Properties

 Label 1050.2.bc.h 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: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 4 x^{15} + 12 x^{14} - 48 x^{13} + 67 x^{12} - 24 x^{11} + 118 x^{10} - 176 x^{9} + 351 x^{8} - 180 x^{7} + 358 x^{6} - 336 x^{5} + 390 x^{4} - 344 x^{3} + 164 x^{2} - 40 x + 4$$ 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.

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

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
 −0.424637 − 3.22544i 0.117630 + 0.893490i 0.339278 − 0.0446668i 2.69978 − 0.355433i 0.792206 − 1.03242i −0.709944 + 0.925217i 0.277956 + 0.213283i −1.09227 − 0.838128i 0.792206 + 1.03242i −0.709944 − 0.925217i 0.277956 − 0.213283i −1.09227 + 0.838128i −0.424637 + 3.22544i 0.117630 − 0.893490i 0.339278 + 0.0446668i 2.69978 + 0.355433i
−0.258819 + 0.965926i −0.965926 + 0.258819i −0.866025 0.500000i 0 1.00000i −2.22701 1.42843i 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 2.55046 + 0.703686i 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 −2.52756 + 0.781940i −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 2.47207 + 0.942805i −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 −2.15951 1.52856i −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 1.38658 + 2.25331i −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 1.86367 1.87796i 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 2.64131 + 0.153213i 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 −2.15951 + 1.52856i −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 1.38658 2.25331i −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 1.86367 + 1.87796i 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 2.64131 0.153213i 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 −2.22701 + 1.42843i 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 2.55046 0.703686i 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 −2.52756 0.781940i −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 2.47207 0.942805i −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.h 16
5.b even 2 1 210.2.u.a 16
5.c odd 4 1 210.2.u.b yes 16
5.c odd 4 1 1050.2.bc.g 16
7.d odd 6 1 1050.2.bc.g 16
15.d odd 2 1 630.2.bv.a 16
15.e even 4 1 630.2.bv.b 16
35.i odd 6 1 210.2.u.b yes 16
35.i odd 6 1 1470.2.m.e 16
35.j even 6 1 1470.2.m.d 16
35.k even 12 1 210.2.u.a 16
35.k even 12 1 inner 1050.2.bc.h 16
35.k even 12 1 1470.2.m.d 16
35.l odd 12 1 1470.2.m.e 16
105.p even 6 1 630.2.bv.b 16
105.w odd 12 1 630.2.bv.a 16

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

$p$ $F_p(T)$
$2$ $$( 1 - T^{4} + T^{8} )^{2}$$
$3$ $$( 1 - T^{4} + T^{8} )^{2}$$
$5$ $$T^{16}$$
$7$ $$5764801 - 6588344 T + 705894 T^{2} + 2016840 T^{3} - 660275 T^{4} - 384160 T^{5} + 233730 T^{6} + 22512 T^{7} - 40872 T^{8} + 3216 T^{9} + 4770 T^{10} - 1120 T^{11} - 275 T^{12} + 120 T^{13} + 6 T^{14} - 8 T^{15} + T^{16}$$
$11$ $$146689 + 38300 T + 470366 T^{2} - 785088 T^{3} + 1355706 T^{4} - 1104284 T^{5} + 799216 T^{6} - 365284 T^{7} + 172687 T^{8} - 57924 T^{9} + 23088 T^{10} - 5596 T^{11} + 1530 T^{12} - 192 T^{13} + 46 T^{14} - 4 T^{15} + T^{16}$$
$13$ $$171295744 - 51933184 T + 7872512 T^{2} - 9624320 T^{3} + 18339968 T^{4} - 7707968 T^{5} + 1764384 T^{6} - 370872 T^{7} + 304225 T^{8} - 126936 T^{9} + 29856 T^{10} - 4424 T^{11} + 1442 T^{12} - 536 T^{13} + 128 T^{14} - 16 T^{15} + T^{16}$$
$17$ $$16384 - 1490944 T + 37621760 T^{2} - 67084288 T^{3} + 61214208 T^{4} - 44632064 T^{5} + 28418176 T^{6} - 13729280 T^{7} + 4810640 T^{8} - 1336352 T^{9} + 319232 T^{10} - 62912 T^{11} + 10108 T^{12} - 1376 T^{13} + 144 T^{14} - 12 T^{15} + T^{16}$$
$19$ $$5893325824 + 9877892096 T + 10678511360 T^{2} + 7275209472 T^{3} + 3759452256 T^{4} + 1413464896 T^{5} + 429754984 T^{6} + 101069264 T^{7} + 20868625 T^{8} + 3538824 T^{9} + 597618 T^{10} + 80816 T^{11} + 11427 T^{12} + 1056 T^{13} + 130 T^{14} + 8 T^{15} + T^{16}$$
$23$ $$82925569024 + 50009674752 T + 34217677056 T^{2} + 21335846912 T^{3} + 7816847584 T^{4} + 1530351552 T^{5} + 10601672 T^{6} - 91314384 T^{7} - 26490111 T^{8} - 2631728 T^{9} + 442600 T^{10} + 220608 T^{11} + 45359 T^{12} + 5968 T^{13} + 536 T^{14} + 32 T^{15} + T^{16}$$
$29$ $$939790336 + 2601994240 T^{2} + 1378025856 T^{4} + 231567136 T^{6} + 18367617 T^{8} + 781240 T^{10} + 18194 T^{12} + 216 T^{14} + T^{16}$$
$31$ $$85229824 - 4503148032 T + 78452558208 T^{2} + 45222688512 T^{3} - 6453955024 T^{4} - 8533768512 T^{5} + 2004625496 T^{6} + 324864744 T^{7} - 72535359 T^{8} - 8844528 T^{9} + 1687622 T^{10} + 207864 T^{11} - 15501 T^{12} - 2544 T^{13} + 86 T^{14} + 24 T^{15} + T^{16}$$
$37$ $$1586310022144 + 371962072064 T + 276663243008 T^{2} + 115153510912 T^{3} + 22979899616 T^{4} + 7448745856 T^{5} + 1118611080 T^{6} + 4832720 T^{7} + 22920081 T^{8} - 13233800 T^{9} + 236184 T^{10} + 27128 T^{11} + 9863 T^{12} + 1112 T^{13} - 184 T^{14} - 8 T^{15} + T^{16}$$
$41$ $$729780649984 + 406165422080 T^{2} + 84574220800 T^{4} + 8100985984 T^{6} + 363545169 T^{8} + 8080772 T^{10} + 90070 T^{12} + 484 T^{14} + T^{16}$$
$43$ $$396169216 + 132481024 T + 22151168 T^{2} - 916025344 T^{3} + 2977536512 T^{4} - 1507826688 T^{5} + 388311040 T^{6} - 22278400 T^{7} + 8241168 T^{8} - 3516480 T^{9} + 806528 T^{10} - 56352 T^{11} + 5416 T^{12} - 1424 T^{13} + 288 T^{14} - 24 T^{15} + T^{16}$$
$47$ $$405330862336 - 308243368960 T + 145954605440 T^{2} - 49680554368 T^{3} + 11779331376 T^{4} - 2116158368 T^{5} + 258739912 T^{6} - 3596552 T^{7} - 5248255 T^{8} + 1555792 T^{9} - 230680 T^{10} - 1112 T^{11} + 4855 T^{12} - 1088 T^{13} + 216 T^{14} - 24 T^{15} + T^{16}$$
$53$ $$7800599041 - 25641706004 T + 2714413304 T^{2} + 30185532368 T^{3} + 30887942350 T^{4} + 19048533764 T^{5} + 8141048288 T^{6} + 2567181404 T^{7} + 622643987 T^{8} + 115468220 T^{9} + 16843616 T^{10} + 2038756 T^{11} + 205902 T^{12} + 16560 T^{13} + 1016 T^{14} + 44 T^{15} + T^{16}$$
$59$ $$1714622464 - 3826761728 T + 9441092608 T^{2} - 1878220800 T^{3} + 5089812672 T^{4} - 2317818368 T^{5} + 1952636304 T^{6} - 387463008 T^{7} + 108046033 T^{8} - 7563320 T^{9} + 2484268 T^{10} - 160816 T^{11} + 33159 T^{12} - 1200 T^{13} + 236 T^{14} - 8 T^{15} + T^{16}$$
$61$ $$4676942565376 - 6948355203072 T + 3048997502976 T^{2} + 582336774144 T^{3} - 274666810624 T^{4} - 40806469632 T^{5} + 22755329536 T^{6} - 694005120 T^{7} - 475624368 T^{8} + 27730560 T^{9} + 8129776 T^{10} - 965664 T^{11} - 7140 T^{12} + 5088 T^{13} - 20 T^{14} - 24 T^{15} + T^{16}$$
$67$ $$13679819493376 + 19302438109184 T + 10857329494016 T^{2} + 2961349593088 T^{3} + 324288706304 T^{4} - 33718560768 T^{5} - 17908703360 T^{6} - 2914198016 T^{7} - 117502128 T^{8} + 50242080 T^{9} + 13764416 T^{10} + 2071968 T^{11} + 216652 T^{12} + 16592 T^{13} + 960 T^{14} + 36 T^{15} + T^{16}$$
$71$ $$( -17030912 - 5972992 T + 328960 T^{2} + 253696 T^{3} + 6240 T^{4} - 3520 T^{5} - 176 T^{6} + 16 T^{7} + T^{8} )^{2}$$
$73$ $$34279432458496 - 11509632167936 T + 4603114833920 T^{2} - 1601335368192 T^{3} + 319406715264 T^{4} - 67951064320 T^{5} + 9857732096 T^{6} - 326701568 T^{7} + 86731184 T^{8} + 27573568 T^{9} - 4010368 T^{10} - 102656 T^{11} + 20440 T^{12} - 2384 T^{13} + 512 T^{14} - 40 T^{15} + T^{16}$$
$79$ $$31950847504 + 80256422016 T + 85961473744 T^{2} + 47131588224 T^{3} + 11641609212 T^{4} - 436207536 T^{5} - 777926444 T^{6} - 27984120 T^{7} + 49267773 T^{8} + 6609828 T^{9} - 832994 T^{10} - 151488 T^{11} + 14399 T^{12} + 1944 T^{13} - 114 T^{14} - 12 T^{15} + T^{16}$$
$83$ $$8632854701584 + 15282725367680 T + 13527489036800 T^{2} + 6608651071808 T^{3} + 1960957443688 T^{4} + 320767035488 T^{5} + 24615111808 T^{6} - 317475888 T^{7} + 128886705 T^{8} + 45923808 T^{9} + 4262656 T^{10} - 437632 T^{11} + 27802 T^{12} + 1280 T^{13} + 128 T^{14} - 16 T^{15} + T^{16}$$
$89$ $$67108864 + 369098752 T + 2319450112 T^{2} - 1352663040 T^{3} + 1884684288 T^{4} + 295174144 T^{5} + 254042112 T^{6} + 40341504 T^{7} + 21479680 T^{8} + 3755008 T^{9} + 980224 T^{10} + 113408 T^{11} + 20208 T^{12} + 1920 T^{13} + 272 T^{14} + 16 T^{15} + T^{16}$$
$97$ $$67108864 - 301989888 T + 679477248 T^{2} - 313262080 T^{3} - 156729344 T^{4} + 386531328 T^{5} + 578642432 T^{6} + 309264096 T^{7} + 93564321 T^{8} + 15942868 T^{9} + 1838152 T^{10} + 261300 T^{11} + 69470 T^{12} + 10732 T^{13} + 968 T^{14} + 44 T^{15} + T^{16}$$