# Properties

 Label 950.2.e.n Level $950$ Weight $2$ Character orbit 950.e Analytic conductor $7.586$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Defining polynomial: $$x^{10} + 10 x^{8} - 12 x^{7} + 85 x^{6} - 70 x^{5} + 186 x^{4} - 110 x^{3} + 285 x^{2} - 150 x + 100$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{2} + \beta_{1} q^{3} + ( -1 - \beta_{5} ) q^{4} + ( -\beta_{1} + \beta_{2} ) q^{6} + ( -1 + \beta_{2} ) q^{7} + q^{8} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{8} ) q^{9} +O(q^{10})$$ $$q + \beta_{5} q^{2} + \beta_{1} q^{3} + ( -1 - \beta_{5} ) q^{4} + ( -\beta_{1} + \beta_{2} ) q^{6} + ( -1 + \beta_{2} ) q^{7} + q^{8} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{8} ) q^{9} + ( -1 + \beta_{4} ) q^{11} -\beta_{2} q^{12} + ( \beta_{3} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{13} + ( -\beta_{1} - \beta_{5} ) q^{14} + \beta_{5} q^{16} + ( \beta_{4} - \beta_{6} + \beta_{9} ) q^{17} + ( 1 - \beta_{2} + \beta_{3} ) q^{18} + ( 2 - \beta_{2} + \beta_{3} + \beta_{5} - \beta_{7} + \beta_{8} ) q^{19} + ( -2 \beta_{1} - 4 \beta_{5} - \beta_{8} ) q^{21} + ( -\beta_{4} - \beta_{5} + \beta_{6} ) q^{22} + ( 3 + \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{23} + \beta_{1} q^{24} + ( -\beta_{3} + \beta_{7} ) q^{26} + ( 4 - \beta_{2} + \beta_{3} - \beta_{4} ) q^{27} + ( 1 + \beta_{1} - \beta_{2} + \beta_{5} ) q^{28} + ( -2 \beta_{3} + \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{29} + ( \beta_{3} - \beta_{4} - \beta_{7} ) q^{31} + ( -1 - \beta_{5} ) q^{32} + ( -\beta_{1} - 2 \beta_{8} + \beta_{9} ) q^{33} + ( \beta_{6} + \beta_{7} - \beta_{9} ) q^{34} + ( \beta_{1} + \beta_{5} + \beta_{8} ) q^{36} + ( 1 + \beta_{2} + \beta_{4} - \beta_{7} ) q^{37} + ( -1 + \beta_{1} - \beta_{3} + \beta_{5} + \beta_{9} ) q^{38} + ( 2 + \beta_{3} - \beta_{7} ) q^{39} + ( -\beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} ) q^{41} + ( 4 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 4 \beta_{5} + \beta_{8} ) q^{42} -2 \beta_{8} q^{43} + ( 1 + \beta_{5} - \beta_{6} ) q^{44} + ( -3 + \beta_{2} - \beta_{3} + \beta_{7} ) q^{46} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{47} + ( -\beta_{1} + \beta_{2} ) q^{48} + ( -2 - 3 \beta_{2} + \beta_{3} ) q^{49} + ( 2 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{51} + ( -\beta_{8} - \beta_{9} ) q^{52} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{53} + ( \beta_{1} + \beta_{4} + 4 \beta_{5} - \beta_{6} + \beta_{8} ) q^{54} + ( -1 + \beta_{2} ) q^{56} + ( 3 \beta_{2} + 2 \beta_{5} + \beta_{6} - \beta_{9} ) q^{57} + ( 2 \beta_{3} - \beta_{7} ) q^{58} + ( -\beta_{1} - 4 \beta_{5} + \beta_{8} + \beta_{9} ) q^{59} + ( 4 \beta_{1} - 4 \beta_{2} - \beta_{7} + \beta_{9} ) q^{61} + ( \beta_{4} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{62} + ( 5 + 5 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} + 5 \beta_{5} - \beta_{6} + 2 \beta_{8} ) q^{63} + q^{64} + ( \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{66} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{67} + ( -\beta_{4} - \beta_{7} ) q^{68} + ( -2 + 4 \beta_{2} - \beta_{7} ) q^{69} + ( -2 \beta_{1} + 4 \beta_{5} - \beta_{8} - \beta_{9} ) q^{71} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{8} ) q^{72} + ( \beta_{1} + \beta_{4} - 4 \beta_{5} - \beta_{6} + 2 \beta_{9} ) q^{73} + ( -\beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} ) q^{74} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{76} + ( 1 - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{7} ) q^{77} + ( 2 \beta_{5} + \beta_{8} + \beta_{9} ) q^{78} + ( 4 \beta_{1} - \beta_{8} + \beta_{9} ) q^{79} + ( 4 \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{9} ) q^{81} + ( 1 + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} ) q^{82} + ( 2 + \beta_{2} - 2 \beta_{7} ) q^{83} + ( -4 + 2 \beta_{2} - \beta_{3} ) q^{84} + ( 2 \beta_{3} + 2 \beta_{8} ) q^{86} + ( -2 - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{7} ) q^{87} + ( -1 + \beta_{4} ) q^{88} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + 2 \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{89} + ( 2 + 2 \beta_{5} ) q^{91} + ( -\beta_{1} - 3 \beta_{5} - \beta_{8} - \beta_{9} ) q^{92} + ( -2 \beta_{5} + \beta_{8} - 2 \beta_{9} ) q^{93} + ( 2 + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{7} ) q^{94} -\beta_{2} q^{96} + ( -5 \beta_{1} - 2 \beta_{5} - \beta_{9} ) q^{97} + ( 3 \beta_{1} - 2 \beta_{5} + \beta_{8} ) q^{98} + ( 3 + 7 \beta_{1} - 7 \beta_{2} + 2 \beta_{3} + 3 \beta_{5} - \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q - 5q^{2} - 5q^{4} - 10q^{7} + 10q^{8} - 5q^{9} + O(q^{10})$$ $$10q - 5q^{2} - 5q^{4} - 10q^{7} + 10q^{8} - 5q^{9} - 6q^{11} - 2q^{13} + 5q^{14} - 5q^{16} + 4q^{17} + 10q^{18} + 11q^{19} + 20q^{21} + 3q^{22} + 13q^{23} + 4q^{26} + 36q^{27} + 5q^{28} + 2q^{29} - 8q^{31} - 5q^{32} + 2q^{33} + 4q^{34} - 5q^{36} + 10q^{37} - 13q^{38} + 16q^{39} + q^{41} + 20q^{42} + 3q^{44} - 26q^{46} - 10q^{47} - 20q^{49} + 4q^{51} - 2q^{52} + 5q^{53} - 18q^{54} - 10q^{56} - 10q^{57} - 4q^{58} + 22q^{59} - 2q^{61} + 4q^{62} + 23q^{63} + 10q^{64} + 2q^{66} + 4q^{67} - 8q^{68} - 24q^{69} - 22q^{71} - 5q^{72} + 26q^{73} - 5q^{74} + 2q^{76} + 10q^{77} - 8q^{78} + 2q^{79} - 5q^{81} + q^{82} + 12q^{83} - 40q^{84} - 20q^{87} - 6q^{88} - q^{89} + 10q^{91} + 13q^{92} + 6q^{93} + 20q^{94} + 8q^{97} + 10q^{98} + 13q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 10 x^{8} - 12 x^{7} + 85 x^{6} - 70 x^{5} + 186 x^{4} - 110 x^{3} + 285 x^{2} - 150 x + 100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-47 \nu^{9} + 582 \nu^{8} - 1940 \nu^{7} + 5464 \nu^{6} - 22892 \nu^{5} + 61110 \nu^{4} - 163092 \nu^{3} + 106700 \nu^{2} - 58200 \nu + 154050$$$$)/224695$$ $$\beta_{3}$$ $$=$$ $$($$$$535 \nu^{9} - 888 \nu^{8} + 2960 \nu^{7} - 13433 \nu^{6} + 34928 \nu^{5} - 93240 \nu^{4} - 61562 \nu^{3} - 162800 \nu^{2} + 88800 \nu - 740030$$$$)/224695$$ $$\beta_{4}$$ $$=$$ $$($$$$864 \nu^{9} - 4962 \nu^{8} + 16540 \nu^{7} - 51681 \nu^{6} + 195172 \nu^{5} - 521010 \nu^{4} + 855387 \nu^{3} - 909700 \nu^{2} + 496200 \nu - 919600$$$$)/224695$$ $$\beta_{5}$$ $$=$$ $$($$$$-3081 \nu^{9} - 94 \nu^{8} - 29646 \nu^{7} + 33092 \nu^{6} - 250957 \nu^{5} + 169886 \nu^{4} - 450846 \nu^{3} + 12726 \nu^{2} - 664685 \nu - 103640$$$$)/449390$$ $$\beta_{6}$$ $$=$$ $$($$$$1752 \nu^{9} - 2572 \nu^{8} + 23553 \nu^{7} - 41134 \nu^{6} + 250962 \nu^{5} - 359938 \nu^{4} + 959337 \nu^{3} - 846025 \nu^{2} + 1605370 \nu - 866100$$$$)/224695$$ $$\beta_{7}$$ $$=$$ $$($$$$-3642 \nu^{9} + 7809 \nu^{8} - 26030 \nu^{7} + 128908 \nu^{6} - 307154 \nu^{5} + 819945 \nu^{4} - 530181 \nu^{3} + 1431650 \nu^{2} - 780900 \nu + 1304860$$$$)/224695$$ $$\beta_{8}$$ $$=$$ $$($$$$5580 \nu^{9} + 1658 \nu^{8} + 54392 \nu^{7} - 47287 \nu^{6} + 444094 \nu^{5} - 185422 \nu^{4} + 800162 \nu^{3} + 19353 \nu^{2} + 957675 \nu + 202580$$$$)/224695$$ $$\beta_{9}$$ $$=$$ $$($$$$6198 \nu^{9} + 11216 \nu^{8} + 67471 \nu^{7} + 27158 \nu^{6} + 427658 \nu^{5} + 323839 \nu^{4} + 785664 \nu^{3} + 288666 \nu^{2} + 1125350 \nu + 318760$$$$)/224695$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{8} - 4 \beta_{5} - \beta_{3} + \beta_{2} - \beta_{1} - 4$$ $$\nu^{3}$$ $$=$$ $$-\beta_{4} + \beta_{3} - 7 \beta_{2} + 4$$ $$\nu^{4}$$ $$=$$ $$-\beta_{9} + 9 \beta_{8} - \beta_{6} + 28 \beta_{5} + \beta_{4} + 13 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$-16 \beta_{8} + 10 \beta_{6} - 54 \beta_{5} - 16 \beta_{3} + 61 \beta_{2} - 61 \beta_{1} - 54$$ $$\nu^{6}$$ $$=$$ $$10 \beta_{7} - 16 \beta_{4} + 81 \beta_{3} - 147 \beta_{2} + 244$$ $$\nu^{7}$$ $$=$$ $$-6 \beta_{9} + 189 \beta_{8} - 91 \beta_{6} + 608 \beta_{5} + 91 \beta_{4} + 573 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$85 \beta_{9} - 761 \beta_{8} - 85 \beta_{7} + 195 \beta_{6} - 2304 \beta_{5} - 761 \beta_{3} + 1571 \beta_{2} - 1571 \beta_{1} - 2304$$ $$\nu^{9}$$ $$=$$ $$110 \beta_{7} - 846 \beta_{4} + 2046 \beta_{3} - 5567 \beta_{2} + 6454$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/950\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
201.1
 −1.58826 + 2.75095i −0.664633 + 1.15118i 0.341187 − 0.590953i 0.741409 − 1.28416i 1.17030 − 2.02701i −1.58826 − 2.75095i −0.664633 − 1.15118i 0.341187 + 0.590953i 0.741409 + 1.28416i 1.17030 + 2.02701i
−0.500000 + 0.866025i −1.58826 + 2.75095i −0.500000 0.866025i 0 −1.58826 2.75095i −4.17652 1.00000 −3.54514 6.14037i 0
201.2 −0.500000 + 0.866025i −0.664633 + 1.15118i −0.500000 0.866025i 0 −0.664633 1.15118i −2.32927 1.00000 0.616527 + 1.06786i 0
201.3 −0.500000 + 0.866025i 0.341187 0.590953i −0.500000 0.866025i 0 0.341187 + 0.590953i −0.317626 1.00000 1.26718 + 2.19482i 0
201.4 −0.500000 + 0.866025i 0.741409 1.28416i −0.500000 0.866025i 0 0.741409 + 1.28416i 0.482818 1.00000 0.400626 + 0.693904i 0
201.5 −0.500000 + 0.866025i 1.17030 2.02701i −0.500000 0.866025i 0 1.17030 + 2.02701i 1.34059 1.00000 −1.23919 2.14634i 0
501.1 −0.500000 0.866025i −1.58826 2.75095i −0.500000 + 0.866025i 0 −1.58826 + 2.75095i −4.17652 1.00000 −3.54514 + 6.14037i 0
501.2 −0.500000 0.866025i −0.664633 1.15118i −0.500000 + 0.866025i 0 −0.664633 + 1.15118i −2.32927 1.00000 0.616527 1.06786i 0
501.3 −0.500000 0.866025i 0.341187 + 0.590953i −0.500000 + 0.866025i 0 0.341187 0.590953i −0.317626 1.00000 1.26718 2.19482i 0
501.4 −0.500000 0.866025i 0.741409 + 1.28416i −0.500000 + 0.866025i 0 0.741409 1.28416i 0.482818 1.00000 0.400626 0.693904i 0
501.5 −0.500000 0.866025i 1.17030 + 2.02701i −0.500000 + 0.866025i 0 1.17030 2.02701i 1.34059 1.00000 −1.23919 + 2.14634i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 501.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.e.n 10
5.b even 2 1 950.2.e.o 10
5.c odd 4 2 190.2.i.a 20
15.e even 4 2 1710.2.t.d 20
19.c even 3 1 inner 950.2.e.n 10
95.i even 6 1 950.2.e.o 10
95.m odd 12 2 190.2.i.a 20
285.v even 12 2 1710.2.t.d 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.i.a 20 5.c odd 4 2
190.2.i.a 20 95.m odd 12 2
950.2.e.n 10 1.a even 1 1 trivial
950.2.e.n 10 19.c even 3 1 inner
950.2.e.o 10 5.b even 2 1
950.2.e.o 10 95.i even 6 1
1710.2.t.d 20 15.e even 4 2
1710.2.t.d 20 285.v even 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$:

 $$T_{3}^{10} + \cdots$$ $$T_{7}^{5} + 5 T_{7}^{4} - 14 T_{7}^{2} + 2 T_{7} + 2$$ $$T_{11}^{5} + 3 T_{11}^{4} - 48 T_{11}^{3} - 176 T_{11}^{2} + 389 T_{11} + 1555$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{5}$$
$3$ $$100 - 150 T + 285 T^{2} - 110 T^{3} + 186 T^{4} - 70 T^{5} + 85 T^{6} - 12 T^{7} + 10 T^{8} + T^{10}$$
$5$ $$T^{10}$$
$7$ $$( 2 + 2 T - 14 T^{2} + 5 T^{4} + T^{5} )^{2}$$
$11$ $$( 1555 + 389 T - 176 T^{2} - 48 T^{3} + 3 T^{4} + T^{5} )^{2}$$
$13$ $$256 + 640 T + 1600 T^{2} + 896 T^{3} + 1088 T^{4} - 144 T^{5} + 744 T^{6} - 56 T^{7} + 32 T^{8} + 2 T^{9} + T^{10}$$
$17$ $$256 - 2496 T + 19856 T^{2} - 41888 T^{3} + 69728 T^{4} - 16912 T^{5} + 4412 T^{6} - 336 T^{7} + 72 T^{8} - 4 T^{9} + T^{10}$$
$19$ $$2476099 - 1433531 T + 267501 T^{2} - 1444 T^{3} - 4351 T^{4} + 543 T^{5} - 229 T^{6} - 4 T^{7} + 39 T^{8} - 11 T^{9} + T^{10}$$
$23$ $$917764 - 783644 T + 529256 T^{2} - 188404 T^{3} + 63218 T^{4} - 15054 T^{5} + 4012 T^{6} - 760 T^{7} + 133 T^{8} - 13 T^{9} + T^{10}$$
$29$ $$4000000 + 2100000 T + 1222500 T^{2} + 233000 T^{3} + 85300 T^{4} + 10640 T^{5} + 4306 T^{6} + 268 T^{7} + 78 T^{8} - 2 T^{9} + T^{10}$$
$31$ $$( 1252 + 342 T - 232 T^{2} - 66 T^{3} + 4 T^{4} + T^{5} )^{2}$$
$37$ $$( -7758 + 2034 T + 434 T^{2} - 104 T^{3} - 5 T^{4} + T^{5} )^{2}$$
$41$ $$6561 - 18225 T + 42687 T^{2} - 32094 T^{3} + 23473 T^{4} + 6445 T^{5} + 3521 T^{6} + 258 T^{7} + 63 T^{8} - T^{9} + T^{10}$$
$43$ $$1327104 + 2101248 T + 3363840 T^{2} + 144384 T^{3} + 161536 T^{4} + 3968 T^{5} + 5920 T^{6} + 64 T^{7} + 88 T^{8} + T^{10}$$
$47$ $$129600 + 28080 T + 151524 T^{2} + 18888 T^{3} + 165076 T^{4} + 27080 T^{5} + 8862 T^{6} + 108 T^{7} + 170 T^{8} + 10 T^{9} + T^{10}$$
$53$ $$62726400 - 17487360 T + 8233344 T^{2} - 647808 T^{3} + 360976 T^{4} - 28240 T^{5} + 9912 T^{6} - 348 T^{7} + 125 T^{8} - 5 T^{9} + T^{10}$$
$59$ $$15376 - 88412 T + 531433 T^{2} + 96410 T^{3} + 141422 T^{4} - 58404 T^{5} + 17937 T^{6} - 2840 T^{7} + 338 T^{8} - 22 T^{9} + T^{10}$$
$61$ $$62853184 + 43714992 T + 31640964 T^{2} + 1708488 T^{3} + 901748 T^{4} + 11144 T^{5} + 21042 T^{6} - 12 T^{7} + 166 T^{8} + 2 T^{9} + T^{10}$$
$67$ $$656100 - 838350 T + 1617165 T^{2} + 409230 T^{3} + 635266 T^{4} - 112502 T^{5} + 33345 T^{6} - 636 T^{7} + 194 T^{8} - 4 T^{9} + T^{10}$$
$71$ $$952576 + 476288 T + 425536 T^{2} + 179584 T^{3} + 126656 T^{4} + 47376 T^{5} + 15864 T^{6} + 2696 T^{7} + 344 T^{8} + 22 T^{9} + T^{10}$$
$73$ $$360620100 - 168916050 T + 63245385 T^{2} - 12829380 T^{3} + 2455726 T^{4} - 324838 T^{5} + 50795 T^{6} - 5364 T^{7} + 534 T^{8} - 26 T^{9} + T^{10}$$
$79$ $$19360000 - 19184000 T + 18024000 T^{2} - 2842240 T^{3} + 965696 T^{4} + 60528 T^{5} + 40136 T^{6} + 872 T^{7} + 216 T^{8} - 2 T^{9} + T^{10}$$
$83$ $$( 792 + 2619 T + 262 T^{2} - 118 T^{3} - 6 T^{4} + T^{5} )^{2}$$
$89$ $$397922704 - 138917872 T + 51369808 T^{2} - 5859296 T^{3} + 1238492 T^{4} - 58644 T^{5} + 22476 T^{6} - 460 T^{7} + 173 T^{8} + T^{9} + T^{10}$$
$97$ $$83905600 + 99083720 T + 107261249 T^{2} + 15576328 T^{3} + 3606750 T^{4} - 53976 T^{5} + 46979 T^{6} - 352 T^{7} + 286 T^{8} - 8 T^{9} + T^{10}$$