# Properties

 Label 2541.2.a.bq Level 2541 Weight 2 Character orbit 2541.a Self dual yes Analytic conductor 20.290 Analytic rank 0 Dimension 10 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$2541 = 3 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 2541.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$20.2899871536$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 231) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 19 x^{8} - x^{7} + 124 x^{6} + 6 x^{5} - 316 x^{4} + 17 x^{3} + 253 x^{2} - 70 x - 11$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ $$\beta_{3}$$ $$=$$ $$($$$$-25 \nu^{9} + 31 \nu^{8} + 558 \nu^{7} - 505 \nu^{6} - 4093 \nu^{5} + 2537 \nu^{4} + 11069 \nu^{3} - 4152 \nu^{2} - 8301 \nu + 385$$$$)/2024$$ $$\beta_{4}$$ $$=$$ $$($$$$25 \nu^{9} - 31 \nu^{8} - 558 \nu^{7} + 505 \nu^{6} + 4093 \nu^{5} - 2537 \nu^{4} - 9045 \nu^{3} + 4152 \nu^{2} - 3843 \nu - 2409$$$$)/2024$$ $$\beta_{5}$$ $$=$$ $$($$$$-31 \nu^{9} - 83 \nu^{8} + 530 \nu^{7} + 993 \nu^{6} - 2687 \nu^{5} - 3169 \nu^{4} + 3727 \nu^{3} + 1976 \nu^{2} + 1365 \nu + 275$$$$)/2024$$ $$\beta_{6}$$ $$=$$ $$($$$$-97 \nu^{9} - 325 \nu^{8} + 2246 \nu^{7} + 5327 \nu^{6} - 16569 \nu^{5} - 27479 \nu^{4} + 43393 \nu^{3} + 43072 \nu^{2} - 29941 \nu - 5995$$$$)/2024$$ $$\beta_{7}$$ $$=$$ $$($$$$-229 \nu^{9} + 203 \nu^{8} + 3654 \nu^{7} - 2197 \nu^{6} - 19033 \nu^{5} + 4861 \nu^{4} + 34681 \nu^{3} + 5848 \nu^{2} - 13617 \nu - 7403$$$$)/2024$$ $$\beta_{8}$$ $$=$$ $$($$$$-285 \nu^{9} + 151 \nu^{8} + 4742 \nu^{7} - 1709 \nu^{6} - 25813 \nu^{5} + 6253 \nu^{4} + 49477 \nu^{3} - 12520 \nu^{2} - 24601 \nu + 9449$$$$)/2024$$ $$\beta_{9}$$ $$=$$ $$($$$$70 \nu^{9} + 65 \nu^{8} - 1360 \nu^{7} - 1116 \nu^{6} + 8981 \nu^{5} + 5850 \nu^{4} - 22543 \nu^{3} - 8412 \nu^{2} + 16260 \nu - 1331$$$$)/506$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{4} + \beta_{3} + 6 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{8} - \beta_{7} - \beta_{5} - \beta_{3} + 8 \beta_{2} + 2 \beta_{1} + 24$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{9} + \beta_{8} + 2 \beta_{6} + \beta_{5} + 9 \beta_{4} + 11 \beta_{3} + 39 \beta_{1} + 15$$ $$\nu^{6}$$ $$=$$ $$\beta_{9} + 11 \beta_{8} - 10 \beta_{7} + 2 \beta_{6} - 14 \beta_{5} + \beta_{4} - 12 \beta_{3} + 58 \beta_{2} + 26 \beta_{1} + 158$$ $$\nu^{7}$$ $$=$$ $$26 \beta_{9} + 13 \beta_{8} - \beta_{7} + 27 \beta_{6} + 10 \beta_{5} + 69 \beta_{4} + 104 \beta_{3} + 3 \beta_{2} + 266 \beta_{1} + 157$$ $$\nu^{8}$$ $$=$$ $$18 \beta_{9} + 97 \beta_{8} - 80 \beta_{7} + 30 \beta_{6} - 145 \beta_{5} + 16 \beta_{4} - 92 \beta_{3} + 412 \beta_{2} + 257 \beta_{1} + 1095$$ $$\nu^{9}$$ $$=$$ $$255 \beta_{9} + 126 \beta_{8} - 21 \beta_{7} + 272 \beta_{6} + 61 \beta_{5} + 509 \beta_{4} + 909 \beta_{3} + 52 \beta_{2} + 1873 \beta_{1} + 1444$$

## 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
 −2.65195 −2.39396 −2.09767 −1.33330 −0.112481 0.473713 0.871604 1.80545 2.63994 2.79866
−2.65195 1.00000 5.03286 1.71311 −2.65195 −1.00000 −8.04301 1.00000 −4.54308
1.2 −2.39396 1.00000 3.73106 −3.93829 −2.39396 −1.00000 −4.14409 1.00000 9.42812
1.3 −2.09767 1.00000 2.40021 3.15947 −2.09767 −1.00000 −0.839503 1.00000 −6.62751
1.4 −1.33330 1.00000 −0.222305 −0.873210 −1.33330 −1.00000 2.96300 1.00000 1.16425
1.5 −0.112481 1.00000 −1.98735 1.06131 −0.112481 −1.00000 0.448501 1.00000 −0.119378
1.6 0.473713 1.00000 −1.77560 3.75881 0.473713 −1.00000 −1.78855 1.00000 1.78060
1.7 0.871604 1.00000 −1.24031 −4.06436 0.871604 −1.00000 −2.82426 1.00000 −3.54252
1.8 1.80545 1.00000 1.25966 2.77715 1.80545 −1.00000 −1.33666 1.00000 5.01400
1.9 2.63994 1.00000 4.96928 3.08369 2.63994 −1.00000 7.83871 1.00000 8.14075
1.10 2.79866 1.00000 5.83249 −1.67767 2.79866 −1.00000 10.7258 1.00000 −4.69523
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.10 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 2541.2.a.bq 10
3.b odd 2 1 7623.2.a.cx 10
11.b odd 2 1 2541.2.a.br 10
11.c even 5 2 231.2.j.g 20
33.d even 2 1 7623.2.a.cy 10
33.h odd 10 2 693.2.m.j 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.g 20 11.c even 5 2
693.2.m.j 20 33.h odd 10 2
2541.2.a.bq 10 1.a even 1 1 trivial
2541.2.a.br 10 11.b odd 2 1
7623.2.a.cx 10 3.b odd 2 1
7623.2.a.cy 10 33.d even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$1$$
$$11$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2541))$$:

 $$T_{2}^{10} - \cdots$$ $$T_{5}^{10} - \cdots$$ $$T_{13}^{10} + \cdots$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2} - T^{3} - 8 T^{5} + 4 T^{6} - 7 T^{7} + 13 T^{8} - 8 T^{9} + 41 T^{10} - 16 T^{11} + 52 T^{12} - 56 T^{13} + 64 T^{14} - 256 T^{15} - 128 T^{17} + 256 T^{18} + 1024 T^{20}$$
$3$ $$( 1 - T )^{10}$$
$5$ $$1 - 5 T + 22 T^{2} - 46 T^{3} + 113 T^{4} - 108 T^{5} + 391 T^{6} - 538 T^{7} + 4002 T^{8} - 9087 T^{9} + 31806 T^{10} - 45435 T^{11} + 100050 T^{12} - 67250 T^{13} + 244375 T^{14} - 337500 T^{15} + 1765625 T^{16} - 3593750 T^{17} + 8593750 T^{18} - 9765625 T^{19} + 9765625 T^{20}$$
$7$ $$( 1 + T )^{10}$$
$11$ 
$13$ $$1 + 6 T + 65 T^{2} + 368 T^{3} + 2385 T^{4} + 10918 T^{5} + 56756 T^{6} + 221994 T^{7} + 984630 T^{8} + 3474482 T^{9} + 13929014 T^{10} + 45168266 T^{11} + 166402470 T^{12} + 487720818 T^{13} + 1621008116 T^{14} + 4053776974 T^{15} + 11511939465 T^{16} + 23091454256 T^{17} + 53022496865 T^{18} + 63626996238 T^{19} + 137858491849 T^{20}$$
$17$ $$1 + 8 T + 136 T^{2} + 954 T^{3} + 9093 T^{4} + 54096 T^{5} + 378365 T^{6} + 1910556 T^{7} + 10713786 T^{8} + 46048194 T^{9} + 215035238 T^{10} + 782819298 T^{11} + 3096284154 T^{12} + 9386561628 T^{13} + 31601423165 T^{14} + 76808584272 T^{15} + 219482914917 T^{16} + 391463094042 T^{17} + 948703011976 T^{18} + 948703011976 T^{19} + 2015993900449 T^{20}$$
$19$ $$1 + 69 T^{2} - 136 T^{3} + 2758 T^{4} - 8362 T^{5} + 89118 T^{6} - 289638 T^{7} + 2274617 T^{8} - 7545660 T^{9} + 46843450 T^{10} - 143367540 T^{11} + 821136737 T^{12} - 1986627042 T^{13} + 11613946878 T^{14} - 20705139838 T^{15} + 129752539798 T^{16} - 121566556504 T^{17} + 1171865849829 T^{18} + 6131066257801 T^{20}$$
$23$ $$1 + 124 T^{2} + 38 T^{3} + 8330 T^{4} + 3634 T^{5} + 381506 T^{6} + 180342 T^{7} + 12965000 T^{8} + 5936040 T^{9} + 338330610 T^{10} + 136528920 T^{11} + 6858485000 T^{12} + 2194221114 T^{13} + 106761020546 T^{14} + 23389670462 T^{15} + 1233138955370 T^{16} + 129383366986 T^{17} + 9710562174844 T^{18} + 41426511213649 T^{20}$$
$29$ $$1 - 14 T + 200 T^{2} - 1928 T^{3} + 18575 T^{4} - 144068 T^{5} + 1110481 T^{6} - 7337936 T^{7} + 48191364 T^{8} - 277661302 T^{9} + 1587859174 T^{10} - 8052177758 T^{11} + 40528937124 T^{12} - 178964921104 T^{13} + 785422112161 T^{14} - 2955000214132 T^{15} + 11048843187575 T^{16} - 33257761523752 T^{17} + 100049282592200 T^{18} - 203100043662166 T^{19} + 420707233300201 T^{20}$$
$31$ $$1 - 26 T + 482 T^{2} - 6400 T^{3} + 71453 T^{4} - 670846 T^{5} + 5624329 T^{6} - 41976980 T^{7} + 287918626 T^{8} - 1800633596 T^{9} + 10465823578 T^{10} - 55819641476 T^{11} + 276689799586 T^{12} - 1250536211180 T^{13} + 5194185942409 T^{14} - 19205751431746 T^{15} + 63414800518493 T^{16} - 176080730310400 T^{17} + 411093480046562 T^{18} - 687430176177446 T^{19} + 819628286980801 T^{20}$$
$37$ $$1 - 24 T + 440 T^{2} - 5840 T^{3} + 65456 T^{4} - 634664 T^{5} + 5522032 T^{6} - 43933344 T^{7} + 323265704 T^{8} - 2198445368 T^{9} + 13921174566 T^{10} - 81342478616 T^{11} + 442550748776 T^{12} - 2225355673632 T^{13} + 10349177015152 T^{14} - 44010113125448 T^{15} + 167942187827504 T^{16} - 554402162456720 T^{17} + 1545490959725240 T^{18} - 3119081755081848 T^{19} + 4808584372417849 T^{20}$$
$41$ $$1 + 19 T + 484 T^{2} + 6388 T^{3} + 94475 T^{4} + 963532 T^{5} + 10457893 T^{6} + 86623352 T^{7} + 748989936 T^{8} + 5154844493 T^{9} + 36776727094 T^{10} + 211348624213 T^{11} + 1259052082416 T^{12} + 5970168043192 T^{13} + 29551506181573 T^{14} + 111631157061932 T^{15} + 448766098168475 T^{16} + 1244090301551828 T^{17} + 3864703810894564 T^{18} + 6220256753485259 T^{19} + 13422659310152401 T^{20}$$
$43$ $$1 - 6 T + 312 T^{2} - 1828 T^{3} + 47319 T^{4} - 259736 T^{5} + 4576717 T^{6} - 22838020 T^{7} + 311402464 T^{8} - 1377461130 T^{9} + 15540959670 T^{10} - 59230828590 T^{11} + 575783155936 T^{12} - 1815782456140 T^{13} + 15646884656317 T^{14} - 38183384951048 T^{15} + 299120578115631 T^{16} - 496884421103596 T^{17} + 3646718486611512 T^{18} - 3015555671621058 T^{19} + 21611482313284249 T^{20}$$
$47$ $$1 - 15 T + 333 T^{2} - 4161 T^{3} + 56109 T^{4} - 576028 T^{5} + 5980092 T^{6} - 51937604 T^{7} + 445599026 T^{8} - 3323250890 T^{9} + 24323199470 T^{10} - 156192791830 T^{11} + 984328248434 T^{12} - 5392317860092 T^{13} + 29180941310652 T^{14} - 132109145692196 T^{15} + 604810992894861 T^{16} - 2108058804246543 T^{17} + 7929158458366413 T^{18} - 16786957096541505 T^{19} + 52599132235830049 T^{20}$$
$53$ $$1 + T + 254 T^{2} - 378 T^{3} + 28817 T^{4} - 120372 T^{5} + 2050339 T^{6} - 14991722 T^{7} + 109654718 T^{8} - 1160429241 T^{9} + 5512758766 T^{10} - 61502749773 T^{11} + 308020102862 T^{12} - 2231922596194 T^{13} + 16178160923059 T^{14} - 50339027883396 T^{15} + 638710394654393 T^{16} - 444040810858386 T^{17} + 15813961364485694 T^{18} + 3299763591802133 T^{19} + 174887470365513049 T^{20}$$
$59$ $$1 - 23 T + 495 T^{2} - 6037 T^{3} + 71081 T^{4} - 556772 T^{5} + 4700924 T^{6} - 26168124 T^{7} + 209482574 T^{8} - 1016636482 T^{9} + 10462754634 T^{10} - 59981552438 T^{11} + 729208840094 T^{12} - 5374383138996 T^{13} + 56962793141564 T^{14} - 398049831802828 T^{15} + 2998234511735921 T^{16} - 15023989013852303 T^{17} + 72681066614138895 T^{18} - 199248903829063597 T^{19} + 511116753300641401 T^{20}$$
$61$ $$1 + 215 T^{2} + 490 T^{3} + 23725 T^{4} + 88654 T^{5} + 1906100 T^{6} + 8267970 T^{7} + 125871770 T^{8} + 598496950 T^{9} + 7588984906 T^{10} + 36508313950 T^{11} + 468368856170 T^{12} + 1876672098570 T^{13} + 26391557530100 T^{14} + 74876840468854 T^{15} + 1222320881714725 T^{16} + 1539943989650290 T^{17} + 41217072294415415 T^{18} + 713342911662882601 T^{20}$$
$67$ $$1 - 38 T + 939 T^{2} - 16920 T^{3} + 253022 T^{4} - 3231648 T^{5} + 37034650 T^{6} - 384823180 T^{7} + 3720320353 T^{8} - 33446868814 T^{9} + 283134556406 T^{10} - 2240940210538 T^{11} + 16700518064617 T^{12} - 115740574086340 T^{13} + 746289713342650 T^{14} - 4363129101786336 T^{15} + 22887960773164718 T^{16} - 102547240362065160 T^{17} + 381297549225685899 T^{18} - 1033848307059207986 T^{19} + 1822837804551761449 T^{20}$$
$71$ $$1 - 26 T + 632 T^{2} - 9130 T^{3} + 126663 T^{4} - 1295526 T^{5} + 13933979 T^{6} - 123894450 T^{7} + 1257395276 T^{8} - 10721089596 T^{9} + 100241082738 T^{10} - 761197361316 T^{11} + 6338529586316 T^{12} - 44343186493950 T^{13} + 354085829408699 T^{14} - 2337426034183626 T^{15} + 16225566262285623 T^{16} - 83038447046109830 T^{17} + 408116231747320952 T^{18} - 1192061018679674806 T^{19} + 3255243551009881201 T^{20}$$
$73$ $$1 - T + 393 T^{2} - 817 T^{3} + 74513 T^{4} - 250396 T^{5} + 9185572 T^{6} - 43089004 T^{7} + 854063166 T^{8} - 4726643438 T^{9} + 66584553318 T^{10} - 345044970974 T^{11} + 4551302611614 T^{12} - 16762355069068 T^{13} + 260854087378852 T^{14} - 519088834600828 T^{15} + 11276367203472257 T^{16} - 9025724590102249 T^{17} + 316938816114373833 T^{18} - 58871586708267913 T^{19} + 4297625829703557649 T^{20}$$
$79$ $$1 + 5 T + 552 T^{2} + 2604 T^{3} + 151027 T^{4} + 658718 T^{5} + 26569829 T^{6} + 105054020 T^{7} + 3304178864 T^{8} + 11590016269 T^{9} + 302495065710 T^{10} + 915611285251 T^{11} + 20621380290224 T^{12} + 51795728966780 T^{13} + 1034896991706149 T^{14} + 2026912437036482 T^{15} + 36712769144970067 T^{16} + 50006978999958036 T^{17} + 837444063068421672 T^{18} + 599257979913091595 T^{19} + 9468276082626847201 T^{20}$$
$83$ $$1 + 6 T + 490 T^{2} + 3578 T^{3} + 118805 T^{4} + 976848 T^{5} + 19084536 T^{6} + 164833504 T^{7} + 2271305170 T^{8} + 19148739052 T^{9} + 211320377724 T^{10} + 1589345341316 T^{11} + 15647021316130 T^{12} + 94249654751648 T^{13} + 905720035624056 T^{14} + 3847843974033264 T^{15} + 38842151058104045 T^{16} + 97092790440885406 T^{17} + 1103623193748130090 T^{18} + 1121641531605242418 T^{19} + 15516041187205853449 T^{20}$$
$89$ $$1 + 9 T + 640 T^{2} + 4050 T^{3} + 178355 T^{4} + 698876 T^{5} + 29313265 T^{6} + 51732294 T^{7} + 3369419368 T^{8} + 1036172651 T^{9} + 317703552166 T^{10} + 92219365939 T^{11} + 26689170813928 T^{12} + 36469663568886 T^{13} + 1839179937126865 T^{14} + 3902565131479324 T^{15} + 88639098149349155 T^{16} + 179136906326892450 T^{17} + 2519416835649331840 T^{18} + 3153207633367366881 T^{19} + 31181719929966183601 T^{20}$$
$97$ $$1 - 24 T + 1069 T^{2} - 19604 T^{3} + 490777 T^{4} - 7233428 T^{5} + 130302052 T^{6} - 1582958620 T^{7} + 22393814246 T^{8} - 226465922980 T^{9} + 2616113341790 T^{10} - 21967194529060 T^{11} + 210703398240614 T^{12} - 1444723592591260 T^{13} + 11535546976384612 T^{14} - 62115907460510996 T^{15} + 408803501663039833 T^{16} - 1583969568908927252 T^{17} + 8378216512388971309 T^{18} - 18245545407709565208 T^{19} + 73742412689492826049 T^{20}$$