# Properties

 Label 1638.2.m.k Level $1638$ Weight $2$ Character orbit 1638.m Analytic conductor $13.079$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1638.m (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.0794958511$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 2 x^{9} + 15 x^{8} + 14 x^{7} + 110 x^{6} + 36 x^{5} + 233 x^{4} + 164 x^{3} + 345 x^{2} + 76 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 546) 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + ( -\beta_{1} - \beta_{2} ) q^{5} + \beta_{4} q^{7} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} + ( -\beta_{1} - \beta_{2} ) q^{5} + \beta_{4} q^{7} + q^{8} + ( -\beta_{1} - \beta_{2} ) q^{10} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{11} + ( \beta_{1} + \beta_{2} + \beta_{7} ) q^{13} + \beta_{4} q^{14} + q^{16} + ( \beta_{2} + \beta_{3} + \beta_{8} - \beta_{9} ) q^{17} + ( 1 + 2 \beta_{1} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{19} + ( -\beta_{1} - \beta_{2} ) q^{20} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{22} + ( \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{8} - \beta_{9} ) q^{23} + ( 1 - 2 \beta_{1} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} ) q^{25} + ( \beta_{1} + \beta_{2} + \beta_{7} ) q^{26} + \beta_{4} q^{28} + ( \beta_{4} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{29} + ( -2 - \beta_{1} - \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{31} + q^{32} + ( \beta_{2} + \beta_{3} + \beta_{8} - \beta_{9} ) q^{34} + ( -\beta_{1} + \beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{35} + ( 1 - \beta_{3} - \beta_{8} + \beta_{9} ) q^{37} + ( 1 + 2 \beta_{1} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{38} + ( -\beta_{1} - \beta_{2} ) q^{40} + ( 2 + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{41} + ( 3 \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{43} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{44} + ( \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{8} - \beta_{9} ) q^{46} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{47} + ( \beta_{2} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{49} + ( 1 - 2 \beta_{1} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} ) q^{50} + ( \beta_{1} + \beta_{2} + \beta_{7} ) q^{52} + ( 3 - \beta_{1} - 2 \beta_{4} + 3 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{53} + ( 1 + \beta_{1} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{55} + \beta_{4} q^{56} + ( \beta_{4} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{58} + ( 4 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{6} + \beta_{8} - 2 \beta_{9} ) q^{59} + ( 3 - \beta_{1} + 3 \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} ) q^{61} + ( -2 - \beta_{1} - \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{62} + q^{64} + ( 3 + 2 \beta_{1} - \beta_{4} + 4 \beta_{5} - \beta_{8} + 2 \beta_{9} ) q^{65} + ( -2 \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{67} + ( \beta_{2} + \beta_{3} + \beta_{8} - \beta_{9} ) q^{68} + ( -\beta_{1} + \beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{70} + ( 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} ) q^{71} + ( 2 + 3 \beta_{1} + 2 \beta_{5} + \beta_{6} + 3 \beta_{7} + \beta_{8} + \beta_{9} ) q^{73} + ( 1 - \beta_{3} - \beta_{8} + \beta_{9} ) q^{74} + ( 1 + 2 \beta_{1} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{76} + ( -3 - 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + \beta_{9} ) q^{77} + ( \beta_{1} + \beta_{2} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} ) q^{79} + ( -\beta_{1} - \beta_{2} ) q^{80} + ( 2 + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{82} + ( -\beta_{3} - \beta_{4} - \beta_{6} - \beta_{9} ) q^{83} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{9} ) q^{85} + ( 3 \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{86} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{88} + ( 3 - 7 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{6} - 4 \beta_{8} + 5 \beta_{9} ) q^{89} + ( 2 \beta_{2} - 2 \beta_{4} - 4 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} ) q^{91} + ( \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{8} - \beta_{9} ) q^{92} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{94} + ( 2 - 2 \beta_{2} + \beta_{4} + \beta_{6} + 3 \beta_{8} - 2 \beta_{9} ) q^{95} + ( 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{97} + ( \beta_{2} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q + 10q^{2} + 10q^{4} + 2q^{5} - 2q^{7} + 10q^{8} + O(q^{10})$$ $$10q + 10q^{2} + 10q^{4} + 2q^{5} - 2q^{7} + 10q^{8} + 2q^{10} - 6q^{11} - 4q^{13} - 2q^{14} + 10q^{16} + 8q^{17} + 3q^{19} + 2q^{20} - 6q^{22} + 12q^{23} - q^{25} - 4q^{26} - 2q^{28} - 10q^{31} + 10q^{32} + 8q^{34} - 16q^{35} - 2q^{37} + 3q^{38} + 2q^{40} + 4q^{41} + 3q^{43} - 6q^{44} + 12q^{46} + 15q^{47} + 4q^{49} - q^{50} - 4q^{52} + 17q^{53} + 3q^{55} - 2q^{56} + 4q^{59} + 11q^{61} - 10q^{62} + 10q^{64} + 4q^{65} - q^{67} + 8q^{68} - 16q^{70} - 18q^{71} + 12q^{73} - 2q^{74} + 3q^{76} - 18q^{77} - 4q^{79} + 2q^{80} + 4q^{82} + q^{85} + 3q^{86} - 6q^{88} + 14q^{89} + 26q^{91} + 12q^{92} + 15q^{94} + 48q^{95} - 6q^{97} + 4q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2 x^{9} + 15 x^{8} + 14 x^{7} + 110 x^{6} + 36 x^{5} + 233 x^{4} + 164 x^{3} + 345 x^{2} + 76 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-503 \nu^{9} - 2241 \nu^{8} + 8466 \nu^{7} - 67528 \nu^{6} + 19422 \nu^{5} - 156870 \nu^{4} + 1003571 \nu^{3} - 301041 \nu^{2} - 66732 \nu + 438544$$$$)/2044008$$ $$\beta_{3}$$ $$=$$ $$($$$$-3064 \nu^{9} + 9207 \nu^{8} - 34782 \nu^{7} - 28346 \nu^{6} - 79794 \nu^{5} + 644490 \nu^{4} + 579550 \nu^{3} + 1236807 \nu^{2} + 274164 \nu + 3620228$$$$)/1022004$$ $$\beta_{4}$$ $$=$$ $$($$$$-3977 \nu^{9} - 10833 \nu^{8} - 12699 \nu^{7} - 334006 \nu^{6} - 625302 \nu^{5} - 1723536 \nu^{4} - 478621 \nu^{3} - 3196425 \nu^{2} - 3391749 \nu - 3118196$$$$)/1022004$$ $$\beta_{5}$$ $$=$$ $$($$$$-27409 \nu^{9} + 54315 \nu^{8} - 413376 \nu^{7} - 375260 \nu^{6} - 3082518 \nu^{5} - 967302 \nu^{4} - 6543167 \nu^{3} - 3491505 \nu^{2} - 9757146 \nu - 2149816$$$$)/2044008$$ $$\beta_{6}$$ $$=$$ $$($$$$-14545 \nu^{9} + 29941 \nu^{8} - 216152 \nu^{7} - 200758 \nu^{6} - 1521222 \nu^{5} - 440214 \nu^{4} - 2887781 \nu^{3} - 2349679 \nu^{2} - 3963994 \nu - 1074620$$$$)/340668$$ $$\beta_{7}$$ $$=$$ $$($$$$45928 \nu^{9} - 96426 \nu^{8} + 695481 \nu^{7} + 561428 \nu^{6} + 5037264 \nu^{5} + 744876 \nu^{4} + 10649492 \nu^{3} + 5556402 \nu^{2} + 16896855 \nu + 165664$$$$)/1022004$$ $$\beta_{8}$$ $$=$$ $$($$$$97465 \nu^{9} - 178473 \nu^{8} + 1399728 \nu^{7} + 1662752 \nu^{6} + 10555542 \nu^{5} + 4653846 \nu^{4} + 20509619 \nu^{3} + 17738031 \nu^{2} + 29957226 \nu + 5377120$$$$)/2044008$$ $$\beta_{9}$$ $$=$$ $$($$$$51344 \nu^{9} - 117222 \nu^{8} + 805587 \nu^{7} + 484042 \nu^{6} + 5520312 \nu^{5} + 367938 \nu^{4} + 11604142 \nu^{3} + 5109630 \nu^{2} + 14145267 \nu + 742892$$$$)/1022004$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{9} + 4 \beta_{5} - \beta_{4} + 2 \beta_{2} + 2 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$-3 \beta_{8} - 3 \beta_{6} - 3 \beta_{4} - 2 \beta_{3} + 11 \beta_{2} - 6$$ $$\nu^{4}$$ $$=$$ $$-16 \beta_{9} - 16 \beta_{8} + 6 \beta_{7} - 18 \beta_{6} - 40 \beta_{5} + 2 \beta_{4} - 37 \beta_{1} - 40$$ $$\nu^{5}$$ $$=$$ $$-65 \beta_{9} + 4 \beta_{8} + 34 \beta_{7} - 4 \beta_{6} - 126 \beta_{5} + 65 \beta_{4} + 34 \beta_{3} - 168 \beta_{2} - 168 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-30 \beta_{9} + 297 \beta_{8} + 267 \beta_{6} + 267 \beta_{4} + 126 \beta_{3} - 657 \beta_{2} + 592$$ $$\nu^{7}$$ $$=$$ $$1080 \beta_{9} + 1080 \beta_{8} - 564 \beta_{7} + 1176 \beta_{6} + 2280 \beta_{5} - 96 \beta_{4} + 2797 \beta_{1} + 2280$$ $$\nu^{8}$$ $$=$$ $$5005 \beta_{9} - 468 \beta_{8} - 2256 \beta_{7} + 468 \beta_{6} + 9784 \beta_{5} - 5005 \beta_{4} - 2256 \beta_{3} + 11402 \beta_{2} + 11402 \beta_{1}$$ $$\nu^{9}$$ $$=$$ $$1788 \beta_{9} - 20451 \beta_{8} - 18663 \beta_{6} - 18663 \beta_{4} - 9542 \beta_{3} + 47603 \beta_{2} - 39726$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times$$.

 $$n$$ $$379$$ $$703$$ $$911$$ $$\chi(n)$$ $$\beta_{5}$$ $$\beta_{5}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 −1.10337 − 1.91109i −0.623307 − 1.07960i −0.114009 − 0.197470i 0.769836 + 1.33339i 2.07085 + 3.58682i −1.10337 + 1.91109i −0.623307 + 1.07960i −0.114009 + 0.197470i 0.769836 − 1.33339i 2.07085 − 3.58682i
1.00000 0 1.00000 −1.10337 + 1.91109i 0 1.19230 2.36187i 1.00000 0 −1.10337 + 1.91109i
289.2 1.00000 0 1.00000 −0.623307 + 1.07960i 0 2.30301 + 1.30235i 1.00000 0 −0.623307 + 1.07960i
289.3 1.00000 0 1.00000 −0.114009 + 0.197470i 0 −2.59452 + 0.518144i 1.00000 0 −0.114009 + 0.197470i
289.4 1.00000 0 1.00000 0.769836 1.33339i 0 −2.22250 + 1.43544i 1.00000 0 0.769836 1.33339i
289.5 1.00000 0 1.00000 2.07085 3.58682i 0 0.321703 2.62612i 1.00000 0 2.07085 3.58682i
1621.1 1.00000 0 1.00000 −1.10337 1.91109i 0 1.19230 + 2.36187i 1.00000 0 −1.10337 1.91109i
1621.2 1.00000 0 1.00000 −0.623307 1.07960i 0 2.30301 1.30235i 1.00000 0 −0.623307 1.07960i
1621.3 1.00000 0 1.00000 −0.114009 0.197470i 0 −2.59452 0.518144i 1.00000 0 −0.114009 0.197470i
1621.4 1.00000 0 1.00000 0.769836 + 1.33339i 0 −2.22250 1.43544i 1.00000 0 0.769836 + 1.33339i
1621.5 1.00000 0 1.00000 2.07085 + 3.58682i 0 0.321703 + 2.62612i 1.00000 0 2.07085 + 3.58682i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1621.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.m.k 10
3.b odd 2 1 546.2.j.e 10
7.c even 3 1 1638.2.p.j 10
13.c even 3 1 1638.2.p.j 10
21.h odd 6 1 546.2.k.e yes 10
39.i odd 6 1 546.2.k.e yes 10
91.h even 3 1 inner 1638.2.m.k 10
273.s odd 6 1 546.2.j.e 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.j.e 10 3.b odd 2 1
546.2.j.e 10 273.s odd 6 1
546.2.k.e yes 10 21.h odd 6 1
546.2.k.e yes 10 39.i odd 6 1
1638.2.m.k 10 1.a even 1 1 trivial
1638.2.m.k 10 91.h even 3 1 inner
1638.2.p.j 10 7.c even 3 1
1638.2.p.j 10 13.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{10} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(1638, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{10}$$
$3$ $$T^{10}$$
$5$ $$16 + 76 T + 345 T^{2} + 164 T^{3} + 233 T^{4} + 36 T^{5} + 110 T^{6} + 14 T^{7} + 15 T^{8} - 2 T^{9} + T^{10}$$
$7$ $$16807 + 4802 T + 1078 T^{3} + 308 T^{4} - 51 T^{5} + 44 T^{6} + 22 T^{7} + 2 T^{9} + T^{10}$$
$11$ $$900 - 630 T + 2391 T^{2} + 2085 T^{3} + 3793 T^{4} + 1062 T^{5} + 555 T^{6} + 58 T^{7} + 48 T^{8} + 6 T^{9} + T^{10}$$
$13$ $$371293 + 114244 T + 52728 T^{2} + 7436 T^{3} + 1118 T^{4} + 15 T^{5} + 86 T^{6} + 44 T^{7} + 24 T^{8} + 4 T^{9} + T^{10}$$
$17$ $$( -28 - 17 T + 71 T^{2} - 20 T^{3} - 4 T^{4} + T^{5} )^{2}$$
$19$ $$12321 + 122211 T + 1200102 T^{2} + 135327 T^{3} + 88183 T^{4} - 804 T^{5} + 3987 T^{6} - 11 T^{7} + 78 T^{8} - 3 T^{9} + T^{10}$$
$23$ $$( -360 - 549 T + 537 T^{2} - 72 T^{3} - 6 T^{4} + T^{5} )^{2}$$
$29$ $$236196 + 196830 T + 177147 T^{2} + 47385 T^{3} + 25029 T^{4} + 2106 T^{5} + 3195 T^{6} + 54 T^{7} + 60 T^{8} + T^{10}$$
$31$ $$1254400 + 1647520 T + 1604961 T^{2} + 675789 T^{3} + 221955 T^{4} + 41274 T^{5} + 7137 T^{6} + 738 T^{7} + 126 T^{8} + 10 T^{9} + T^{10}$$
$37$ $$( -565 + 368 T + 17 T^{2} - 38 T^{3} + T^{4} + T^{5} )^{2}$$
$41$ $$770884 - 194038 T + 160347 T^{2} - 28125 T^{3} + 19689 T^{4} - 3174 T^{5} + 1311 T^{6} - 126 T^{7} + 48 T^{8} - 4 T^{9} + T^{10}$$
$43$ $$308025 - 213120 T + 277881 T^{2} - 22980 T^{3} + 92728 T^{4} - 22221 T^{5} + 10725 T^{6} - 164 T^{7} + 111 T^{8} - 3 T^{9} + T^{10}$$
$47$ $$271854144 - 17164008 T + 19550241 T^{2} - 614784 T^{3} + 1063294 T^{4} - 45738 T^{5} + 18675 T^{6} - 1430 T^{7} + 279 T^{8} - 15 T^{9} + T^{10}$$
$53$ $$256 - 66704 T + 17364273 T^{2} - 4244362 T^{3} + 1078286 T^{4} - 131550 T^{5} + 21575 T^{6} - 2206 T^{7} + 279 T^{8} - 17 T^{9} + T^{10}$$
$59$ $$( -1268 + 1867 T - 440 T^{2} - 167 T^{3} - 2 T^{4} + T^{5} )^{2}$$
$61$ $$6400 + 81280 T + 1064256 T^{2} - 407680 T^{3} + 150992 T^{4} - 25632 T^{5} + 5480 T^{6} - 712 T^{7} + 129 T^{8} - 11 T^{9} + T^{10}$$
$67$ $$400 + 14380 T + 520941 T^{2} - 137601 T^{3} + 138084 T^{4} + 25845 T^{5} + 18249 T^{6} + 261 T^{7} + 138 T^{8} + T^{9} + T^{10}$$
$71$ $$202500 - 2544750 T + 32555925 T^{2} + 7241610 T^{3} + 1686319 T^{4} + 192492 T^{5} + 28812 T^{6} + 2726 T^{7} + 315 T^{8} + 18 T^{9} + T^{10}$$
$73$ $$37941975369 - 3240866106 T + 887285502 T^{2} - 50704044 T^{3} + 11876944 T^{4} - 622851 T^{5} + 90666 T^{6} - 3100 T^{7} + 408 T^{8} - 12 T^{9} + T^{10}$$
$79$ $$817216 - 709640 T + 580065 T^{2} - 159768 T^{3} + 60951 T^{4} - 4344 T^{5} + 4416 T^{6} - 204 T^{7} + 87 T^{8} + 4 T^{9} + T^{10}$$
$83$ $$( -486 + 405 T + 27 T^{2} - 60 T^{3} + T^{5} )^{2}$$
$89$ $$( -453250 + 50575 T + 3770 T^{2} - 464 T^{3} - 7 T^{4} + T^{5} )^{2}$$
$97$ $$1734489 + 1457919 T + 1797027 T^{2} + 159624 T^{3} + 449455 T^{4} + 93495 T^{5} + 60546 T^{6} - 590 T^{7} + 279 T^{8} + 6 T^{9} + T^{10}$$