# Properties

 Label 6042.2.a.bd Level 6042 Weight 2 Character orbit 6042.a Self dual yes Analytic conductor 48.246 Analytic rank 1 Dimension 11 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6042 = 2 \cdot 3 \cdot 19 \cdot 53$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6042.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$48.2456129013$$ Analytic rank: $$1$$ Dimension: $$11$$ Coefficient field: $$\mathbb{Q}[x]/(x^{11} - \cdots)$$ Defining polynomial: $$x^{11} - 2 x^{10} - 35 x^{9} + 77 x^{8} + 394 x^{7} - 994 x^{6} - 1477 x^{5} + 4683 x^{4} + 563 x^{3} - 6194 x^{2} + 1258 x + 1534$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{10}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + \beta_{1} q^{5} + q^{6} + \beta_{5} q^{7} - q^{8} + q^{9} +O(q^{10})$$ $$q - q^{2} - q^{3} + q^{4} + \beta_{1} q^{5} + q^{6} + \beta_{5} q^{7} - q^{8} + q^{9} -\beta_{1} q^{10} + ( -1 - \beta_{9} ) q^{11} - q^{12} + ( -1 + \beta_{8} ) q^{13} -\beta_{5} q^{14} -\beta_{1} q^{15} + q^{16} + ( -1 + \beta_{3} - \beta_{6} ) q^{17} - q^{18} + q^{19} + \beta_{1} q^{20} -\beta_{5} q^{21} + ( 1 + \beta_{9} ) q^{22} + ( -1 + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} ) q^{23} + q^{24} + ( 2 - \beta_{1} - \beta_{3} + \beta_{9} - \beta_{10} ) q^{25} + ( 1 - \beta_{8} ) q^{26} - q^{27} + \beta_{5} q^{28} + ( \beta_{2} - \beta_{5} + \beta_{7} - \beta_{10} ) q^{29} + \beta_{1} q^{30} + ( 2 - \beta_{4} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{31} - q^{32} + ( 1 + \beta_{9} ) q^{33} + ( 1 - \beta_{3} + \beta_{6} ) q^{34} + ( -2 - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{35} + q^{36} + ( -\beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{10} ) q^{37} - q^{38} + ( 1 - \beta_{8} ) q^{39} -\beta_{1} q^{40} + ( 1 - \beta_{1} - \beta_{3} - \beta_{5} + \beta_{9} ) q^{41} + \beta_{5} q^{42} + ( -\beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{9} ) q^{43} + ( -1 - \beta_{9} ) q^{44} + \beta_{1} q^{45} + ( 1 - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} ) q^{46} + ( -3 - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{47} - q^{48} + ( 3 - \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} ) q^{49} + ( -2 + \beta_{1} + \beta_{3} - \beta_{9} + \beta_{10} ) q^{50} + ( 1 - \beta_{3} + \beta_{6} ) q^{51} + ( -1 + \beta_{8} ) q^{52} + q^{53} + q^{54} + ( -2 \beta_{1} + \beta_{2} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} ) q^{55} -\beta_{5} q^{56} - q^{57} + ( -\beta_{2} + \beta_{5} - \beta_{7} + \beta_{10} ) q^{58} + ( -1 + \beta_{2} - \beta_{3} - \beta_{5} + \beta_{7} - \beta_{8} ) q^{59} -\beta_{1} q^{60} + ( 2 - \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} ) q^{61} + ( -2 + \beta_{4} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{62} + \beta_{5} q^{63} + q^{64} + ( 1 + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{65} + ( -1 - \beta_{9} ) q^{66} + ( -3 - 2 \beta_{1} - \beta_{2} - 2 \beta_{5} ) q^{67} + ( -1 + \beta_{3} - \beta_{6} ) q^{68} + ( 1 - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} ) q^{69} + ( 2 + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{70} + ( 1 - 2 \beta_{1} - \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{9} ) q^{71} - q^{72} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{9} - \beta_{10} ) q^{73} + ( \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{10} ) q^{74} + ( -2 + \beta_{1} + \beta_{3} - \beta_{9} + \beta_{10} ) q^{75} + q^{76} + ( -1 - \beta_{2} + 2 \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{7} + \beta_{9} + \beta_{10} ) q^{77} + ( -1 + \beta_{8} ) q^{78} + ( -\beta_{1} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} ) q^{79} + \beta_{1} q^{80} + q^{81} + ( -1 + \beta_{1} + \beta_{3} + \beta_{5} - \beta_{9} ) q^{82} + ( -1 + \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{83} -\beta_{5} q^{84} + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{85} + ( \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{9} ) q^{86} + ( -\beta_{2} + \beta_{5} - \beta_{7} + \beta_{10} ) q^{87} + ( 1 + \beta_{9} ) q^{88} + ( -2 - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} ) q^{89} -\beta_{1} q^{90} + ( -3 + \beta_{2} + 2 \beta_{4} + \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} ) q^{91} + ( -1 + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} ) q^{92} + ( -2 + \beta_{4} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{93} + ( 3 + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{94} + \beta_{1} q^{95} + q^{96} + ( -1 - 2 \beta_{1} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} ) q^{97} + ( -3 + \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} ) q^{98} + ( -1 - \beta_{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$11q - 11q^{2} - 11q^{3} + 11q^{4} + 2q^{5} + 11q^{6} - 2q^{7} - 11q^{8} + 11q^{9} + O(q^{10})$$ $$11q - 11q^{2} - 11q^{3} + 11q^{4} + 2q^{5} + 11q^{6} - 2q^{7} - 11q^{8} + 11q^{9} - 2q^{10} - 6q^{11} - 11q^{12} - 7q^{13} + 2q^{14} - 2q^{15} + 11q^{16} - 16q^{17} - 11q^{18} + 11q^{19} + 2q^{20} + 2q^{21} + 6q^{22} - 11q^{23} + 11q^{24} + 19q^{25} + 7q^{26} - 11q^{27} - 2q^{28} - 7q^{29} + 2q^{30} + 12q^{31} - 11q^{32} + 6q^{33} + 16q^{34} - 5q^{35} + 11q^{36} + 3q^{37} - 11q^{38} + 7q^{39} - 2q^{40} + 11q^{41} - 2q^{42} + 7q^{43} - 6q^{44} + 2q^{45} + 11q^{46} - 37q^{47} - 11q^{48} + 23q^{49} - 19q^{50} + 16q^{51} - 7q^{52} + 11q^{53} + 11q^{54} - 11q^{55} + 2q^{56} - 11q^{57} + 7q^{58} - 16q^{59} - 2q^{60} + 24q^{61} - 12q^{62} - 2q^{63} + 11q^{64} - 7q^{65} - 6q^{66} - 28q^{67} - 16q^{68} + 11q^{69} + 5q^{70} + 4q^{71} - 11q^{72} - 7q^{73} - 3q^{74} - 19q^{75} + 11q^{76} - 13q^{77} - 7q^{78} + 5q^{79} + 2q^{80} + 11q^{81} - 11q^{82} - 11q^{83} + 2q^{84} - 11q^{85} - 7q^{86} + 7q^{87} + 6q^{88} - 10q^{89} - 2q^{90} - 24q^{91} - 11q^{92} - 12q^{93} + 37q^{94} + 2q^{95} + 11q^{96} - 24q^{97} - 23q^{98} - 6q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{11} - 2 x^{10} - 35 x^{9} + 77 x^{8} + 394 x^{7} - 994 x^{6} - 1477 x^{5} + 4683 x^{4} + 563 x^{3} - 6194 x^{2} + 1258 x + 1534$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-61365 \nu^{10} - 19979 \nu^{9} + 2101724 \nu^{8} + 162971 \nu^{7} - 23813959 \nu^{6} + 5613147 \nu^{5} + 103864180 \nu^{4} - 45917643 \nu^{3} - 141974290 \nu^{2} + 50557192 \nu + 40606678$$$$)/34480$$ $$\beta_{3}$$ $$=$$ $$($$$$-84083 \nu^{10} - 26993 \nu^{9} + 2879648 \nu^{8} + 209481 \nu^{7} - 32622537 \nu^{6} + 7851333 \nu^{5} + 142207024 \nu^{4} - 63537677 \nu^{3} - 194086526 \nu^{2} + 69706224 \nu + 55455562$$$$)/34480$$ $$\beta_{4}$$ $$=$$ $$($$$$-77556 \nu^{10} - 25031 \nu^{9} + 2656416 \nu^{8} + 198432 \nu^{7} - 30099019 \nu^{6} + 7177316 \nu^{5} + 131247773 \nu^{4} - 58352059 \nu^{3} - 179277372 \nu^{2} + 64132198 \nu + 51286814$$$$)/17240$$ $$\beta_{5}$$ $$=$$ $$($$$$-50505 \nu^{10} - 16163 \nu^{9} + 1730043 \nu^{8} + 124862 \nu^{7} - 19605108 \nu^{6} + 4717169 \nu^{5} + 85498490 \nu^{4} - 38143726 \nu^{3} - 116783530 \nu^{2} + 41822964 \nu + 33382236$$$$)/8620$$ $$\beta_{6}$$ $$=$$ $$($$$$210451 \nu^{10} + 67875 \nu^{9} - 7208340 \nu^{8} - 537653 \nu^{7} + 81674783 \nu^{6} - 19471453 \nu^{5} - 356110518 \nu^{4} + 158228347 \nu^{3} + 486159382 \nu^{2} - 173556980 \nu - 138742022$$$$)/34480$$ $$\beta_{7}$$ $$=$$ $$($$$$192480 \nu^{10} + 61607 \nu^{9} - 6593832 \nu^{8} - 477308 \nu^{7} + 74728307 \nu^{6} - 17946496 \nu^{5} - 325910105 \nu^{4} + 145209819 \nu^{3} + 445157380 \nu^{2} - 159257926 \nu - 127192774$$$$)/17240$$ $$\beta_{8}$$ $$=$$ $$($$$$107137 \nu^{10} + 34407 \nu^{9} - 3669944 \nu^{8} - 269099 \nu^{7} + 41587847 \nu^{6} - 9957687 \nu^{5} - 181362420 \nu^{4} + 80709251 \nu^{3} + 247718074 \nu^{2} - 88503368 \nu - 70791910$$$$)/6896$$ $$\beta_{9}$$ $$=$$ $$($$$$647795 \nu^{10} + 207847 \nu^{9} - 22190632 \nu^{8} - 1620833 \nu^{7} + 251475487 \nu^{6} - 60275181 \nu^{5} - 1096742850 \nu^{4} + 488288139 \nu^{3} + 1498189590 \nu^{2} - 535480556 \nu - 428293574$$$$)/34480$$ $$\beta_{10}$$ $$=$$ $$($$$$365939 \nu^{10} + 117420 \nu^{9} - 12535140 \nu^{8} - 915157 \nu^{7} + 142049012 \nu^{6} - 34063257 \nu^{5} - 619474937 \nu^{4} + 275912908 \nu^{3} + 846120818 \nu^{2} - 302610630 \nu - 241753888$$$$)/17240$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{10} + \beta_{9} - \beta_{3} - \beta_{1} + 7$$ $$\nu^{3}$$ $$=$$ $$\beta_{10} - \beta_{9} + \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{3} - \beta_{2} + 11 \beta_{1} - 4$$ $$\nu^{4}$$ $$=$$ $$-14 \beta_{10} + 16 \beta_{9} - 3 \beta_{8} + \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} - 17 \beta_{3} - 2 \beta_{2} - 17 \beta_{1} + 80$$ $$\nu^{5}$$ $$=$$ $$19 \beta_{10} - 17 \beta_{9} + 5 \beta_{8} + 17 \beta_{7} - 23 \beta_{6} + 48 \beta_{5} + 6 \beta_{4} - 26 \beta_{3} - 19 \beta_{2} + 143 \beta_{1} - 73$$ $$\nu^{6}$$ $$=$$ $$-201 \beta_{10} + 245 \beta_{9} - 64 \beta_{8} + 20 \beta_{7} + 38 \beta_{6} + 39 \beta_{5} + 72 \beta_{4} - 268 \beta_{3} - 57 \beta_{2} - 253 \beta_{1} + 1076$$ $$\nu^{7}$$ $$=$$ $$304 \beta_{10} - 218 \beta_{9} + 86 \beta_{8} + 257 \beta_{7} - 406 \beta_{6} + 896 \beta_{5} + 137 \beta_{4} - 516 \beta_{3} - 294 \beta_{2} + 2027 \beta_{1} - 997$$ $$\nu^{8}$$ $$=$$ $$-3020 \beta_{10} + 3714 \beta_{9} - 1050 \beta_{8} + 372 \beta_{7} + 593 \beta_{6} + 607 \beta_{5} + 1353 \beta_{4} - 4251 \beta_{3} - 1204 \beta_{2} - 3619 \beta_{1} + 15505$$ $$\nu^{9}$$ $$=$$ $$4602 \beta_{10} - 2352 \beta_{9} + 1003 \beta_{8} + 3834 \beta_{7} - 6611 \beta_{6} + 15577 \beta_{5} + 2417 \beta_{4} - 9378 \beta_{3} - 4348 \beta_{2} + 29997 \beta_{1} - 11958$$ $$\nu^{10}$$ $$=$$ $$-46515 \beta_{10} + 56273 \beta_{9} - 15844 \beta_{8} + 6913 \beta_{7} + 8667 \beta_{6} + 9242 \beta_{5} + 23602 \beta_{4} - 67999 \beta_{3} - 22857 \beta_{2} - 50937 \beta_{1} + 230908$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −3.92287 −3.61806 −2.56652 −1.23681 −0.425735 0.864610 1.67126 2.32043 2.32145 2.52811 4.06414
−1.00000 −1.00000 1.00000 −3.92287 1.00000 −2.55808 −1.00000 1.00000 3.92287
1.2 −1.00000 −1.00000 1.00000 −3.61806 1.00000 4.03393 −1.00000 1.00000 3.61806
1.3 −1.00000 −1.00000 1.00000 −2.56652 1.00000 −0.973492 −1.00000 1.00000 2.56652
1.4 −1.00000 −1.00000 1.00000 −1.23681 1.00000 2.50061 −1.00000 1.00000 1.23681
1.5 −1.00000 −1.00000 1.00000 −0.425735 1.00000 −2.24729 −1.00000 1.00000 0.425735
1.6 −1.00000 −1.00000 1.00000 0.864610 1.00000 2.90002 −1.00000 1.00000 −0.864610
1.7 −1.00000 −1.00000 1.00000 1.67126 1.00000 −4.35376 −1.00000 1.00000 −1.67126
1.8 −1.00000 −1.00000 1.00000 2.32043 1.00000 −0.928685 −1.00000 1.00000 −2.32043
1.9 −1.00000 −1.00000 1.00000 2.32145 1.00000 −4.55788 −1.00000 1.00000 −2.32145
1.10 −1.00000 −1.00000 1.00000 2.52811 1.00000 0.198222 −1.00000 1.00000 −2.52811
1.11 −1.00000 −1.00000 1.00000 4.06414 1.00000 3.98638 −1.00000 1.00000 −4.06414
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.11 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6042.2.a.bd 11

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.bd 11 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$19$$ $$-1$$
$$53$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6042))$$:

 $$T_{5}^{11} - \cdots$$ $$T_{7}^{11} + \cdots$$ $$T_{11}^{11} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{11}$$
$3$ $$( 1 + T )^{11}$$
$5$ $$1 - 2 T + 20 T^{2} - 23 T^{3} + 194 T^{4} - 164 T^{5} + 1438 T^{6} - 1237 T^{7} + 9238 T^{8} - 8784 T^{9} + 51703 T^{10} - 49206 T^{11} + 258515 T^{12} - 219600 T^{13} + 1154750 T^{14} - 773125 T^{15} + 4493750 T^{16} - 2562500 T^{17} + 15156250 T^{18} - 8984375 T^{19} + 39062500 T^{20} - 19531250 T^{21} + 48828125 T^{22}$$
$7$ $$1 + 2 T + 29 T^{2} + 54 T^{3} + 475 T^{4} + 940 T^{5} + 5943 T^{6} + 11549 T^{7} + 59778 T^{8} + 109110 T^{9} + 504906 T^{10} + 846114 T^{11} + 3534342 T^{12} + 5346390 T^{13} + 20503854 T^{14} + 27729149 T^{15} + 99884001 T^{16} + 110590060 T^{17} + 391182925 T^{18} + 311299254 T^{19} + 1170254603 T^{20} + 564950498 T^{21} + 1977326743 T^{22}$$
$11$ $$1 + 6 T + 60 T^{2} + 219 T^{3} + 1401 T^{4} + 3360 T^{5} + 18260 T^{6} + 14422 T^{7} + 107359 T^{8} - 386534 T^{9} - 114283 T^{10} - 7259618 T^{11} - 1257113 T^{12} - 46770614 T^{13} + 142894829 T^{14} + 211152502 T^{15} + 2940791260 T^{16} + 5952444960 T^{17} + 27301526571 T^{18} + 46944594939 T^{19} + 141476861460 T^{20} + 155624547606 T^{21} + 285311670611 T^{22}$$
$13$ $$1 + 7 T + 78 T^{2} + 393 T^{3} + 2835 T^{4} + 12608 T^{5} + 73565 T^{6} + 297340 T^{7} + 1467640 T^{8} + 5385045 T^{9} + 23374393 T^{10} + 77946622 T^{11} + 303867109 T^{12} + 910072605 T^{13} + 3224405080 T^{14} + 8492327740 T^{15} + 27314169545 T^{16} + 60856407872 T^{17} + 177892045695 T^{18} + 320582173353 T^{19} + 827150951094 T^{20} + 965009442943 T^{21} + 1792160394037 T^{22}$$
$17$ $$1 + 16 T + 183 T^{2} + 1334 T^{3} + 7533 T^{4} + 29240 T^{5} + 84251 T^{6} + 169239 T^{7} + 941322 T^{8} + 8789472 T^{9} + 64280790 T^{10} + 311087510 T^{11} + 1092773430 T^{12} + 2540157408 T^{13} + 4624714986 T^{14} + 14135010519 T^{15} + 119624372107 T^{16} + 705782517560 T^{17} + 3091081223709 T^{18} + 9305660426294 T^{19} + 21701581398951 T^{20} + 32255902407184 T^{21} + 34271896307633 T^{22}$$
$19$ $$( 1 - T )^{11}$$
$23$ $$1 + 11 T + 194 T^{2} + 1716 T^{3} + 17508 T^{4} + 130270 T^{5} + 989465 T^{6} + 6361430 T^{7} + 39447060 T^{8} + 222510119 T^{9} + 1178474742 T^{10} + 5856349228 T^{11} + 27104919066 T^{12} + 117707852951 T^{13} + 479952379020 T^{14} + 1780188932630 T^{15} + 6368536126495 T^{16} + 19284635260030 T^{17} + 59611683926076 T^{18} + 134381650742196 T^{19} + 349423616323822 T^{20} + 455691623350139 T^{21} + 952809757913927 T^{22}$$
$29$ $$1 + 7 T + 163 T^{2} + 1160 T^{3} + 14128 T^{4} + 96020 T^{5} + 832765 T^{6} + 5264185 T^{7} + 37177784 T^{8} + 214662965 T^{9} + 1322471407 T^{10} + 6929329166 T^{11} + 38351670803 T^{12} + 180531553565 T^{13} + 906728973976 T^{14} + 3723258030985 T^{15} + 17080966996985 T^{16} + 57114935282420 T^{17} + 243706252493552 T^{18} + 580285839034760 T^{19} + 2364664794066647 T^{20} + 2944950633101407 T^{21} + 12200509765705829 T^{22}$$
$31$ $$1 - 12 T + 207 T^{2} - 1927 T^{3} + 20712 T^{4} - 167111 T^{5} + 1390266 T^{6} - 10015042 T^{7} + 69660249 T^{8} - 450767853 T^{9} + 2718340377 T^{10} - 15758444366 T^{11} + 84268551687 T^{12} - 433187906733 T^{13} + 2075248477959 T^{14} - 9249101602882 T^{15} + 39802135244166 T^{16} - 148311627635591 T^{17} + 569841263467032 T^{18} - 1643521029148807 T^{19} + 5473001787258897 T^{20} - 9835539443769612 T^{21} + 25408476896404831 T^{22}$$
$37$ $$1 - 3 T + 98 T^{2} - 283 T^{3} + 6431 T^{4} - 9606 T^{5} + 319403 T^{6} - 268054 T^{7} + 15530426 T^{8} - 14199723 T^{9} + 679980329 T^{10} - 789107974 T^{11} + 25159272173 T^{12} - 19439420787 T^{13} + 786662668178 T^{14} - 502376352694 T^{15} + 22148667897671 T^{16} - 24646367884854 T^{17} + 610506901842323 T^{18} - 994031685459643 T^{19} + 12736250499917546 T^{20} - 14425753117253547 T^{21} + 177917621779460413 T^{22}$$
$41$ $$1 - 11 T + 311 T^{2} - 2705 T^{3} + 45635 T^{4} - 343947 T^{5} + 4370628 T^{6} - 29198142 T^{7} + 303092829 T^{8} - 1804913570 T^{9} + 16003133244 T^{10} - 84591371122 T^{11} + 656128463004 T^{12} - 3034059711170 T^{13} + 20889460867509 T^{14} - 82506970936062 T^{15} + 506364356064228 T^{16} - 1633784103379227 T^{17} + 8887611288559435 T^{18} - 21599222744772305 T^{19} + 101815781596521871 T^{20} - 147649252411676411 T^{21} + 550329031716248441 T^{22}$$
$43$ $$1 - 7 T + 254 T^{2} - 1828 T^{3} + 28016 T^{4} - 189850 T^{5} + 1708741 T^{6} - 9323822 T^{7} + 60537376 T^{8} - 170629807 T^{9} + 1406280154 T^{10} + 57487268 T^{11} + 60470046622 T^{12} - 315494513143 T^{13} + 4813145153632 T^{14} - 31876291977422 T^{15} + 251199353900263 T^{16} - 1200110774852650 T^{17} + 7615270208773712 T^{18} - 21366030107454628 T^{19} + 127658523431958122 T^{20} - 151280376192989743 T^{21} + 929293739471222707 T^{22}$$
$47$ $$1 + 37 T + 961 T^{2} + 18079 T^{3} + 283801 T^{4} + 3757391 T^{5} + 43918102 T^{6} + 455166870 T^{7} + 4274135363 T^{8} + 36384289722 T^{9} + 284133579722 T^{10} + 2029934153562 T^{11} + 13354278246934 T^{12} + 80372895995898 T^{13} + 443753555792749 T^{14} + 2221069127368470 T^{15} + 10072397410616714 T^{16} + 40501726664246639 T^{17} + 143780148210519863 T^{18} + 430484251557977119 T^{19} + 1075484384651759087 T^{20} + 1946167892725711813 T^{21} + 2472159215084012303 T^{22}$$
$53$ $$( 1 - T )^{11}$$
$59$ $$1 + 16 T + 506 T^{2} + 6032 T^{3} + 103517 T^{4} + 951448 T^{5} + 11403524 T^{6} + 81585588 T^{7} + 768371536 T^{8} + 4414384312 T^{9} + 39019869652 T^{10} + 220404658440 T^{11} + 2302172309468 T^{12} + 15366471790072 T^{13} + 157807377692144 T^{14} + 988602022193268 T^{15} + 8152656401829676 T^{16} + 40132584371662168 T^{17} + 257617735754008423 T^{18} + 885681199629264272 T^{19} + 4383475884239399134 T^{20} + 8177868052810262416 T^{21} + 30155888444737842659 T^{22}$$
$61$ $$1 - 24 T + 661 T^{2} - 10794 T^{3} + 178276 T^{4} - 2258942 T^{5} + 28235013 T^{6} - 296228966 T^{7} + 3054104252 T^{8} - 27508310034 T^{9} + 243772105121 T^{10} - 1918589268944 T^{11} + 14870098412381 T^{12} - 102358421636514 T^{13} + 693223637223212 T^{14} - 4101539162830406 T^{15} + 23847187538486913 T^{16} - 116381537499786062 T^{17} + 560275621834479796 T^{18} - 2069288736492651114 T^{19} + 7729830567363367201 T^{20} - 17120229879909182424 T^{21} + 43513917611435838661 T^{22}$$
$67$ $$1 + 28 T + 766 T^{2} + 12685 T^{3} + 199539 T^{4} + 2291678 T^{5} + 25195860 T^{6} + 207387576 T^{7} + 1695219223 T^{8} + 9914344278 T^{9} + 71263761365 T^{10} + 409476997078 T^{11} + 4774672011455 T^{12} + 44505491463942 T^{13} + 509859219167149 T^{14} + 4179092137872696 T^{15} + 34017563178457020 T^{16} + 207301484332289582 T^{17} + 1209348333014546097 T^{18} + 5150968489805991085 T^{19} + 20840205347561929402 T^{20} + 51039458527449320572 T^{21} +$$$$12\!\cdots\!83$$$$T^{22}$$
$71$ $$1 - 4 T + 396 T^{2} - 1631 T^{3} + 75168 T^{4} - 311736 T^{5} + 8876715 T^{6} - 37764052 T^{7} + 746510386 T^{8} - 3382350308 T^{9} + 52530128596 T^{10} - 254577659802 T^{11} + 3729639130316 T^{12} - 17050427902628 T^{13} + 267184278763646 T^{14} - 959648042691412 T^{15} + 16015629743461965 T^{16} - 39933470108396856 T^{17} + 683661992065934688 T^{18} - 1053224009461836191 T^{19} + 18156006284505816276 T^{20} - 13020974204039524804 T^{21} +$$$$23\!\cdots\!71$$$$T^{22}$$
$73$ $$1 + 7 T + 263 T^{2} + 1437 T^{3} + 33449 T^{4} + 124687 T^{5} + 2748471 T^{6} + 9384756 T^{7} + 211304504 T^{8} + 1024533594 T^{9} + 17686597072 T^{10} + 95260794270 T^{11} + 1291121586256 T^{12} + 5459739522426 T^{13} + 82201044232568 T^{14} + 266510562614196 T^{15} + 5697777154284303 T^{16} + 18869410673296543 T^{17} + 369524433065275553 T^{18} + 1158883152051794397 T^{19} + 15483227304274461119 T^{20} + 30083380807924903543 T^{21} +$$$$31\!\cdots\!77$$$$T^{22}$$
$79$ $$1 - 5 T + 461 T^{2} - 3121 T^{3} + 96102 T^{4} - 774834 T^{5} + 12572030 T^{6} - 104861325 T^{7} + 1243408761 T^{8} - 9358180061 T^{9} + 106300984409 T^{10} - 721408297564 T^{11} + 8397777768311 T^{12} - 58404401760701 T^{13} + 613049012114679 T^{14} - 4084357102517325 T^{15} + 38684845359919970 T^{16} - 188352425511158514 T^{17} + 1845534061387852218 T^{18} - 4734896595718376881 T^{19} + 55251585747987045059 T^{20} - 47341380413134236005 T^{21} +$$$$74\!\cdots\!79$$$$T^{22}$$
$83$ $$1 + 11 T + 458 T^{2} + 5971 T^{3} + 121471 T^{4} + 1515064 T^{5} + 23086287 T^{6} + 253101862 T^{7} + 3233145964 T^{8} + 31458378941 T^{9} + 343494397731 T^{10} + 2987492467822 T^{11} + 28510035011673 T^{12} + 216716772524549 T^{13} + 1848670831317668 T^{14} + 12011789412493702 T^{15} + 90937822788962541 T^{16} + 495335589837930616 T^{17} + 3296243249760981317 T^{18} + 13448436918102213811 T^{19} + 85618636912533504574 T^{20} +$$$$17\!\cdots\!39$$$$T^{21} +$$$$12\!\cdots\!67$$$$T^{22}$$
$89$ $$1 + 10 T + 577 T^{2} + 4523 T^{3} + 146272 T^{4} + 828871 T^{5} + 21565126 T^{6} + 69737170 T^{7} + 2123459053 T^{8} + 1110902383 T^{9} + 170312612667 T^{10} - 194269945114 T^{11} + 15157822527363 T^{12} + 8799457775743 T^{13} + 1496972805134357 T^{14} + 4375466326797970 T^{15} + 120420945609175574 T^{16} + 411933379620135031 T^{17} + 6469805817838817888 T^{18} + 17805191168190512363 T^{19} +$$$$20\!\cdots\!93$$$$T^{20} +$$$$31\!\cdots\!10$$$$T^{21} +$$$$27\!\cdots\!89$$$$T^{22}$$
$97$ $$1 + 24 T + 814 T^{2} + 15031 T^{3} + 315107 T^{4} + 4768796 T^{5} + 76720544 T^{6} + 988481484 T^{7} + 13176065299 T^{8} + 147130773772 T^{9} + 1681670485851 T^{10} + 16398179468410 T^{11} + 163122037127547 T^{12} + 1384353450420748 T^{13} + 12025439044634227 T^{14} + 87509555060333004 T^{15} + 658825416030139808 T^{16} + 3972273565217395484 T^{17} + 25460105027044753091 T^{18} +$$$$11\!\cdots\!91$$$$T^{19} +$$$$61\!\cdots\!38$$$$T^{20} +$$$$17\!\cdots\!76$$$$T^{21} +$$$$71\!\cdots\!53$$$$T^{22}$$