# Properties

 Label 567.2.e.f Level $567$ Weight $2$ Character orbit 567.e Analytic conductor $4.528$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$567 = 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 567.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.52751779461$$ 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.

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

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 $$\left(\mathbb{Z}/567\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$407$$ $$\chi(n)$$ $$-1 - \beta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
163.1
 −1.02682 + 1.77851i −0.335166 + 0.580525i 0.247934 − 0.429435i 0.920620 − 1.59456i 1.19343 − 2.06709i −1.02682 − 1.77851i −0.335166 − 0.580525i 0.247934 + 0.429435i 0.920620 + 1.59456i 1.19343 + 2.06709i
−1.02682 + 1.77851i 0 −1.10873 1.92038i 0.0731228 0.126652i 0 2.25011 1.39176i 0.446582 0 0.150168 + 0.260099i
163.2 −0.335166 + 0.580525i 0 0.775327 + 1.34291i −0.712469 + 1.23403i 0 0.145107 2.64177i −2.38012 0 −0.477591 0.827212i
163.3 0.247934 0.429435i 0 0.877057 + 1.51911i 1.84629 3.19787i 0 1.68284 + 2.04158i 1.86155 0 −0.915516 1.58572i
163.4 0.920620 1.59456i 0 −0.695084 1.20392i −0.667377 + 1.15593i 0 0.640804 + 2.56698i 1.12285 0 1.22880 + 2.12835i
163.5 1.19343 2.06709i 0 −1.84857 3.20182i 1.46043 2.52954i 0 −2.21886 1.44106i −4.05086 0 −3.48586 6.03769i
487.1 −1.02682 1.77851i 0 −1.10873 + 1.92038i 0.0731228 + 0.126652i 0 2.25011 + 1.39176i 0.446582 0 0.150168 0.260099i
487.2 −0.335166 0.580525i 0 0.775327 1.34291i −0.712469 1.23403i 0 0.145107 + 2.64177i −2.38012 0 −0.477591 + 0.827212i
487.3 0.247934 + 0.429435i 0 0.877057 1.51911i 1.84629 + 3.19787i 0 1.68284 2.04158i 1.86155 0 −0.915516 + 1.58572i
487.4 0.920620 + 1.59456i 0 −0.695084 + 1.20392i −0.667377 1.15593i 0 0.640804 2.56698i 1.12285 0 1.22880 2.12835i
487.5 1.19343 + 2.06709i 0 −1.84857 + 3.20182i 1.46043 + 2.52954i 0 −2.21886 + 1.44106i −4.05086 0 −3.48586 + 6.03769i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 487.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.b 10 9.c even 3 1
63.2.g.b 10 63.h even 3 1
63.2.h.b yes 10 9.c even 3 1
63.2.h.b yes 10 63.g even 3 1
189.2.g.b 10 9.d odd 6 1
189.2.g.b 10 63.j odd 6 1
189.2.h.b 10 9.d odd 6 1
189.2.h.b 10 63.n odd 6 1
441.2.f.e 10 63.g even 3 1
441.2.f.e 10 63.h even 3 1
441.2.f.f 10 63.k odd 6 1
441.2.f.f 10 63.t odd 6 1
441.2.g.f 10 63.l odd 6 1
441.2.g.f 10 63.t odd 6 1
441.2.h.f 10 63.k odd 6 1
441.2.h.f 10 63.l odd 6 1
567.2.e.e 10 3.b odd 2 1
567.2.e.e 10 21.h odd 6 1
567.2.e.f 10 1.a even 1 1 trivial
567.2.e.f 10 7.c even 3 1 inner
1008.2.q.i 10 36.f odd 6 1
1008.2.q.i 10 252.bl odd 6 1
1008.2.t.i 10 36.f odd 6 1
1008.2.t.i 10 252.u odd 6 1
1323.2.f.e 10 63.j odd 6 1
1323.2.f.e 10 63.n odd 6 1
1323.2.f.f 10 63.i even 6 1
1323.2.f.f 10 63.s even 6 1
1323.2.g.f 10 63.i even 6 1
1323.2.g.f 10 63.o even 6 1
1323.2.h.f 10 63.o even 6 1
1323.2.h.f 10 63.s even 6 1
3024.2.q.i 10 36.h even 6 1
3024.2.q.i 10 252.o even 6 1
3024.2.t.i 10 36.h even 6 1
3024.2.t.i 10 252.bb even 6 1
3969.2.a.z 5 7.c even 3 1
3969.2.a.ba 5 7.d odd 6 1
3969.2.a.bb 5 21.g even 6 1
3969.2.a.bc 5 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(567, [\chi])$$.

## 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
$5$ $$1 - 4 T - 4 T^{2} + 44 T^{3} - 41 T^{4} - 119 T^{5} + 222 T^{6} - 456 T^{7} + 1623 T^{8} + 2021 T^{9} - 16541 T^{10} + 10105 T^{11} + 40575 T^{12} - 57000 T^{13} + 138750 T^{14} - 371875 T^{15} - 640625 T^{16} + 3437500 T^{17} - 1562500 T^{18} - 7812500 T^{19} + 9765625 T^{20}$$
$7$ $$1 - 5 T + 21 T^{2} - 43 T^{3} + 83 T^{4} - 87 T^{5} + 581 T^{6} - 2107 T^{7} + 7203 T^{8} - 12005 T^{9} + 16807 T^{10}$$
$11$ $$1 - 4 T - 31 T^{2} + 134 T^{3} + 607 T^{4} - 2492 T^{5} - 8385 T^{6} + 27495 T^{7} + 98940 T^{8} - 135733 T^{9} - 1043873 T^{10} - 1493063 T^{11} + 11971740 T^{12} + 36595845 T^{13} - 122764785 T^{14} - 401339092 T^{15} + 1075337527 T^{16} + 2611280914 T^{17} - 6645125311 T^{18} - 9431790764 T^{19} + 25937424601 T^{20}$$
$13$ $$( 1 - 8 T + 78 T^{2} - 403 T^{3} + 2174 T^{4} - 7779 T^{5} + 28262 T^{6} - 68107 T^{7} + 171366 T^{8} - 228488 T^{9} + 371293 T^{10} )^{2}$$
$17$ $$1 - 12 T + 14 T^{2} + 192 T^{3} + 1185 T^{4} - 11847 T^{5} - 6180 T^{6} + 65736 T^{7} + 1002861 T^{8} - 2436261 T^{9} - 7749777 T^{10} - 41416437 T^{11} + 289826829 T^{12} + 322960968 T^{13} - 516159780 T^{14} - 16821045879 T^{15} + 28603019265 T^{16} + 78785025216 T^{17} + 97660604174 T^{18} - 1423054517964 T^{19} + 2015993900449 T^{20}$$
$19$ $$1 - T - 53 T^{2} + 190 T^{3} + 1262 T^{4} - 7007 T^{5} - 13111 T^{6} + 116110 T^{7} + 67964 T^{8} - 721616 T^{9} - 440023 T^{10} - 13710704 T^{11} + 24535004 T^{12} + 796398490 T^{13} - 1708638631 T^{14} - 17350025693 T^{15} + 59371901822 T^{16} + 169835630410 T^{17} - 900128841173 T^{18} - 322687697779 T^{19} + 6131066257801 T^{20}$$
$23$ $$1 - 3 T - 43 T^{2} + 294 T^{3} + 6 T^{4} - 5127 T^{5} + 21792 T^{6} - 135027 T^{7} + 502362 T^{8} + 3271749 T^{9} - 33095343 T^{10} + 75250227 T^{11} + 265749498 T^{12} - 1642873509 T^{13} + 6098295072 T^{14} - 32999130561 T^{15} + 888215334 T^{16} + 1001018681418 T^{17} - 3367372367083 T^{18} - 5403457984389 T^{19} + 41426511213649 T^{20}$$
$29$ $$( 1 + 7 T + 125 T^{2} + 647 T^{3} + 6535 T^{4} + 25761 T^{5} + 189515 T^{6} + 544127 T^{7} + 3048625 T^{8} + 4950967 T^{9} + 20511149 T^{10} )^{2}$$
$31$ $$1 + 3 T - 125 T^{2} - 214 T^{3} + 9282 T^{4} + 8387 T^{5} - 503981 T^{6} - 245082 T^{7} + 21459514 T^{8} + 3619498 T^{9} - 734820027 T^{10} + 112204438 T^{11} + 20622592954 T^{12} - 7301237862 T^{13} - 465437037101 T^{14} + 240112689437 T^{15} + 8237809167042 T^{16} - 5887699419754 T^{17} - 106611379680125 T^{18} + 79318866482013 T^{19} + 819628286980801 T^{20}$$
$37$ $$1 - 89 T^{2} + 560 T^{3} + 4503 T^{4} - 45352 T^{5} + 27130 T^{6} + 2296536 T^{7} - 9801827 T^{8} - 33131096 T^{9} + 610977105 T^{10} - 1225850552 T^{11} - 13418701163 T^{12} + 116326438008 T^{13} + 50845987930 T^{14} - 3144887137864 T^{15} + 11553466019727 T^{16} + 53161851194480 T^{17} - 312610671398969 T^{18} + 4808584372417849 T^{20}$$
$41$ $$( 1 + 5 T + 161 T^{2} + 769 T^{3} + 11569 T^{4} + 46293 T^{5} + 474329 T^{6} + 1292689 T^{7} + 11096281 T^{8} + 14128805 T^{9} + 115856201 T^{10} )^{2}$$
$43$ $$( 1 - 7 T + 126 T^{2} - 474 T^{3} + 5544 T^{4} - 14049 T^{5} + 238392 T^{6} - 876426 T^{7} + 10017882 T^{8} - 23931607 T^{9} + 147008443 T^{10} )^{2}$$
$47$ $$1 - 27 T + 281 T^{2} - 1758 T^{3} + 13050 T^{4} - 78783 T^{5} - 25248 T^{6} + 1518381 T^{7} + 9454350 T^{8} - 53043051 T^{9} - 242331903 T^{10} - 2493023397 T^{11} + 20884659150 T^{12} + 157642870563 T^{13} - 123202185888 T^{14} - 18068487686481 T^{15} + 140668760043450 T^{16} - 890643445773954 T^{17} + 6690971551954841 T^{18} - 30216522773774709 T^{19} + 52599132235830049 T^{20}$$
$53$ $$1 + 21 T + 41 T^{2} - 924 T^{3} + 12966 T^{4} + 177027 T^{5} - 601755 T^{6} - 3783942 T^{7} + 110973258 T^{8} + 340111866 T^{9} - 4044436041 T^{10} + 18025928898 T^{11} + 311723881722 T^{12} - 563341933134 T^{13} - 4748136394155 T^{14} + 74031893539311 T^{15} + 287383106398614 T^{16} - 1085433093209388 T^{17} + 2552647306865801 T^{18} + 69295035427844793 T^{19} + 174887470365513049 T^{20}$$
$59$ $$1 - 30 T + 299 T^{2} - 1644 T^{3} + 26547 T^{4} - 344442 T^{5} + 1635267 T^{6} - 9620487 T^{7} + 170035344 T^{8} - 1056366303 T^{9} + 3109579647 T^{10} - 62325611877 T^{11} + 591893032464 T^{12} - 1975845999573 T^{13} + 19815120570387 T^{14} - 246249955396158 T^{15} + 1119766626567627 T^{16} - 4091343041042436 T^{17} + 43902300843691979 T^{18} - 259889874559648170 T^{19} + 511116753300641401 T^{20}$$
$61$ $$1 + 14 T - 143 T^{2} - 2072 T^{3} + 23777 T^{4} + 251656 T^{5} - 2164351 T^{6} - 13562879 T^{7} + 202896254 T^{8} + 466067647 T^{9} - 12461386219 T^{10} + 28430126467 T^{11} + 754976961134 T^{12} - 3078515838299 T^{13} - 29967259814191 T^{14} + 212547726724456 T^{15} + 1224999941181497 T^{16} - 6511763156235512 T^{17} - 27414145758611183 T^{18} + 163718045299677974 T^{19} + 713342911662882601 T^{20}$$
$67$ $$1 + 2 T - 128 T^{2} - 128 T^{3} + 6161 T^{4} - 2183 T^{5} + 29300 T^{6} + 394018 T^{7} - 17169907 T^{8} - 2850929 T^{9} + 1197895103 T^{10} - 191012243 T^{11} - 77075712523 T^{12} + 118506035734 T^{13} + 590427845300 T^{14} - 2947323108581 T^{15} + 557314092543209 T^{16} - 775771085481344 T^{17} - 51976662727250048 T^{18} + 54413068792589894 T^{19} + 1822837804551761449 T^{20}$$
$71$ $$( 1 + 3 T + 187 T^{2} + 285 T^{3} + 15679 T^{4} + 10143 T^{5} + 1113209 T^{6} + 1436685 T^{7} + 66929357 T^{8} + 76235043 T^{9} + 1804229351 T^{10} )^{2}$$
$73$ $$1 - 15 T - 134 T^{2} + 2501 T^{3} + 16563 T^{4} - 235276 T^{5} - 2002535 T^{6} + 9021201 T^{7} + 288508378 T^{8} - 238799411 T^{9} - 25271949561 T^{10} - 17432357003 T^{11} + 1537461146362 T^{12} + 3509400549417 T^{13} - 56868471540935 T^{14} - 487743992114668 T^{15} + 2506548790024707 T^{16} + 27629543696261597 T^{17} - 108065652313806854 T^{18} - 883073800624018695 T^{19} + 4297625829703557649 T^{20}$$
$79$ $$1 + 4 T - 284 T^{2} - 1776 T^{3} + 44175 T^{4} + 312399 T^{5} - 4187754 T^{6} - 29772300 T^{7} + 295992489 T^{8} + 1067553919 T^{9} - 20151634301 T^{10} + 84336759601 T^{11} + 1847289123849 T^{12} - 14678905019700 T^{13} - 163113357508074 T^{14} + 961269341991201 T^{15} + 10738388347640175 T^{16} - 34106142359418384 T^{17} - 430858902013463324 T^{18} + 479406383930473276 T^{19} + 9468276082626847201 T^{20}$$
$83$ $$( 1 + 9 T + 229 T^{2} + 882 T^{3} + 19849 T^{4} + 37179 T^{5} + 1647467 T^{6} + 6076098 T^{7} + 130939223 T^{8} + 427124889 T^{9} + 3939040643 T^{10} )^{2}$$
$89$ $$1 - 28 T + 104 T^{2} + 1736 T^{3} + 31273 T^{4} - 611939 T^{5} - 1780638 T^{6} + 18973932 T^{7} + 740914101 T^{8} - 3271180573 T^{9} - 40614588329 T^{10} - 291135070997 T^{11} + 5868780594021 T^{12} + 13376033868108 T^{13} - 111721218529758 T^{14} - 3417103755161611 T^{15} + 15542095912223353 T^{16} + 76785597378638344 T^{17} + 409405235793016424 T^{18} - 9809979303809585852 T^{19} + 31181719929966183601 T^{20}$$
$97$ $$( 1 - 12 T + 341 T^{2} - 2813 T^{3} + 54712 T^{4} - 367945 T^{5} + 5307064 T^{6} - 26467517 T^{7} + 311221493 T^{8} - 1062351372 T^{9} + 8587340257 T^{10} )^{2}$$