# Properties

 Label 693.2.m.i Level 693 Weight 2 Character orbit 693.m Analytic conductor 5.534 Analytic rank 0 Dimension 16 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$693 = 3^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 693.m (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.53363286007$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$5$$ Twist minimal: no (minimal twist has level 77) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{5} + \beta_{6} ) q^{2} + ( \beta_{2} + \beta_{7} + \beta_{11} ) q^{4} + ( -\beta_{1} + \beta_{3} - \beta_{6} + \beta_{7} - \beta_{13} ) q^{5} + ( -1 + \beta_{7} - \beta_{8} + \beta_{10} ) q^{7} + ( \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} + \beta_{8} - \beta_{10} + \beta_{12} + \beta_{13} - \beta_{15} ) q^{8} +O(q^{10})$$ $$q + ( \beta_{5} + \beta_{6} ) q^{2} + ( \beta_{2} + \beta_{7} + \beta_{11} ) q^{4} + ( -\beta_{1} + \beta_{3} - \beta_{6} + \beta_{7} - \beta_{13} ) q^{5} + ( -1 + \beta_{7} - \beta_{8} + \beta_{10} ) q^{7} + ( \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} + \beta_{8} - \beta_{10} + \beta_{12} + \beta_{13} - \beta_{15} ) q^{8} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{7} + 2 \beta_{8} ) q^{10} + ( 1 - \beta_{3} + \beta_{5} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{13} + \beta_{14} ) q^{11} + ( \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{13} + \beta_{15} ) q^{13} -\beta_{6} q^{14} + ( 1 - \beta_{1} + \beta_{4} + \beta_{5} - \beta_{7} - 2 \beta_{10} - \beta_{11} - \beta_{14} - \beta_{15} ) q^{16} + ( 1 + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{17} + ( 2 + \beta_{2} - \beta_{6} + \beta_{8} + \beta_{10} - \beta_{12} - \beta_{15} ) q^{19} + ( -\beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - 3 \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{12} + \beta_{14} ) q^{20} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{22} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{23} + ( 2 - 3 \beta_{1} + 3 \beta_{5} + 2 \beta_{8} - \beta_{9} - 5 \beta_{10} + \beta_{11} - \beta_{13} - \beta_{15} ) q^{25} + ( 1 - 2 \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} ) q^{26} + ( -\beta_{2} - \beta_{7} - \beta_{9} - \beta_{10} ) q^{28} + ( -\beta_{2} - \beta_{3} - \beta_{4} - \beta_{7} + 3 \beta_{8} - 2 \beta_{9} - 4 \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{15} ) q^{29} + ( -3 \beta_{1} + 2 \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{13} + 2 \beta_{15} ) q^{31} + ( -3 + 2 \beta_{1} - \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} ) q^{32} + ( 2 \beta_{2} + \beta_{3} + 2 \beta_{7} - 2 \beta_{8} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{34} + ( -\beta_{1} + \beta_{3} + \beta_{4} - \beta_{8} - \beta_{14} ) q^{35} + ( \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{12} - 2 \beta_{14} + \beta_{15} ) q^{37} + ( \beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} - 2 \beta_{10} + \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{38} + ( 1 + 3 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 2 \beta_{7} - \beta_{8} - \beta_{10} + 3 \beta_{11} - 2 \beta_{15} ) q^{40} + ( 2 + 2 \beta_{2} + \beta_{9} - \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{41} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} ) q^{43} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - \beta_{7} + 4 \beta_{8} + \beta_{10} + \beta_{12} - \beta_{13} - 2 \beta_{14} ) q^{44} + ( \beta_{3} - 3 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} - 3 \beta_{10} - 2 \beta_{12} + \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{46} + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{5} + 2 \beta_{6} + 2 \beta_{9} + 4 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + \beta_{15} ) q^{47} -\beta_{7} q^{49} + ( -5 + 3 \beta_{1} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + 4 \beta_{7} + 4 \beta_{10} + 4 \beta_{11} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{50} + ( -2 \beta_{1} - \beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{8} + 3 \beta_{10} + \beta_{12} - 2 \beta_{13} - \beta_{15} ) q^{52} + ( 2 \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{7} - 5 \beta_{8} - 3 \beta_{12} - 2 \beta_{13} + 2 \beta_{15} ) q^{53} + ( -1 + \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} - 4 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} + \beta_{10} - 4 \beta_{11} + \beta_{12} + 2 \beta_{13} + 2 \beta_{14} ) q^{55} + ( \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{56} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{5} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - \beta_{15} ) q^{58} + ( -1 + \beta_{3} + 3 \beta_{4} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{14} + \beta_{15} ) q^{59} + ( -2 \beta_{1} - \beta_{2} + 3 \beta_{4} + 3 \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{14} - \beta_{15} ) q^{61} + ( 5 \beta_{2} + \beta_{3} + \beta_{5} + 5 \beta_{7} + \beta_{8} + 3 \beta_{9} + 2 \beta_{11} - 3 \beta_{12} + \beta_{14} + \beta_{15} ) q^{62} + ( 2 \beta_{1} - 4 \beta_{3} - 2 \beta_{4} + \beta_{7} - \beta_{9} - 2 \beta_{10} + \beta_{12} + 3 \beta_{13} + \beta_{14} - 3 \beta_{15} ) q^{64} + ( -3 + 4 \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} ) q^{65} + ( -4 + 2 \beta_{1} + \beta_{4} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{13} - \beta_{14} ) q^{67} + ( 4 \beta_{1} - 2 \beta_{3} - 4 \beta_{4} - 5 \beta_{5} - 5 \beta_{6} + \beta_{7} + \beta_{9} - 2 \beta_{10} + \beta_{12} + 3 \beta_{13} - \beta_{14} - 3 \beta_{15} ) q^{68} + ( -1 + \beta_{2} - 2 \beta_{4} + 2 \beta_{7} + \beta_{9} - \beta_{12} - \beta_{14} ) q^{70} + ( 3 - \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{7} + 3 \beta_{9} - 3 \beta_{10} + \beta_{11} + 3 \beta_{13} - 3 \beta_{14} - 3 \beta_{15} ) q^{71} + ( -1 - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{7} + 6 \beta_{8} - 4 \beta_{10} - 2 \beta_{11} - \beta_{14} + 2 \beta_{15} ) q^{73} + ( -\beta_{2} + \beta_{8} + 2 \beta_{9} + 5 \beta_{10} - 2 \beta_{11} + \beta_{12} + 2 \beta_{15} ) q^{74} + ( 5 + 3 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{11} - 2 \beta_{12} ) q^{76} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} + \beta_{12} - \beta_{15} ) q^{77} + ( -\beta_{1} + 4 \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} + 5 \beta_{10} - \beta_{12} - 4 \beta_{13} + 4 \beta_{15} ) q^{79} + ( 8 - \beta_{1} - 2 \beta_{2} + \beta_{5} - 4 \beta_{6} + 10 \beta_{8} + \beta_{9} - 6 \beta_{10} - \beta_{11} + 2 \beta_{12} + \beta_{13} - \beta_{15} ) q^{80} + ( 1 + 6 \beta_{1} + 2 \beta_{2} - \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + 2 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} ) q^{82} + ( 2 + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - 4 \beta_{7} + \beta_{9} + \beta_{10} - 2 \beta_{11} + 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{83} + ( 3 \beta_{1} - \beta_{2} - 3 \beta_{5} - 3 \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{85} + ( -\beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} + 5 \beta_{7} - 2 \beta_{8} - \beta_{9} + 4 \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{86} + ( 3 - 3 \beta_{1} - \beta_{2} + 5 \beta_{4} + 4 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{88} + ( -3 + 4 \beta_{1} + 3 \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} + 3 \beta_{11} - 3 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{89} + ( 1 - \beta_{1} - \beta_{2} + \beta_{5} + 2 \beta_{8} - \beta_{10} + \beta_{12} + \beta_{13} ) q^{91} + ( 3 - 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{14} - 2 \beta_{15} ) q^{92} + ( -2 + 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 4 \beta_{6} + 6 \beta_{7} + 4 \beta_{9} + 7 \beta_{10} - \beta_{11} + 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{94} + ( -1 - 3 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - \beta_{5} - 2 \beta_{7} + 7 \beta_{8} - 2 \beta_{9} - 5 \beta_{10} - \beta_{11} + 2 \beta_{12} + 2 \beta_{14} + 2 \beta_{15} ) q^{95} + ( -2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} - \beta_{9} + 4 \beta_{10} - \beta_{12} - \beta_{13} + 3 \beta_{14} + \beta_{15} ) q^{97} -\beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 3q^{2} - 11q^{4} + 5q^{5} - 4q^{7} + 5q^{8} + O(q^{10})$$ $$16q + 3q^{2} - 11q^{4} + 5q^{5} - 4q^{7} + 5q^{8} + 12q^{10} + 3q^{11} - 7q^{13} - 2q^{14} + 17q^{16} + 5q^{17} + 19q^{19} - q^{20} - 33q^{22} - 32q^{23} + 7q^{25} + 27q^{26} + 4q^{28} - 3q^{29} - 7q^{31} - 32q^{32} - 24q^{34} + 5q^{35} + 4q^{37} + 5q^{38} - 10q^{40} + 10q^{41} - 8q^{43} + 38q^{44} - 42q^{46} + 23q^{47} - 4q^{49} - 52q^{50} + 33q^{52} - 4q^{53} - 12q^{55} + 20q^{58} - 17q^{59} - 7q^{61} - 79q^{62} + 7q^{64} + 8q^{65} - 38q^{67} + 2q^{68} - 18q^{70} + 14q^{71} - 35q^{73} + 29q^{74} + 52q^{76} + 3q^{77} + 15q^{79} + 87q^{80} + 19q^{82} - 5q^{83} + 6q^{85} + 52q^{86} + 55q^{88} - 74q^{89} + 13q^{91} + 55q^{92} - 24q^{94} - 32q^{95} + 20q^{97} - 2q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 3 x^{15} + 14 x^{14} - 32 x^{13} + 86 x^{12} - 145 x^{11} + 245 x^{10} - 245 x^{9} + 640 x^{8} - 1175 x^{7} + 2135 x^{6} - 2300 x^{5} + 1850 x^{4} - 925 x^{3} + 700 x^{2} - 200 x + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-33722118880975 \nu^{15} - 215936836813390 \nu^{14} + 154302737507927 \nu^{13} - 3234633146529285 \nu^{12} + 4494612950061958 \nu^{11} - 20835947263746904 \nu^{10} + 25956864481307657 \nu^{9} - 65098048812157320 \nu^{8} + 30869277708222543 \nu^{7} - 160665000409949210 \nu^{6} + 100524107363786610 \nu^{5} - 560336279122558890 \nu^{4} + 546963632352958070 \nu^{3} - 434474896057227745 \nu^{2} + 120133073008741205 \nu + 88528161377575350$$$$)/ 454580475630153760$$ $$\beta_{2}$$ $$=$$ $$($$$$-686277383720070 \nu^{15} - 1360333162376645 \nu^{14} + 444742620384542 \nu^{13} - 24553779304461089 \nu^{12} + 45808084826141647 \nu^{11} - 179218800902324756 \nu^{10} + 292393929070812298 \nu^{9} - 592228758909440833 \nu^{8} + 274072637813433960 \nu^{7} - 1226946693965489055 \nu^{6} + 2313198962618101340 \nu^{5} - 5164313069616781590 \nu^{4} + 5505326380247646890 \nu^{3} - 4265354995237286060 \nu^{2} + 1164454349991851775 \nu - 1569079812936964435$$$$)/ 454580475630153760$$ $$\beta_{3}$$ $$=$$ $$($$$$-874936745197272 \nu^{15} - 514706296938285 \nu^{14} - 5527222230184928 \nu^{13} - 8597109141630243 \nu^{12} - 9687442744709873 \nu^{11} - 69609203132183412 \nu^{10} + 41469078152400808 \nu^{9} - 248365491806816213 \nu^{8} - 292096370375405502 \nu^{7} - 583922916320059255 \nu^{6} + 395306407846606840 \nu^{5} - 2070346335621533450 \nu^{4} + 1021707328353270230 \nu^{3} - 1173049633586345730 \nu^{2} + 368306587037923705 \nu - 928639412535219835$$$$)/ 454580475630153760$$ $$\beta_{4}$$ $$=$$ $$($$$$-1565624412275039 \nu^{15} + 4832412997019746 \nu^{14} - 21672367495495401 \nu^{13} + 49577176484430299 \nu^{12} - 129795369286809034 \nu^{11} + 214474805906670840 \nu^{10} - 347621794857624183 \nu^{9} + 312652728515985000 \nu^{8} - 887685879699924177 \nu^{7} + 1770758763681867190 \nu^{6} - 3159142418183242750 \nu^{5} + 2957115228664641670 \nu^{4} - 1979053855141622330 \nu^{3} + 366777001484809055 \nu^{2} - 739274040704445675 \nu + 95214593589332550$$$$)/ 454580475630153760$$ $$\beta_{5}$$ $$=$$ $$($$$$1676337845170790 \nu^{15} - 5194753922189191 \nu^{14} + 23996414540642882 \nu^{13} - 55793399026072099 \nu^{12} + 149577475159066613 \nu^{11} - 254981761232364844 \nu^{10} + 433123005758956374 \nu^{9} - 436889234085257675 \nu^{8} + 1101433166283370416 \nu^{7} - 2030901870569891045 \nu^{6} + 3769863137668677300 \nu^{5} - 4085868252644216690 \nu^{4} + 3585858111334770990 \nu^{3} - 1476295694818421740 \nu^{2} + 1479871527393288525 \nu - 189874590198959025$$$$)/ 454580475630153760$$ $$\beta_{6}$$ $$=$$ $$($$$$-1676337845170790 \nu^{15} + 5194753922189191 \nu^{14} - 23996414540642882 \nu^{13} + 55793399026072099 \nu^{12} - 149577475159066613 \nu^{11} + 254981761232364844 \nu^{10} - 433123005758956374 \nu^{9} + 436889234085257675 \nu^{8} - 1101433166283370416 \nu^{7} + 2030901870569891045 \nu^{6} - 3769863137668677300 \nu^{5} + 4085868252644216690 \nu^{4} - 3585858111334770990 \nu^{3} + 1476295694818421740 \nu^{2} - 1025291051763134765 \nu + 189874590198959025$$$$)/ 454580475630153760$$ $$\beta_{7}$$ $$=$$ $$($$$$-3541126455103014 \nu^{15} + 10589657246428067 \nu^{14} - 49791707208255586 \nu^{13} + 113470349300804375 \nu^{12} - 307771508285388489 \nu^{11} + 517957948939998988 \nu^{10} - 888411928763985334 \nu^{9} + 893532845981546087 \nu^{8} - 2331418980078086280 \nu^{7} + 4191692862454263993 \nu^{6} - 7720969982054884100 \nu^{5} + 8245114954100718810 \nu^{4} - 7111420221063134790 \nu^{3} + 3822505603323246020 \nu^{2} - 2913263414629337545 \nu + 828358364029344005$$$$)/ 454580475630153760$$ $$\beta_{8}$$ $$=$$ $$($$$$-3808583743573302 \nu^{15} + 9860126818444867 \nu^{14} - 48487759413006482 \nu^{13} + 100202312298850263 \nu^{12} - 277961025462873673 \nu^{11} + 422449273531319756 \nu^{10} - 718628211268788150 \nu^{9} + 585481222317834807 \nu^{8} - 2124840867370928280 \nu^{7} + 3587400018998705673 \nu^{6} - 6360567528847132580 \nu^{5} + 5600600192035351850 \nu^{4} - 4088764696945967030 \nu^{3} + 1543886107663682020 \nu^{2} - 2299231619016502345 \nu + 22442708010214725$$$$)/ 454580475630153760$$ $$\beta_{9}$$ $$=$$ $$($$$$4147870816217860 \nu^{15} - 9098179878471885 \nu^{14} + 48642002600512068 \nu^{13} - 88958591156351927 \nu^{12} + 262233539631762351 \nu^{11} - 349871456206210068 \nu^{10} + 623455379026927044 \nu^{9} - 381247122554708893 \nu^{8} + 2150910817199786310 \nu^{7} - 3112048315103081495 \nu^{6} + 5535211606668989440 \nu^{5} - 3712550967753264370 \nu^{4} + 2767110524781279550 \nu^{3} - 1335895153189756830 \nu^{2} + 2975718288695695725 \nu - 29045982804284875$$$$)/ 454580475630153760$$ $$\beta_{10}$$ $$=$$ $$($$$$7327526319488073 \nu^{15} - 21838143406687493 \nu^{14} + 102438964384476967 \nu^{13} - 232311097915978782 \nu^{12} + 627185674080861763 \nu^{11} - 1047203823403574964 \nu^{10} + 1775572940212630785 \nu^{9} - 1735699601854553351 \nu^{8} + 4630478387715251365 \nu^{7} - 8426965416523262079 \nu^{6} + 15544790585628961210 \nu^{5} - 16343113922700919960 \nu^{4} + 12987506946195918920 \nu^{3} - 5718121221686276935 \nu^{2} + 4454224626365266160 \nu - 845040850217512955$$$$)/ 454580475630153760$$ $$\beta_{11}$$ $$=$$ $$($$$$-5481281688546819 \nu^{15} + 16493950936678582 \nu^{14} - 76461534295292341 \nu^{13} + 174584775915853027 \nu^{12} - 466393479528723526 \nu^{11} + 782325573976058616 \nu^{10} - 1307780222870847083 \nu^{9} + 1276995934654789444 \nu^{8} - 3395227103875767205 \nu^{7} + 6340299488823316290 \nu^{6} - 11480502622702818310 \nu^{5} + 12158198218393584390 \nu^{4} - 9161804501671007210 \nu^{3} + 3888216659798809835 \nu^{2} - 2965788298433688075 \nu + 864066394282347650$$$$)/ 227290237815076880$$ $$\beta_{12}$$ $$=$$ $$($$$$-11668286039317543 \nu^{15} + 32209452556198329 \nu^{14} - 155422622027191961 \nu^{13} + 335017774059801892 \nu^{12} - 918061503999498391 \nu^{11} + 1456783386223745044 \nu^{10} - 2469198487594261567 \nu^{9} + 2179375478325730279 \nu^{8} - 6793193349504836855 \nu^{7} + 11856118418058363915 \nu^{6} - 21771993760224740470 \nu^{5} + 21003963276182506660 \nu^{4} - 15299497235328621900 \nu^{3} + 5270338292583504165 \nu^{2} - 5226852994856502950 \nu + 59418569180970775$$$$)/ 454580475630153760$$ $$\beta_{13}$$ $$=$$ $$($$$$-16296167766296901 \nu^{15} + 46657645630438849 \nu^{14} - 220693813391828059 \nu^{13} + 490255203026234886 \nu^{12} - 1324081568271118439 \nu^{11} + 2171662124721910708 \nu^{10} - 3646287371621077981 \nu^{9} + 3462410165728884427 \nu^{8} - 9848220683260194529 \nu^{7} + 17797250626139187779 \nu^{6} - 31784433905471092610 \nu^{5} + 32902654497697219800 \nu^{4} - 24876887445936951080 \nu^{3} + 11522918100224582795 \nu^{2} - 8819030598088243120 \nu + 1667167131638041055$$$$)/ 454580475630153760$$ $$\beta_{14}$$ $$=$$ $$($$$$16576781827677814 \nu^{15} - 45453688208119129 \nu^{14} + 219284175557671186 \nu^{13} - 470575312000685073 \nu^{12} + 1289205279920317419 \nu^{11} - 2038073676823909444 \nu^{10} + 3445236405584454006 \nu^{9} - 3032211154151816061 \nu^{8} + 9585093068819660548 \nu^{7} - 16816185325336642299 \nu^{6} + 30431971572347844100 \nu^{5} - 29120858031490839590 \nu^{4} + 20842243045981534410 \nu^{3} - 8417551544734639360 \nu^{2} + 7973236770827366535 \nu - 1025213227739747135$$$$)/ 454580475630153760$$ $$\beta_{15}$$ $$=$$ $$($$$$-31742657223153564 \nu^{15} + 90930749048576029 \nu^{14} - 431347940936808556 \nu^{13} + 955931937354509231 \nu^{12} - 2590293343054165503 \nu^{11} + 4236554416589668948 \nu^{10} - 7143632052250437628 \nu^{9} + 6742540241983255677 \nu^{8} - 19234203982822940782 \nu^{7} + 34610408690995495111 \nu^{6} - 62595008084751532400 \nu^{5} + 64110835668533354210 \nu^{4} - 48322167102593217550 \nu^{3} + 21980961042064417270 \nu^{2} - 18197077383887239845 \nu + 3492178872565443195$$$$)/ 454580475630153760$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{6} + \beta_{5}$$ $$\nu^{2}$$ $$=$$ $$\beta_{11} + 2 \beta_{10} - 2 \beta_{8} + 3 \beta_{7} + \beta_{2} - 2$$ $$\nu^{3}$$ $$=$$ $$-\beta_{15} + \beta_{13} + \beta_{12} - \beta_{10} + \beta_{8} - 5 \beta_{6} - \beta_{5} - \beta_{2} + \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-\beta_{15} - \beta_{14} - \beta_{11} - 8 \beta_{10} - 6 \beta_{9} - 15 \beta_{7} + \beta_{5} + \beta_{4} - 6 \beta_{2} - \beta_{1} + 1$$ $$\nu^{5}$$ $$=$$ $$-7 \beta_{14} - 7 \beta_{13} - 7 \beta_{12} + 7 \beta_{11} - 8 \beta_{8} + 8 \beta_{7} + 10 \beta_{6} - 10 \beta_{4} - \beta_{3} + 8 \beta_{2} - 18 \beta_{1} - 3$$ $$\nu^{6}$$ $$=$$ $$7 \beta_{15} + \beta_{14} - 7 \beta_{13} + 11 \beta_{12} + 54 \beta_{10} + 35 \beta_{9} + 40 \beta_{8} + 47 \beta_{7} - 10 \beta_{6} - 10 \beta_{5} - 2 \beta_{4} + 6 \beta_{3} + 2 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$42 \beta_{15} + 54 \beta_{14} + 13 \beta_{12} - 45 \beta_{11} + 12 \beta_{10} - 13 \beta_{9} + 30 \beta_{8} - 86 \beta_{7} + 76 \beta_{5} + 165 \beta_{4} + 42 \beta_{3} - 58 \beta_{2} + 28$$ $$\nu^{8}$$ $$=$$ $$-9 \beta_{15} + 38 \beta_{13} - 210 \beta_{12} + 89 \beta_{11} - 234 \beta_{10} - 89 \beta_{9} - 308 \beta_{8} + 80 \beta_{6} + 28 \beta_{5} + 210 \beta_{2} - 28 \beta_{1} - 98$$ $$\nu^{9}$$ $$=$$ $$-346 \beta_{15} - 346 \beta_{14} + 248 \beta_{13} + 117 \beta_{11} + 38 \beta_{10} + 290 \beta_{9} + 614 \beta_{7} - 524 \beta_{6} - 1000 \beta_{5} - 1000 \beta_{4} - 248 \beta_{3} + 290 \beta_{2} + 476 \beta_{1} - 94$$ $$\nu^{10}$$ $$=$$ $$-56 \beta_{14} - 56 \beta_{13} + 1290 \beta_{12} - 1290 \beta_{11} + 1992 \beta_{8} - 1992 \beta_{7} - 272 \beta_{6} + 272 \beta_{4} - 131 \beta_{3} - 1931 \beta_{2} + 328 \beta_{1} + 1406$$ $$\nu^{11}$$ $$=$$ $$2174 \beta_{15} + 1477 \beta_{14} - 2174 \beta_{13} - 913 \beta_{12} - 447 \beta_{10} - 1890 \beta_{9} - 1625 \beta_{8} - 2621 \beta_{7} + 6160 \beta_{6} + 6160 \beta_{5} + 3461 \beta_{4} + 697 \beta_{3} - 3461 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$284 \beta_{15} + 848 \beta_{14} - 4374 \beta_{12} + 8050 \beta_{11} + 8579 \beta_{10} + 4374 \beta_{9} - 8295 \beta_{8} + 21093 \beta_{7} - 2273 \beta_{5} - 4375 \beta_{4} + 284 \beta_{3} + 12424 \beta_{2} - 8669$$ $$\nu^{13}$$ $$=$$ $$-8898 \beta_{15} + 13556 \beta_{13} + 12425 \beta_{12} - 6647 \beta_{11} + 2237 \beta_{10} + 6647 \beta_{9} + 17819 \beta_{8} - 38325 \beta_{6} - 22394 \beta_{5} - 12425 \beta_{2} + 22394 \beta_{1} + 5394$$ $$\nu^{14}$$ $$=$$ $$-3382 \beta_{15} - 3382 \beta_{14} + 1131 \beta_{13} - 29041 \beta_{11} - 85364 \beta_{10} - 50750 \beta_{9} - 133987 \beta_{7} + 17557 \beta_{6} + 31350 \beta_{5} + 31350 \beta_{4} - 1131 \beta_{3} - 50750 \beta_{2} - 13793 \beta_{1} + 31232$$ $$\nu^{15}$$ $$=$$ $$-54132 \beta_{14} - 54132 \beta_{13} - 82100 \beta_{12} + 82100 \beta_{11} - 120596 \beta_{8} + 120596 \beta_{7} + 143446 \beta_{6} - 143446 \beta_{4} - 30172 \beta_{3} + 128698 \beta_{2} - 96554 \beta_{1} - 67213$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/693\mathbb{Z}\right)^\times$$.

 $$n$$ $$155$$ $$199$$ $$442$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1 + \beta_{7} - \beta_{8} + \beta_{10}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1
 −0.788594 + 2.42704i −0.206962 + 0.636964i 0.435488 − 1.34029i 0.751051 − 2.31150i −1.38112 + 1.00344i 0.183009 − 0.132964i 0.901622 − 0.655067i 1.60551 − 1.16647i −0.788594 − 2.42704i −0.206962 − 0.636964i 0.435488 + 1.34029i 0.751051 + 2.31150i −1.38112 − 1.00344i 0.183009 + 0.132964i 0.901622 + 0.655067i 1.60551 + 1.16647i
−0.788594 + 2.42704i 0 −3.65062 2.65233i −1.05961 3.26115i 0 −0.809017 0.587785i 5.18703 3.76860i 0 8.75055
64.2 −0.206962 + 0.636964i 0 1.25514 + 0.911915i −0.662464 2.03885i 0 −0.809017 0.587785i −1.92429 + 1.39808i 0 1.43578
64.3 0.435488 1.34029i 0 0.0112975 + 0.00820814i 0.565930 + 1.74175i 0 −0.809017 0.587785i 2.29616 1.66826i 0 2.58091
64.4 0.751051 2.31150i 0 −3.16091 2.29654i −0.388938 1.19703i 0 −0.809017 0.587785i −3.74989 + 2.72445i 0 −3.05904
190.1 −1.38112 + 1.00344i 0 0.282562 0.869638i 3.28976 + 2.39015i 0 0.309017 0.951057i −0.572703 1.76260i 0 −6.94194
190.2 0.183009 0.132964i 0 −0.602221 + 1.85345i −2.01892 1.46683i 0 0.309017 0.951057i 0.276036 + 0.849550i 0 −0.564516
190.3 0.901622 0.655067i 0 −0.234224 + 0.720867i 2.79603 + 2.03143i 0 0.309017 0.951057i 0.949813 + 2.92322i 0 3.85168
190.4 1.60551 1.16647i 0 0.598967 1.84343i −0.0217822 0.0158257i 0 0.309017 0.951057i 0.0378378 + 0.116453i 0 −0.0534317
379.1 −0.788594 2.42704i 0 −3.65062 + 2.65233i −1.05961 + 3.26115i 0 −0.809017 + 0.587785i 5.18703 + 3.76860i 0 8.75055
379.2 −0.206962 0.636964i 0 1.25514 0.911915i −0.662464 + 2.03885i 0 −0.809017 + 0.587785i −1.92429 1.39808i 0 1.43578
379.3 0.435488 + 1.34029i 0 0.0112975 0.00820814i 0.565930 1.74175i 0 −0.809017 + 0.587785i 2.29616 + 1.66826i 0 2.58091
379.4 0.751051 + 2.31150i 0 −3.16091 + 2.29654i −0.388938 + 1.19703i 0 −0.809017 + 0.587785i −3.74989 2.72445i 0 −3.05904
631.1 −1.38112 1.00344i 0 0.282562 + 0.869638i 3.28976 2.39015i 0 0.309017 + 0.951057i −0.572703 + 1.76260i 0 −6.94194
631.2 0.183009 + 0.132964i 0 −0.602221 1.85345i −2.01892 + 1.46683i 0 0.309017 + 0.951057i 0.276036 0.849550i 0 −0.564516
631.3 0.901622 + 0.655067i 0 −0.234224 0.720867i 2.79603 2.03143i 0 0.309017 + 0.951057i 0.949813 2.92322i 0 3.85168
631.4 1.60551 + 1.16647i 0 0.598967 + 1.84343i −0.0217822 + 0.0158257i 0 0.309017 + 0.951057i 0.0378378 0.116453i 0 −0.0534317
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 631.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 693.2.m.i 16
3.b odd 2 1 77.2.f.b 16
11.c even 5 1 inner 693.2.m.i 16
11.c even 5 1 7623.2.a.ct 8
11.d odd 10 1 7623.2.a.cw 8
21.c even 2 1 539.2.f.e 16
21.g even 6 2 539.2.q.f 32
21.h odd 6 2 539.2.q.g 32
33.d even 2 1 847.2.f.x 16
33.f even 10 1 847.2.a.o 8
33.f even 10 2 847.2.f.v 16
33.f even 10 1 847.2.f.x 16
33.h odd 10 1 77.2.f.b 16
33.h odd 10 1 847.2.a.p 8
33.h odd 10 2 847.2.f.w 16
231.r odd 10 1 5929.2.a.bs 8
231.u even 10 1 539.2.f.e 16
231.u even 10 1 5929.2.a.bt 8
231.z odd 30 2 539.2.q.g 32
231.bc even 30 2 539.2.q.f 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.b 16 3.b odd 2 1
77.2.f.b 16 33.h odd 10 1
539.2.f.e 16 21.c even 2 1
539.2.f.e 16 231.u even 10 1
539.2.q.f 32 21.g even 6 2
539.2.q.f 32 231.bc even 30 2
539.2.q.g 32 21.h odd 6 2
539.2.q.g 32 231.z odd 30 2
693.2.m.i 16 1.a even 1 1 trivial
693.2.m.i 16 11.c even 5 1 inner
847.2.a.o 8 33.f even 10 1
847.2.a.p 8 33.h odd 10 1
847.2.f.v 16 33.f even 10 2
847.2.f.w 16 33.h odd 10 2
847.2.f.x 16 33.d even 2 1
847.2.f.x 16 33.f even 10 1
5929.2.a.bs 8 231.r odd 10 1
5929.2.a.bt 8 231.u even 10 1
7623.2.a.ct 8 11.c even 5 1
7623.2.a.cw 8 11.d odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(693, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 3 T + 6 T^{2} - 12 T^{3} + 18 T^{4} - 17 T^{5} + 15 T^{6} + T^{7} - 58 T^{8} + 119 T^{9} - 185 T^{10} + 292 T^{11} - 242 T^{12} - 33 T^{13} + 312 T^{14} - 878 T^{15} + 1785 T^{16} - 1756 T^{17} + 1248 T^{18} - 264 T^{19} - 3872 T^{20} + 9344 T^{21} - 11840 T^{22} + 15232 T^{23} - 14848 T^{24} + 512 T^{25} + 15360 T^{26} - 34816 T^{27} + 73728 T^{28} - 98304 T^{29} + 98304 T^{30} - 98304 T^{31} + 65536 T^{32}$$
$3$ 
$5$ $$1 - 5 T - T^{2} + 66 T^{3} - 156 T^{4} - 267 T^{5} + 1789 T^{6} - 1241 T^{7} - 10249 T^{8} + 25094 T^{9} + 25181 T^{10} - 188691 T^{11} + 137316 T^{12} + 851484 T^{13} - 1968914 T^{14} - 1709452 T^{15} + 12417791 T^{16} - 8547260 T^{17} - 49222850 T^{18} + 106435500 T^{19} + 85822500 T^{20} - 589659375 T^{21} + 393453125 T^{22} + 1960468750 T^{23} - 4003515625 T^{24} - 2423828125 T^{25} + 17470703125 T^{26} - 13037109375 T^{27} - 38085937500 T^{28} + 80566406250 T^{29} - 6103515625 T^{30} - 152587890625 T^{31} + 152587890625 T^{32}$$
$7$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$
$11$ $$1 - 3 T + 19 T^{2} + 76 T^{3} - 286 T^{4} + 2245 T^{5} + 1052 T^{6} - 7668 T^{7} + 114073 T^{8} - 84348 T^{9} + 127292 T^{10} + 2988095 T^{11} - 4187326 T^{12} + 12239876 T^{13} + 33659659 T^{14} - 58461513 T^{15} + 214358881 T^{16}$$
$13$ $$1 + 7 T - 13 T^{2} - 114 T^{3} + 302 T^{4} + 619 T^{5} - 3203 T^{6} + 11659 T^{7} - 57039 T^{8} - 415732 T^{9} + 983197 T^{10} + 802283 T^{11} - 7060468 T^{12} + 40515806 T^{13} - 160505050 T^{14} - 179149458 T^{15} + 5116354083 T^{16} - 2328942954 T^{17} - 27125353450 T^{18} + 89013225782 T^{19} - 201654026548 T^{20} + 297882061919 T^{21} + 4745704128373 T^{22} - 26086566469444 T^{23} - 46528464595119 T^{24} + 123637858189807 T^{25} - 441560749392347 T^{26} + 1109347283908903 T^{27} + 7036021706989262 T^{28} - 34527762151516842 T^{29} - 51185893014090757 T^{30} + 358301251098635299 T^{31} + 665416609183179841 T^{32}$$
$17$ $$1 - 5 T + 13 T^{2} - 163 T^{3} + 1572 T^{4} - 7139 T^{5} + 24321 T^{6} - 178695 T^{7} + 1262499 T^{8} - 5104293 T^{9} + 17336797 T^{10} - 107294017 T^{11} + 637972982 T^{12} - 2309382396 T^{13} + 7576691718 T^{14} - 42485576390 T^{15} + 218631742615 T^{16} - 722254798630 T^{17} + 2189663906502 T^{18} - 11345995711548 T^{19} + 53284141429622 T^{20} - 152342161095569 T^{21} + 418468133826493 T^{22} - 2094488816223189 T^{23} + 8806886793505059 T^{24} - 21191060590631415 T^{25} + 49030987652820129 T^{26} - 244667067740191987 T^{27} + 915882156925184292 T^{28} - 1614446219363667731 T^{29} + 2188911745272212077 T^{30} - 14312115257549078965 T^{31} + 48661191875666868481 T^{32}$$
$19$ $$1 - 19 T + 113 T^{2} + 15 T^{3} - 2596 T^{4} + 2631 T^{5} + 106245 T^{6} - 756527 T^{7} + 1484429 T^{8} + 6206619 T^{9} - 26235105 T^{10} - 133461535 T^{11} + 1444663142 T^{12} - 5677361136 T^{13} + 11109732198 T^{14} - 13150474478 T^{15} + 38443302841 T^{16} - 249859015082 T^{17} + 4010613323478 T^{18} - 38941020031824 T^{19} + 188269945328582 T^{20} - 330463973351965 T^{21} - 1234253627852505 T^{22} + 5547921318840441 T^{23} + 25210893501388589 T^{24} - 244121955937653533 T^{25} + 651395134560067245 T^{26} + 306485871161214189 T^{27} - 5745765529895753956 T^{28} + 630794751933855885 T^{29} + 90287755493465905673 T^{30} -$$$$28\!\cdots\!81$$$$T^{31} +$$$$28\!\cdots\!81$$$$T^{32}$$
$23$ $$( 1 + 16 T + 224 T^{2} + 2234 T^{3} + 19567 T^{4} + 141712 T^{5} + 926670 T^{6} + 5230592 T^{7} + 26816927 T^{8} + 120303616 T^{9} + 490208430 T^{10} + 1724209904 T^{11} + 5475648847 T^{12} + 14378790262 T^{13} + 33160039136 T^{14} + 54477207152 T^{15} + 78310985281 T^{16} )^{2}$$
$29$ $$1 + 3 T + 75 T^{3} + 2441 T^{4} + 473 T^{5} - 37550 T^{6} + 114715 T^{7} + 3177286 T^{8} - 2623287 T^{9} - 24972736 T^{10} + 263064707 T^{11} + 3471423716 T^{12} - 694532172 T^{13} - 11302302466 T^{14} + 87801320252 T^{15} + 2485262854281 T^{16} + 2546238287308 T^{17} - 9505236373906 T^{18} - 16938945142908 T^{19} + 2455272037276196 T^{20} + 5395759401918343 T^{21} - 14854365761976256 T^{22} - 45251376273007683 T^{23} + 1589425924451203846 T^{24} + 1664187250621812335 T^{25} - 15797556610422547550 T^{26} + 5770841119178857117 T^{27} +$$$$86\!\cdots\!81$$$$T^{28} +$$$$76\!\cdots\!75$$$$T^{29} +$$$$25\!\cdots\!47$$$$T^{31} +$$$$25\!\cdots\!21$$$$T^{32}$$
$31$ $$1 + 7 T + 13 T^{3} + 77 T^{4} - 8801 T^{5} - 22254 T^{6} + 56119 T^{7} - 947868 T^{8} + 3202093 T^{9} + 57615842 T^{10} - 102279009 T^{11} - 159564504 T^{12} + 5764151836 T^{13} - 36979480986 T^{14} - 181717554252 T^{15} + 841105835367 T^{16} - 5633244181812 T^{17} - 35537281227546 T^{18} + 171719847346276 T^{19} - 147361170298584 T^{20} - 2928161192791359 T^{21} + 51134271858914402 T^{22} + 88097949056534323 T^{23} - 808428121877125788 T^{24} + 1483765156034695849 T^{25} - 18240007898470745454 T^{26} -$$$$22\!\cdots\!31$$$$T^{27} + 60650034351718331597 T^{28} +$$$$31\!\cdots\!83$$$$T^{29} +$$$$16\!\cdots\!57$$$$T^{31} +$$$$72\!\cdots\!81$$$$T^{32}$$
$37$ $$1 - 4 T - 64 T^{2} + 392 T^{3} + 2308 T^{4} - 10820 T^{5} - 47691 T^{6} - 31178 T^{7} + 2748163 T^{8} + 12045860 T^{9} - 124307808 T^{10} - 75116270 T^{11} + 4767089611 T^{12} - 456957494 T^{13} - 122520873497 T^{14} + 217199596514 T^{15} + 4709853143282 T^{16} + 8036385071018 T^{17} - 167731075817393 T^{18} - 23146267943582 T^{19} + 8934293432441371 T^{20} - 5208859396880390 T^{21} - 318939825830501472 T^{22} + 1143536101481319380 T^{23} + 9652866073525897123 T^{24} - 4051947123330910706 T^{25} -$$$$22\!\cdots\!59$$$$T^{26} -$$$$19\!\cdots\!60$$$$T^{27} +$$$$15\!\cdots\!48$$$$T^{28} +$$$$95\!\cdots\!24$$$$T^{29} -$$$$57\!\cdots\!96$$$$T^{30} -$$$$13\!\cdots\!72$$$$T^{31} +$$$$12\!\cdots\!41$$$$T^{32}$$
$41$ $$1 - 10 T + 32 T^{2} - 192 T^{3} + 5438 T^{4} - 48920 T^{5} + 140400 T^{6} - 769008 T^{7} + 14682039 T^{8} - 96831496 T^{9} + 182717296 T^{10} - 1985070190 T^{11} + 32206741548 T^{12} - 131499578440 T^{13} + 27334729480 T^{14} - 4391234846920 T^{15} + 62202943867805 T^{16} - 180040628723720 T^{17} + 45949680255880 T^{18} - 9063082445663240 T^{19} + 91008554203418028 T^{20} - 229982690931748190 T^{21} + 867926202633652336 T^{22} - 18858347692290955976 T^{23} +$$$$11\!\cdots\!19$$$$T^{24} -$$$$25\!\cdots\!88$$$$T^{25} +$$$$18\!\cdots\!00$$$$T^{26} -$$$$26\!\cdots\!20$$$$T^{27} +$$$$12\!\cdots\!78$$$$T^{28} -$$$$17\!\cdots\!32$$$$T^{29} +$$$$12\!\cdots\!52$$$$T^{30} -$$$$15\!\cdots\!10$$$$T^{31} +$$$$63\!\cdots\!41$$$$T^{32}$$
$43$ $$( 1 + 4 T + 239 T^{2} + 936 T^{3} + 27462 T^{4} + 98508 T^{5} + 2007760 T^{6} + 6271668 T^{7} + 102278657 T^{8} + 269681724 T^{9} + 3712348240 T^{10} + 7832075556 T^{11} + 93887113062 T^{12} + 137599902648 T^{13} + 1510805768711 T^{14} + 1087274444428 T^{15} + 11688200277601 T^{16} )^{2}$$
$47$ $$1 - 23 T + 186 T^{2} - 322 T^{3} - 4127 T^{4} + 30138 T^{5} - 70920 T^{6} - 1094344 T^{7} + 18527582 T^{8} - 119631636 T^{9} + 262096090 T^{10} + 2200503287 T^{11} - 30693701772 T^{12} + 88025997822 T^{13} + 1410058928022 T^{14} - 15969608141898 T^{15} + 104939392199145 T^{16} - 750571582669206 T^{17} + 3114820172000598 T^{18} + 9139123171873506 T^{19} - 149775473356494732 T^{20} + 504674441760538009 T^{21} + 2825190190998963610 T^{22} - 60608152736413767468 T^{23} +$$$$44\!\cdots\!02$$$$T^{24} -$$$$12\!\cdots\!48$$$$T^{25} -$$$$37\!\cdots\!80$$$$T^{26} +$$$$74\!\cdots\!14$$$$T^{27} -$$$$47\!\cdots\!07$$$$T^{28} -$$$$17\!\cdots\!94$$$$T^{29} +$$$$47\!\cdots\!34$$$$T^{30} -$$$$27\!\cdots\!89$$$$T^{31} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$1 + 4 T - 22 T^{2} + 440 T^{3} + 660 T^{4} - 49868 T^{5} + 140113 T^{6} + 2285596 T^{7} - 15873165 T^{8} + 76638860 T^{9} + 1323931162 T^{10} - 10376469922 T^{11} - 35381234389 T^{12} + 446229246850 T^{13} - 3132652820265 T^{14} - 9035271835264 T^{15} + 292112805840914 T^{16} - 478869407268992 T^{17} - 8799621772124385 T^{18} + 66433271583287450 T^{19} - 279174957702951109 T^{20} - 4339392954630461546 T^{21} + 29344088384504601898 T^{22} + 90028522586408265820 T^{23} -$$$$98\!\cdots\!65$$$$T^{24} +$$$$75\!\cdots\!68$$$$T^{25} +$$$$24\!\cdots\!37$$$$T^{26} -$$$$46\!\cdots\!96$$$$T^{27} +$$$$32\!\cdots\!60$$$$T^{28} +$$$$11\!\cdots\!20$$$$T^{29} -$$$$30\!\cdots\!18$$$$T^{30} +$$$$29\!\cdots\!28$$$$T^{31} +$$$$38\!\cdots\!21$$$$T^{32}$$
$59$ $$1 + 17 T - 50 T^{2} - 1985 T^{3} + 451 T^{4} + 76307 T^{5} - 426220 T^{6} + 1429235 T^{7} + 60743716 T^{8} - 309308083 T^{9} - 1984484046 T^{10} + 33011892753 T^{11} - 101353409674 T^{12} - 2740413120218 T^{13} + 8464635562844 T^{14} + 86256705224328 T^{15} - 317724345351369 T^{16} + 5089145608235352 T^{17} + 29465396394259964 T^{18} - 562823306217252622 T^{19} - 1228135853600750314 T^{20} + 23601004285101705147 T^{21} - 83706596062330791486 T^{22} -$$$$76\!\cdots\!77$$$$T^{23} +$$$$89\!\cdots\!36$$$$T^{24} +$$$$12\!\cdots\!65$$$$T^{25} -$$$$21\!\cdots\!20$$$$T^{26} +$$$$23\!\cdots\!13$$$$T^{27} +$$$$80\!\cdots\!31$$$$T^{28} -$$$$20\!\cdots\!15$$$$T^{29} -$$$$30\!\cdots\!50$$$$T^{30} +$$$$62\!\cdots\!83$$$$T^{31} +$$$$21\!\cdots\!41$$$$T^{32}$$
$61$ $$1 + 7 T - 20 T^{2} + 403 T^{3} + 10067 T^{4} + 48739 T^{5} - 116314 T^{6} + 1172749 T^{7} + 26082242 T^{8} + 24039263 T^{9} - 379016678 T^{10} - 5077729829 T^{11} - 113601259714 T^{12} - 664603591174 T^{13} + 819501759744 T^{14} - 46004665392052 T^{15} - 871290620425703 T^{16} - 2806284588915172 T^{17} + 3049366048007424 T^{18} - 150852387728265694 T^{19} - 1572904979399749474 T^{20} - 4288631831050762529 T^{21} - 19527081139622592758 T^{22} + 75549221576474692523 T^{23} +$$$$50\!\cdots\!02$$$$T^{24} +$$$$13\!\cdots\!09$$$$T^{25} -$$$$82\!\cdots\!14$$$$T^{26} +$$$$21\!\cdots\!79$$$$T^{27} +$$$$26\!\cdots\!07$$$$T^{28} +$$$$65\!\cdots\!43$$$$T^{29} -$$$$19\!\cdots\!20$$$$T^{30} +$$$$42\!\cdots\!07$$$$T^{31} +$$$$36\!\cdots\!61$$$$T^{32}$$
$67$ $$( 1 + 19 T + 520 T^{2} + 7751 T^{3} + 119005 T^{4} + 1415171 T^{5} + 15702408 T^{6} + 150444400 T^{7} + 1308159665 T^{8} + 10079774800 T^{9} + 70488109512 T^{10} + 425631075473 T^{11} + 2398084154605 T^{12} + 10464819704357 T^{13} + 47038358727880 T^{14} + 115153520501137 T^{15} + 406067677556641 T^{16} )^{2}$$
$71$ $$1 - 14 T + 140 T^{2} - 763 T^{3} + 10437 T^{4} - 25566 T^{5} + 32310 T^{6} + 3862236 T^{7} + 10586088 T^{8} + 330657408 T^{9} - 2081383949 T^{10} + 39603959100 T^{11} + 15217861524 T^{12} + 1721196695007 T^{13} + 1593568125167 T^{14} + 22773634065006 T^{15} + 1780879231377332 T^{16} + 1616928018615426 T^{17} + 8033176918966847 T^{18} + 616035230306650377 T^{19} + 386711442550061844 T^{20} + 71454625424023544100 T^{21} -$$$$26\!\cdots\!29$$$$T^{22} +$$$$30\!\cdots\!28$$$$T^{23} +$$$$68\!\cdots\!68$$$$T^{24} +$$$$17\!\cdots\!16$$$$T^{25} +$$$$10\!\cdots\!10$$$$T^{26} -$$$$59\!\cdots\!86$$$$T^{27} +$$$$17\!\cdots\!17$$$$T^{28} -$$$$88\!\cdots\!93$$$$T^{29} +$$$$11\!\cdots\!40$$$$T^{30} -$$$$82\!\cdots\!14$$$$T^{31} +$$$$41\!\cdots\!21$$$$T^{32}$$
$73$ $$1 + 35 T + 622 T^{2} + 8881 T^{3} + 123217 T^{4} + 1506367 T^{5} + 15880754 T^{6} + 159131205 T^{7} + 1535260024 T^{8} + 13342640169 T^{9} + 108277262748 T^{10} + 880406179959 T^{11} + 6960680213462 T^{12} + 52467783616198 T^{13} + 405906809006112 T^{14} + 3284917386947140 T^{15} + 27500270697009885 T^{16} + 239798969247141220 T^{17} + 2163077385193570848 T^{18} + 20410859779022497366 T^{19} +$$$$19\!\cdots\!42$$$$T^{20} +$$$$18\!\cdots\!87$$$$T^{21} +$$$$16\!\cdots\!72$$$$T^{22} +$$$$14\!\cdots\!93$$$$T^{23} +$$$$12\!\cdots\!44$$$$T^{24} +$$$$93\!\cdots\!65$$$$T^{25} +$$$$68\!\cdots\!46$$$$T^{26} +$$$$47\!\cdots\!59$$$$T^{27} +$$$$28\!\cdots\!57$$$$T^{28} +$$$$14\!\cdots\!73$$$$T^{29} +$$$$75\!\cdots\!98$$$$T^{30} +$$$$31\!\cdots\!95$$$$T^{31} +$$$$65\!\cdots\!61$$$$T^{32}$$
$79$ $$1 - 15 T + 94 T^{2} - 2495 T^{3} + 38855 T^{4} - 414785 T^{5} + 5676295 T^{6} - 62322940 T^{7} + 690670255 T^{8} - 8423294280 T^{9} + 83337602737 T^{10} - 856819337855 T^{11} + 8989269459608 T^{12} - 82967566755485 T^{13} + 808287092858990 T^{14} - 7591629044633355 T^{15} + 65763672844350810 T^{16} - 599738694526035045 T^{17} + 5044519746532956590 T^{18} - 40906246145557568915 T^{19} +$$$$35\!\cdots\!48$$$$T^{20} -$$$$26\!\cdots\!45$$$$T^{21} +$$$$20\!\cdots\!77$$$$T^{22} -$$$$16\!\cdots\!20$$$$T^{23} +$$$$10\!\cdots\!55$$$$T^{24} -$$$$74\!\cdots\!60$$$$T^{25} +$$$$53\!\cdots\!95$$$$T^{26} -$$$$31\!\cdots\!15$$$$T^{27} +$$$$22\!\cdots\!55$$$$T^{28} -$$$$11\!\cdots\!05$$$$T^{29} +$$$$34\!\cdots\!14$$$$T^{30} -$$$$43\!\cdots\!85$$$$T^{31} +$$$$23\!\cdots\!21$$$$T^{32}$$
$83$ $$1 + 5 T - 318 T^{2} - 3741 T^{3} + 48517 T^{4} + 994133 T^{5} - 3255786 T^{6} - 170309105 T^{7} - 263285806 T^{8} + 21750909411 T^{9} + 119250152558 T^{10} - 2097202277489 T^{11} - 20999477370338 T^{12} + 141328121959992 T^{13} + 2545394345691772 T^{14} - 4513257769614610 T^{15} - 237031046695965905 T^{16} - 374600394878012630 T^{17} + 17535221647470617308 T^{18} + 80809582871137945704 T^{19} -$$$$99\!\cdots\!98$$$$T^{20} -$$$$82\!\cdots\!27$$$$T^{21} +$$$$38\!\cdots\!02$$$$T^{22} +$$$$59\!\cdots\!97$$$$T^{23} -$$$$59\!\cdots\!46$$$$T^{24} -$$$$31\!\cdots\!15$$$$T^{25} -$$$$50\!\cdots\!14$$$$T^{26} +$$$$12\!\cdots\!11$$$$T^{27} +$$$$51\!\cdots\!37$$$$T^{28} -$$$$33\!\cdots\!83$$$$T^{29} -$$$$23\!\cdots\!22$$$$T^{30} +$$$$30\!\cdots\!35$$$$T^{31} +$$$$50\!\cdots\!81$$$$T^{32}$$
$89$ $$( 1 + 37 T + 1032 T^{2} + 20901 T^{3} + 357823 T^{4} + 5152639 T^{5} + 65606528 T^{6} + 731972999 T^{7} + 7339853952 T^{8} + 65145596911 T^{9} + 519669308288 T^{10} + 3632450763191 T^{11} + 22450616901343 T^{12} + 116712426543549 T^{13} + 512884692271752 T^{14} + 1636559391134573 T^{15} + 3936588805702081 T^{16} )^{2}$$
$97$ $$1 - 20 T + 123 T^{2} + 28 T^{3} + 4997 T^{4} - 239296 T^{5} + 2683386 T^{6} - 13918670 T^{7} + 12108314 T^{8} + 81926108 T^{9} + 9118270907 T^{10} - 123570735168 T^{11} + 816960697702 T^{12} + 3552108922056 T^{13} - 86512314985382 T^{14} - 43450963654390 T^{15} + 5424229096525225 T^{16} - 4214743474475830 T^{17} - 813994371697459238 T^{18} + 3241913906219615688 T^{19} + 72324943172816412262 T^{20} -$$$$10\!\cdots\!76$$$$T^{21} +$$$$75\!\cdots\!03$$$$T^{22} +$$$$66\!\cdots\!04$$$$T^{23} +$$$$94\!\cdots\!54$$$$T^{24} -$$$$10\!\cdots\!90$$$$T^{25} +$$$$19\!\cdots\!14$$$$T^{26} -$$$$17\!\cdots\!88$$$$T^{27} +$$$$34\!\cdots\!77$$$$T^{28} +$$$$18\!\cdots\!56$$$$T^{29} +$$$$80\!\cdots\!87$$$$T^{30} -$$$$12\!\cdots\!60$$$$T^{31} +$$$$61\!\cdots\!21$$$$T^{32}$$