Properties

Label 1856.4.a.bg
Level $1856$
Weight $4$
Character orbit 1856.a
Self dual yes
Analytic conductor $109.508$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.507544971\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 4 x^{8} - 138 x^{7} + 394 x^{6} + 5872 x^{5} - 10822 x^{4} - 85158 x^{3} + 30654 x^{2} + 439999 x + 396802\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 928)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( -1 - \beta_{1} + \beta_{4} ) q^{5} + ( -1 - \beta_{6} ) q^{7} + ( 5 + 2 \beta_{1} - \beta_{4} + \beta_{5} + \beta_{8} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( -1 - \beta_{1} + \beta_{4} ) q^{5} + ( -1 - \beta_{6} ) q^{7} + ( 5 + 2 \beta_{1} - \beta_{4} + \beta_{5} + \beta_{8} ) q^{9} + ( 7 + \beta_{2} - \beta_{4} + \beta_{5} ) q^{11} + ( -6 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{13} + ( -18 - 6 \beta_{1} - \beta_{2} + 3 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{8} ) q^{15} + ( -9 - \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{17} + ( 5 + \beta_{1} - 2 \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{19} + ( -6 - 6 \beta_{1} + \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{8} ) q^{21} + ( -4 - 3 \beta_{1} + 3 \beta_{2} - \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{8} ) q^{23} + ( 7 + 13 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - \beta_{6} - 4 \beta_{7} - \beta_{8} ) q^{25} + ( 37 + 10 \beta_{1} - \beta_{2} - 2 \beta_{3} - 7 \beta_{4} - \beta_{5} - 2 \beta_{7} + 4 \beta_{8} ) q^{27} + 29 q^{29} + ( 4 + 4 \beta_{1} - 6 \beta_{2} + 6 \beta_{4} + \beta_{5} + 4 \beta_{6} - \beta_{7} + 3 \beta_{8} ) q^{31} + ( -14 + 19 \beta_{1} - 7 \beta_{2} - 3 \beta_{3} + \beta_{4} - 5 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} ) q^{33} + ( 56 + 3 \beta_{1} - 4 \beta_{4} - \beta_{5} + 2 \beta_{6} - 5 \beta_{7} - \beta_{8} ) q^{35} + ( 11 - 3 \beta_{1} + 3 \beta_{2} + \beta_{3} - 8 \beta_{4} - 2 \beta_{7} + 2 \beta_{8} ) q^{37} + ( -58 - 20 \beta_{1} + 8 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - \beta_{7} + \beta_{8} ) q^{39} + ( -48 + 14 \beta_{1} + 8 \beta_{3} - 6 \beta_{4} - \beta_{6} + \beta_{7} ) q^{41} + ( 3 - 26 \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - 8 \beta_{5} + 7 \beta_{6} + 5 \beta_{7} - 5 \beta_{8} ) q^{43} + ( -101 - 54 \beta_{1} + 6 \beta_{2} + 2 \beta_{4} - 4 \beta_{5} + 6 \beta_{6} + 3 \beta_{7} - 13 \beta_{8} ) q^{45} + ( -122 - 20 \beta_{1} - 12 \beta_{2} - \beta_{3} - \beta_{4} + 4 \beta_{6} + 5 \beta_{7} - \beta_{8} ) q^{47} + ( -43 + 16 \beta_{1} + \beta_{2} - 9 \beta_{3} - 6 \beta_{4} - 5 \beta_{5} - 3 \beta_{6} + \beta_{7} - 7 \beta_{8} ) q^{49} + ( 51 - 48 \beta_{1} - 9 \beta_{2} - 9 \beta_{3} + 16 \beta_{4} - 3 \beta_{5} + 12 \beta_{6} + 11 \beta_{7} - 7 \beta_{8} ) q^{51} + ( -18 - \beta_{1} - 2 \beta_{2} - 8 \beta_{3} - 21 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - 7 \beta_{7} + \beta_{8} ) q^{53} + ( -60 - 18 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + 6 \beta_{5} + \beta_{6} + 4 \beta_{8} ) q^{55} + ( -4 + 37 \beta_{1} + 18 \beta_{2} - 14 \beta_{4} + 3 \beta_{5} - 9 \beta_{6} - 6 \beta_{7} + 6 \beta_{8} ) q^{57} + ( 76 - 45 \beta_{1} + 4 \beta_{2} + 16 \beta_{3} + 16 \beta_{4} + \beta_{5} + 2 \beta_{6} + 5 \beta_{7} - 7 \beta_{8} ) q^{59} + ( -113 - 39 \beta_{1} + 4 \beta_{3} + 12 \beta_{4} - 11 \beta_{5} + 20 \beta_{6} + 12 \beta_{7} - 3 \beta_{8} ) q^{61} + ( -179 - 29 \beta_{1} + 3 \beta_{2} + 9 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 8 \beta_{6} - 10 \beta_{8} ) q^{63} + ( -76 + 20 \beta_{1} - 4 \beta_{2} - 8 \beta_{3} - 19 \beta_{4} - 7 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} + 3 \beta_{8} ) q^{65} + ( 153 - 53 \beta_{1} + 17 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} + 6 \beta_{5} - 2 \beta_{6} - 14 \beta_{7} - 12 \beta_{8} ) q^{67} + ( -19 + 27 \beta_{1} - 30 \beta_{2} - 10 \beta_{3} - 18 \beta_{4} - \beta_{5} + 6 \beta_{6} + 9 \beta_{8} ) q^{69} + ( -196 - 14 \beta_{1} - 12 \beta_{2} - 20 \beta_{3} - 28 \beta_{4} - 4 \beta_{5} - 12 \beta_{6} - 4 \beta_{8} ) q^{71} + ( -87 - 9 \beta_{1} - 21 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} + 14 \beta_{5} - 4 \beta_{6} + 18 \beta_{7} ) q^{73} + ( 352 + 32 \beta_{1} - 7 \beta_{2} + 9 \beta_{3} - 14 \beta_{4} + 19 \beta_{5} - 21 \beta_{6} - 11 \beta_{7} + 25 \beta_{8} ) q^{75} + ( -137 + 7 \beta_{1} - 2 \beta_{2} + 10 \beta_{3} - 12 \beta_{4} + \beta_{5} - 14 \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{77} + ( -304 - 48 \beta_{1} + 21 \beta_{2} + 10 \beta_{3} - 11 \beta_{4} + \beta_{5} - 3 \beta_{6} - 10 \beta_{7} - 10 \beta_{8} ) q^{79} + ( 19 + 85 \beta_{1} + 5 \beta_{2} + \beta_{3} - 24 \beta_{4} - 8 \beta_{5} - 6 \beta_{6} - 17 \beta_{7} + 15 \beta_{8} ) q^{81} + ( 333 - 50 \beta_{1} + 3 \beta_{2} + 17 \beta_{3} + 6 \beta_{4} + 25 \beta_{5} - 10 \beta_{6} - 7 \beta_{7} + 11 \beta_{8} ) q^{83} + ( -18 + 68 \beta_{1} - 9 \beta_{2} - 11 \beta_{3} - 20 \beta_{4} + 3 \beta_{5} + 8 \beta_{7} + 27 \beta_{8} ) q^{85} + 29 \beta_{1} q^{87} + ( -44 + 4 \beta_{1} - 18 \beta_{2} - 6 \beta_{3} - 2 \beta_{4} + 20 \beta_{5} + 7 \beta_{6} - 9 \beta_{7} + 2 \beta_{8} ) q^{89} + ( 349 - 2 \beta_{1} - \beta_{2} - 35 \beta_{3} + 6 \beta_{4} - 13 \beta_{5} - 6 \beta_{6} - \beta_{7} + \beta_{8} ) q^{91} + ( 165 + 57 \beta_{1} + 27 \beta_{2} - 7 \beta_{3} + 15 \beta_{4} + 7 \beta_{5} + 11 \beta_{6} - 23 \beta_{7} + 15 \beta_{8} ) q^{93} + ( -472 - 56 \beta_{1} - 3 \beta_{2} - 10 \beta_{3} + 9 \beta_{4} - 18 \beta_{5} + 15 \beta_{6} + 15 \beta_{7} - 5 \beta_{8} ) q^{95} + ( -229 - 91 \beta_{1} + 11 \beta_{2} + 13 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} + 15 \beta_{6} + 15 \beta_{7} - 10 \beta_{8} ) q^{97} + ( 469 - 55 \beta_{1} + 23 \beta_{2} + 2 \beta_{3} + 29 \beta_{4} + 3 \beta_{5} + 12 \beta_{6} + 2 \beta_{7} + 12 \beta_{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 4 q^{3} - 10 q^{5} - 12 q^{7} + 49 q^{9} + O(q^{10}) \) \( 9 q + 4 q^{3} - 10 q^{5} - 12 q^{7} + 49 q^{9} + 64 q^{11} - 70 q^{13} - 170 q^{15} - 66 q^{17} + 42 q^{19} - 76 q^{21} - 40 q^{23} + 111 q^{25} + 322 q^{27} + 261 q^{29} + 64 q^{31} - 52 q^{33} + 496 q^{35} + 54 q^{37} - 590 q^{39} - 378 q^{41} - 32 q^{43} - 1046 q^{45} - 1164 q^{47} - 351 q^{49} + 376 q^{51} - 278 q^{53} - 614 q^{55} + 28 q^{57} + 640 q^{59} - 1054 q^{61} - 1660 q^{63} - 708 q^{65} + 1184 q^{67} - 188 q^{69} - 1988 q^{71} - 750 q^{73} + 3126 q^{75} - 1260 q^{77} - 2916 q^{79} + 293 q^{81} + 2832 q^{83} - 56 q^{85} + 116 q^{87} - 370 q^{89} + 3016 q^{91} + 1696 q^{93} - 4412 q^{95} - 2234 q^{97} + 4118 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 4 x^{8} - 138 x^{7} + 394 x^{6} + 5872 x^{5} - 10822 x^{4} - 85158 x^{3} + 30654 x^{2} + 439999 x + 396802\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 2129 \nu^{8} - 11757 \nu^{7} - 264339 \nu^{6} + 1175057 \nu^{5} + 9556105 \nu^{4} - 33394123 \nu^{3} - 97616725 \nu^{2} + 180274991 \nu + 369231982 \)\()/4740120\)
\(\beta_{3}\)\(=\)\((\)\( -39 \nu^{8} + 317 \nu^{7} + 4409 \nu^{6} - 34357 \nu^{5} - 127235 \nu^{4} + 1014083 \nu^{3} + 248575 \nu^{2} - 4511931 \nu - 860902 \)\()/83160\)
\(\beta_{4}\)\(=\)\((\)\( 347 \nu^{8} - 2165 \nu^{7} - 43679 \nu^{6} + 235173 \nu^{5} + 1598859 \nu^{4} - 7355187 \nu^{3} - 16395629 \nu^{2} + 46817579 \nu + 78704822 \)\()/316008\)
\(\beta_{5}\)\(=\)\((\)\( -3196 \nu^{8} + 17103 \nu^{7} + 412971 \nu^{6} - 1812133 \nu^{5} - 15686240 \nu^{4} + 55607237 \nu^{3} + 176498495 \nu^{2} - 336420739 \nu - 787744538 \)\()/2370060\)
\(\beta_{6}\)\(=\)\((\)\( -8339 \nu^{8} + 49617 \nu^{7} + 1041729 \nu^{6} - 5309147 \nu^{5} - 37053595 \nu^{4} + 163032163 \nu^{3} + 342770455 \nu^{2} - 954029561 \nu - 1519251202 \)\()/4740120\)
\(\beta_{7}\)\(=\)\((\)\( 1333 \nu^{8} - 6999 \nu^{7} - 171603 \nu^{6} + 734299 \nu^{5} + 6418745 \nu^{4} - 22288601 \nu^{3} - 67067285 \nu^{2} + 132639157 \nu + 259520594 \)\()/677160\)
\(\beta_{8}\)\(=\)\((\)\( 11597 \nu^{8} - 66681 \nu^{7} - 1481127 \nu^{6} + 7151861 \nu^{5} + 55355365 \nu^{4} - 221542279 \nu^{3} - 594191305 \nu^{2} + 1365624923 \nu + 2604377566 \)\()/4740120\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} + \beta_{5} - \beta_{4} + 2 \beta_{1} + 32\)
\(\nu^{3}\)\(=\)\(4 \beta_{8} - 2 \beta_{7} - \beta_{5} - 7 \beta_{4} - 2 \beta_{3} - \beta_{2} + 64 \beta_{1} + 37\)
\(\nu^{4}\)\(=\)\(96 \beta_{8} - 17 \beta_{7} - 6 \beta_{6} + 73 \beta_{5} - 105 \beta_{4} + \beta_{3} + 5 \beta_{2} + 247 \beta_{1} + 1882\)
\(\nu^{5}\)\(=\)\(580 \beta_{8} - 258 \beta_{7} - 96 \beta_{6} + 88 \beta_{5} - 880 \beta_{4} - 114 \beta_{3} - 108 \beta_{2} + 4905 \beta_{1} + 5092\)
\(\nu^{6}\)\(=\)\(8663 \beta_{8} - 2414 \beta_{7} - 1284 \beta_{6} + 5625 \beta_{5} - 10253 \beta_{4} + 250 \beta_{3} + 578 \beta_{2} + 27656 \beta_{1} + 134664\)
\(\nu^{7}\)\(=\)\(64308 \beta_{8} - 26950 \beta_{7} - 14772 \beta_{6} + 19853 \beta_{5} - 92853 \beta_{4} - 5062 \beta_{3} - 8711 \beta_{2} + 409818 \beta_{1} + 611739\)
\(\nu^{8}\)\(=\)\(788310 \beta_{8} - 261219 \beta_{7} - 161082 \beta_{6} + 461973 \beta_{5} - 984441 \beta_{4} + 30147 \beta_{3} + 47367 \beta_{2} + 2891949 \beta_{1} + 10714546\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−7.63776
−7.02483
−2.38920
−1.95108
−1.51339
3.39022
4.87969
6.50540
9.74094
0 −7.63776 0 7.65015 0 −5.64550 0 31.3353 0
1.2 0 −7.02483 0 −2.19045 0 −5.10018 0 22.3483 0
1.3 0 −2.38920 0 3.17647 0 28.0947 0 −21.2917 0
1.4 0 −1.95108 0 −18.0628 0 −22.0180 0 −23.1933 0
1.5 0 −1.51339 0 6.33195 0 −1.77839 0 −24.7097 0
1.6 0 3.39022 0 19.3767 0 7.92881 0 −15.5064 0
1.7 0 4.87969 0 −10.4780 0 23.3159 0 −3.18861 0
1.8 0 6.50540 0 1.74434 0 −26.0581 0 15.3202 0
1.9 0 9.74094 0 −17.5483 0 −10.7391 0 67.8859 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(29\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1856.4.a.bg 9
4.b odd 2 1 1856.4.a.bf 9
8.b even 2 1 928.4.a.d 9
8.d odd 2 1 928.4.a.e yes 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
928.4.a.d 9 8.b even 2 1
928.4.a.e yes 9 8.d odd 2 1
1856.4.a.bf 9 4.b odd 2 1
1856.4.a.bg 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1856))\):

\(T_{3}^{9} - \cdots\)
\(T_{5}^{9} + \cdots\)
\(T_{7}^{9} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \)
$3$ \( 396802 + 439999 T + 30654 T^{2} - 85158 T^{3} - 10822 T^{4} + 5872 T^{5} + 394 T^{6} - 138 T^{7} - 4 T^{8} + T^{9} \)
$5$ \( -37835002 + 21340421 T + 7153684 T^{2} - 4664192 T^{3} + 256400 T^{4} + 85426 T^{5} - 4532 T^{6} - 568 T^{7} + 10 T^{8} + T^{9} \)
$7$ \( -1638662144 - 1487601664 T - 333385728 T^{2} + 2377728 T^{3} + 6862272 T^{4} + 407984 T^{5} - 17936 T^{6} - 1296 T^{7} + 12 T^{8} + T^{9} \)
$11$ \( -86950571182 - 7611950305 T + 4744765818 T^{2} + 194135674 T^{3} - 57032042 T^{4} - 353264 T^{5} + 166366 T^{6} - 2346 T^{7} - 64 T^{8} + T^{9} \)
$13$ \( 119297288098 - 226795092763 T + 3164810548 T^{2} + 4260812080 T^{3} + 163319480 T^{4} - 8030670 T^{5} - 464084 T^{6} - 4424 T^{7} + 70 T^{8} + T^{9} \)
$17$ \( 239296714517504 - 51317500090368 T - 5087419003392 T^{2} - 5818930176 T^{3} + 6228527456 T^{4} + 60278224 T^{5} - 1664624 T^{6} - 20248 T^{7} + 66 T^{8} + T^{9} \)
$19$ \( -7185256532770816 + 291320927862784 T + 9106050934784 T^{2} - 375232573440 T^{3} - 3943967360 T^{4} + 150418304 T^{5} + 683296 T^{6} - 21968 T^{7} - 42 T^{8} + T^{9} \)
$23$ \( 6098656614344704 + 5864262198060544 T - 194297165183232 T^{2} - 4997296434304 T^{3} + 49549841280 T^{4} + 976003056 T^{5} - 2920464 T^{6} - 57936 T^{7} + 40 T^{8} + T^{9} \)
$29$ \( ( -29 + T )^{9} \)
$31$ \( -12758051507405198134 + 296389651358752019 T + 3978999936378202 T^{2} - 68414581089482 T^{3} - 373556406450 T^{4} + 4818076956 T^{5} + 11174478 T^{6} - 130150 T^{7} - 64 T^{8} + T^{9} \)
$37$ \( -25254224946266112 + 18996778594467840 T + 643023040675840 T^{2} - 14946135703552 T^{3} - 140139343872 T^{4} + 2152877056 T^{5} + 5416832 T^{6} - 84912 T^{7} - 54 T^{8} + T^{9} \)
$41$ \( 7195943999639650304 + 470480190202445824 T - 7910794886578176 T^{2} - 209843777882624 T^{3} + 1650630884576 T^{4} + 8891846864 T^{5} - 50242704 T^{6} - 156896 T^{7} + 378 T^{8} + T^{9} \)
$43$ \( -301147619879488942 + 135612509813604831 T + 7183359855544538 T^{2} - 214191868216102 T^{3} - 884960515338 T^{4} + 19040723664 T^{5} - 1687618 T^{6} - 286474 T^{7} + 32 T^{8} + T^{9} \)
$47$ \( -\)\(20\!\cdots\!22\)\( - 19797456013250996893 T - 526026885030837310 T^{2} - 6508445889899146 T^{3} - 42822974136098 T^{4} - 147460747156 T^{5} - 191239514 T^{6} + 287418 T^{7} + 1164 T^{8} + T^{9} \)
$53$ \( \)\(25\!\cdots\!86\)\( + 70618695749096928165 T - 339270618728759508 T^{2} - 4784243344582512 T^{3} + 13083097172096 T^{4} + 87469601842 T^{5} - 113093612 T^{6} - 526184 T^{7} + 278 T^{8} + T^{9} \)
$59$ \( \)\(70\!\cdots\!64\)\( - \)\(20\!\cdots\!44\)\( T + 9388931627936776960 T^{2} + 19926810207922304 T^{3} - 169050168240640 T^{4} + 111210062128 T^{5} + 645566448 T^{6} - 810688 T^{7} - 640 T^{8} + T^{9} \)
$61$ \( -19677805022242614272 + 1753888944260889600 T - 21738616774602752 T^{2} - 1477704057509632 T^{3} + 36729597829024 T^{4} - 116990060144 T^{5} - 751633968 T^{6} - 474792 T^{7} + 1054 T^{8} + T^{9} \)
$67$ \( -\)\(33\!\cdots\!96\)\( + \)\(66\!\cdots\!16\)\( T + \)\(12\!\cdots\!88\)\( T^{2} - 132156925990256640 T^{3} - 804213518086144 T^{4} + 693866615552 T^{5} + 1715286912 T^{6} - 1418080 T^{7} - 1184 T^{8} + T^{9} \)
$71$ \( -\)\(55\!\cdots\!76\)\( - \)\(38\!\cdots\!20\)\( T + 24165839263433233152 T^{2} + 180773731697343360 T^{3} + 61975210970048 T^{4} - 1195053889856 T^{5} - 1989966448 T^{6} + 101832 T^{7} + 1988 T^{8} + T^{9} \)
$73$ \( -\)\(61\!\cdots\!76\)\( - \)\(81\!\cdots\!64\)\( T - \)\(30\!\cdots\!72\)\( T^{2} - 67682859783397376 T^{3} + 1385987060290048 T^{4} + 1311626488320 T^{5} - 1878708736 T^{6} - 2222496 T^{7} + 750 T^{8} + T^{9} \)
$79$ \( \)\(16\!\cdots\!86\)\( + \)\(21\!\cdots\!19\)\( T + 91816991968152759414 T^{2} + 54607180844381654 T^{3} - 591576151806630 T^{4} - 1425470582404 T^{5} - 282009950 T^{6} + 2484218 T^{7} + 2916 T^{8} + T^{9} \)
$83$ \( \)\(77\!\cdots\!36\)\( + \)\(71\!\cdots\!36\)\( T + 10483375340186959104 T^{2} + 20955029717489280 T^{3} - 265599544047616 T^{4} - 763628379152 T^{5} + 1193069584 T^{6} + 1632448 T^{7} - 2832 T^{8} + T^{9} \)
$89$ \( -\)\(37\!\cdots\!00\)\( + \)\(31\!\cdots\!80\)\( T + 33495561857322903552 T^{2} - 193837785498127360 T^{3} - 527920974525472 T^{4} + 1947502642320 T^{5} + 32738192 T^{6} - 2644400 T^{7} + 370 T^{8} + T^{9} \)
$97$ \( \)\(21\!\cdots\!88\)\( + \)\(38\!\cdots\!68\)\( T - 15636631825925518336 T^{2} - 90322325413091840 T^{3} + 770744614850016 T^{4} - 20812204912 T^{5} - 2654309136 T^{6} - 466192 T^{7} + 2234 T^{8} + T^{9} \)
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