# Properties

 Label 1323.2.g.f Level $1323$ Weight $2$ Character orbit 1323.g Analytic conductor $10.564$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1323.g (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.5642081874$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: 10.0.991381711347.1 Defining polynomial: $$x^{10} - 2 x^{9} + 9 x^{8} - 8 x^{7} + 40 x^{6} - 36 x^{5} + 90 x^{4} - 3 x^{3} + 36 x^{2} - 9 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 63) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( -\beta_{3} + \beta_{6} + \beta_{7} ) q^{4} + ( -1 + \beta_{2} - \beta_{9} ) q^{5} + ( 1 - \beta_{4} - \beta_{8} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( -\beta_{3} + \beta_{6} + \beta_{7} ) q^{4} + ( -1 + \beta_{2} - \beta_{9} ) q^{5} + ( 1 - \beta_{4} - \beta_{8} ) q^{8} + ( 2 - \beta_{2} - \beta_{4} + 2 \beta_{6} ) q^{10} + ( 1 - \beta_{3} - \beta_{4} - \beta_{8} ) q^{11} + ( 1 + \beta_{1} + \beta_{2} + \beta_{6} ) q^{13} + ( \beta_{2} + \beta_{4} + \beta_{7} ) q^{16} + ( 3 - \beta_{1} + 3 \beta_{6} + \beta_{7} ) q^{17} + ( -\beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{19} + ( \beta_{3} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{20} + ( -1 + \beta_{1} + \beta_{2} - \beta_{6} - \beta_{7} ) q^{22} + ( \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{8} - \beta_{9} ) q^{23} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{5} + 2 \beta_{9} ) q^{25} + ( \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{26} + ( 2 \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{29} + ( -\beta_{3} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{31} + ( -\beta_{3} - 3 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{32} + ( -\beta_{3} - 4 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} ) q^{34} + ( -2 \beta_{5} - 2 \beta_{8} ) q^{37} + ( -5 + 2 \beta_{1} - 3 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{8} ) q^{38} + ( -1 - \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{8} ) q^{40} + ( \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{7} ) q^{41} + ( \beta_{3} - 2 \beta_{5} + 3 \beta_{6} - \beta_{7} + \beta_{8} - 3 \beta_{9} ) q^{43} + ( -\beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} ) q^{44} + ( 4 - 5 \beta_{1} - 3 \beta_{2} - \beta_{4} + 4 \beta_{6} + 2 \beta_{7} ) q^{46} + ( 4 + \beta_{1} + \beta_{2} + \beta_{4} + 4 \beta_{6} - 2 \beta_{7} ) q^{47} + ( -6 + 3 \beta_{1} + 2 \beta_{2} + \beta_{4} - 6 \beta_{6} - \beta_{7} ) q^{50} + ( -1 - 2 \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{8} ) q^{52} + ( 5 - \beta_{1} - 2 \beta_{2} + 5 \beta_{6} ) q^{53} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{9} ) q^{55} + ( 3 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{8} + \beta_{9} ) q^{58} + ( \beta_{3} - \beta_{5} - 6 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{59} + ( 2 + \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{61} + ( -3 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{8} - 2 \beta_{9} ) q^{62} + ( -6 + \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{9} ) q^{64} + ( 1 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{65} + ( -2 \beta_{3} + 4 \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} ) q^{67} + ( -7 + 2 \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{5} - \beta_{9} ) q^{68} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 3 \beta_{8} - 2 \beta_{9} ) q^{71} + ( -4 + 3 \beta_{4} - 4 \beta_{6} + \beta_{7} ) q^{73} + ( -10 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{8} - 2 \beta_{9} ) q^{74} + ( -3 + 5 \beta_{1} - \beta_{2} - 3 \beta_{6} - \beta_{7} ) q^{76} + ( -3 + 4 \beta_{1} + \beta_{2} - 3 \beta_{6} - 2 \beta_{7} ) q^{79} + ( 3 + 2 \beta_{1} - \beta_{2} + \beta_{4} + 3 \beta_{6} ) q^{80} + ( -4 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} - 3 \beta_{9} ) q^{82} + ( 2 \beta_{3} - \beta_{6} - 2 \beta_{7} + 4 \beta_{8} + 2 \beta_{9} ) q^{83} + ( -2 + \beta_{2} + \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{85} + ( 1 - 2 \beta_{1} - 4 \beta_{2} - \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 3 \beta_{8} + 4 \beta_{9} ) q^{86} + ( 4 + \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{8} - \beta_{9} ) q^{88} + ( 2 \beta_{5} - 7 \beta_{6} - 2 \beta_{8} - \beta_{9} ) q^{89} + ( -2 \beta_{5} + 5 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{92} + ( -4 \beta_{5} + 3 \beta_{6} + 4 \beta_{8} - 2 \beta_{9} ) q^{94} + ( -2 \beta_{3} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{95} + ( 4 \beta_{3} - \beta_{5} - 2 \beta_{6} - 4 \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q - 2q^{2} - 4q^{4} - 8q^{5} + 6q^{8} + O(q^{10})$$ $$10q - 2q^{2} - 4q^{4} - 8q^{5} + 6q^{8} + 7q^{10} + 8q^{11} + 8q^{13} + 2q^{16} + 12q^{17} - q^{19} + 5q^{20} - q^{22} + 6q^{23} + 2q^{25} + 11q^{26} - 7q^{29} + 3q^{31} + 2q^{32} - 3q^{34} - 40q^{38} - 6q^{40} + 5q^{41} - 7q^{43} + 10q^{44} + 3q^{46} + 27q^{47} - 19q^{50} - 20q^{52} + 21q^{53} - 4q^{55} + 20q^{58} + 30q^{59} + 14q^{61} - 12q^{62} - 50q^{64} + 11q^{65} - 2q^{67} - 54q^{68} + 6q^{71} - 15q^{73} - 72q^{74} - 5q^{76} - 4q^{79} + 20q^{80} + 5q^{82} + 9q^{83} - 6q^{85} - 16q^{86} + 36q^{88} + 28q^{89} - 27q^{92} + 3q^{94} + 14q^{95} + 12q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2 x^{9} + 9 x^{8} - 8 x^{7} + 40 x^{6} - 36 x^{5} + 90 x^{4} - 3 x^{3} + 36 x^{2} - 9 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{9} - 9 \nu^{8} - 3 \nu^{7} - 61 \nu^{6} - 72 \nu^{5} - 282 \nu^{4} - 204 \nu^{3} - 387 \nu^{2} - 873 \nu - 117$$$$)/189$$ $$\beta_{3}$$ $$=$$ $$($$$$7 \nu^{9} - 12 \nu^{8} + 48 \nu^{7} - 23 \nu^{6} + 204 \nu^{5} - 240 \nu^{4} + 303 \nu^{3} - 108 \nu^{2} + 36 \nu - 1557$$$$)/567$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{9} - \nu^{8} + 12 \nu^{7} + 8 \nu^{6} + 68 \nu^{5} + 30 \nu^{4} + 123 \nu^{3} + 204 \nu^{2} + 270 \nu + 63$$$$)/63$$ $$\beta_{5}$$ $$=$$ $$($$$$16 \nu^{9} - 39 \nu^{8} + 156 \nu^{7} - 176 \nu^{6} + 663 \nu^{5} - 780 \nu^{4} + 1680 \nu^{3} - 351 \nu^{2} + 684 \nu - 180$$$$)/567$$ $$\beta_{6}$$ $$=$$ $$($$$$20 \nu^{9} - 24 \nu^{8} + 141 \nu^{7} - 4 \nu^{6} + 624 \nu^{5} - 57 \nu^{4} + 1020 \nu^{3} + 1620 \nu^{2} + 369 \nu - 63$$$$)/567$$ $$\beta_{7}$$ $$=$$ $$($$$$-53 \nu^{9} + 60 \nu^{8} - 375 \nu^{7} - 11 \nu^{6} - 1668 \nu^{5} - 69 \nu^{4} - 2757 \nu^{3} - 4401 \nu^{2} - 1071 \nu - 1368$$$$)/567$$ $$\beta_{8}$$ $$=$$ $$($$$$-82 \nu^{9} + 165 \nu^{8} - 732 \nu^{7} + 632 \nu^{6} - 3264 \nu^{5} + 2850 \nu^{4} - 7260 \nu^{3} - 432 \nu^{2} - 2898 \nu + 720$$$$)/567$$ $$\beta_{9}$$ $$=$$ $$($$$$-91 \nu^{9} + 174 \nu^{8} - 813 \nu^{7} + 704 \nu^{6} - 3633 \nu^{5} + 3174 \nu^{4} - 8070 \nu^{3} + 648 \nu^{2} - 3222 \nu + 801$$$$)/567$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} + 3 \beta_{6} - \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$\beta_{8} + 4 \beta_{5} + \beta_{4} - 4 \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$-5 \beta_{7} - 14 \beta_{6} + \beta_{4} + \beta_{2} - 14$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{9} - 7 \beta_{8} - \beta_{7} - 9 \beta_{6} - 17 \beta_{5} + \beta_{3}$$ $$\nu^{6}$$ $$=$$ $$9 \beta_{9} - 10 \beta_{8} - \beta_{5} - 10 \beta_{4} + 24 \beta_{3} - 9 \beta_{2} + \beta_{1} + 70$$ $$\nu^{7}$$ $$=$$ $$11 \beta_{7} + 65 \beta_{6} - 43 \beta_{4} - 19 \beta_{2} + 75 \beta_{1} + 65$$ $$\nu^{8}$$ $$=$$ $$-62 \beta_{9} + 73 \beta_{8} + 118 \beta_{7} + 360 \beta_{6} + 14 \beta_{5} - 118 \beta_{3}$$ $$\nu^{9}$$ $$=$$ $$-135 \beta_{9} + 253 \beta_{8} + 343 \beta_{5} + 253 \beta_{4} - 87 \beta_{3} + 135 \beta_{2} - 343 \beta_{1} - 430$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1081$$ $$\chi(n)$$ $$-1 - \beta_{6}$$ $$\beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 1.19343 − 2.06709i 0.920620 − 1.59456i 0.247934 − 0.429435i −0.335166 + 0.580525i −1.02682 + 1.77851i 1.19343 + 2.06709i 0.920620 + 1.59456i 0.247934 + 0.429435i −0.335166 − 0.580525i −1.02682 − 1.77851i
−1.19343 + 2.06709i 0 −1.84857 3.20182i −2.92087 0 0 4.05086 0 3.48586 6.03769i
361.2 −0.920620 + 1.59456i 0 −0.695084 1.20392i 1.33475 0 0 −1.12285 0 −1.22880 + 2.12835i
361.3 −0.247934 + 0.429435i 0 0.877057 + 1.51911i −3.69258 0 0 −1.86155 0 0.915516 1.58572i
361.4 0.335166 0.580525i 0 0.775327 + 1.34291i 1.42494 0 0 2.38012 0 0.477591 0.827212i
361.5 1.02682 1.77851i 0 −1.10873 1.92038i −0.146246 0 0 −0.446582 0 −0.150168 + 0.260099i
667.1 −1.19343 2.06709i 0 −1.84857 + 3.20182i −2.92087 0 0 4.05086 0 3.48586 + 6.03769i
667.2 −0.920620 1.59456i 0 −0.695084 + 1.20392i 1.33475 0 0 −1.12285 0 −1.22880 2.12835i
667.3 −0.247934 0.429435i 0 0.877057 1.51911i −3.69258 0 0 −1.86155 0 0.915516 + 1.58572i
667.4 0.335166 + 0.580525i 0 0.775327 1.34291i 1.42494 0 0 2.38012 0 0.477591 + 0.827212i
667.5 1.02682 + 1.77851i 0 −1.10873 + 1.92038i −0.146246 0 0 −0.446582 0 −0.150168 0.260099i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 667.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.g.f 10
3.b odd 2 1 441.2.g.f 10
7.b odd 2 1 189.2.g.b 10
7.c even 3 1 1323.2.f.f 10
7.c even 3 1 1323.2.h.f 10
7.d odd 6 1 189.2.h.b 10
7.d odd 6 1 1323.2.f.e 10
9.c even 3 1 1323.2.h.f 10
9.d odd 6 1 441.2.h.f 10
21.c even 2 1 63.2.g.b 10
21.g even 6 1 63.2.h.b yes 10
21.g even 6 1 441.2.f.e 10
21.h odd 6 1 441.2.f.f 10
21.h odd 6 1 441.2.h.f 10
28.d even 2 1 3024.2.t.i 10
28.f even 6 1 3024.2.q.i 10
63.g even 3 1 inner 1323.2.g.f 10
63.g even 3 1 3969.2.a.bb 5
63.h even 3 1 1323.2.f.f 10
63.i even 6 1 441.2.f.e 10
63.i even 6 1 567.2.e.f 10
63.j odd 6 1 441.2.f.f 10
63.k odd 6 1 189.2.g.b 10
63.k odd 6 1 3969.2.a.bc 5
63.l odd 6 1 189.2.h.b 10
63.l odd 6 1 567.2.e.e 10
63.n odd 6 1 441.2.g.f 10
63.n odd 6 1 3969.2.a.ba 5
63.o even 6 1 63.2.h.b yes 10
63.o even 6 1 567.2.e.f 10
63.s even 6 1 63.2.g.b 10
63.s even 6 1 3969.2.a.z 5
63.t odd 6 1 567.2.e.e 10
63.t odd 6 1 1323.2.f.e 10
84.h odd 2 1 1008.2.t.i 10
84.j odd 6 1 1008.2.q.i 10
252.n even 6 1 3024.2.t.i 10
252.s odd 6 1 1008.2.q.i 10
252.bi even 6 1 3024.2.q.i 10
252.bn odd 6 1 1008.2.t.i 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.b 10 21.c even 2 1
63.2.g.b 10 63.s even 6 1
63.2.h.b yes 10 21.g even 6 1
63.2.h.b yes 10 63.o even 6 1
189.2.g.b 10 7.b odd 2 1
189.2.g.b 10 63.k odd 6 1
189.2.h.b 10 7.d odd 6 1
189.2.h.b 10 63.l odd 6 1
441.2.f.e 10 21.g even 6 1
441.2.f.e 10 63.i even 6 1
441.2.f.f 10 21.h odd 6 1
441.2.f.f 10 63.j odd 6 1
441.2.g.f 10 3.b odd 2 1
441.2.g.f 10 63.n odd 6 1
441.2.h.f 10 9.d odd 6 1
441.2.h.f 10 21.h odd 6 1
567.2.e.e 10 63.l odd 6 1
567.2.e.e 10 63.t odd 6 1
567.2.e.f 10 63.i even 6 1
567.2.e.f 10 63.o even 6 1
1008.2.q.i 10 84.j odd 6 1
1008.2.q.i 10 252.s odd 6 1
1008.2.t.i 10 84.h odd 2 1
1008.2.t.i 10 252.bn odd 6 1
1323.2.f.e 10 7.d odd 6 1
1323.2.f.e 10 63.t odd 6 1
1323.2.f.f 10 7.c even 3 1
1323.2.f.f 10 63.h even 3 1
1323.2.g.f 10 1.a even 1 1 trivial
1323.2.g.f 10 63.g even 3 1 inner
1323.2.h.f 10 7.c even 3 1
1323.2.h.f 10 9.c even 3 1
3024.2.q.i 10 28.f even 6 1
3024.2.q.i 10 252.bi even 6 1
3024.2.t.i 10 28.d even 2 1
3024.2.t.i 10 252.n even 6 1
3969.2.a.z 5 63.s even 6 1
3969.2.a.ba 5 63.n odd 6 1
3969.2.a.bb 5 63.g even 3 1
3969.2.a.bc 5 63.k odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1323, [\chi])$$:

 $$T_{2}^{10} + \cdots$$ $$T_{5}^{5} + 4 T_{5}^{4} - 5 T_{5}^{3} - 18 T_{5}^{2} + 18 T_{5} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T - T^{2} - 4 T^{3} - 2 T^{4} - 2 T^{5} + 6 T^{6} + 21 T^{7} + 6 T^{8} - 13 T^{9} - 5 T^{10} - 26 T^{11} + 24 T^{12} + 168 T^{13} + 96 T^{14} - 64 T^{15} - 128 T^{16} - 512 T^{17} - 256 T^{18} + 1024 T^{19} + 1024 T^{20}$$
$3$ 1
$5$ $$( 1 + 4 T + 20 T^{2} + 62 T^{3} + 193 T^{4} + 423 T^{5} + 965 T^{6} + 1550 T^{7} + 2500 T^{8} + 2500 T^{9} + 3125 T^{10} )^{2}$$
$7$ 1
$11$ $$( 1 - 4 T + 47 T^{2} - 161 T^{3} + 958 T^{4} - 2589 T^{5} + 10538 T^{6} - 19481 T^{7} + 62557 T^{8} - 58564 T^{9} + 161051 T^{10} )^{2}$$
$13$ $$1 - 8 T - 14 T^{2} + 182 T^{3} + 686 T^{4} - 4429 T^{5} - 12871 T^{6} + 43199 T^{7} + 305249 T^{8} - 358672 T^{9} - 3841969 T^{10} - 4662736 T^{11} + 51587081 T^{12} + 94908203 T^{13} - 367608631 T^{14} - 1644456697 T^{15} + 3311190974 T^{16} + 11420230094 T^{17} - 11420230094 T^{18} - 84835994984 T^{19} + 137858491849 T^{20}$$
$17$ $$1 - 12 T + 14 T^{2} + 192 T^{3} + 1185 T^{4} - 11847 T^{5} - 6180 T^{6} + 65736 T^{7} + 1002861 T^{8} - 2436261 T^{9} - 7749777 T^{10} - 41416437 T^{11} + 289826829 T^{12} + 322960968 T^{13} - 516159780 T^{14} - 16821045879 T^{15} + 28603019265 T^{16} + 78785025216 T^{17} + 97660604174 T^{18} - 1423054517964 T^{19} + 2015993900449 T^{20}$$
$19$ $$1 + T - 53 T^{2} - 190 T^{3} + 1262 T^{4} + 7007 T^{5} - 13111 T^{6} - 116110 T^{7} + 67964 T^{8} + 721616 T^{9} - 440023 T^{10} + 13710704 T^{11} + 24535004 T^{12} - 796398490 T^{13} - 1708638631 T^{14} + 17350025693 T^{15} + 59371901822 T^{16} - 169835630410 T^{17} - 900128841173 T^{18} + 322687697779 T^{19} + 6131066257801 T^{20}$$
$23$ $$( 1 - 3 T + 52 T^{2} - 225 T^{3} + 2023 T^{4} - 5565 T^{5} + 46529 T^{6} - 119025 T^{7} + 632684 T^{8} - 839523 T^{9} + 6436343 T^{10} )^{2}$$
$29$ $$1 + 7 T - 76 T^{2} - 419 T^{3} + 4561 T^{4} + 15146 T^{5} - 199563 T^{6} - 341373 T^{7} + 6918636 T^{8} + 2570041 T^{9} - 219913241 T^{10} + 74531189 T^{11} + 5818572876 T^{12} - 8325746097 T^{13} - 141147118203 T^{14} + 310661862754 T^{15} + 2712989167081 T^{16} - 7227698173471 T^{17} - 38018727385036 T^{18} + 101550021831083 T^{19} + 420707233300201 T^{20}$$
$31$ $$1 - 3 T - 125 T^{2} + 214 T^{3} + 9282 T^{4} - 8387 T^{5} - 503981 T^{6} + 245082 T^{7} + 21459514 T^{8} - 3619498 T^{9} - 734820027 T^{10} - 112204438 T^{11} + 20622592954 T^{12} + 7301237862 T^{13} - 465437037101 T^{14} - 240112689437 T^{15} + 8237809167042 T^{16} + 5887699419754 T^{17} - 106611379680125 T^{18} - 79318866482013 T^{19} + 819628286980801 T^{20}$$
$37$ $$1 - 89 T^{2} + 560 T^{3} + 4503 T^{4} - 45352 T^{5} + 27130 T^{6} + 2296536 T^{7} - 9801827 T^{8} - 33131096 T^{9} + 610977105 T^{10} - 1225850552 T^{11} - 13418701163 T^{12} + 116326438008 T^{13} + 50845987930 T^{14} - 3144887137864 T^{15} + 11553466019727 T^{16} + 53161851194480 T^{17} - 312610671398969 T^{18} + 4808584372417849 T^{20}$$
$41$ $$1 - 5 T - 136 T^{2} + 733 T^{3} + 10507 T^{4} - 54412 T^{5} - 554055 T^{6} + 2345451 T^{7} + 23706084 T^{8} - 41392439 T^{9} - 952045937 T^{10} - 1697089999 T^{11} + 39849927204 T^{12} + 161650828371 T^{13} - 1565627010855 T^{14} - 6303967608812 T^{15} + 49909345260187 T^{16} + 142754882754773 T^{17} - 1085949831160456 T^{18} - 1636909671969805 T^{19} + 13422659310152401 T^{20}$$
$43$ $$1 + 7 T - 77 T^{2} - 66 T^{3} + 7014 T^{4} - 3843 T^{5} - 95427 T^{6} + 1632678 T^{7} - 3708600 T^{8} - 15416324 T^{9} + 670279801 T^{10} - 662901932 T^{11} - 6857201400 T^{12} + 129809329746 T^{13} - 326245923027 T^{14} - 564953446449 T^{15} + 44338040425686 T^{16} - 17940028333062 T^{17} - 899991421375277 T^{18} + 3518148283557901 T^{19} + 21611482313284249 T^{20}$$
$47$ $$1 - 27 T + 281 T^{2} - 1758 T^{3} + 13050 T^{4} - 78783 T^{5} - 25248 T^{6} + 1518381 T^{7} + 9454350 T^{8} - 53043051 T^{9} - 242331903 T^{10} - 2493023397 T^{11} + 20884659150 T^{12} + 157642870563 T^{13} - 123202185888 T^{14} - 18068487686481 T^{15} + 140668760043450 T^{16} - 890643445773954 T^{17} + 6690971551954841 T^{18} - 30216522773774709 T^{19} + 52599132235830049 T^{20}$$
$53$ $$1 - 21 T + 41 T^{2} + 924 T^{3} + 12966 T^{4} - 177027 T^{5} - 601755 T^{6} + 3783942 T^{7} + 110973258 T^{8} - 340111866 T^{9} - 4044436041 T^{10} - 18025928898 T^{11} + 311723881722 T^{12} + 563341933134 T^{13} - 4748136394155 T^{14} - 74031893539311 T^{15} + 287383106398614 T^{16} + 1085433093209388 T^{17} + 2552647306865801 T^{18} - 69295035427844793 T^{19} + 174887470365513049 T^{20}$$
$59$ $$1 - 30 T + 299 T^{2} - 1644 T^{3} + 26547 T^{4} - 344442 T^{5} + 1635267 T^{6} - 9620487 T^{7} + 170035344 T^{8} - 1056366303 T^{9} + 3109579647 T^{10} - 62325611877 T^{11} + 591893032464 T^{12} - 1975845999573 T^{13} + 19815120570387 T^{14} - 246249955396158 T^{15} + 1119766626567627 T^{16} - 4091343041042436 T^{17} + 43902300843691979 T^{18} - 259889874559648170 T^{19} + 511116753300641401 T^{20}$$
$61$ $$1 - 14 T - 143 T^{2} + 2072 T^{3} + 23777 T^{4} - 251656 T^{5} - 2164351 T^{6} + 13562879 T^{7} + 202896254 T^{8} - 466067647 T^{9} - 12461386219 T^{10} - 28430126467 T^{11} + 754976961134 T^{12} + 3078515838299 T^{13} - 29967259814191 T^{14} - 212547726724456 T^{15} + 1224999941181497 T^{16} + 6511763156235512 T^{17} - 27414145758611183 T^{18} - 163718045299677974 T^{19} + 713342911662882601 T^{20}$$
$67$ $$1 + 2 T - 128 T^{2} - 128 T^{3} + 6161 T^{4} - 2183 T^{5} + 29300 T^{6} + 394018 T^{7} - 17169907 T^{8} - 2850929 T^{9} + 1197895103 T^{10} - 191012243 T^{11} - 77075712523 T^{12} + 118506035734 T^{13} + 590427845300 T^{14} - 2947323108581 T^{15} + 557314092543209 T^{16} - 775771085481344 T^{17} - 51976662727250048 T^{18} + 54413068792589894 T^{19} + 1822837804551761449 T^{20}$$
$71$ $$( 1 - 3 T + 187 T^{2} - 285 T^{3} + 15679 T^{4} - 10143 T^{5} + 1113209 T^{6} - 1436685 T^{7} + 66929357 T^{8} - 76235043 T^{9} + 1804229351 T^{10} )^{2}$$
$73$ $$1 + 15 T - 134 T^{2} - 2501 T^{3} + 16563 T^{4} + 235276 T^{5} - 2002535 T^{6} - 9021201 T^{7} + 288508378 T^{8} + 238799411 T^{9} - 25271949561 T^{10} + 17432357003 T^{11} + 1537461146362 T^{12} - 3509400549417 T^{13} - 56868471540935 T^{14} + 487743992114668 T^{15} + 2506548790024707 T^{16} - 27629543696261597 T^{17} - 108065652313806854 T^{18} + 883073800624018695 T^{19} + 4297625829703557649 T^{20}$$
$79$ $$1 + 4 T - 284 T^{2} - 1776 T^{3} + 44175 T^{4} + 312399 T^{5} - 4187754 T^{6} - 29772300 T^{7} + 295992489 T^{8} + 1067553919 T^{9} - 20151634301 T^{10} + 84336759601 T^{11} + 1847289123849 T^{12} - 14678905019700 T^{13} - 163113357508074 T^{14} + 961269341991201 T^{15} + 10738388347640175 T^{16} - 34106142359418384 T^{17} - 430858902013463324 T^{18} + 479406383930473276 T^{19} + 9468276082626847201 T^{20}$$
$83$ $$1 - 9 T - 148 T^{2} - 297 T^{3} + 24654 T^{4} + 118125 T^{5} - 807174 T^{6} - 21382137 T^{7} - 37648479 T^{8} + 452536146 T^{9} + 15509586612 T^{10} + 37560500118 T^{11} - 259360371831 T^{12} - 12226027968819 T^{13} - 38307122794854 T^{14} + 465299175954375 T^{15} + 8060387965039326 T^{16} - 8059407143919219 T^{17} - 333339250356578068 T^{18} - 1682462297407863627 T^{19} + 15516041187205853449 T^{20}$$
$89$ $$1 - 28 T + 104 T^{2} + 1736 T^{3} + 31273 T^{4} - 611939 T^{5} - 1780638 T^{6} + 18973932 T^{7} + 740914101 T^{8} - 3271180573 T^{9} - 40614588329 T^{10} - 291135070997 T^{11} + 5868780594021 T^{12} + 13376033868108 T^{13} - 111721218529758 T^{14} - 3417103755161611 T^{15} + 15542095912223353 T^{16} + 76785597378638344 T^{17} + 409405235793016424 T^{18} - 9809979303809585852 T^{19} + 31181719929966183601 T^{20}$$
$97$ $$1 - 12 T - 197 T^{2} + 1534 T^{3} + 27813 T^{4} - 14090 T^{5} - 4545035 T^{6} + 6881349 T^{7} + 472663750 T^{8} - 908843245 T^{9} - 38512186359 T^{10} - 88157794765 T^{11} + 4447293223750 T^{12} + 6280421435877 T^{13} - 402368680669835 T^{14} - 120995624221130 T^{15} + 23167450373090277 T^{16} + 123944568389425342 T^{17} - 1543974418092261317 T^{18} - 9122772703854782604 T^{19} + 73742412689492826049 T^{20}$$