# Properties

 Label 1323.2.h.f Level $1323$ Weight $2$ Character orbit 1323.h 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.h (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: $$\Z[a_1, \ldots, a_{13}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{5} ) q^{2} + ( 1 + \beta_{3} ) q^{4} + ( -\beta_{6} + \beta_{9} ) q^{5} + ( 1 - \beta_{4} - \beta_{8} ) q^{8} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{5} ) q^{2} + ( 1 + \beta_{3} ) q^{4} + ( -\beta_{6} + \beta_{9} ) q^{5} + ( 1 - \beta_{4} - \beta_{8} ) q^{8} + ( -2 \beta_{6} - \beta_{8} + \beta_{9} ) q^{10} + ( -1 + \beta_{4} - \beta_{6} + \beta_{7} ) q^{11} + ( 1 + \beta_{1} + \beta_{2} + \beta_{6} ) q^{13} + ( -\beta_{2} - \beta_{3} - \beta_{4} - \beta_{8} + \beta_{9} ) 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} + ( \beta_{3} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{20} + ( -1 + \beta_{1} + \beta_{2} - \beta_{6} - \beta_{7} ) q^{22} + ( -2 \beta_{3} + 2 \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{23} + ( -\beta_{1} + 2 \beta_{2} + \beta_{7} ) q^{25} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{26} + ( 2 \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{29} + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{8} - \beta_{9} ) q^{31} + ( 1 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{8} + 2 \beta_{9} ) q^{32} + ( -\beta_{3} - 4 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} ) q^{34} + ( 2 \beta_{1} - 2 \beta_{4} ) q^{37} + ( 5 - 2 \beta_{1} + \beta_{4} + 5 \beta_{6} + 3 \beta_{7} ) q^{38} + ( 2 \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{7} - \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} + ( 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} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} + \beta_{9} ) q^{47} + ( -6 + 3 \beta_{1} + 2 \beta_{2} + \beta_{4} - 6 \beta_{6} - \beta_{7} ) q^{50} + ( 1 + 2 \beta_{1} + \beta_{4} + \beta_{6} + \beta_{7} ) 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} + ( -2 \beta_{3} - 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} ) q^{58} + ( -6 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{8} + \beta_{9} ) q^{59} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{9} ) 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_{3} - \beta_{4} + \beta_{5} - \beta_{8} + \beta_{9} ) q^{65} + ( -1 + 4 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - 4 \beta_{5} + \beta_{8} + \beta_{9} ) q^{67} + ( 3 \beta_{3} + 2 \beta_{5} - 7 \beta_{6} - 3 \beta_{7} + \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} + ( \beta_{3} + 4 \beta_{6} - \beta_{7} + 3 \beta_{8} ) q^{73} + ( 10 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + 10 \beta_{6} + 4 \beta_{7} ) q^{74} + ( -3 + 5 \beta_{1} - \beta_{2} - 3 \beta_{6} - \beta_{7} ) q^{76} + ( 3 - 4 \beta_{1} - \beta_{2} + 2 \beta_{3} + 4 \beta_{5} + \beta_{9} ) 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} + ( 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} + ( \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{7} + 3 \beta_{8} - 4 \beta_{9} ) q^{86} + ( -4 - \beta_{2} + 2 \beta_{4} - 4 \beta_{6} + \beta_{7} ) q^{88} + ( 7 - 2 \beta_{1} + \beta_{2} - 2 \beta_{4} + 7 \beta_{6} ) q^{89} + ( -2 \beta_{5} + 5 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{92} + ( 3 - 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{4} + 4 \beta_{5} - 4 \beta_{8} + 2 \beta_{9} ) q^{94} + ( -2 + 2 \beta_{3} - \beta_{4} - \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})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q + 4q^{2} + 8q^{4} + 4q^{5} + 6q^{8} + O(q^{10})$$ $$10q + 4q^{2} + 8q^{4} + 4q^{5} + 6q^{8} + 7q^{10} - 4q^{11} + 8q^{13} - 4q^{16} + 12q^{17} - q^{19} + 5q^{20} - q^{22} - 3q^{23} - q^{25} + 11q^{26} - 7q^{29} - 6q^{31} - 4q^{32} - 3q^{34} + 20q^{38} + 3q^{40} + 5q^{41} - 7q^{43} + 10q^{44} + 3q^{46} - 54q^{47} - 19q^{50} + 10q^{52} + 21q^{53} - 4q^{55} - 10q^{58} - 60q^{59} - 28q^{61} - 12q^{62} - 50q^{64} - 22q^{65} + 4q^{67} + 27q^{68} + 6q^{71} - 15q^{73} + 36q^{74} - 5q^{76} + 8q^{79} + 20q^{80} + 5q^{82} + 9q^{83} - 6q^{85} + 8q^{86} - 18q^{88} + 28q^{89} - 27q^{92} - 6q^{94} - 28q^{95} + 12q^{97} + O(q^{100})$$

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 $$\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
226.1
 −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
−2.05365 0 2.21746 0.0731228 + 0.126652i 0 0 −0.446582 0 −0.150168 0.260099i
226.2 −0.670333 0 −1.55065 −0.712469 1.23403i 0 0 2.38012 0 0.477591 + 0.827212i
226.3 0.495868 0 −1.75411 1.84629 + 3.19787i 0 0 −1.86155 0 0.915516 + 1.58572i
226.4 1.84124 0 1.39017 −0.667377 1.15593i 0 0 −1.12285 0 −1.22880 2.12835i
226.5 2.38687 0 3.69714 1.46043 + 2.52954i 0 0 4.05086 0 3.48586 + 6.03769i
802.1 −2.05365 0 2.21746 0.0731228 0.126652i 0 0 −0.446582 0 −0.150168 + 0.260099i
802.2 −0.670333 0 −1.55065 −0.712469 + 1.23403i 0 0 2.38012 0 0.477591 0.827212i
802.3 0.495868 0 −1.75411 1.84629 3.19787i 0 0 −1.86155 0 0.915516 1.58572i
802.4 1.84124 0 1.39017 −0.667377 + 1.15593i 0 0 −1.12285 0 −1.22880 + 2.12835i
802.5 2.38687 0 3.69714 1.46043 2.52954i 0 0 4.05086 0 3.48586 6.03769i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 802.5 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.h 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.h.f 10
3.b odd 2 1 441.2.h.f 10
7.b odd 2 1 189.2.h.b 10
7.c even 3 1 1323.2.f.f 10
7.c even 3 1 1323.2.g.f 10
7.d odd 6 1 189.2.g.b 10
7.d odd 6 1 1323.2.f.e 10
9.c even 3 1 1323.2.g.f 10
9.d odd 6 1 441.2.g.f 10
21.c even 2 1 63.2.h.b yes 10
21.g even 6 1 63.2.g.b 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.g.f 10
28.d even 2 1 3024.2.q.i 10
28.f even 6 1 3024.2.t.i 10
63.g even 3 1 1323.2.f.f 10
63.h even 3 1 inner 1323.2.h.f 10
63.h even 3 1 3969.2.a.bb 5
63.i even 6 1 63.2.h.b yes 10
63.i even 6 1 3969.2.a.z 5
63.j odd 6 1 441.2.h.f 10
63.j odd 6 1 3969.2.a.ba 5
63.k odd 6 1 567.2.e.e 10
63.k odd 6 1 1323.2.f.e 10
63.l odd 6 1 189.2.g.b 10
63.l odd 6 1 567.2.e.e 10
63.n odd 6 1 441.2.f.f 10
63.o even 6 1 63.2.g.b 10
63.o even 6 1 567.2.e.f 10
63.s even 6 1 441.2.f.e 10
63.s even 6 1 567.2.e.f 10
63.t odd 6 1 189.2.h.b 10
63.t odd 6 1 3969.2.a.bc 5
84.h odd 2 1 1008.2.q.i 10
84.j odd 6 1 1008.2.t.i 10
252.r odd 6 1 1008.2.q.i 10
252.s odd 6 1 1008.2.t.i 10
252.bi even 6 1 3024.2.t.i 10
252.bj even 6 1 3024.2.q.i 10

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

## Hecke kernels

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

 $$T_{2}^{5} - 2 T_{2}^{4} - 5 T_{2}^{3} + 9 T_{2}^{2} + 3 T_{2} - 3$$ $$T_{5}^{10} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -3 + 3 T + 9 T^{2} - 5 T^{3} - 2 T^{4} + T^{5} )^{2}$$
$3$ $$T^{10}$$
$5$ $$9 - 54 T + 378 T^{2} + 294 T^{3} + 402 T^{4} + 51 T^{5} + 79 T^{6} - 16 T^{7} + 21 T^{8} - 4 T^{9} + T^{10}$$
$7$ $$T^{10}$$
$11$ $$225 + 180 T + 369 T^{2} + 60 T^{3} + 261 T^{4} + 39 T^{5} + 112 T^{6} - 2 T^{7} + 24 T^{8} + 4 T^{9} + T^{10}$$
$13$ $$25 + 115 T + 594 T^{2} - 169 T^{3} + 428 T^{4} - 204 T^{5} + 296 T^{6} - 130 T^{7} + 51 T^{8} - 8 T^{9} + T^{10}$$
$17$ $$81 + 162 T + 864 T^{2} - 1890 T^{3} + 2898 T^{4} - 2259 T^{5} + 1287 T^{6} - 420 T^{7} + 99 T^{8} - 12 T^{9} + T^{10}$$
$19$ $$185761 - 111629 T + 86907 T^{2} - 23428 T^{3} + 13166 T^{4} - 2835 T^{5} + 1376 T^{6} - 133 T^{7} + 42 T^{8} + T^{9} + T^{10}$$
$23$ $$2595321 - 1739880 T + 1084239 T^{2} - 258066 T^{3} + 75474 T^{4} - 4878 T^{5} + 3042 T^{6} - 87 T^{7} + 72 T^{8} + 3 T^{9} + T^{10}$$
$29$ $$81 - 1215 T + 19710 T^{2} + 22635 T^{3} + 24462 T^{4} + 5199 T^{5} + 1690 T^{6} + 190 T^{7} + 69 T^{8} + 7 T^{9} + T^{10}$$
$31$ $$( 285 + 93 T - 64 T^{2} - 21 T^{3} + 3 T^{4} + T^{5} )^{2}$$
$37$ $$82944 + 110592 T + 228096 T^{2} - 52224 T^{3} + 115264 T^{4} + 27168 T^{5} + 8832 T^{6} + 560 T^{7} + 96 T^{8} + T^{10}$$
$41$ $$2025 - 7695 T + 31536 T^{2} + 4761 T^{3} + 9900 T^{4} - 579 T^{5} + 2020 T^{6} + 118 T^{7} + 69 T^{8} - 5 T^{9} + T^{10}$$
$43$ $$687241 + 1214485 T + 1541055 T^{2} + 921888 T^{3} + 408318 T^{4} + 84651 T^{5} + 14496 T^{6} + 837 T^{7} + 138 T^{8} + 7 T^{9} + T^{10}$$
$47$ $$( -6615 - 4032 T + 93 T^{2} + 213 T^{3} + 27 T^{4} + T^{5} )^{2}$$
$53$ $$178929 - 178929 T + 267759 T^{2} - 25380 T^{3} + 110088 T^{4} - 45693 T^{5} + 14238 T^{6} - 2415 T^{7} + 306 T^{8} - 21 T^{9} + T^{10}$$
$59$ $$( -5625 - 108 T + 1113 T^{2} + 306 T^{3} + 30 T^{4} + T^{5} )^{2}$$
$61$ $$( -1 - 14 T - 7 T^{2} + 34 T^{3} + 14 T^{4} + T^{5} )^{2}$$
$67$ $$( -7121 + 6784 T + 340 T^{2} - 203 T^{3} - 2 T^{4} + T^{5} )^{2}$$
$71$ $$( 81 + 1053 T + 567 T^{2} - 168 T^{3} - 3 T^{4} + T^{5} )^{2}$$
$73$ $$772641 - 1052163 T + 1048686 T^{2} - 533637 T^{3} + 211336 T^{4} - 34167 T^{5} + 5394 T^{6} + 784 T^{7} + 231 T^{8} + 15 T^{9} + T^{10}$$
$79$ $$( 193 - 22 T - 224 T^{2} - 95 T^{3} - 4 T^{4} + T^{5} )^{2}$$
$83$ $$218123361 + 40275063 T + 38540043 T^{2} - 11237130 T^{3} + 3795093 T^{4} - 455571 T^{5} + 56277 T^{6} - 2538 T^{7} + 267 T^{8} - 9 T^{9} + T^{10}$$
$89$ $$7080921 - 4502412 T + 3980484 T^{2} - 540030 T^{3} + 648528 T^{4} - 190791 T^{5} + 45157 T^{6} - 5740 T^{7} + 549 T^{8} - 28 T^{9} + T^{10}$$
$97$ $$2307745521 - 121346514 T + 94916553 T^{2} - 9179814 T^{3} + 3183925 T^{4} - 252807 T^{5} + 40326 T^{6} - 1958 T^{7} + 288 T^{8} - 12 T^{9} + T^{10}$$