 1134.2.k.b 16
12.b even 2 1 1008.2.df.c 16
21.c even 2 1 882.2.t.a 16
21.g even 6 1 126.2.l.a 16
21.g even 6 1 882.2.m.b 16
21.h odd 6 1 882.2.l.b 16
21.h odd 6 1 882.2.m.a 16
28.f even 6 1 3024.2.ca.c 16
36.f odd 6 1 1008.2.ca.c 16
36.h even 6 1 3024.2.ca.c 16
63.g even 3 1 882.2.t.a 16
63.h even 3 1 882.2.m.b 16
63.i even 6 1 1134.2.k.a 16
63.i even 6 1 2646.2.m.a 16
63.j odd 6 1 2646.2.m.b 16
63.k odd 6 1 126.2.t.a yes 16
63.l odd 6 1 882.2.l.b 16
63.n odd 6 1 2646.2.t.b 16
63.o even 6 1 2646.2.l.a 16
63.s even 6 1 inner 378.2.t.a 16
63.t odd 6 1 882.2.m.a 16
63.t odd 6 1 1134.2.k.b 16
84.j odd 6 1 1008.2.ca.c 16
252.n even 6 1 1008.2.df.c 16
252.bn odd 6 1 3024.2.df.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.l.a 16 9.c even 3 1
126.2.l.a 16 21.g even 6 1
126.2.t.a yes 16 3.b odd 2 1
126.2.t.a yes 16 63.k odd 6 1
378.2.l.a 16 7.d odd 6 1
378.2.l.a 16 9.d odd 6 1
378.2.t.a 16 1.a even 1 1 trivial
378.2.t.a 16 63.s even 6 1 inner
882.2.l.b 16 21.h odd 6 1
882.2.l.b 16 63.l odd 6 1
882.2.m.a 16 21.h odd 6 1
882.2.m.a 16 63.t odd 6 1
882.2.m.b 16 21.g even 6 1
882.2.m.b 16 63.h even 3 1
882.2.t.a 16 21.c even 2 1
882.2.t.a 16 63.g even 3 1
1008.2.ca.c 16 36.f odd 6 1
1008.2.ca.c 16 84.j odd 6 1
1008.2.df.c 16 12.b even 2 1
1008.2.df.c 16 252.n even 6 1
1134.2.k.a 16 9.c even 3 1
1134.2.k.a 16 63.i even 6 1
1134.2.k.b 16 9.d odd 6 1
1134.2.k.b 16 63.t odd 6 1
2646.2.l.a 16 7.c even 3 1
2646.2.l.a 16 63.o even 6 1
2646.2.m.a 16 7.c even 3 1
2646.2.m.a 16 63.i even 6 1
2646.2.m.b 16 7.d odd 6 1
2646.2.m.b 16 63.j odd 6 1
2646.2.t.b 16 7.b odd 2 1
2646.2.t.b 16 63.n odd 6 1
3024.2.ca.c 16 28.f even 6 1
3024.2.ca.c 16 36.h even 6 1
3024.2.df.c 16 4.b odd 2 1
3024.2.df.c 16 252.bn odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(378, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{4}$$
$3$ 
$5$ $$( 1 + 16 T^{2} - 12 T^{3} + 133 T^{4} - 138 T^{5} + 943 T^{6} - 696 T^{7} + 5539 T^{8} - 3480 T^{9} + 23575 T^{10} - 17250 T^{11} + 83125 T^{12} - 37500 T^{13} + 250000 T^{14} + 390625 T^{16} )^{2}$$
$7$ $$1 - 2 T - 3 T^{2} + 26 T^{3} - 67 T^{4} - 84 T^{5} + 355 T^{6} + 40 T^{7} - 675 T^{8} + 280 T^{9} + 17395 T^{10} - 28812 T^{11} - 160867 T^{12} + 436982 T^{13} - 352947 T^{14} - 1647086 T^{15} + 5764801 T^{16}$$
$11$ $$1 - 68 T^{2} + 2622 T^{4} - 71710 T^{6} + 1545179 T^{8} - 27618072 T^{10} + 422760301 T^{12} - 5636332820 T^{14} + 66055843863 T^{16} - 681996271220 T^{18} + 6189633566941 T^{20} - 48927099250392 T^{22} + 331222841384699 T^{24} - 1859972718137710 T^{26} + 8228959203762462 T^{28} - 25822988683660388 T^{30} + 45949729863572161 T^{32}$$
$13$ $$1 + 6 T + 47 T^{2} + 210 T^{3} + 888 T^{4} + 2658 T^{5} + 9181 T^{6} + 12564 T^{7} + 21641 T^{8} - 214770 T^{9} - 1221396 T^{10} - 6263526 T^{11} - 19011605 T^{12} - 72302586 T^{13} - 179114935 T^{14} - 742674348 T^{15} - 2320143696 T^{16} - 9654766524 T^{17} - 30270424015 T^{18} - 158848781442 T^{19} - 542990450405 T^{20} - 2325603359118 T^{21} - 5895445205364 T^{22} - 13476498996090 T^{23} + 17653228533161 T^{24} + 133234930122372 T^{25} + 1265678813665669 T^{26} + 4763562327350346 T^{27} + 20688699588763128 T^{28} + 63603772384373130 T^{29} + 185056690127866583 T^{30} + 307115358084544542 T^{31} + 665416609183179841 T^{32}$$
$17$ $$1 + 18 T + 95 T^{2} - 42 T^{3} - 849 T^{4} + 7584 T^{5} + 29152 T^{6} - 139356 T^{7} - 154873 T^{8} + 2564070 T^{9} - 13186653 T^{10} - 89037906 T^{11} + 222290986 T^{12} + 933503742 T^{13} - 5890959001 T^{14} - 7644891330 T^{15} + 98422426836 T^{16} - 129963152610 T^{17} - 1702487151289 T^{18} + 4586303884446 T^{19} + 18565965441706 T^{20} - 126421094099442 T^{21} - 318293746666557 T^{22} + 1052137081279110 T^{23} - 1080356482159993 T^{24} - 16525932117115932 T^{25} + 58770254185889248 T^{26} + 259918061597088672 T^{27} - 494646279408067089 T^{28} - 415992277382049354 T^{29} + 15995893523143088255 T^{30} + 51523614927176684274 T^{31} + 48661191875666868481 T^{32}$$
$19$ $$1 + 80 T^{2} + 3483 T^{4} - 882 T^{5} + 107770 T^{6} - 26190 T^{7} + 2596658 T^{8} + 477594 T^{9} + 51326154 T^{10} + 58243698 T^{11} + 858124540 T^{12} + 2331197604 T^{13} + 13169269781 T^{14} + 60976736712 T^{15} + 222780924306 T^{16} + 1158557997528 T^{17} + 4754106390941 T^{18} + 15989684365836 T^{19} + 111831648177340 T^{20} + 144217162374102 T^{21} + 2414684133271674 T^{22} + 426907779315966 T^{23} + 44100504838916978 T^{24} - 8451190804832010 T^{25} + 660745010603213770 T^{26} - 102744408348229158 T^{27} + 7708975863107438763 T^{28} + 63920534862630729680 T^{30} +$$$$28\!\cdots\!81$$$$T^{32}$$
$23$ $$1 - 224 T^{2} + 24654 T^{4} - 1769830 T^{6} + 93146507 T^{8} - 3841107576 T^{10} + 129779286901 T^{12} - 3708814203716 T^{14} + 91488095720271 T^{16} - 1961962713765764 T^{18} + 36317565425662741 T^{20} - 568621774757795064 T^{22} + 7294394738653563467 T^{24} - 73317882341252409670 T^{26} +$$$$54\!\cdots\!34$$$$T^{28} -$$$$25\!\cdots\!16$$$$T^{30} +$$$$61\!\cdots\!61$$$$T^{32}$$
$29$ $$1 + 6 T + 196 T^{2} + 1104 T^{3} + 19989 T^{4} + 108858 T^{5} + 1435478 T^{6} + 7594740 T^{7} + 81322766 T^{8} + 416245488 T^{9} + 3844091772 T^{10} + 18872529852 T^{11} + 155981060614 T^{12} + 727660406082 T^{13} + 5509975202215 T^{14} + 24244157784798 T^{15} + 170557455776958 T^{16} + 703080575759142 T^{17} + 4633889145062815 T^{18} + 17746909643933898 T^{19} + 110322440532130534 T^{20} + 387097271801319948 T^{21} + 2286555434049814812 T^{22} + 7180183182179343792 T^{23} + 40681421983566770126 T^{24} +$$$$11\!\cdots\!60$$$$T^{25} +$$$$60\!\cdots\!78$$$$T^{26} +$$$$13\!\cdots\!82$$$$T^{27} +$$$$70\!\cdots\!49$$$$T^{28} +$$$$11\!\cdots\!56$$$$T^{29} +$$$$58\!\cdots\!76$$$$T^{30} +$$$$51\!\cdots\!94$$$$T^{31} +$$$$25\!\cdots\!21$$$$T^{32}$$
$31$ $$1 - 6 T + 164 T^{2} - 912 T^{3} + 14145 T^{4} - 82056 T^{5} + 882730 T^{6} - 5457276 T^{7} + 44537675 T^{8} - 289281588 T^{9} + 1950112812 T^{10} - 12838947000 T^{11} + 76837130221 T^{12} - 493000882878 T^{13} + 2756895706292 T^{14} - 16890347347524 T^{15} + 89911869890409 T^{16} - 523600767773244 T^{17} + 2649376773746612 T^{18} - 14686989301818498 T^{19} + 70960703338828141 T^{20} - 367568152343997000 T^{21} + 1730732299015260972 T^{22} - 7958892700061288268 T^{23} + 37985783835960089675 T^{24} -$$$$14\!\cdots\!96$$$$T^{25} +$$$$72\!\cdots\!30$$$$T^{26} -$$$$20\!\cdots\!36$$$$T^{27} +$$$$11\!\cdots\!45$$$$T^{28} -$$$$22\!\cdots\!92$$$$T^{29} +$$$$12\!\cdots\!44$$$$T^{30} -$$$$14\!\cdots\!06$$$$T^{31} +$$$$72\!\cdots\!81$$$$T^{32}$$
$37$ $$1 + 2 T - 116 T^{2} - 656 T^{3} + 3854 T^{4} + 48538 T^{5} + 98460 T^{6} - 747702 T^{7} - 5251023 T^{8} - 50514234 T^{9} - 417738792 T^{10} + 801423570 T^{11} + 29966075382 T^{12} + 116541321036 T^{13} - 322717550496 T^{14} - 3343790405826 T^{15} - 14327838321804 T^{16} - 123720245015562 T^{17} - 441800326629024 T^{18} + 5903167534436508 T^{19} + 56161249804004502 T^{20} + 55573881576866490 T^{21} - 1071803450698157928 T^{22} - 4795411055555611122 T^{23} - 18444110399566611183 T^{24} - 97172652768258663054 T^{25} +$$$$47\!\cdots\!40$$$$T^{26} +$$$$86\!\cdots\!94$$$$T^{27} +$$$$25\!\cdots\!74$$$$T^{28} -$$$$15\!\cdots\!32$$$$T^{29} -$$$$10\!\cdots\!24$$$$T^{30} +$$$$66\!\cdots\!86$$$$T^{31} +$$$$12\!\cdots\!41$$$$T^{32}$$
$41$ $$1 + 6 T - 223 T^{2} - 1686 T^{3} + 25980 T^{4} + 231654 T^{5} - 1971341 T^{6} - 20408106 T^{7} + 109216031 T^{8} + 1268388768 T^{9} - 4836103872 T^{10} - 57438167556 T^{11} + 192545389345 T^{12} + 1824668193534 T^{13} - 7677147470143 T^{14} - 27964729912410 T^{15} + 313424888729076 T^{16} - 1146553926408810 T^{17} - 12905284897310383 T^{18} + 125757956566556814 T^{19} + 544087251940916545 T^{20} - 6654567885439614756 T^{21} - 22971997512303721152 T^{22} +$$$$24\!\cdots\!08$$$$T^{23} +$$$$87\!\cdots\!51$$$$T^{24} -$$$$66\!\cdots\!66$$$$T^{25} -$$$$26\!\cdots\!41$$$$T^{26} +$$$$12\!\cdots\!14$$$$T^{27} +$$$$58\!\cdots\!80$$$$T^{28} -$$$$15\!\cdots\!06$$$$T^{29} -$$$$84\!\cdots\!03$$$$T^{30} +$$$$93\!\cdots\!06$$$$T^{31} +$$$$63\!\cdots\!41$$$$T^{32}$$
$43$ $$1 + 2 T - 209 T^{2} - 602 T^{3} + 20774 T^{4} + 72052 T^{5} - 1457073 T^{6} - 4933350 T^{7} + 91147851 T^{8} + 235722522 T^{9} - 5466637218 T^{10} - 9144255228 T^{11} + 301177025103 T^{12} + 290829011802 T^{13} - 14773623661707 T^{14} - 4808085837138 T^{15} + 658882369500660 T^{16} - 206747690996934 T^{17} - 27316430150496243 T^{18} + 23122942241341614 T^{19} + 1029664314599161503 T^{20} - 1344282723462890004 T^{21} - 34556598512153357682 T^{22} + 64073768536679251854 T^{23} +$$$$10\!\cdots\!51$$$$T^{24} -$$$$24\!\cdots\!50$$$$T^{25} -$$$$31\!\cdots\!77$$$$T^{26} +$$$$66\!\cdots\!64$$$$T^{27} +$$$$83\!\cdots\!74$$$$T^{28} -$$$$10\!\cdots\!86$$$$T^{29} -$$$$15\!\cdots\!41$$$$T^{30} +$$$$63\!\cdots\!14$$$$T^{31} +$$$$13\!\cdots\!01$$$$T^{32}$$
$47$ $$1 - 18 T - 55 T^{2} + 1722 T^{3} + 12957 T^{4} - 149790 T^{5} - 1485620 T^{6} + 10153800 T^{7} + 96840002 T^{8} - 357022848 T^{9} - 6832958010 T^{10} + 20400467106 T^{11} + 370230517390 T^{12} - 853239339180 T^{13} - 20043242784151 T^{14} + 7467517424682 T^{15} + 1132351959046185 T^{16} + 350973318960054 T^{17} - 44275523310189559 T^{18} - 88585867911685140 T^{19} + 1806606821328152590 T^{20} + 4678745271228839742 T^{21} - 73653925723805335290 T^{22} -$$$$18\!\cdots\!24$$$$T^{23} +$$$$23\!\cdots\!22$$$$T^{24} +$$$$11\!\cdots\!00$$$$T^{25} -$$$$78\!\cdots\!80$$$$T^{26} -$$$$37\!\cdots\!70$$$$T^{27} +$$$$15\!\cdots\!37$$$$T^{28} +$$$$94\!\cdots\!94$$$$T^{29} -$$$$14\!\cdots\!95$$$$T^{30} -$$$$21\!\cdots\!74$$$$T^{31} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$1 + 36 T + 766 T^{2} + 12024 T^{3} + 153999 T^{4} + 1662408 T^{5} + 15312632 T^{6} + 121481640 T^{7} + 835010600 T^{8} + 5038052724 T^{9} + 28358007246 T^{10} + 182105031078 T^{11} + 1622318702806 T^{12} + 17289317216124 T^{13} + 177226593270661 T^{14} + 1605696832118286 T^{15} + 12594715934262750 T^{16} + 85101932102269158 T^{17} + 497829500497286749 T^{18} + 2573981679184892748 T^{19} + 12800874900435389686 T^{20} + 76155503249444531454 T^{21} +$$$$62\!\cdots\!34$$$$T^{22} +$$$$59\!\cdots\!88$$$$T^{23} +$$$$51\!\cdots\!00$$$$T^{24} +$$$$40\!\cdots\!20$$$$T^{25} +$$$$26\!\cdots\!68$$$$T^{26} +$$$$15\!\cdots\!76$$$$T^{27} +$$$$75\!\cdots\!59$$$$T^{28} +$$$$31\!\cdots\!52$$$$T^{29} +$$$$10\!\cdots\!54$$$$T^{30} +$$$$26\!\cdots\!52$$$$T^{31} +$$$$38\!\cdots\!21$$$$T^{32}$$
$59$ $$1 + 30 T + 200 T^{2} - 1272 T^{3} + 1905 T^{4} + 286356 T^{5} - 330074 T^{6} - 21958908 T^{7} + 129465491 T^{8} + 1641971784 T^{9} - 13037705556 T^{10} - 82617584088 T^{11} + 950143418149 T^{12} + 1952014889118 T^{13} - 70581518213368 T^{14} - 49596128462484 T^{15} + 4103530612541721 T^{16} - 2926171579286556 T^{17} - 245694264900734008 T^{18} + 400902865912165722 T^{19} + 11513230799485384789 T^{20} - 59065318389186954312 T^{21} -$$$$54\!\cdots\!96$$$$T^{22} +$$$$40\!\cdots\!96$$$$T^{23} +$$$$19\!\cdots\!11$$$$T^{24} -$$$$19\!\cdots\!12$$$$T^{25} -$$$$16\!\cdots\!74$$$$T^{26} +$$$$86\!\cdots\!04$$$$T^{27} +$$$$33\!\cdots\!05$$$$T^{28} -$$$$13\!\cdots\!88$$$$T^{29} +$$$$12\!\cdots\!00$$$$T^{30} +$$$$10\!\cdots\!70$$$$T^{31} +$$$$21\!\cdots\!41$$$$T^{32}$$
$61$ $$1 + 60 T + 2036 T^{2} + 50160 T^{3} + 994527 T^{4} + 16748022 T^{5} + 247695826 T^{6} + 3290908266 T^{7} + 39930949007 T^{8} + 448092194298 T^{9} + 4697101745298 T^{10} + 46372941826116 T^{11} + 434191490396749 T^{12} + 3878280141400200 T^{13} + 33210009451501436 T^{14} + 273658668372668376 T^{15} + 2175265354887641205 T^{16} + 16693178770732770936 T^{17} +$$$$12\!\cdots\!56$$$$T^{18} +$$$$88\!\cdots\!00$$$$T^{19} +$$$$60\!\cdots\!09$$$$T^{20} +$$$$39\!\cdots\!16$$$$T^{21} +$$$$24\!\cdots\!78$$$$T^{22} +$$$$14\!\cdots\!58$$$$T^{23} +$$$$76\!\cdots\!67$$$$T^{24} +$$$$38\!\cdots\!06$$$$T^{25} +$$$$17\!\cdots\!26$$$$T^{26} +$$$$72\!\cdots\!42$$$$T^{27} +$$$$26\!\cdots\!67$$$$T^{28} +$$$$81\!\cdots\!60$$$$T^{29} +$$$$20\!\cdots\!76$$$$T^{30} +$$$$36\!\cdots\!60$$$$T^{31} +$$$$36\!\cdots\!61$$$$T^{32}$$
$67$ $$1 - 14 T - 239 T^{2} + 5102 T^{3} + 20453 T^{4} - 910354 T^{5} + 617796 T^{6} + 105135588 T^{7} - 361795050 T^{8} - 8998911156 T^{9} + 52479815982 T^{10} + 604126066254 T^{11} - 5379501807402 T^{12} - 30548371355232 T^{13} + 457126004374545 T^{14} + 769012940073186 T^{15} - 33126987823382439 T^{16} + 51523866984903462 T^{17} + 2052038633637332505 T^{18} - 9187819813913642016 T^{19} -$$$$10\!\cdots\!42$$$$T^{20} +$$$$81\!\cdots\!78$$$$T^{21} +$$$$47\!\cdots\!58$$$$T^{22} -$$$$54\!\cdots\!88$$$$T^{23} -$$$$14\!\cdots\!50$$$$T^{24} +$$$$28\!\cdots\!36$$$$T^{25} +$$$$11\!\cdots\!04$$$$T^{26} -$$$$11\!\cdots\!82$$$$T^{27} +$$$$16\!\cdots\!33$$$$T^{28} +$$$$27\!\cdots\!74$$$$T^{29} -$$$$87\!\cdots\!31$$$$T^{30} -$$$$34\!\cdots\!02$$$$T^{31} +$$$$16\!\cdots\!81$$$$T^{32}$$
$71$ $$1 - 650 T^{2} + 199389 T^{4} - 38930632 T^{6} + 5566228364 T^{8} - 640511863116 T^{10} + 63163988645884 T^{12} - 5475157404521894 T^{14} + 416179213677825948 T^{16} - 27600268476194867654 T^{18} +$$$$16\!\cdots\!04$$$$T^{20} -$$$$82\!\cdots\!36$$$$T^{22} +$$$$35\!\cdots\!04$$$$T^{24} -$$$$12\!\cdots\!32$$$$T^{26} +$$$$32\!\cdots\!49$$$$T^{28} -$$$$53\!\cdots\!50$$$$T^{30} +$$$$41\!\cdots\!21$$$$T^{32}$$
$73$ $$1 + 434 T^{2} + 101253 T^{4} + 70380 T^{5} + 16361620 T^{6} + 29833866 T^{7} + 2027081504 T^{8} + 6817588704 T^{9} + 204710173188 T^{10} + 1064159524062 T^{11} + 17706469884892 T^{12} + 124668495857988 T^{13} + 1383495353685575 T^{14} + 11427374402737884 T^{15} + 102528286806790386 T^{16} + 834198331399865532 T^{17} + 7372646739790429175 T^{18} + 48498164253186917796 T^{19} +$$$$50\!\cdots\!72$$$$T^{20} +$$$$22\!\cdots\!66$$$$T^{21} +$$$$30\!\cdots\!32$$$$T^{22} +$$$$75\!\cdots\!88$$$$T^{23} +$$$$16\!\cdots\!24$$$$T^{24} +$$$$17\!\cdots\!58$$$$T^{25} +$$$$70\!\cdots\!80$$$$T^{26} +$$$$22\!\cdots\!60$$$$T^{27} +$$$$23\!\cdots\!13$$$$T^{28} +$$$$52\!\cdots\!06$$$$T^{30} +$$$$65\!\cdots\!61$$$$T^{32}$$
$79$ $$1 + 16 T - 227 T^{2} - 5356 T^{3} + 15491 T^{4} + 818138 T^{5} + 465174 T^{6} - 79955730 T^{7} - 161428446 T^{8} + 5129891178 T^{9} + 6184544046 T^{10} - 147661653486 T^{11} + 2275304241738 T^{12} - 3934261822074 T^{13} - 453211437102513 T^{14} + 289666157971230 T^{15} + 45106559543464353 T^{16} + 22883626479727170 T^{17} - 2828492578956783633 T^{18} - 1939744514493542886 T^{19} + 88623284515338680778 T^{20} -$$$$45\!\cdots\!14$$$$T^{21} +$$$$15\!\cdots\!66$$$$T^{22} +$$$$98\!\cdots\!02$$$$T^{23} -$$$$24\!\cdots\!06$$$$T^{24} -$$$$95\!\cdots\!70$$$$T^{25} +$$$$44\!\cdots\!74$$$$T^{26} +$$$$61\!\cdots\!02$$$$T^{27} +$$$$91\!\cdots\!31$$$$T^{28} -$$$$25\!\cdots\!84$$$$T^{29} -$$$$83\!\cdots\!87$$$$T^{30} +$$$$46\!\cdots\!84$$$$T^{31} +$$$$23\!\cdots\!21$$$$T^{32}$$
$83$ $$1 - 487 T^{2} + 312 T^{3} + 123774 T^{4} - 132990 T^{5} - 22183883 T^{6} + 29138634 T^{7} + 3170469341 T^{8} - 4110229572 T^{9} - 386467088226 T^{10} + 393115428402 T^{11} + 41656592194789 T^{12} - 25282823866380 T^{13} - 4030130568645907 T^{14} + 780079655467782 T^{15} + 351851707607703156 T^{16} + 64746611403825906 T^{17} - 27763569487401653323 T^{18} - 14456390010085821060 T^{19} +$$$$19\!\cdots\!69$$$$T^{20} +$$$$15\!\cdots\!86$$$$T^{21} -$$$$12\!\cdots\!94$$$$T^{22} -$$$$11\!\cdots\!44$$$$T^{23} +$$$$71\!\cdots\!81$$$$T^{24} +$$$$54\!\cdots\!02$$$$T^{25} -$$$$34\!\cdots\!67$$$$T^{26} -$$$$17\!\cdots\!30$$$$T^{27} +$$$$13\!\cdots\!14$$$$T^{28} +$$$$27\!\cdots\!56$$$$T^{29} -$$$$35\!\cdots\!23$$$$T^{30} +$$$$50\!\cdots\!81$$$$T^{32}$$
$89$ $$1 + 24 T - 10 T^{2} - 3636 T^{3} + 1197 T^{4} + 543420 T^{5} + 1792912 T^{6} - 58468350 T^{7} - 544062388 T^{8} + 3510821196 T^{9} + 61278252588 T^{10} - 296095488138 T^{11} - 6613804791944 T^{12} + 16297764313308 T^{13} + 597103648531739 T^{14} + 218011613471172 T^{15} - 39896652829458150 T^{16} + 19403033598934308 T^{17} + 4729658000019904619 T^{18} + 11489418610188427452 T^{19} -$$$$41\!\cdots\!04$$$$T^{20} -$$$$16\!\cdots\!62$$$$T^{21} +$$$$30\!\cdots\!68$$$$T^{22} +$$$$15\!\cdots\!84$$$$T^{23} -$$$$21\!\cdots\!28$$$$T^{24} -$$$$20\!\cdots\!50$$$$T^{25} +$$$$55\!\cdots\!12$$$$T^{26} +$$$$15\!\cdots\!80$$$$T^{27} +$$$$29\!\cdots\!37$$$$T^{28} -$$$$79\!\cdots\!84$$$$T^{29} -$$$$19\!\cdots\!10$$$$T^{30} +$$$$41\!\cdots\!76$$$$T^{31} +$$$$15\!\cdots\!61$$$$T^{32}$$
$97$ $$1 + 6 T + 395 T^{2} + 2298 T^{3} + 68379 T^{4} + 571872 T^{5} + 8922340 T^{6} + 109039644 T^{7} + 1233764039 T^{8} + 14418445794 T^{9} + 171203907675 T^{10} + 1605167799666 T^{11} + 21002714173786 T^{12} + 194363495836338 T^{13} + 2133943680044999 T^{14} + 22669145285986746 T^{15} + 200280216945639852 T^{16} + 2198907092740714362 T^{17} + 20078276085543395591 T^{18} +$$$$17\!\cdots\!74$$$$T^{19} +$$$$18\!\cdots\!66$$$$T^{20} +$$$$13\!\cdots\!62$$$$T^{21} +$$$$14\!\cdots\!75$$$$T^{22} +$$$$11\!\cdots\!22$$$$T^{23} +$$$$96\!\cdots\!79$$$$T^{24} +$$$$82\!\cdots\!48$$$$T^{25} +$$$$65\!\cdots\!60$$$$T^{26} +$$$$40\!\cdots\!16$$$$T^{27} +$$$$47\!\cdots\!39$$$$T^{28} +$$$$15\!\cdots\!46$$$$T^{29} +$$$$25\!\cdots\!55$$$$T^{30} +$$$$37\!\cdots\!58$$$$T^{31} +$$$$61\!\cdots\!21$$$$T^{32}$$