Properties

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

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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
1.17030 2.02701i
0.741409 1.28416i
0.341187 0.590953i
−0.664633 + 1.15118i
−1.58826 + 2.75095i
1.17030 + 2.02701i
0.741409 + 1.28416i
0.341187 + 0.590953i
−0.664633 1.15118i
−1.58826 2.75095i
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
201.2 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
201.3 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 0.664633 1.15118i −0.500000 0.866025i 0 −0.664633 1.15118i 2.32927 −1.00000 0.616527 + 1.06786i 0
201.5 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
501.1 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
501.2 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
501.3 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 0.664633 + 1.15118i −0.500000 + 0.866025i 0 −0.664633 + 1.15118i 2.32927 −1.00000 0.616527 1.06786i 0
501.5 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
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 501.5
Significant digits:
Format:

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.o 10
5.b even 2 1 950.2.e.n 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.o 10
95.i even 6 1 950.2.e.n 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 5.b even 2 1
950.2.e.n 10 95.i even 6 1
950.2.e.o 10 1.a even 1 1 trivial
950.2.e.o 10 19.c even 3 1 inner
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

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