Properties

Label 1110.2.i.p
Level $1110$
Weight $2$
Character orbit 1110.i
Analytic conductor $8.863$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 2 x^{9} + 15 x^{8} - 6 x^{7} + 123 x^{6} - 62 x^{5} + 458 x^{4} + 100 x^{3} + 844 x^{2} - 312 x + 144\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{6} ) q^{2} + \beta_{6} q^{3} + \beta_{6} q^{4} + \beta_{6} q^{5} + q^{6} -\beta_{3} q^{7} + q^{8} + ( -1 - \beta_{6} ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{6} ) q^{2} + \beta_{6} q^{3} + \beta_{6} q^{4} + \beta_{6} q^{5} + q^{6} -\beta_{3} q^{7} + q^{8} + ( -1 - \beta_{6} ) q^{9} + q^{10} + ( -\beta_{4} - \beta_{5} ) q^{11} + ( -1 - \beta_{6} ) q^{12} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{6} ) q^{13} -\beta_{4} q^{14} + ( -1 - \beta_{6} ) q^{15} + ( -1 - \beta_{6} ) q^{16} -\beta_{8} q^{17} + \beta_{6} q^{18} + ( -\beta_{1} - \beta_{2} + \beta_{6} + \beta_{9} ) q^{19} + ( -1 - \beta_{6} ) q^{20} + ( \beta_{3} + \beta_{4} ) q^{21} + ( \beta_{3} + \beta_{4} + \beta_{5} - \beta_{9} ) q^{22} + ( -1 - 2 \beta_{2} + \beta_{4} - \beta_{7} ) q^{23} + \beta_{6} q^{24} + ( -1 - \beta_{6} ) q^{25} + ( -1 - 2 \beta_{2} ) q^{26} + q^{27} + ( \beta_{3} + \beta_{4} ) q^{28} + ( 2 + \beta_{2} - \beta_{4} + \beta_{5} ) q^{29} + \beta_{6} q^{30} + ( -1 + \beta_{2} - \beta_{7} ) q^{31} + \beta_{6} q^{32} + ( -\beta_{3} + \beta_{9} ) q^{33} + ( \beta_{7} + \beta_{8} ) q^{34} + ( \beta_{3} + \beta_{4} ) q^{35} + q^{36} + ( -\beta_{1} - \beta_{2} - \beta_{5} + \beta_{8} ) q^{37} + ( 1 + \beta_{2} - \beta_{5} ) q^{38} + ( 1 - 2 \beta_{1} + \beta_{6} ) q^{39} + \beta_{6} q^{40} + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{6} - \beta_{9} ) q^{41} -\beta_{3} q^{42} + ( 1 + \beta_{2} - 2 \beta_{4} - \beta_{7} ) q^{43} + ( -\beta_{3} + \beta_{9} ) q^{44} + q^{45} + ( 1 - 2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{8} ) q^{46} + ( 4 + 2 \beta_{2} + \beta_{4} - \beta_{5} ) q^{47} + q^{48} + ( -2 + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{8} + \beta_{9} ) q^{49} + \beta_{6} q^{50} -\beta_{7} q^{51} + ( 1 - 2 \beta_{1} + \beta_{6} ) q^{52} + ( -1 + 2 \beta_{1} - \beta_{5} - \beta_{6} + \beta_{9} ) q^{53} + ( -1 - \beta_{6} ) q^{54} + ( -\beta_{3} + \beta_{9} ) q^{55} -\beta_{3} q^{56} + ( -1 + \beta_{1} + \beta_{5} - \beta_{6} - \beta_{9} ) q^{57} + ( -2 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{9} ) q^{58} + ( -2 + 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{8} + \beta_{9} ) q^{59} + q^{60} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{6} + 3 \beta_{9} ) q^{61} + ( 1 + \beta_{1} + \beta_{6} - \beta_{8} ) q^{62} -\beta_{4} q^{63} + q^{64} + ( 1 - 2 \beta_{1} + \beta_{6} ) q^{65} + ( -\beta_{4} - \beta_{5} ) q^{66} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{6} + 2 \beta_{9} ) q^{67} -\beta_{7} q^{68} + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{69} -\beta_{3} q^{70} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{71} + ( -1 - \beta_{6} ) q^{72} + ( -1 + \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{73} + ( \beta_{2} + \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{74} + q^{75} + ( -1 + \beta_{1} + \beta_{5} - \beta_{6} - \beta_{9} ) q^{76} + ( -2 \beta_{1} - 2 \beta_{2} + 7 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{77} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{6} ) q^{78} + ( \beta_{1} + \beta_{2} + 3 \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{79} + q^{80} + \beta_{6} q^{81} + ( 1 - 2 \beta_{2} + \beta_{5} ) q^{82} + ( -1 + \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{8} + 2 \beta_{9} ) q^{83} -\beta_{4} q^{84} -\beta_{7} q^{85} + ( -1 + \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{8} ) q^{86} + ( -\beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{6} - \beta_{9} ) q^{87} + ( -\beta_{4} - \beta_{5} ) q^{88} + ( -5 - \beta_{3} - \beta_{4} - 5 \beta_{6} + \beta_{8} ) q^{89} + ( -1 - \beta_{6} ) q^{90} + ( -4 + 2 \beta_{1} + \beta_{3} + \beta_{4} - 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{8} + 2 \beta_{9} ) q^{91} + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{92} + ( -\beta_{1} - \beta_{2} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{93} + ( -4 + 2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - 4 \beta_{6} - \beta_{9} ) q^{94} + ( -1 + \beta_{1} + \beta_{5} - \beta_{6} - \beta_{9} ) q^{95} + ( -1 - \beta_{6} ) q^{96} + ( 1 + \beta_{2} - 2 \beta_{4} + \beta_{7} ) q^{97} + ( -\beta_{3} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{98} + ( \beta_{3} + \beta_{4} + \beta_{5} - \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 5q^{2} - 5q^{3} - 5q^{4} - 5q^{5} + 10q^{6} + 10q^{8} - 5q^{9} + O(q^{10}) \) \( 10q - 5q^{2} - 5q^{3} - 5q^{4} - 5q^{5} + 10q^{6} + 10q^{8} - 5q^{9} + 10q^{10} - 2q^{11} - 5q^{12} + q^{13} - 5q^{15} - 5q^{16} - 5q^{18} - 2q^{19} - 5q^{20} + q^{22} - 2q^{23} - 5q^{24} - 5q^{25} - 2q^{26} + 10q^{27} + 18q^{29} - 5q^{30} - 14q^{31} - 5q^{32} + q^{33} + 10q^{36} + 4q^{38} + q^{39} - 5q^{40} - 10q^{41} + 6q^{43} + q^{44} + 10q^{45} + q^{46} + 30q^{47} + 10q^{48} - 11q^{49} - 5q^{50} + q^{52} - 2q^{53} - 5q^{54} + q^{55} - 2q^{57} - 9q^{58} - 7q^{59} + 10q^{60} - 6q^{61} + 7q^{62} + 10q^{64} + q^{65} - 2q^{66} - 5q^{67} + q^{69} + 3q^{71} - 5q^{72} - 18q^{73} - 3q^{74} + 10q^{75} - 2q^{76} - 32q^{77} + q^{78} - 15q^{79} + 10q^{80} - 5q^{81} + 20q^{82} - 5q^{83} - 3q^{86} - 9q^{87} - 2q^{88} - 25q^{89} - 5q^{90} - 18q^{91} + q^{92} + 7q^{93} - 15q^{94} - 2q^{95} - 5q^{96} + 6q^{97} - 11q^{98} + q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} + 15 x^{8} - 6 x^{7} + 123 x^{6} - 62 x^{5} + 458 x^{4} + 100 x^{3} + 844 x^{2} - 312 x + 144\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 52206 \nu^{9} - 298647 \nu^{8} + 1382625 \nu^{7} - 3088861 \nu^{6} + 9390789 \nu^{5} - 22088817 \nu^{4} + 68253733 \nu^{3} - 53999802 \nu^{2} + 20573460 \nu - 57589572 \)\()/ 131893214 \)
\(\beta_{3}\)\(=\)\((\)\(1026853 \nu^{9} + 4535866 \nu^{8} - 8787045 \nu^{7} + 105828882 \nu^{6} - 91335981 \nu^{5} + 696971194 \nu^{4} - 1318853566 \nu^{3} + 2363790868 \nu^{2} - 3507217508 \nu + 1346033064\)\()/ 1582718568 \)
\(\beta_{4}\)\(=\)\((\)\(223413 \nu^{9} - 1789641 \nu^{8} + 8285375 \nu^{7} - 21741437 \nu^{6} + 56274267 \nu^{5} - 132367151 \nu^{4} + 354916248 \nu^{3} - 323593606 \nu^{2} + 123286380 \nu - 204447812\)\()/ 263786428 \)
\(\beta_{5}\)\(=\)\((\)\(-246441 \nu^{9} + 898182 \nu^{8} - 4158250 \nu^{7} + 6058312 \nu^{6} - 28242834 \nu^{5} + 66432202 \nu^{4} - 127474135 \nu^{3} + 162404612 \nu^{2} - 61874760 \nu + 709537978\)\()/ 131893214 \)
\(\beta_{6}\)\(=\)\((\)\(4799131 \nu^{9} - 8971790 \nu^{8} + 68403201 \nu^{7} - 12203286 \nu^{6} + 553226781 \nu^{5} - 184856654 \nu^{4} + 1932936194 \nu^{3} + 1298957896 \nu^{2} + 3402468940 \nu - 1250447352\)\()/ 1582718568 \)
\(\beta_{7}\)\(=\)\((\)\(1710145 \nu^{9} - 7020351 \nu^{8} + 32501625 \nu^{7} - 63953489 \nu^{6} + 220751037 \nu^{5} - 519245961 \nu^{4} + 850207980 \nu^{3} - 1269383466 \nu^{2} + 483624180 \nu - 2060561952\)\()/ 263786428 \)
\(\beta_{8}\)\(=\)\((\)\(3473146 \nu^{9} - 3300175 \nu^{8} + 47030658 \nu^{7} + 24599493 \nu^{6} + 448160006 \nu^{5} + 141819053 \nu^{4} + 1655609196 \nu^{3} + 1332828340 \nu^{2} + 4213451412 \nu + 320551920\)\()/ 263786428 \)
\(\beta_{9}\)\(=\)\((\)\(-24622127 \nu^{9} + 48442714 \nu^{8} - 358607505 \nu^{7} + 98082762 \nu^{6} - 2878823373 \nu^{5} + 1189349074 \nu^{4} - 10483725766 \nu^{3} - 4264073288 \nu^{2} - 18841944788 \nu + 6943311624\)\()/ 1582718568 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9} + 5 \beta_{6} + \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{5} - 2 \beta_{4} + 9 \beta_{2} - 3\)
\(\nu^{4}\)\(=\)\(-11 \beta_{9} - 2 \beta_{8} - 39 \beta_{6} + 11 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} - 17 \beta_{1} - 39\)
\(\nu^{5}\)\(=\)\(-19 \beta_{9} - 4 \beta_{8} - 4 \beta_{7} - 53 \beta_{6} - 30 \beta_{3} - 93 \beta_{2} - 93 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-30 \beta_{7} - 119 \beta_{5} + 76 \beta_{4} - 233 \beta_{2} + 363\)
\(\nu^{7}\)\(=\)\(279 \beta_{9} + 76 \beta_{8} + 745 \beta_{6} - 279 \beta_{5} + 374 \beta_{4} + 374 \beta_{3} + 1029 \beta_{1} + 745\)
\(\nu^{8}\)\(=\)\(1327 \beta_{9} + 374 \beta_{8} + 374 \beta_{7} + 3763 \beta_{6} + 1084 \beta_{3} + 2985 \beta_{2} + 2985 \beta_{1}\)
\(\nu^{9}\)\(=\)\(1084 \beta_{7} + 3695 \beta_{5} - 4486 \beta_{4} + 11813 \beta_{2} - 9729\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1110\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(\beta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
1.10642 1.91637i
0.201664 0.349293i
−1.29048 + 2.23517i
1.73363 3.00274i
−0.751235 + 1.30118i
1.10642 + 1.91637i
0.201664 + 0.349293i
−1.29048 2.23517i
1.73363 + 3.00274i
−0.751235 1.30118i
−0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.00000 −2.44097 4.22788i 1.00000 −0.500000 + 0.866025i 1.00000
121.2 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.00000 −0.330923 0.573175i 1.00000 −0.500000 + 0.866025i 1.00000
121.3 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.00000 −0.301577 0.522347i 1.00000 −0.500000 + 0.866025i 1.00000
121.4 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.00000 0.980808 + 1.69881i 1.00000 −0.500000 + 0.866025i 1.00000
121.5 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.00000 2.09266 + 3.62459i 1.00000 −0.500000 + 0.866025i 1.00000
211.1 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 −2.44097 + 4.22788i 1.00000 −0.500000 0.866025i 1.00000
211.2 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 −0.330923 + 0.573175i 1.00000 −0.500000 0.866025i 1.00000
211.3 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 −0.301577 + 0.522347i 1.00000 −0.500000 0.866025i 1.00000
211.4 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 0.980808 1.69881i 1.00000 −0.500000 0.866025i 1.00000
211.5 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 2.09266 3.62459i 1.00000 −0.500000 0.866025i 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 211.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.i.p 10
37.c even 3 1 inner 1110.2.i.p 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.i.p 10 1.a even 1 1 trivial
1110.2.i.p 10 37.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1110, [\chi])\):

\(T_{7}^{10} + \cdots\)
\( T_{11}^{5} + T_{11}^{4} - 41 T_{11}^{3} - T_{11}^{2} + 404 T_{11} - 492 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{5} \)
$3$ \( ( 1 + T + T^{2} )^{5} \)
$5$ \( ( 1 + T + T^{2} )^{5} \)
$7$ \( 256 + 672 T + 1572 T^{2} + 1240 T^{3} + 1110 T^{4} - 260 T^{5} + 487 T^{6} - 24 T^{7} + 23 T^{8} + T^{10} \)
$11$ \( ( -492 + 404 T - T^{2} - 41 T^{3} + T^{4} + T^{5} )^{2} \)
$13$ \( 9409 - 48209 T + 250307 T^{2} + 7198 T^{3} + 25909 T^{4} - 803 T^{5} + 2037 T^{6} - 18 T^{7} + 51 T^{8} - T^{9} + T^{10} \)
$17$ \( 5184 - 27936 T + 154144 T^{2} + 11768 T^{3} + 23064 T^{4} - 2722 T^{5} + 2421 T^{6} - 100 T^{7} + 53 T^{8} + T^{10} \)
$19$ \( 30976 + 41536 T + 52880 T^{2} + 20320 T^{3} + 10996 T^{4} - 1520 T^{5} + 1941 T^{6} - 126 T^{7} + 51 T^{8} + 2 T^{9} + T^{10} \)
$23$ \( ( 6144 + 1952 T - 196 T^{2} - 88 T^{3} + T^{4} + T^{5} )^{2} \)
$29$ \( ( 192 - 574 T + 435 T^{2} - 37 T^{3} - 9 T^{4} + T^{5} )^{2} \)
$31$ \( ( 4064 + 536 T - 356 T^{2} - 50 T^{3} + 7 T^{4} + T^{5} )^{2} \)
$37$ \( 69343957 + 1266325 T^{2} - 243682 T^{3} + 71965 T^{4} + 1886 T^{5} + 1945 T^{6} - 178 T^{7} + 25 T^{8} + T^{10} \)
$41$ \( 2359296 - 2187264 T + 3152128 T^{2} + 1195968 T^{3} + 449264 T^{4} + 66616 T^{5} + 11244 T^{6} + 964 T^{7} + 150 T^{8} + 10 T^{9} + T^{10} \)
$43$ \( ( -24064 + 5280 T + 696 T^{2} - 164 T^{3} - 3 T^{4} + T^{5} )^{2} \)
$47$ \( ( -6792 + 446 T + 1071 T^{2} - 55 T^{3} - 15 T^{4} + T^{5} )^{2} \)
$53$ \( 331776 - 36864 T + 195328 T^{2} + 106496 T^{3} + 104336 T^{4} + 25400 T^{5} + 6204 T^{6} + 516 T^{7} + 78 T^{8} + 2 T^{9} + T^{10} \)
$59$ \( 337971456 - 256383264 T + 181787572 T^{2} - 17835950 T^{3} + 3716127 T^{4} - 59535 T^{5} + 40620 T^{6} - 179 T^{7} + 272 T^{8} + 7 T^{9} + T^{10} \)
$61$ \( 43992545536 + 6223523968 T + 1346059264 T^{2} + 80948960 T^{3} + 14055136 T^{4} + 630680 T^{5} + 106148 T^{6} + 2340 T^{7} + 386 T^{8} + 6 T^{9} + T^{10} \)
$67$ \( 3063401104 + 735464224 T + 221015388 T^{2} + 17557216 T^{3} + 3756509 T^{4} + 127233 T^{5} + 55752 T^{6} + 331 T^{7} + 280 T^{8} + 5 T^{9} + T^{10} \)
$71$ \( 82944 + 565632 T + 3814096 T^{2} + 347592 T^{3} + 204052 T^{4} - 1728 T^{5} + 6950 T^{6} - 24 T^{7} + 101 T^{8} - 3 T^{9} + T^{10} \)
$73$ \( ( -2048 - 3072 T - 1440 T^{2} - 192 T^{3} + 9 T^{4} + T^{5} )^{2} \)
$79$ \( 262144 + 1622016 T + 10544128 T^{2} - 3070976 T^{3} + 1198144 T^{4} - 25088 T^{5} + 16612 T^{6} + 934 T^{7} + 295 T^{8} + 15 T^{9} + T^{10} \)
$83$ \( 56430144 - 17067264 T + 9654160 T^{2} - 1616096 T^{3} + 845020 T^{4} - 148636 T^{5} + 33942 T^{6} - 2186 T^{7} + 223 T^{8} + 5 T^{9} + T^{10} \)
$89$ \( 5308416 + 3409920 T + 2383936 T^{2} + 622176 T^{3} + 304416 T^{4} + 85304 T^{5} + 25624 T^{6} + 3882 T^{7} + 463 T^{8} + 25 T^{9} + T^{10} \)
$97$ \( ( 128 + 912 T - 172 T^{2} - 108 T^{3} - 3 T^{4} + T^{5} )^{2} \)
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