Properties

 Label 1050.2.bc.g Level 1050 Weight 2 Character orbit 1050.bc Analytic conductor 8.384 Analytic rank 0 Dimension 16 CM no Inner twists 2

Related objects

Newspace parameters

 Level: $$N$$ = $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1050.bc (of order $$12$$, degree $$4$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$8.38429221223$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 210) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{6} q^{2} -\beta_{12} q^{3} + ( -\beta_{5} + \beta_{13} ) q^{4} -\beta_{13} q^{6} + ( 2 + \beta_{2} - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} - 2 \beta_{12} + \beta_{14} - \beta_{15} ) q^{7} + ( -\beta_{2} + \beta_{15} ) q^{8} + \beta_{5} q^{9} +O(q^{10})$$ $$q -\beta_{6} q^{2} -\beta_{12} q^{3} + ( -\beta_{5} + \beta_{13} ) q^{4} -\beta_{13} q^{6} + ( 2 + \beta_{2} - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} - 2 \beta_{12} + \beta_{14} - \beta_{15} ) q^{7} + ( -\beta_{2} + \beta_{15} ) q^{8} + \beta_{5} q^{9} + ( 1 - 2 \beta_{1} - \beta_{3} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{10} - \beta_{11} - 2 \beta_{12} - 2 \beta_{15} ) q^{11} -\beta_{15} q^{12} + ( -1 + 2 \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{11} + \beta_{15} ) q^{13} + ( -2 + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{8} - \beta_{9} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{14} + ( 1 + \beta_{14} ) q^{16} + ( 1 - \beta_{1} - \beta_{3} + 2 \beta_{4} + 3 \beta_{5} - \beta_{7} - \beta_{8} + \beta_{11} - 2 \beta_{13} + 3 \beta_{14} ) q^{17} + \beta_{2} q^{18} + ( 2 + \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} - 3 \beta_{6} - 3 \beta_{8} + 2 \beta_{10} - 2 \beta_{11} - \beta_{12} + 4 \beta_{13} - \beta_{15} ) q^{19} + ( 2 - \beta_{1} + \beta_{4} + 2 \beta_{5} - \beta_{7} + \beta_{9} + \beta_{11} - \beta_{13} + 2 \beta_{14} ) q^{21} + ( -3 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{11} + \beta_{13} - 3 \beta_{14} ) q^{22} + ( 1 + \beta_{2} + 2 \beta_{3} + \beta_{4} + 3 \beta_{5} - 2 \beta_{6} - \beta_{8} - 2 \beta_{9} - \beta_{11} + \beta_{12} - 3 \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{23} -\beta_{14} q^{24} + ( -\beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} - \beta_{12} + \beta_{14} ) q^{26} + ( -\beta_{2} + \beta_{15} ) q^{27} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{10} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{28} + ( \beta_{2} - \beta_{4} - 2 \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - 5 \beta_{12} - \beta_{13} - \beta_{14} - 4 \beta_{15} ) q^{29} + ( -2 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} + 5 \beta_{5} - 5 \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{10} - \beta_{11} - 2 \beta_{12} - 4 \beta_{13} - 2 \beta_{14} + 3 \beta_{15} ) q^{31} + \beta_{12} q^{32} + ( -2 \beta_{1} - 2 \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{10} - 2 \beta_{13} + \beta_{14} ) q^{33} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{9} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{34} - q^{36} + ( 1 - 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 4 \beta_{4} - 5 \beta_{5} + 4 \beta_{7} - 3 \beta_{10} - 4 \beta_{11} - 4 \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{37} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + 4 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 3 \beta_{11} + 5 \beta_{12} - \beta_{13} + 2 \beta_{14} + 4 \beta_{15} ) q^{38} + ( 3 - \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{9} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{39} + ( 2 + \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - \beta_{4} + 3 \beta_{5} - 6 \beta_{6} - \beta_{7} - 4 \beta_{8} - 2 \beta_{9} + 4 \beta_{10} - \beta_{11} - 4 \beta_{12} + 2 \beta_{13} - \beta_{14} - 3 \beta_{15} ) q^{41} + ( \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} + \beta_{12} - \beta_{15} ) q^{42} + ( 1 + \beta_{3} + 3 \beta_{4} + \beta_{5} - 2 \beta_{7} + 3 \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} + 3 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{43} + ( 2 - \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{14} + \beta_{15} ) q^{44} + ( 1 + \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{5} - 3 \beta_{6} - \beta_{7} + 2 \beta_{10} + 2 \beta_{11} + \beta_{13} + \beta_{14} - 3 \beta_{15} ) q^{46} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + \beta_{10} + 2 \beta_{12} - \beta_{13} - \beta_{14} + 3 \beta_{15} ) q^{47} + ( -\beta_{6} - \beta_{12} ) q^{48} + ( -1 - 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + \beta_{7} + \beta_{8} + 3 \beta_{9} + \beta_{10} - 3 \beta_{12} - \beta_{13} - 4 \beta_{15} ) q^{49} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} + 3 \beta_{12} - 2 \beta_{13} + 4 \beta_{15} ) q^{51} + ( -1 + 2 \beta_{1} + 2 \beta_{3} + \beta_{5} - \beta_{9} + \beta_{10} + \beta_{12} ) q^{52} + ( 4 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 6 \beta_{6} - \beta_{9} + \beta_{10} + \beta_{11} - 3 \beta_{12} - 3 \beta_{13} + 4 \beta_{14} ) q^{53} + ( 1 + \beta_{14} ) q^{54} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{8} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} ) q^{56} + ( -2 \beta_{2} + \beta_{3} + 3 \beta_{4} - 4 \beta_{6} - \beta_{7} - 3 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - 3 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{57} + ( -1 - \beta_{1} + \beta_{3} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{10} - 3 \beta_{13} - 4 \beta_{14} - \beta_{15} ) q^{58} + ( -3 + \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 5 \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} - 3 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} + \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{59} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} + 2 \beta_{15} ) q^{61} + ( 3 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + \beta_{11} + 2 \beta_{13} + 4 \beta_{14} - 4 \beta_{15} ) q^{62} + ( -1 + \beta_{1} - \beta_{3} - \beta_{7} - \beta_{9} - \beta_{13} + 2 \beta_{15} ) q^{63} + \beta_{13} q^{64} + ( -1 + \beta_{1} + 2 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{8} - \beta_{9} - 2 \beta_{11} - \beta_{12} + \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{66} + ( 5 - \beta_{1} + 2 \beta_{2} + \beta_{3} + 7 \beta_{5} - \beta_{7} + \beta_{8} + \beta_{11} - 2 \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{67} + ( -2 - \beta_{1} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{68} + ( -1 + \beta_{2} + \beta_{4} - 2 \beta_{5} - 2 \beta_{8} - \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - 3 \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{69} + ( -2 - 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{4} - 2 \beta_{6} - 4 \beta_{7} + 2 \beta_{12} - 2 \beta_{15} ) q^{71} + \beta_{6} q^{72} + ( 4 - 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 6 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 4 \beta_{11} - 6 \beta_{12} + 9 \beta_{13} + 5 \beta_{14} ) q^{73} + ( -2 - \beta_{2} + 3 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - 5 \beta_{6} - 4 \beta_{7} - 3 \beta_{8} - 4 \beta_{9} - 3 \beta_{11} - \beta_{13} - 4 \beta_{14} - \beta_{15} ) q^{74} + ( 4 - 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 2 \beta_{9} - \beta_{11} - 2 \beta_{12} + 2 \beta_{13} + 3 \beta_{14} - \beta_{15} ) q^{76} + ( -6 + 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 9 \beta_{6} - 4 \beta_{8} - 4 \beta_{9} - \beta_{10} - 5 \beta_{11} - 4 \beta_{12} - 2 \beta_{13} - 5 \beta_{14} - 3 \beta_{15} ) q^{77} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{9} - 2 \beta_{10} + \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{78} + ( 3 - 2 \beta_{1} - 5 \beta_{2} + \beta_{3} + 2 \beta_{4} + 3 \beta_{5} - \beta_{6} - 3 \beta_{7} + \beta_{8} + 3 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} - 2 \beta_{12} + 4 \beta_{14} + 2 \beta_{15} ) q^{79} -\beta_{14} q^{81} + ( -1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} + 4 \beta_{8} + \beta_{9} + 2 \beta_{10} + 3 \beta_{11} + 3 \beta_{12} - \beta_{13} + 2 \beta_{15} ) q^{82} + ( 4 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + \beta_{7} - 3 \beta_{8} + 3 \beta_{9} - \beta_{10} + \beta_{11} - 3 \beta_{12} + 2 \beta_{13} + 6 \beta_{14} + \beta_{15} ) q^{83} + ( -1 - \beta_{3} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{14} ) q^{84} + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{86} + ( -2 + \beta_{2} + \beta_{4} + 4 \beta_{5} + \beta_{6} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{14} ) q^{87} + ( 1 - \beta_{1} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{13} + \beta_{14} ) q^{88} + ( 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{12} + 4 \beta_{13} + 2 \beta_{15} ) q^{89} + ( -2 - 3 \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + 4 \beta_{8} - 2 \beta_{10} - \beta_{11} - 4 \beta_{12} - 4 \beta_{13} - \beta_{14} - 3 \beta_{15} ) q^{91} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + 2 \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{92} + ( 1 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + 4 \beta_{6} + 2 \beta_{7} - \beta_{8} - 2 \beta_{10} + \beta_{11} + 3 \beta_{12} - 3 \beta_{13} - \beta_{14} + 4 \beta_{15} ) q^{93} + ( -2 \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - \beta_{11} - 2 \beta_{12} + \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{94} -\beta_{5} q^{96} + ( 1 + 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 4 \beta_{5} + 4 \beta_{6} - 3 \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} + 3 \beta_{11} + 3 \beta_{12} - 2 \beta_{14} + 5 \beta_{15} ) q^{97} + ( -6 + 3 \beta_{2} + \beta_{4} - 4 \beta_{5} + 2 \beta_{6} + \beta_{8} - \beta_{9} - 3 \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} - 5 \beta_{14} - \beta_{15} ) q^{98} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} - 2 \beta_{10} + \beta_{12} - \beta_{13} + \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 4q^{7} + O(q^{10})$$ $$16q + 4q^{7} + 4q^{11} - 16q^{13} - 16q^{14} + 8q^{16} - 12q^{17} + 8q^{19} + 8q^{21} - 4q^{22} + 40q^{23} + 8q^{24} - 12q^{26} + 4q^{28} - 24q^{31} - 4q^{33} - 16q^{34} - 16q^{36} + 8q^{37} + 20q^{38} + 12q^{39} - 8q^{42} + 24q^{43} - 4q^{46} - 52q^{49} + 8q^{51} - 8q^{52} + 28q^{53} + 8q^{54} + 8q^{56} + 8q^{57} + 12q^{58} - 8q^{59} + 24q^{61} + 8q^{62} + 4q^{63} + 84q^{67} - 12q^{68} + 8q^{69} - 32q^{71} - 16q^{73} + 24q^{74} - 44q^{77} + 16q^{78} - 12q^{79} + 8q^{81} - 36q^{82} - 16q^{83} - 4q^{84} - 8q^{86} - 48q^{87} + 4q^{88} + 16q^{89} + 8q^{91} - 8q^{92} + 32q^{93} - 8q^{94} + 44q^{97} - 24q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{15} + 12 x^{14} - 48 x^{13} + 67 x^{12} - 24 x^{11} + 118 x^{10} - 176 x^{9} + 351 x^{8} - 180 x^{7} + 358 x^{6} - 336 x^{5} + 390 x^{4} - 344 x^{3} + 164 x^{2} - 40 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$1434138866353 \nu^{15} - 74224360591111 \nu^{14} + 217326371958939 \nu^{13} - 648847305484636 \nu^{12} + 2680829374033629 \nu^{11} - 1607350163227359 \nu^{10} - 900494454550979 \nu^{9} - 8792384561334166 \nu^{8} + 3536477625795241 \nu^{7} - 17495309067416083 \nu^{6} - 6622116382602757 \nu^{5} - 26263502656698172 \nu^{4} - 5327429661771194 \nu^{3} - 19428846797347660 \nu^{2} + 4742345058394342 \nu + 4680253552809794$$$$)/ 5173992472766390$$ $$\beta_{2}$$ $$=$$ $$($$$$-123434729989149 \nu^{15} + 492304781090243 \nu^{14} - 1406992399278677 \nu^{13} + 5707540667520213 \nu^{12} - 7621279603788347 \nu^{11} + 281604145705947 \nu^{10} - 12957947975492223 \nu^{9} + 22625006932641203 \nu^{8} - 34533205664857133 \nu^{7} + 18681773772251579 \nu^{6} - 26694324268699259 \nu^{5} + 48096185658956821 \nu^{4} - 21876042039069938 \nu^{3} + 47788976778038450 \nu^{2} - 814448920872776 \nu + 195044141171618$$$$)/ 5173992472766390$$ $$\beta_{3}$$ $$=$$ $$($$$$-137809854283897 \nu^{15} + 666808683532056 \nu^{14} - 2027562782554100 \nu^{13} + 7694802116157034 \nu^{12} - 13885376435295071 \nu^{11} + 7288708077624688 \nu^{10} - 15122432396762870 \nu^{9} + 37940274269902486 \nu^{8} - 58228159495736079 \nu^{7} + 55051784324946342 \nu^{6} - 44775296775045806 \nu^{5} + 85474895099571132 \nu^{4} - 62657114582598388 \nu^{3} + 77038886286558388 \nu^{2} - 37418608795158304 \nu + 6950822659807744$$$$)/ 1034798494553278$$ $$\beta_{4}$$ $$=$$ $$($$$$3909951128714 \nu^{15} - 13126607646508 \nu^{14} + 38145193554947 \nu^{13} - 162143292202288 \nu^{12} + 154868117754112 \nu^{11} + 18595969360728 \nu^{10} + 464520812336313 \nu^{9} - 395501573216208 \nu^{8} + 1073827229563468 \nu^{7} + 11545660383276 \nu^{6} + 1336974192211299 \nu^{5} - 488205468817916 \nu^{4} + 1087029641011808 \nu^{3} - 622728293466680 \nu^{2} + 162118602709476 \nu - 16575048540328$$$$)/ 28585593772190$$ $$\beta_{5}$$ $$=$$ $$($$$$-4143762135082 \nu^{15} + 12665097411614 \nu^{14} - 36598537974476 \nu^{13} + 160755388928989 \nu^{12} - 115488770848206 \nu^{11} - 55417826512144 \nu^{10} - 507559901300404 \nu^{9} + 264781323438119 \nu^{8} - 1058958936197574 \nu^{7} - 327950045248708 \nu^{6} - 1495012504742632 \nu^{5} + 55329885176253 \nu^{4} - 1127861763864064 \nu^{3} + 338424533456400 \nu^{2} - 56848696686768 \nu + 3631882693804$$$$)/ 28585593772190$$ $$\beta_{6}$$ $$=$$ $$($$$$-1439070217869327 \nu^{15} + 5626426049162414 \nu^{14} - 16762149286150656 \nu^{13} + 67570735976932179 \nu^{12} - 90330349700000641 \nu^{11} + 26412913789425376 \nu^{10} - 167480504119187474 \nu^{9} + 237626790891811969 \nu^{8} - 482812364930441289 \nu^{7} + 215899963434715562 \nu^{6} - 494473747233645422 \nu^{5} + 437389184714757453 \nu^{4} - 517428552172387524 \nu^{3} + 446449271988400340 \nu^{2} - 192208665603330138 \nu + 35411484066617244$$$$)/ 5173992472766390$$ $$\beta_{7}$$ $$=$$ $$($$$$6100180981 \nu^{15} - 22259671750 \nu^{14} + 65510725326 \nu^{13} - 270219726184 \nu^{12} + 314950092835 \nu^{11} - 40514369326 \nu^{10} + 709289788986 \nu^{9} - 820031528032 \nu^{8} + 1863520118983 \nu^{7} - 458012963070 \nu^{6} + 2053184268060 \nu^{5} - 1324040796852 \nu^{4} + 1933409293532 \nu^{3} - 1441782411844 \nu^{2} + 520681411390 \nu - 77929949288$$$$)/ 18265555126$$ $$\beta_{8}$$ $$=$$ $$($$$$-6072104849199 \nu^{15} + 21780610045393 \nu^{14} - 63731636791632 \nu^{13} + 264606089742658 \nu^{12} - 295959094070367 \nu^{11} + 17152993349497 \nu^{10} - 701312474278478 \nu^{9} + 777595182487398 \nu^{8} - 1797695843305223 \nu^{7} + 332786417052279 \nu^{6} - 1990877398734524 \nu^{5} + 1207350248071826 \nu^{4} - 1824887125776248 \nu^{3} + 1306467148621370 \nu^{2} - 411232893748306 \nu + 65633529574178$$$$)/ 16530327389030$$ $$\beta_{9}$$ $$=$$ $$($$$$2035054902038187 \nu^{15} - 7578221328661024 \nu^{14} + 22354204007810251 \nu^{13} - 91579556495364929 \nu^{12} + 111269852929486391 \nu^{11} - 19058949252444566 \nu^{10} + 235208674633202009 \nu^{9} - 292701695999447459 \nu^{8} + 636133145212162719 \nu^{7} - 188961643415479902 \nu^{6} + 683022296693330837 \nu^{5} - 490835043167996523 \nu^{4} + 666044503208027234 \nu^{3} - 504441981276855820 \nu^{2} + 198734847296148758 \nu - 28123111426177514$$$$)/ 5173992472766390$$ $$\beta_{10}$$ $$=$$ $$($$$$30320067 \nu^{15} - 111020724 \nu^{14} + 326080511 \nu^{13} - 1344589864 \nu^{12} + 1575363211 \nu^{11} - 188871116 \nu^{10} + 3515753069 \nu^{9} - 4158557204 \nu^{8} + 9212296429 \nu^{7} - 2344575592 \nu^{6} + 10021703527 \nu^{5} - 6825888288 \nu^{4} + 9461086554 \nu^{3} - 7254602740 \nu^{2} + 2513736318 \nu - 330526984$$$$)/63923990$$ $$\beta_{11}$$ $$=$$ $$($$$$567889621787826 \nu^{15} - 2163110819836755 \nu^{14} + 6376823282252541 \nu^{13} - 25955396623934578 \nu^{12} + 32832120300673190 \nu^{11} - 6275666808832427 \nu^{10} + 64640506440221435 \nu^{9} - 87228624024570764 \nu^{8} + 179418986889806458 \nu^{7} - 65452477393295353 \nu^{6} + 182753898461327581 \nu^{5} - 154497881200594756 \nu^{4} + 181717723715357082 \nu^{3} - 157676074754561598 \nu^{2} + 53424782974620022 \nu - 7740432548067096$$$$)/ 1034798494553278$$ $$\beta_{12}$$ $$=$$ $$($$$$5281395788857158 \nu^{15} - 19095159515314451 \nu^{14} + 56052725193796619 \nu^{13} - 232054361068567251 \nu^{12} + 264944456866733364 \nu^{11} - 25993531049108449 \nu^{10} + 615506034577358971 \nu^{9} - 694365286936506611 \nu^{8} + 1588081348228614396 \nu^{7} - 344839767257990343 \nu^{6} + 1765864153109925593 \nu^{5} - 1105265950762762657 \nu^{4} + 1638034897211160076 \nu^{3} - 1197019598040248580 \nu^{2} + 413425841726845612 \nu - 60388611162910706$$$$)/ 5173992472766390$$ $$\beta_{13}$$ $$=$$ $$($$$$-19482487322 \nu^{15} + 71829768307 \nu^{14} - 211530176114 \nu^{13} + 869648666130 \nu^{12} - 1035106924390 \nu^{11} + 152629602893 \nu^{10} - 2258419134670 \nu^{9} + 2719627979686 \nu^{8} - 6018321521990 \nu^{7} + 1643327598977 \nu^{6} - 6516717498206 \nu^{5} + 4492931472132 \nu^{4} - 6274129258728 \nu^{3} + 4768566345236 \nu^{2} - 1753345508964 \nu + 258618081490$$$$)/ 18265555126$$ $$\beta_{14}$$ $$=$$ $$($$$$-82631746 \nu^{15} + 300206917 \nu^{14} - 880560228 \nu^{13} + 3640243297 \nu^{12} - 4191737118 \nu^{11} + 407798693 \nu^{10} - 9561674912 \nu^{9} + 11027434227 \nu^{8} - 24845185642 \nu^{7} + 5661417851 \nu^{6} - 27237589476 \nu^{5} + 17742563129 \nu^{4} - 25400492652 \nu^{3} + 18964234070 \nu^{2} - 6297003604 \nu + 791533522$$$$)/63923990$$ $$\beta_{15}$$ $$=$$ $$($$$$6956104161848134 \nu^{15} - 25667361154231553 \nu^{14} + 75476948193017452 \nu^{13} - 310349129733580443 \nu^{12} + 369420728986424472 \nu^{11} - 50736196725114107 \nu^{10} + 803087087863156658 \nu^{9} - 975207215422026373 \nu^{8} + 2135058242437940538 \nu^{7} - 584968875881439509 \nu^{6} + 2300137181821316574 \nu^{5} - 1620912261036340171 \nu^{4} + 2200212896950517088 \nu^{3} - 1699651856766889730 \nu^{2} + 607998971285547366 \nu - 83437215315298168$$$$)/ 5173992472766390$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{6} - \beta_{5} - \beta_{3}$$ $$\nu^{2}$$ $$=$$ $$\beta_{14} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + \beta_{6} + \beta_{5} + \beta_{4} + 3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{15} + 4 \beta_{14} + 2 \beta_{13} + 3 \beta_{12} + 2 \beta_{11} - 2 \beta_{8} + \beta_{6} - 5 \beta_{1} + 10$$ $$\nu^{4}$$ $$=$$ $$-2 \beta_{15} - 6 \beta_{14} - 11 \beta_{12} - 4 \beta_{11} - 4 \beta_{10} + 9 \beta_{9} - 10 \beta_{8} - 9 \beta_{7} - 3 \beta_{6} - 15 \beta_{5} + 9 \beta_{4} - 10 \beta_{3} - 20 \beta_{2} - 9 \beta_{1} + 17$$ $$\nu^{5}$$ $$=$$ $$-23 \beta_{15} - 16 \beta_{14} - 22 \beta_{13} - 21 \beta_{12} - 2 \beta_{11} + 21 \beta_{10} - 2 \beta_{9} + 5 \beta_{8} + 22 \beta_{6} - 2 \beta_{5} + 39 \beta_{4} - 21 \beta_{3} + 14 \beta_{2} + 7 \beta_{1} - 37$$ $$\nu^{6}$$ $$=$$ $$28 \beta_{15} + 154 \beta_{14} + 23 \beta_{13} + 140 \beta_{12} + 89 \beta_{11} + 95 \beta_{10} - 56 \beta_{9} + 12 \beta_{8} + 89 \beta_{7} + 84 \beta_{6} + 121 \beta_{5} - 12 \beta_{4} + 56 \beta_{3} + 173 \beta_{2} - 39 \beta_{1} + 93$$ $$\nu^{7}$$ $$=$$ $$163 \beta_{15} + 184 \beta_{14} + 240 \beta_{13} + 50 \beta_{12} - 94 \beta_{10} + 200 \beta_{9} - 351 \beta_{8} - 35 \beta_{7} - 219 \beta_{6} - 128 \beta_{5} - 94 \beta_{4} + 35 \beta_{3} - 440 \beta_{2} - 400 \beta_{1} + 985$$ $$\nu^{8}$$ $$=$$ $$-645 \beta_{15} - 1488 \beta_{14} - 329 \beta_{13} - 1794 \beta_{12} - 914 \beta_{11} - 242 \beta_{10} + 914 \beta_{9} - 672 \beta_{8} - 663 \beta_{7} - 645 \beta_{6} - 1243 \beta_{5} + 914 \beta_{4} - 914 \beta_{3} - 2063 \beta_{2} - 251 \beta_{1} + 9$$ $$\nu^{9}$$ $$=$$ $$-1828 \beta_{15} - 186 \beta_{14} - 1819 \beta_{13} - 195 \beta_{12} + 477 \beta_{11} + 3325 \beta_{10} - 1299 \beta_{9} + 1776 \beta_{8} + 1915 \beta_{7} + 2296 \beta_{6} + 2491 \beta_{5} + 1915 \beta_{4} + 4267 \beta_{2} + 1776 \beta_{1} - 5566$$ $$\nu^{10}$$ $$=$$ $$6400 \beta_{15} + 16412 \beta_{14} + 6539 \beta_{13} + 14863 \beta_{12} + 7481 \beta_{11} + 6042 \beta_{10} - 3465 \beta_{9} - 1493 \beta_{8} + 8974 \beta_{7} + 1665 \beta_{6} + 11899 \beta_{5} - 7481 \beta_{4} + 9507 \beta_{3} + 10946 \beta_{2} - 6042 \beta_{1} + 17408$$ $$\nu^{11}$$ $$=$$ $$8014 \beta_{15} - 12665 \beta_{14} + 18481 \beta_{13} - 24749 \beta_{12} - 18707 \beta_{11} - 21849 \beta_{10} + 32416 \beta_{9} - 37414 \beta_{8} - 15992 \beta_{7} - 38812 \beta_{6} - 32352 \beta_{5} - 5857 \beta_{4} - 5857 \beta_{3} - 77618 \beta_{2} - 32416 \beta_{1} + 77085$$ $$\nu^{12}$$ $$=$$ $$-91679 \beta_{15} - 170862 \beta_{14} - 70257 \beta_{13} - 181429 \beta_{12} - 89750 \beta_{11} + 7240 \beta_{10} + 55995 \beta_{9} - 7240 \beta_{8} - 43472 \beta_{7} - 31804 \beta_{6} - 82510 \beta_{5} + 99467 \beta_{4} - 82510 \beta_{3} - 121181 \beta_{2} + 43472 \beta_{1} - 166538$$ $$\nu^{13}$$ $$=$$ $$-55995 \beta_{15} + 241852 \beta_{14} - 92227 \beta_{13} + 278084 \beta_{12} + 185857 \beta_{11} + 372602 \beta_{10} - 253893 \beta_{9} + 253893 \beta_{8} + 322709 \beta_{7} + 281000 \beta_{6} + 467745 \beta_{5} + 186301 \beta_{3} + 711477 \beta_{2} + 185857 \beta_{1} - 469625$$ $$\nu^{14}$$ $$=$$ $$880388 \beta_{15} + 1459906 \beta_{14} + 949204 \beta_{13} + 1273161 \beta_{12} + 510702 \beta_{11} + 13265 \beta_{10} + 13265 \beta_{9} - 533645 \beta_{8} + 533645 \beta_{7} - 400341 \beta_{6} + 656706 \beta_{5} - 1044347 \beta_{4} + 911043 \beta_{3} + 57243 \beta_{2} - 911043 \beta_{1} + 2650257$$ $$\nu^{15}$$ $$=$$ $$-533645 \beta_{15} - 4295841 \beta_{14} + 533645 \beta_{13} - 5220149 \beta_{12} - 3264759 \beta_{11} - 2855121 \beta_{10} + 3769764 \beta_{9} - 3264759 \beta_{8} - 2855121 \beta_{7} - 4161149 \beta_{6} - 4943250 \beta_{5} + 765039 \beta_{4} - 2090082 \beta_{3} - 9354046 \beta_{2} - 1884882 \beta_{1} + 3840272$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$701$$ $$\chi(n)$$ $$-\beta_{13}$$ $$1 + \beta_{14}$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
157.1
 −1.09227 − 0.838128i 0.277956 + 0.213283i −0.709944 + 0.925217i 0.792206 − 1.03242i 0.339278 − 0.0446668i 2.69978 − 0.355433i −0.424637 − 3.22544i 0.117630 + 0.893490i 0.339278 + 0.0446668i 2.69978 + 0.355433i −0.424637 + 3.22544i 0.117630 − 0.893490i −1.09227 + 0.838128i 0.277956 − 0.213283i −0.709944 − 0.925217i 0.792206 + 1.03242i
−0.258819 + 0.965926i 0.965926 0.258819i −0.866025 0.500000i 0 1.00000i −0.153213 + 2.64131i 0.707107 0.707107i 0.866025 0.500000i 0
157.2 −0.258819 + 0.965926i 0.965926 0.258819i −0.866025 0.500000i 0 1.00000i 1.87796 + 1.86367i 0.707107 0.707107i 0.866025 0.500000i 0
157.3 0.258819 0.965926i −0.965926 + 0.258819i −0.866025 0.500000i 0 1.00000i −2.25331 + 1.38658i −0.707107 + 0.707107i 0.866025 0.500000i 0
157.4 0.258819 0.965926i −0.965926 + 0.258819i −0.866025 0.500000i 0 1.00000i 1.52856 2.15951i −0.707107 + 0.707107i 0.866025 0.500000i 0
493.1 −0.965926 0.258819i 0.258819 + 0.965926i 0.866025 + 0.500000i 0 1.00000i 0.781940 + 2.52756i −0.707107 0.707107i −0.866025 + 0.500000i 0
493.2 −0.965926 0.258819i 0.258819 + 0.965926i 0.866025 + 0.500000i 0 1.00000i 0.942805 2.47207i −0.707107 0.707107i −0.866025 + 0.500000i 0
493.3 0.965926 + 0.258819i −0.258819 0.965926i 0.866025 + 0.500000i 0 1.00000i −1.42843 + 2.22701i 0.707107 + 0.707107i −0.866025 + 0.500000i 0
493.4 0.965926 + 0.258819i −0.258819 0.965926i 0.866025 + 0.500000i 0 1.00000i 0.703686 2.55046i 0.707107 + 0.707107i −0.866025 + 0.500000i 0
607.1 −0.965926 + 0.258819i 0.258819 0.965926i 0.866025 0.500000i 0 1.00000i 0.781940 2.52756i −0.707107 + 0.707107i −0.866025 0.500000i 0
607.2 −0.965926 + 0.258819i 0.258819 0.965926i 0.866025 0.500000i 0 1.00000i 0.942805 + 2.47207i −0.707107 + 0.707107i −0.866025 0.500000i 0
607.3 0.965926 0.258819i −0.258819 + 0.965926i 0.866025 0.500000i 0 1.00000i −1.42843 2.22701i 0.707107 0.707107i −0.866025 0.500000i 0
607.4 0.965926 0.258819i −0.258819 + 0.965926i 0.866025 0.500000i 0 1.00000i 0.703686 + 2.55046i 0.707107 0.707107i −0.866025 0.500000i 0
943.1 −0.258819 0.965926i 0.965926 + 0.258819i −0.866025 + 0.500000i 0 1.00000i −0.153213 2.64131i 0.707107 + 0.707107i 0.866025 + 0.500000i 0
943.2 −0.258819 0.965926i 0.965926 + 0.258819i −0.866025 + 0.500000i 0 1.00000i 1.87796 1.86367i 0.707107 + 0.707107i 0.866025 + 0.500000i 0
943.3 0.258819 + 0.965926i −0.965926 0.258819i −0.866025 + 0.500000i 0 1.00000i −2.25331 1.38658i −0.707107 0.707107i 0.866025 + 0.500000i 0
943.4 0.258819 + 0.965926i −0.965926 0.258819i −0.866025 + 0.500000i 0 1.00000i 1.52856 + 2.15951i −0.707107 0.707107i 0.866025 + 0.500000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 943.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.k even 12 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.bc.g 16
5.b even 2 1 210.2.u.b yes 16
5.c odd 4 1 210.2.u.a 16
5.c odd 4 1 1050.2.bc.h 16
7.d odd 6 1 1050.2.bc.h 16
15.d odd 2 1 630.2.bv.b 16
15.e even 4 1 630.2.bv.a 16
35.i odd 6 1 210.2.u.a 16
35.i odd 6 1 1470.2.m.d 16
35.j even 6 1 1470.2.m.e 16
35.k even 12 1 210.2.u.b yes 16
35.k even 12 1 inner 1050.2.bc.g 16
35.k even 12 1 1470.2.m.e 16
35.l odd 12 1 1470.2.m.d 16
105.p even 6 1 630.2.bv.a 16
105.w odd 12 1 630.2.bv.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.u.a 16 5.c odd 4 1
210.2.u.a 16 35.i odd 6 1
210.2.u.b yes 16 5.b even 2 1
210.2.u.b yes 16 35.k even 12 1
630.2.bv.a 16 15.e even 4 1
630.2.bv.a 16 105.p even 6 1
630.2.bv.b 16 15.d odd 2 1
630.2.bv.b 16 105.w odd 12 1
1050.2.bc.g 16 1.a even 1 1 trivial
1050.2.bc.g 16 35.k even 12 1 inner
1050.2.bc.h 16 5.c odd 4 1
1050.2.bc.h 16 7.d odd 6 1
1470.2.m.d 16 35.i odd 6 1
1470.2.m.d 16 35.l odd 12 1
1470.2.m.e 16 35.j even 6 1
1470.2.m.e 16 35.k even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1050, [\chi])$$:

 $$T_{11}^{16} - \cdots$$ $$T_{13}^{16} + \cdots$$ $$T_{17}^{16} + \cdots$$

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{4} + T^{8} )^{2}$$
$3$ $$( 1 - T^{4} + T^{8} )^{2}$$
$5$ 1
$7$ $$1 - 4 T + 34 T^{2} - 80 T^{3} + 445 T^{4} - 704 T^{5} + 3926 T^{6} - 5044 T^{7} + 29688 T^{8} - 35308 T^{9} + 192374 T^{10} - 241472 T^{11} + 1068445 T^{12} - 1344560 T^{13} + 4000066 T^{14} - 3294172 T^{15} + 5764801 T^{16}$$
$11$ $$1 - 4 T - 42 T^{2} + 72 T^{3} + 1354 T^{4} - 52 T^{5} - 25972 T^{6} - 39092 T^{7} + 336125 T^{8} + 1014028 T^{9} - 2401652 T^{10} - 15137644 T^{11} - 1718882 T^{12} + 141984000 T^{13} + 309093362 T^{14} - 600162532 T^{15} - 4715588236 T^{16} - 6601787852 T^{17} + 37400296802 T^{18} + 188980704000 T^{19} - 25166151362 T^{20} - 2437932703844 T^{21} - 4254673018772 T^{22} + 19760537034788 T^{23} + 72051378876125 T^{24} - 92176891136572 T^{25} - 673646791737172 T^{26} - 14836206871772 T^{27} + 4249432022080234 T^{28} + 2485635274363032 T^{29} - 15949493010496122 T^{30} - 16708992677662604 T^{31} + 45949729863572161 T^{32}$$
$13$ $$1 + 16 T + 128 T^{2} + 744 T^{3} + 3314 T^{4} + 10768 T^{5} + 24864 T^{6} + 28864 T^{7} - 4655 T^{8} + 572216 T^{9} + 6801728 T^{10} + 43610328 T^{11} + 198012786 T^{12} + 641713232 T^{13} + 1520861600 T^{14} + 2393411960 T^{15} + 3297590084 T^{16} + 31114355480 T^{17} + 257025610400 T^{18} + 1409843970704 T^{19} + 5655443180946 T^{20} + 16192209514104 T^{21} + 32830641925952 T^{22} + 35905705403672 T^{23} - 3797226506255 T^{24} + 306088269902272 T^{25} + 3427713541333536 T^{26} + 19297983122990416 T^{27} + 77209854095902034 T^{28} + 225339079304636232 T^{29} + 503984177369508992 T^{30} + 818974288225452112 T^{31} + 665416609183179841 T^{32}$$
$17$ $$1 + 12 T - 760 T^{3} - 4000 T^{4} + 9644 T^{5} + 164912 T^{6} + 417780 T^{7} - 2033646 T^{8} - 15406336 T^{9} - 21694928 T^{10} + 141136884 T^{11} + 768607008 T^{12} + 1405601268 T^{13} - 1667112976 T^{14} - 24205669888 T^{15} - 123458993805 T^{16} - 411496388096 T^{17} - 481795650064 T^{18} + 6905719029684 T^{19} + 64194825915168 T^{20} + 200394192705588 T^{21} - 523662821550032 T^{22} - 6321815470032128 T^{23} - 14186221216859886 T^{24} + 49543643042916660 T^{25} + 332461586110845488 T^{26} + 330518167990812652 T^{27} - 2330488948919044000 T^{28} - 7527479305008512120 T^{29} + 34349076618117789516 T^{31} + 48661191875666868481 T^{32}$$
$19$ $$1 - 8 T - 22 T^{2} - 144 T^{3} + 3371 T^{4} + 4000 T^{5} - 7722 T^{6} - 652040 T^{7} - 370487 T^{8} + 2996624 T^{9} + 89400308 T^{10} + 59920656 T^{11} - 371517522 T^{12} - 10373644960 T^{13} - 8723924296 T^{14} + 39539357168 T^{15} + 977996314130 T^{16} + 751247786192 T^{17} - 3149336670856 T^{18} - 71152830780640 T^{19} - 48416534984562 T^{20} + 148369476400944 T^{21} + 4205916251531348 T^{22} + 2678597506009136 T^{23} - 6292189320370967 T^{24} - 210405286459819160 T^{25} - 47344093642739322 T^{26} + 465961035592876000 T^{27} + 7461084592172028731 T^{28} - 6055629618565016496 T^{29} - 17578147087223450662 T^{30} -$$$$12\!\cdots\!92$$$$T^{31} +$$$$28\!\cdots\!81$$$$T^{32}$$
$23$ $$1 - 40 T + 776 T^{2} - 9784 T^{3} + 91799 T^{4} - 705368 T^{5} + 4787160 T^{6} - 29946728 T^{7} + 172895985 T^{8} - 902021104 T^{9} + 4178360768 T^{10} - 17091640336 T^{11} + 61289409778 T^{12} - 182133009408 T^{13} + 342869112800 T^{14} + 372347426016 T^{15} - 5577275049586 T^{16} + 8563990798368 T^{17} + 181377760671200 T^{18} - 2216012325467136 T^{19} + 17151289721685298 T^{20} - 110007659635131248 T^{21} + 618547350853602752 T^{22} - 3071224408630233488 T^{23} + 13539654936478996785 T^{24} - 53938628839308543064 T^{25} +$$$$19\!\cdots\!40$$$$T^{26} -$$$$67\!\cdots\!36$$$$T^{27} +$$$$20\!\cdots\!79$$$$T^{28} -$$$$49\!\cdots\!72$$$$T^{29} +$$$$89\!\cdots\!84$$$$T^{30} -$$$$10\!\cdots\!80$$$$T^{31} +$$$$61\!\cdots\!61$$$$T^{32}$$
$29$ $$1 - 248 T^{2} + 31418 T^{4} - 2677416 T^{6} + 171374865 T^{8} - 8752284864 T^{10} + 370628380738 T^{12} - 13351487849248 T^{14} + 415501762604420 T^{16} - 11228601281217568 T^{18} + 262138411756753378 T^{20} - 5206063149142513344 T^{22} + 85729661487925625265 T^{24} -$$$$11\!\cdots\!16$$$$T^{26} +$$$$11\!\cdots\!38$$$$T^{28} -$$$$73\!\cdots\!88$$$$T^{30} +$$$$25\!\cdots\!21$$$$T^{32}$$
$31$ $$1 + 24 T + 334 T^{2} + 3408 T^{3} + 27899 T^{4} + 199680 T^{5} + 1273586 T^{6} + 7589688 T^{7} + 43727785 T^{8} + 250780944 T^{9} + 1474786764 T^{10} + 8526426768 T^{11} + 48339918958 T^{12} + 259935770400 T^{13} + 1349761682520 T^{14} + 7075762017840 T^{15} + 37942723845666 T^{16} + 219348622553040 T^{17} + 1297120976901720 T^{18} + 7743746535986400 T^{19} + 44642930296011118 T^{20} + 244104359431513968 T^{21} + 1308878681740078284 T^{22} + 6899639338664300784 T^{23} + 37295035913646998185 T^{24} +$$$$20\!\cdots\!48$$$$T^{25} +$$$$10\!\cdots\!86$$$$T^{26} +$$$$50\!\cdots\!80$$$$T^{27} +$$$$21\!\cdots\!39$$$$T^{28} +$$$$83\!\cdots\!28$$$$T^{29} +$$$$25\!\cdots\!14$$$$T^{30} +$$$$56\!\cdots\!24$$$$T^{31} +$$$$72\!\cdots\!81$$$$T^{32}$$
$37$ $$1 - 8 T + 248 T^{2} - 1752 T^{3} + 30199 T^{4} - 206976 T^{5} + 2540200 T^{6} - 17745904 T^{7} + 169033089 T^{8} - 1209867128 T^{9} + 9485737632 T^{10} - 68571970520 T^{11} + 465855584354 T^{12} - 3322795052152 T^{13} + 20345020109280 T^{14} - 139940970795928 T^{15} + 794780046531614 T^{16} - 5177815919449336 T^{17} + 27852332529604320 T^{18} - 168309537776655256 T^{19} + 873088367828476994 T^{20} - 4755051775144147640 T^{21} + 24337807551267523488 T^{22} -$$$$11\!\cdots\!24$$$$T^{23} +$$$$59\!\cdots\!69$$$$T^{24} -$$$$23\!\cdots\!08$$$$T^{25} +$$$$12\!\cdots\!00$$$$T^{26} -$$$$36\!\cdots\!88$$$$T^{27} +$$$$19\!\cdots\!19$$$$T^{28} -$$$$42\!\cdots\!44$$$$T^{29} +$$$$22\!\cdots\!72$$$$T^{30} -$$$$26\!\cdots\!44$$$$T^{31} +$$$$12\!\cdots\!41$$$$T^{32}$$
$41$ $$1 - 172 T^{2} + 13974 T^{4} - 791464 T^{6} + 43993793 T^{8} - 2592908552 T^{10} + 134337801798 T^{12} - 5711936189732 T^{14} + 227906401150852 T^{16} - 9601764734939492 T^{18} + 379606521146518278 T^{20} - 12316585909380369032 T^{22} +$$$$35\!\cdots\!53$$$$T^{24} -$$$$10\!\cdots\!64$$$$T^{26} +$$$$31\!\cdots\!94$$$$T^{28} -$$$$65\!\cdots\!92$$$$T^{30} +$$$$63\!\cdots\!41$$$$T^{32}$$
$43$ $$1 - 24 T + 288 T^{2} - 2456 T^{3} + 18144 T^{4} - 133752 T^{5} + 1000544 T^{6} - 6828856 T^{7} + 39963644 T^{8} - 227672888 T^{9} + 1517218592 T^{10} - 11117129912 T^{11} + 81274307360 T^{12} - 589957036568 T^{13} + 4310182316384 T^{14} - 31623166948824 T^{15} + 218467749390534 T^{16} - 1359796178799432 T^{17} + 7969527102994016 T^{18} - 46905714106411976 T^{19} + 277860683276675360 T^{20} - 1634311958991847016 T^{21} + 9590889544724607008 T^{22} - 61885728202879567016 T^{23} +$$$$46\!\cdots\!44$$$$T^{24} -$$$$34\!\cdots\!08$$$$T^{25} +$$$$21\!\cdots\!56$$$$T^{26} -$$$$12\!\cdots\!64$$$$T^{27} +$$$$72\!\cdots\!44$$$$T^{28} -$$$$42\!\cdots\!08$$$$T^{29} +$$$$21\!\cdots\!12$$$$T^{30} -$$$$76\!\cdots\!68$$$$T^{31} +$$$$13\!\cdots\!01$$$$T^{32}$$
$47$ $$1 + 72 T^{2} - 616 T^{3} + 3239 T^{4} - 50872 T^{5} + 298520 T^{6} - 2523744 T^{7} + 15099153 T^{8} - 113209408 T^{9} + 814540096 T^{10} - 3210267840 T^{11} + 24971489874 T^{12} - 159883078560 T^{13} + 379015320032 T^{14} - 3041257505584 T^{15} + 39450787699278 T^{16} - 142939102762448 T^{17} + 837244841950688 T^{18} - 16599540865334880 T^{19} + 121852904679850194 T^{20} - 736258900236674880 T^{21} + 8780103088888331584 T^{22} - 57354503546728915904 T^{23} +$$$$35\!\cdots\!33$$$$T^{24} -$$$$28\!\cdots\!48$$$$T^{25} +$$$$15\!\cdots\!80$$$$T^{26} -$$$$12\!\cdots\!16$$$$T^{27} +$$$$37\!\cdots\!99$$$$T^{28} -$$$$33\!\cdots\!32$$$$T^{29} +$$$$18\!\cdots\!68$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$1 - 28 T + 344 T^{2} - 2912 T^{3} + 25298 T^{4} - 232644 T^{5} + 1917568 T^{6} - 14158268 T^{7} + 104375121 T^{8} - 770687892 T^{9} + 5270701040 T^{10} - 39047642852 T^{11} + 318443399442 T^{12} - 2209044273928 T^{13} + 14320519756920 T^{14} - 111315852463604 T^{15} + 873920983850628 T^{16} - 5899740180571012 T^{17} + 40226339997188280 T^{18} - 328875884369578856 T^{19} + 2512671592872511602 T^{20} - 16329548252980066036 T^{21} +$$$$11\!\cdots\!60$$$$T^{22} -$$$$90\!\cdots\!04$$$$T^{23} +$$$$64\!\cdots\!81$$$$T^{24} -$$$$46\!\cdots\!44$$$$T^{25} +$$$$33\!\cdots\!32$$$$T^{26} -$$$$21\!\cdots\!68$$$$T^{27} +$$$$12\!\cdots\!18$$$$T^{28} -$$$$75\!\cdots\!76$$$$T^{29} +$$$$47\!\cdots\!36$$$$T^{30} -$$$$20\!\cdots\!96$$$$T^{31} +$$$$38\!\cdots\!21$$$$T^{32}$$
$59$ $$1 + 8 T - 236 T^{2} - 1632 T^{3} + 30563 T^{4} + 163648 T^{5} - 2682716 T^{6} - 9836488 T^{7} + 178469613 T^{8} + 332320192 T^{9} - 10946443072 T^{10} - 3817987168 T^{11} + 756189977838 T^{12} - 175777223888 T^{13} - 56160273686696 T^{14} + 6297587928448 T^{15} + 3675723794639094 T^{16} + 371557687778432 T^{17} - 195493912703388776 T^{18} - 36100950464893552 T^{19} + 9163026946045045518 T^{20} - 2729571799673395232 T^{21} -$$$$46\!\cdots\!52$$$$T^{22} +$$$$82\!\cdots\!48$$$$T^{23} +$$$$26\!\cdots\!73$$$$T^{24} -$$$$85\!\cdots\!32$$$$T^{25} -$$$$13\!\cdots\!16$$$$T^{26} +$$$$49\!\cdots\!32$$$$T^{27} +$$$$54\!\cdots\!03$$$$T^{28} -$$$$17\!\cdots\!28$$$$T^{29} -$$$$14\!\cdots\!96$$$$T^{30} +$$$$29\!\cdots\!92$$$$T^{31} +$$$$21\!\cdots\!41$$$$T^{32}$$
$61$ $$1 - 24 T + 468 T^{2} - 6624 T^{3} + 79480 T^{4} - 829512 T^{5} + 7576872 T^{6} - 61686168 T^{7} + 432435906 T^{8} - 2460066384 T^{9} + 8493304812 T^{10} + 36652670904 T^{11} - 1139116635712 T^{12} + 15272951044440 T^{13} - 160222534806948 T^{14} + 1465410452960880 T^{15} - 12014486118540493 T^{16} + 89390037630613680 T^{17} - 596188052016653508 T^{18} + 3466669701018035640 T^{19} - 15772027818523273792 T^{20} + 30956710267288726104 T^{21} +$$$$43\!\cdots\!32$$$$T^{22} -$$$$77\!\cdots\!64$$$$T^{23} +$$$$82\!\cdots\!86$$$$T^{24} -$$$$72\!\cdots\!88$$$$T^{25} +$$$$54\!\cdots\!72$$$$T^{26} -$$$$36\!\cdots\!32$$$$T^{27} +$$$$21\!\cdots\!80$$$$T^{28} -$$$$10\!\cdots\!44$$$$T^{29} +$$$$46\!\cdots\!88$$$$T^{30} -$$$$14\!\cdots\!24$$$$T^{31} +$$$$36\!\cdots\!61$$$$T^{32}$$
$67$ $$1 - 84 T + 3216 T^{2} - 71776 T^{3} + 957024 T^{4} - 5741436 T^{5} - 40997200 T^{6} + 1158667636 T^{7} - 7828301806 T^{8} - 48210145144 T^{9} + 1314996417536 T^{10} - 8000602781236 T^{11} - 45952509042976 T^{12} + 1085658642853028 T^{13} - 5561139754106368 T^{14} - 34677137718483048 T^{15} + 641528682100762995 T^{16} - 2323368227138364216 T^{17} - 24963956356183485952 T^{18} +$$$$32\!\cdots\!64$$$$T^{19} -$$$$92\!\cdots\!96$$$$T^{20} -$$$$10\!\cdots\!52$$$$T^{21} +$$$$11\!\cdots\!84$$$$T^{22} -$$$$29\!\cdots\!12$$$$T^{23} -$$$$31\!\cdots\!46$$$$T^{24} +$$$$31\!\cdots\!92$$$$T^{25} -$$$$74\!\cdots\!00$$$$T^{26} -$$$$70\!\cdots\!88$$$$T^{27} +$$$$78\!\cdots\!64$$$$T^{28} -$$$$39\!\cdots\!12$$$$T^{29} +$$$$11\!\cdots\!64$$$$T^{30} -$$$$20\!\cdots\!12$$$$T^{31} +$$$$16\!\cdots\!81$$$$T^{32}$$
$71$ $$( 1 + 16 T + 392 T^{2} + 4432 T^{3} + 72412 T^{4} + 697872 T^{5} + 8835896 T^{6} + 71051216 T^{7} + 737387398 T^{8} + 5044636336 T^{9} + 44541751736 T^{10} + 249776065392 T^{11} + 1840110644572 T^{12} + 7996344483632 T^{13} + 50215311297032 T^{14} + 145521922534256 T^{15} + 645753531245761 T^{16} )^{2}$$
$73$ $$1 + 16 T + 416 T^{2} + 4440 T^{3} + 68404 T^{4} + 584824 T^{5} + 7229216 T^{6} + 53644504 T^{7} + 544685226 T^{8} + 2736110216 T^{9} + 18336160640 T^{10} - 72833715600 T^{11} - 1347659375280 T^{12} - 28574375923760 T^{13} - 284065502536896 T^{14} - 3320804755103832 T^{15} - 26931854637922413 T^{16} - 242418747122579736 T^{17} - 1513785063019118784 T^{18} - 11115917998733343920 T^{19} - 38271155725110882480 T^{20} -$$$$15\!\cdots\!00$$$$T^{21} +$$$$27\!\cdots\!60$$$$T^{22} +$$$$30\!\cdots\!52$$$$T^{23} +$$$$43\!\cdots\!06$$$$T^{24} +$$$$31\!\cdots\!52$$$$T^{25} +$$$$31\!\cdots\!84$$$$T^{26} +$$$$18\!\cdots\!48$$$$T^{27} +$$$$15\!\cdots\!84$$$$T^{28} +$$$$74\!\cdots\!20$$$$T^{29} +$$$$50\!\cdots\!44$$$$T^{30} +$$$$14\!\cdots\!12$$$$T^{31} +$$$$65\!\cdots\!61$$$$T^{32}$$
$79$ $$1 + 12 T + 518 T^{2} + 5640 T^{3} + 137639 T^{4} + 1484376 T^{5} + 26147402 T^{6} + 279843540 T^{7} + 3950719429 T^{8} + 41115674928 T^{9} + 499791767516 T^{10} + 4982987471808 T^{11} + 54520483633690 T^{12} + 515559074594184 T^{13} + 5198103978036912 T^{14} + 46390749479828928 T^{15} + 436708099187124858 T^{16} + 3664869208906485312 T^{17} + 32441366926928367792 T^{18} +$$$$25\!\cdots\!76$$$$T^{19} +$$$$21\!\cdots\!90$$$$T^{20} +$$$$15\!\cdots\!92$$$$T^{21} +$$$$12\!\cdots\!36$$$$T^{22} +$$$$78\!\cdots\!52$$$$T^{23} +$$$$59\!\cdots\!69$$$$T^{24} +$$$$33\!\cdots\!60$$$$T^{25} +$$$$24\!\cdots\!02$$$$T^{26} +$$$$11\!\cdots\!04$$$$T^{27} +$$$$81\!\cdots\!99$$$$T^{28} +$$$$26\!\cdots\!60$$$$T^{29} +$$$$19\!\cdots\!58$$$$T^{30} +$$$$34\!\cdots\!88$$$$T^{31} +$$$$23\!\cdots\!21$$$$T^{32}$$
$83$ $$1 + 16 T + 128 T^{2} + 48 T^{3} - 16022 T^{4} - 55056 T^{5} + 1171072 T^{6} + 29664624 T^{7} + 191234977 T^{8} - 1519529248 T^{9} - 27088145920 T^{10} - 210706506976 T^{11} + 1209312464722 T^{12} + 24316714882016 T^{13} + 89859490185984 T^{14} - 1324590918805248 T^{15} - 28693493560058396 T^{16} - 109941046260835584 T^{17} + 619042027891243776 T^{18} + 13903981452243282592 T^{19} + 57391939140077851762 T^{20} -$$$$82\!\cdots\!68$$$$T^{21} -$$$$88\!\cdots\!80$$$$T^{22} -$$$$41\!\cdots\!96$$$$T^{23} +$$$$43\!\cdots\!57$$$$T^{24} +$$$$55\!\cdots\!72$$$$T^{25} +$$$$18\!\cdots\!28$$$$T^{26} -$$$$70\!\cdots\!52$$$$T^{27} -$$$$17\!\cdots\!42$$$$T^{28} +$$$$42\!\cdots\!24$$$$T^{29} +$$$$94\!\cdots\!12$$$$T^{30} +$$$$97\!\cdots\!12$$$$T^{31} +$$$$50\!\cdots\!81$$$$T^{32}$$
$89$ $$1 - 16 T - 440 T^{2} + 6624 T^{3} + 131636 T^{4} - 1629968 T^{5} - 29449232 T^{6} + 280328720 T^{7} + 5329291050 T^{8} - 36595065152 T^{9} - 800017753480 T^{10} + 3635275214288 T^{11} + 102057968083920 T^{12} - 258547916350832 T^{13} - 11209896420041672 T^{14} + 8808353629704256 T^{15} + 1069443829226298003 T^{16} + 783943473043678784 T^{17} - 88793589543150083912 T^{18} -$$$$18\!\cdots\!08$$$$T^{19} +$$$$64\!\cdots\!20$$$$T^{20} +$$$$20\!\cdots\!12$$$$T^{21} -$$$$39\!\cdots\!80$$$$T^{22} -$$$$16\!\cdots\!08$$$$T^{23} +$$$$20\!\cdots\!50$$$$T^{24} +$$$$98\!\cdots\!80$$$$T^{25} -$$$$91\!\cdots\!32$$$$T^{26} -$$$$45\!\cdots\!52$$$$T^{27} +$$$$32\!\cdots\!56$$$$T^{28} +$$$$14\!\cdots\!56$$$$T^{29} -$$$$86\!\cdots\!40$$$$T^{30} -$$$$27\!\cdots\!84$$$$T^{31} +$$$$15\!\cdots\!61$$$$T^{32}$$
$97$ $$1 - 44 T + 968 T^{2} - 15000 T^{3} + 231654 T^{4} - 3860388 T^{5} + 58116000 T^{6} - 743419588 T^{7} + 9111376513 T^{8} - 116138978204 T^{9} + 1430090287664 T^{10} - 15964265880436 T^{11} + 171574947065342 T^{12} - 1888109943974960 T^{13} + 20331243756190952 T^{14} - 203613182755137748 T^{15} + 1985416913924716484 T^{16} - 19750478727248361556 T^{17} +$$$$19\!\cdots\!68$$$$T^{18} -$$$$17\!\cdots\!80$$$$T^{19} +$$$$15\!\cdots\!02$$$$T^{20} -$$$$13\!\cdots\!52$$$$T^{21} +$$$$11\!\cdots\!56$$$$T^{22} -$$$$93\!\cdots\!52$$$$T^{23} +$$$$71\!\cdots\!93$$$$T^{24} -$$$$56\!\cdots\!96$$$$T^{25} +$$$$42\!\cdots\!00$$$$T^{26} -$$$$27\!\cdots\!64$$$$T^{27} +$$$$16\!\cdots\!14$$$$T^{28} -$$$$10\!\cdots\!00$$$$T^{29} +$$$$63\!\cdots\!92$$$$T^{30} -$$$$27\!\cdots\!92$$$$T^{31} +$$$$61\!\cdots\!21$$$$T^{32}$$