# Properties

 Label 504.2.r.f Level 504 Weight 2 Character orbit 504.r Analytic conductor 4.024 Analytic rank 0 Dimension 10 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 504.r (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: 10.0.6095158642368.1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{3} + ( -\beta_{2} - \beta_{7} ) q^{5} + ( 1 - \beta_{2} ) q^{7} + ( \beta_{3} - \beta_{7} ) q^{9} +O(q^{10})$$ $$q + \beta_{6} q^{3} + ( -\beta_{2} - \beta_{7} ) q^{5} + ( 1 - \beta_{2} ) q^{7} + ( \beta_{3} - \beta_{7} ) q^{9} + ( -\beta_{1} - \beta_{3} + \beta_{7} - \beta_{8} ) q^{11} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} ) q^{13} + ( 1 + \beta_{2} + \beta_{5} - \beta_{6} - \beta_{9} ) q^{15} + ( -\beta_{1} - \beta_{3} - \beta_{5} + \beta_{8} ) q^{17} + ( -1 - \beta_{1} + \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{8} - 2 \beta_{9} ) q^{19} -\beta_{1} q^{21} + ( \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{23} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{9} ) q^{25} + ( -3 + 3 \beta_{1} + 3 \beta_{2} + \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{27} + ( -3 + 3 \beta_{2} - \beta_{3} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{29} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{9} ) q^{31} + ( 5 - 2 \beta_{1} - 4 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{9} ) q^{33} + ( -1 + \beta_{1} + \beta_{6} - \beta_{8} ) q^{35} + ( -1 + 2 \beta_{1} - \beta_{4} - 2 \beta_{5} - \beta_{9} ) q^{37} + ( -2 - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{39} + ( -2 \beta_{1} - 4 \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{41} + ( -3 + 3 \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{43} + ( -3 - 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{7} + \beta_{8} ) q^{45} + ( 1 - 3 \beta_{1} - \beta_{2} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{9} ) q^{47} -\beta_{2} q^{49} + ( 3 - \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{51} + ( 6 - 2 \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{53} + ( 6 + 5 \beta_{1} + 3 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - \beta_{8} + 2 \beta_{9} ) q^{55} + ( -3 + \beta_{1} + 3 \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{7} + \beta_{9} ) q^{57} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{9} ) q^{59} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{61} + ( \beta_{4} - \beta_{8} ) q^{63} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{9} ) q^{65} + ( 2 \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{9} ) q^{67} + ( 2 - \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{69} + ( -1 + 2 \beta_{1} - \beta_{4} + \beta_{5} + 3 \beta_{6} + 2 \beta_{9} ) q^{71} + ( 5 - 2 \beta_{4} - \beta_{5} - \beta_{6} + 4 \beta_{8} + \beta_{9} ) q^{73} + ( -3 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{7} + \beta_{8} ) q^{75} + ( -\beta_{4} - \beta_{6} + \beta_{7} ) q^{77} + ( 1 - 5 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{6} - 3 \beta_{7} + 3 \beta_{8} + \beta_{9} ) q^{79} + ( -2 \beta_{3} + \beta_{4} + 3 \beta_{5} - \beta_{7} + 2 \beta_{8} + 3 \beta_{9} ) q^{81} + ( 4 - 3 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} + \beta_{7} - \beta_{8} + 3 \beta_{9} ) q^{83} + ( -\beta_{1} - 3 \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} - 3 \beta_{6} + \beta_{7} - \beta_{9} ) q^{85} + ( 3 - \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{87} + ( 3 \beta_{1} + \beta_{3} - 3 \beta_{4} + \beta_{6} + 3 \beta_{8} + 2 \beta_{9} ) q^{89} + ( -1 + \beta_{1} + \beta_{3} - \beta_{6} - \beta_{8} - \beta_{9} ) q^{91} + ( -5 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} - 2 \beta_{8} ) q^{93} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - 4 \beta_{6} + \beta_{9} ) q^{95} + ( -7 + 7 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{97} + ( 3 + 6 \beta_{2} + \beta_{3} - \beta_{4} + 3 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q - 3q^{5} + 5q^{7} + O(q^{10})$$ $$10q - 3q^{5} + 5q^{7} + 4q^{11} - 3q^{13} + 15q^{15} + 2q^{19} + 8q^{23} - 10q^{25} - 9q^{27} - 9q^{29} - 3q^{31} + 30q^{33} - 6q^{35} - 6q^{37} - 18q^{39} - 12q^{41} - 5q^{43} - 9q^{45} + 3q^{47} - 5q^{49} + 9q^{51} + 60q^{53} + 44q^{55} - 21q^{57} + 7q^{59} - 14q^{61} + 6q^{63} - 11q^{65} - 8q^{67} + 21q^{69} - 18q^{71} + 30q^{73} - 51q^{75} - 4q^{77} - 3q^{79} - 12q^{81} + 20q^{83} - 21q^{85} + 9q^{87} - 24q^{89} - 6q^{91} - 39q^{93} - 12q^{95} - 37q^{97} + 60q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 4 x^{9} + 6 x^{8} - 7 x^{7} + 25 x^{6} - 66 x^{5} + 75 x^{4} - 63 x^{3} + 162 x^{2} - 324 x + 243$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{9} - \nu^{8} + 3 \nu^{7} + 2 \nu^{6} + 31 \nu^{5} + 27 \nu^{4} + 156 \nu^{3} - 243 \nu^{2} + 81 \nu - 81$$$$)/648$$ $$\beta_{2}$$ $$=$$ $$($$$$-2 \nu^{9} + 3 \nu^{8} - 4 \nu^{7} + 5 \nu^{6} - 33 \nu^{5} + 64 \nu^{4} - 39 \nu^{3} + 57 \nu^{2} - 180 \nu + 405$$$$)/108$$ $$\beta_{3}$$ $$=$$ $$($$$$11 \nu^{9} - 59 \nu^{8} + 45 \nu^{7} - 86 \nu^{6} + 461 \nu^{5} - 939 \nu^{4} + 600 \nu^{3} - 441 \nu^{2} + 3375 \nu - 3807$$$$)/648$$ $$\beta_{4}$$ $$=$$ $$($$$$5 \nu^{9} - 11 \nu^{8} + 3 \nu^{7} - 20 \nu^{6} + 101 \nu^{5} - 141 \nu^{4} + 54 \nu^{3} - 201 \nu^{2} + 657 \nu - 495$$$$)/72$$ $$\beta_{5}$$ $$=$$ $$($$$$55 \nu^{9} - 127 \nu^{8} + 129 \nu^{7} - 214 \nu^{6} + 1129 \nu^{5} - 1935 \nu^{4} + 912 \nu^{3} - 2133 \nu^{2} + 6075 \nu - 8019$$$$)/648$$ $$\beta_{6}$$ $$=$$ $$($$$$29 \nu^{9} - 74 \nu^{8} + 51 \nu^{7} - 77 \nu^{6} + 566 \nu^{5} - 1017 \nu^{4} + 393 \nu^{3} - 702 \nu^{2} + 3321 \nu - 4050$$$$)/324$$ $$\beta_{7}$$ $$=$$ $$($$$$71 \nu^{9} - 155 \nu^{8} + 153 \nu^{7} - 290 \nu^{6} + 1277 \nu^{5} - 2271 \nu^{4} + 1428 \nu^{3} - 2169 \nu^{2} + 6939 \nu - 9639$$$$)/648$$ $$\beta_{8}$$ $$=$$ $$($$$$25 \nu^{9} - 49 \nu^{8} + 27 \nu^{7} - 94 \nu^{6} + 439 \nu^{5} - 645 \nu^{4} + 300 \nu^{3} - 891 \nu^{2} + 2241 \nu - 2457$$$$)/216$$ $$\beta_{9}$$ $$=$$ $$($$$$-89 \nu^{9} + 167 \nu^{8} - 111 \nu^{7} + 308 \nu^{6} - 1577 \nu^{5} + 2265 \nu^{4} - 798 \nu^{3} + 2853 \nu^{2} - 7965 \nu + 8667$$$$)/648$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{8} + \beta_{7} + 2 \beta_{4} - \beta_{3}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-3 \beta_{9} - 2 \beta_{8} + \beta_{7} - 3 \beta_{5} - \beta_{4} + 2 \beta_{3}$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{8} + \beta_{7} - 3 \beta_{5} - \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 6 \beta_{1} + 6$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$-6 \beta_{9} - 8 \beta_{8} + 4 \beta_{7} - 3 \beta_{5} + 2 \beta_{4} - \beta_{3} + 12 \beta_{2} + 12 \beta_{1} - 24$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-18 \beta_{9} - 2 \beta_{8} - 8 \beta_{7} - 3 \beta_{6} - 6 \beta_{5} - 13 \beta_{4} + 17 \beta_{3} + 18 \beta_{1}$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$3 \beta_{9} - 2 \beta_{8} - 11 \beta_{7} + 15 \beta_{6} + 3 \beta_{5} - \beta_{4} - 7 \beta_{3} - 21 \beta_{2} + 33 \beta_{1} + 33$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$-21 \beta_{9} - 59 \beta_{8} + 25 \beta_{7} - 3 \beta_{6} + 21 \beta_{5} + 14 \beta_{4} - 25 \beta_{3} + 48 \beta_{2} + 21 \beta_{1} - 24$$$$)/3$$ $$\nu^{8}$$ $$=$$ $$($$$$-42 \beta_{9} - 29 \beta_{8} - 17 \beta_{7} - 42 \beta_{6} - 9 \beta_{5} + 2 \beta_{4} + 32 \beta_{3} - 180 \beta_{2} - 27 \beta_{1} + 216$$$$)/3$$ $$\nu^{9}$$ $$=$$ $$($$$$6 \beta_{9} - 50 \beta_{8} + 76 \beta_{7} + 30 \beta_{6} - 72 \beta_{5} + 62 \beta_{4} - 124 \beta_{3} - 138 \beta_{2} - 30 \beta_{1} + 177$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/504\mathbb{Z}\right)^\times$$.

 $$n$$ $$73$$ $$127$$ $$253$$ $$281$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
169.1
 −1.28430 − 1.16214i 1.72987 − 0.0867982i −0.902451 + 1.47837i 1.34147 + 1.09565i 1.11541 − 1.32509i −1.28430 + 1.16214i 1.72987 + 0.0867982i −0.902451 − 1.47837i 1.34147 − 1.09565i 1.11541 + 1.32509i
0 −1.58317 + 0.702538i 0 −1.53571 2.65993i 0 0.500000 0.866025i 0 2.01288 2.22448i 0
169.2 0 −1.25506 1.19366i 0 0.846320 + 1.46587i 0 0.500000 0.866025i 0 0.150339 + 2.99623i 0
169.3 0 0.468714 1.66743i 0 −0.553143 0.958072i 0 0.500000 0.866025i 0 −2.56061 1.56309i 0
169.4 0 0.742385 + 1.56488i 0 −1.76217 3.05216i 0 0.500000 0.866025i 0 −1.89773 + 2.32349i 0
169.5 0 1.62713 + 0.593666i 0 1.50470 + 2.60622i 0 0.500000 0.866025i 0 2.29512 + 1.93195i 0
337.1 0 −1.58317 0.702538i 0 −1.53571 + 2.65993i 0 0.500000 + 0.866025i 0 2.01288 + 2.22448i 0
337.2 0 −1.25506 + 1.19366i 0 0.846320 1.46587i 0 0.500000 + 0.866025i 0 0.150339 2.99623i 0
337.3 0 0.468714 + 1.66743i 0 −0.553143 + 0.958072i 0 0.500000 + 0.866025i 0 −2.56061 + 1.56309i 0
337.4 0 0.742385 1.56488i 0 −1.76217 + 3.05216i 0 0.500000 + 0.866025i 0 −1.89773 2.32349i 0
337.5 0 1.62713 0.593666i 0 1.50470 2.60622i 0 0.500000 + 0.866025i 0 2.29512 1.93195i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 337.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.r.f 10
3.b odd 2 1 1512.2.r.f 10
4.b odd 2 1 1008.2.r.n 10
9.c even 3 1 inner 504.2.r.f 10
9.c even 3 1 4536.2.a.bd 5
9.d odd 6 1 1512.2.r.f 10
9.d odd 6 1 4536.2.a.bc 5
12.b even 2 1 3024.2.r.n 10
36.f odd 6 1 1008.2.r.n 10
36.f odd 6 1 9072.2.a.cn 5
36.h even 6 1 3024.2.r.n 10
36.h even 6 1 9072.2.a.cm 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.r.f 10 1.a even 1 1 trivial
504.2.r.f 10 9.c even 3 1 inner
1008.2.r.n 10 4.b odd 2 1
1008.2.r.n 10 36.f odd 6 1
1512.2.r.f 10 3.b odd 2 1
1512.2.r.f 10 9.d odd 6 1
3024.2.r.n 10 12.b even 2 1
3024.2.r.n 10 36.h even 6 1
4536.2.a.bc 5 9.d odd 6 1
4536.2.a.bd 5 9.c even 3 1
9072.2.a.cm 5 36.h even 6 1
9072.2.a.cn 5 36.f odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{10} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 + 3 T^{3} + 3 T^{4} - 18 T^{5} + 9 T^{6} + 27 T^{7} + 243 T^{10}$$
$5$ $$1 + 3 T - 3 T^{2} - 16 T^{3} - 25 T^{4} - 63 T^{5} - 19 T^{6} + 277 T^{7} + 775 T^{8} + 451 T^{9} - 1814 T^{10} + 2255 T^{11} + 19375 T^{12} + 34625 T^{13} - 11875 T^{14} - 196875 T^{15} - 390625 T^{16} - 1250000 T^{17} - 1171875 T^{18} + 5859375 T^{19} + 9765625 T^{20}$$
$7$ $$( 1 - T + T^{2} )^{5}$$
$11$ $$1 - 4 T - 6 T^{2} - 10 T^{3} + 191 T^{4} + 240 T^{5} + 1529 T^{6} - 16178 T^{7} - 5837 T^{8} + 14002 T^{9} + 502605 T^{10} + 154022 T^{11} - 706277 T^{12} - 21532918 T^{13} + 22386089 T^{14} + 38652240 T^{15} + 338368151 T^{16} - 194871710 T^{17} - 1286153286 T^{18} - 9431790764 T^{19} + 25937424601 T^{20}$$
$13$ $$1 + 3 T - 31 T^{2} - 106 T^{3} + 402 T^{4} + 1426 T^{5} - 5437 T^{6} - 15135 T^{7} + 85123 T^{8} + 101952 T^{9} - 1104756 T^{10} + 1325376 T^{11} + 14385787 T^{12} - 33251595 T^{13} - 155286157 T^{14} + 529463818 T^{15} + 1940377218 T^{16} - 6651342802 T^{17} - 25287652351 T^{18} + 31813498119 T^{19} + 137858491849 T^{20}$$
$17$ $$( 1 + 46 T^{2} - 9 T^{3} + 1261 T^{4} - 198 T^{5} + 21437 T^{6} - 2601 T^{7} + 225998 T^{8} + 1419857 T^{10} )^{2}$$
$19$ $$( 1 - T + 42 T^{2} - 66 T^{3} + 1281 T^{4} - 1209 T^{5} + 24339 T^{6} - 23826 T^{7} + 288078 T^{8} - 130321 T^{9} + 2476099 T^{10} )^{2}$$
$23$ $$1 - 8 T + 9 T^{2} - 222 T^{3} + 2650 T^{4} - 9138 T^{5} + 54826 T^{6} - 441030 T^{7} + 1653286 T^{8} - 8929410 T^{9} + 60424408 T^{10} - 205376430 T^{11} + 874588294 T^{12} - 5366012010 T^{13} + 15342562666 T^{14} - 58815302334 T^{15} + 392295105850 T^{16} - 755871249234 T^{17} + 704798867529 T^{18} - 14409221291704 T^{19} + 41426511213649 T^{20}$$
$29$ $$1 + 9 T - 52 T^{2} - 657 T^{3} + 1944 T^{4} + 29007 T^{5} - 46389 T^{6} - 788049 T^{7} + 920775 T^{8} + 10497078 T^{9} - 9302769 T^{10} + 304415262 T^{11} + 774371775 T^{12} - 19219727061 T^{13} - 32810058309 T^{14} + 594966899043 T^{15} + 1156336536024 T^{16} - 11333168735013 T^{17} - 26012813473972 T^{18} + 130564313782821 T^{19} + 420707233300201 T^{20}$$
$31$ $$1 + 3 T - 106 T^{2} - 157 T^{3} + 6498 T^{4} + 2989 T^{5} - 283525 T^{6} + 3627 T^{7} + 9910699 T^{8} - 942522 T^{9} - 316500783 T^{10} - 29218182 T^{11} + 9524181739 T^{12} + 108051957 T^{13} - 261841291525 T^{14} + 85572532339 T^{15} + 5766998919138 T^{16} - 4319480415427 T^{17} - 90406449968746 T^{18} + 79318866482013 T^{19} + 819628286980801 T^{20}$$
$37$ $$( 1 + 3 T + 103 T^{2} + 197 T^{3} + 6112 T^{4} + 11312 T^{5} + 226144 T^{6} + 269693 T^{7} + 5217259 T^{8} + 5622483 T^{9} + 69343957 T^{10} )^{2}$$
$41$ $$1 + 12 T - 30 T^{2} - 790 T^{3} - 277 T^{4} + 13968 T^{5} - 140563 T^{6} - 610562 T^{7} + 9173539 T^{8} + 24360490 T^{9} - 273097535 T^{10} + 998780090 T^{11} + 15420719059 T^{12} - 42080543602 T^{13} - 397197443443 T^{14} + 1618279415568 T^{15} - 1315778874757 T^{16} - 153855876365990 T^{17} - 239547756873630 T^{18} + 3928583212727532 T^{19} + 13422659310152401 T^{20}$$
$43$ $$1 + 5 T - 88 T^{2} + 275 T^{3} + 6540 T^{4} - 38663 T^{5} - 127543 T^{6} + 2251353 T^{7} - 2752001 T^{8} - 27649602 T^{9} + 456453449 T^{10} - 1188932886 T^{11} - 5088449849 T^{12} + 178998322971 T^{13} - 436044135943 T^{14} - 5683787431709 T^{15} + 41341714340460 T^{16} + 74750118054425 T^{17} - 1028561624428888 T^{18} + 2512963059684215 T^{19} + 21611482313284249 T^{20}$$
$47$ $$1 - 3 T - 24 T^{2} + 751 T^{3} - 2770 T^{4} - 11169 T^{5} + 300305 T^{6} - 1566601 T^{7} - 499337 T^{8} + 58566752 T^{9} - 599985995 T^{10} + 2752637344 T^{11} - 1103035433 T^{12} - 162649215623 T^{13} + 1465392602705 T^{14} - 2561554383183 T^{15} - 29858426461330 T^{16} + 380473963467713 T^{17} - 571470879882264 T^{18} - 3357391419308301 T^{19} + 52599132235830049 T^{20}$$
$53$ $$( 1 - 30 T + 580 T^{2} - 7683 T^{3} + 79903 T^{4} - 646182 T^{5} + 4234859 T^{6} - 21581547 T^{7} + 86348660 T^{8} - 236714430 T^{9} + 418195493 T^{10} )^{2}$$
$59$ $$1 - 7 T - 84 T^{2} + 891 T^{3} - 548 T^{4} - 21051 T^{5} + 190525 T^{6} - 1336983 T^{7} + 4615039 T^{8} + 49687002 T^{9} - 884146943 T^{10} + 2931533118 T^{11} + 16064950759 T^{12} - 274588231557 T^{13} + 2308660204525 T^{14} - 15049871418249 T^{15} - 23114932435268 T^{16} + 2217388472973729 T^{17} - 12333756758762964 T^{18} - 60640970730584573 T^{19} + 511116753300641401 T^{20}$$
$61$ $$1 + 14 T - 67 T^{2} - 1032 T^{3} + 10040 T^{4} + 70122 T^{5} - 675608 T^{6} - 3001836 T^{7} + 20748698 T^{8} - 6195900 T^{9} - 1395441908 T^{10} - 377949900 T^{11} + 77205905258 T^{12} - 681359737116 T^{13} - 9354360946328 T^{14} + 59224781818722 T^{15} + 517264558584440 T^{16} - 3243310606773672 T^{17} - 12844389970817827 T^{18} + 163718045299677974 T^{19} + 713342911662882601 T^{20}$$
$67$ $$1 + 8 T - 142 T^{2} - 1462 T^{3} + 7779 T^{4} + 110116 T^{5} - 343663 T^{6} - 6410154 T^{7} + 10809007 T^{8} + 209041554 T^{9} + 61061861 T^{10} + 14005784118 T^{11} + 48521632423 T^{12} - 1927937147502 T^{13} - 6925194696223 T^{14} + 148670376282412 T^{15} + 703675754892651 T^{16} - 8860760366982226 T^{17} - 57661610213043022 T^{18} + 217652275170359576 T^{19} + 1822837804551761449 T^{20}$$
$71$ $$( 1 + 9 T + 306 T^{2} + 2174 T^{3} + 40411 T^{4} + 221049 T^{5} + 2869181 T^{6} + 10959134 T^{7} + 109520766 T^{8} + 228705129 T^{9} + 1804229351 T^{10} )^{2}$$
$73$ $$( 1 - 15 T + 247 T^{2} - 2719 T^{3} + 26596 T^{4} - 243028 T^{5} + 1941508 T^{6} - 14489551 T^{7} + 96087199 T^{8} - 425973615 T^{9} + 2073071593 T^{10} )^{2}$$
$79$ $$1 + 3 T - 271 T^{2} - 1532 T^{3} + 38715 T^{4} + 263753 T^{5} - 3446203 T^{6} - 24313053 T^{7} + 236347015 T^{8} + 845708523 T^{9} - 15972358530 T^{10} + 66810973317 T^{11} + 1475041720615 T^{12} - 11987283338067 T^{13} - 134229885992443 T^{14} + 811582856405447 T^{15} + 9411130840495515 T^{16} - 29420388566795588 T^{17} - 411136487484678031 T^{18} + 359554787947854957 T^{19} + 9468276082626847201 T^{20}$$
$83$ $$1 - 20 T + 98 T^{2} - 318 T^{3} + 879 T^{4} + 169800 T^{5} - 2124519 T^{6} + 6011178 T^{7} + 51057123 T^{8} - 1398046658 T^{9} + 18343755493 T^{10} - 116037872614 T^{11} + 351732520347 T^{12} + 3437113435086 T^{13} - 100826104672599 T^{14} + 668849101181400 T^{15} + 287380588191351 T^{16} - 8629264214701386 T^{17} + 220724638749626018 T^{18} - 3738805105350808060 T^{19} + 15516041187205853449 T^{20}$$
$89$ $$( 1 + 12 T + 292 T^{2} + 2463 T^{3} + 41383 T^{4} + 291726 T^{5} + 3683087 T^{6} + 19509423 T^{7} + 205850948 T^{8} + 752906892 T^{9} + 5584059449 T^{10} )^{2}$$
$97$ $$1 + 37 T + 608 T^{2} + 4491 T^{3} - 11038 T^{4} - 630495 T^{5} - 6947105 T^{6} - 31739955 T^{7} + 128335391 T^{8} + 3636093960 T^{9} + 40545974467 T^{10} + 352701114120 T^{11} + 1207507693919 T^{12} - 28968199949715 T^{13} - 615022210681505 T^{14} - 5414275095337215 T^{15} - 9194344990406302 T^{16} + 362865095591205483 T^{17} + 4765159625381192288 T^{18} + 28128549170218913029 T^{19} + 73742412689492826049 T^{20}$$