Properties

Label 147.7.d.a
Level $147$
Weight $7$
Character orbit 147.d
Analytic conductor $33.818$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 147.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(33.8179502921\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - x^{7} + 212 x^{6} - 787 x^{5} + 38792 x^{4} - 92833 x^{3} + 1563109 x^{2} + 3107772 x + 38787984\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{3} ) q^{2} -9 \beta_{1} q^{3} + ( 43 - \beta_{5} ) q^{4} + ( -3 \beta_{1} - \beta_{4} - \beta_{6} ) q^{5} + ( -9 \beta_{1} + 9 \beta_{2} ) q^{6} + ( -61 + 16 \beta_{3} + 3 \beta_{5} - 2 \beta_{7} ) q^{8} -243 q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{3} ) q^{2} -9 \beta_{1} q^{3} + ( 43 - \beta_{5} ) q^{4} + ( -3 \beta_{1} - \beta_{4} - \beta_{6} ) q^{5} + ( -9 \beta_{1} + 9 \beta_{2} ) q^{6} + ( -61 + 16 \beta_{3} + 3 \beta_{5} - 2 \beta_{7} ) q^{8} -243 q^{9} + ( -9 \beta_{1} + 34 \beta_{2} - 14 \beta_{4} - 3 \beta_{6} ) q^{10} + ( 265 - \beta_{3} - 9 \beta_{5} - \beta_{7} ) q^{11} + ( -387 \beta_{1} - 9 \beta_{6} ) q^{12} + ( -864 \beta_{1} + 14 \beta_{2} + 17 \beta_{4} - 5 \beta_{6} ) q^{13} + ( -81 - 9 \beta_{3} + 27 \beta_{5} + 9 \beta_{7} ) q^{15} + ( -933 - 194 \beta_{3} + 5 \beta_{5} - 10 \beta_{7} ) q^{16} + ( -626 \beta_{1} - 70 \beta_{2} + 54 \beta_{4} - 24 \beta_{6} ) q^{17} + ( -243 - 243 \beta_{3} ) q^{18} + ( -1954 \beta_{1} + 360 \beta_{2} - 27 \beta_{4} + \beta_{6} ) q^{19} + ( -3945 \beta_{1} + 24 \beta_{2} - 66 \beta_{4} - 45 \beta_{6} ) q^{20} + ( -169 + 596 \beta_{3} + 11 \beta_{5} - 26 \beta_{7} ) q^{22} + ( 4144 - 1306 \beta_{3} + 54 \beta_{5} - 4 \beta_{7} ) q^{23} + ( 549 \beta_{1} + 126 \beta_{2} - 54 \beta_{4} + 27 \beta_{6} ) q^{24} + ( -5198 - 1399 \beta_{3} + 81 \beta_{5} + \beta_{7} ) q^{25} + ( -1366 \beta_{1} + 1093 \beta_{2} + 106 \beta_{4} + 103 \beta_{6} ) q^{26} + 2187 \beta_{1} q^{27} + ( 3949 + 405 \beta_{3} - 183 \beta_{5} + 61 \beta_{7} ) q^{29} + ( -243 - 1044 \beta_{3} + 81 \beta_{5} + 126 \beta_{7} ) q^{30} + ( -3401 \beta_{1} + 122 \beta_{2} + 44 \beta_{4} + 282 \beta_{6} ) q^{31} + ( -17073 - 1978 \beta_{3} - 183 \beta_{5} + 58 \beta_{7} ) q^{32} + ( -2385 \beta_{1} - 18 \beta_{2} - 27 \beta_{4} - 81 \beta_{6} ) q^{33} + ( 10238 \beta_{1} + 1752 \beta_{2} + 288 \beta_{4} + 466 \beta_{6} ) q^{34} + ( -10449 + 243 \beta_{5} ) q^{36} + ( 11970 - 2827 \beta_{3} - 475 \beta_{5} - 71 \beta_{7} ) q^{37} + ( -41396 \beta_{1} + 1451 \beta_{2} - 210 \beta_{4} - 525 \beta_{6} ) q^{38} + ( -23328 - 225 \beta_{3} + 135 \beta_{5} - 153 \beta_{7} ) q^{39} + ( -7149 \beta_{1} + 3056 \beta_{2} + 98 \beta_{4} - 93 \beta_{6} ) q^{40} + ( -6362 \beta_{1} - 340 \beta_{2} - 256 \beta_{4} - 436 \beta_{6} ) q^{41} + ( -57678 + 6317 \beta_{3} - 1051 \beta_{5} - 257 \beta_{7} ) q^{43} + ( 47319 - 1186 \beta_{3} - 495 \beta_{5} - 122 \beta_{7} ) q^{44} + ( 729 \beta_{1} + 243 \beta_{4} + 243 \beta_{6} ) q^{45} + ( -132004 + 3440 \beta_{3} + 1076 \beta_{5} + 76 \beta_{7} ) q^{46} + ( 6688 \beta_{1} - 1702 \beta_{2} - 770 \beta_{4} - 56 \beta_{6} ) q^{47} + ( 8397 \beta_{1} - 1836 \beta_{2} - 270 \beta_{4} + 45 \beta_{6} ) q^{48} + ( -150284 - 6793 \beta_{3} + 1173 \beta_{5} + 170 \beta_{7} ) q^{50} + ( -16902 + 2376 \beta_{3} + 648 \beta_{5} - 486 \beta_{7} ) q^{51} + ( -61172 \beta_{1} - 3804 \beta_{2} + 378 \beta_{4} - 446 \beta_{6} ) q^{52} + ( 3055 - 1947 \beta_{3} - 2463 \beta_{5} - 149 \beta_{7} ) q^{53} + ( 2187 \beta_{1} - 2187 \beta_{2} ) q^{54} + ( -44475 \beta_{1} + 1474 \beta_{2} - 371 \beta_{4} - 345 \beta_{6} ) q^{55} + ( -52758 - 9963 \beta_{3} - 27 \beta_{5} + 243 \beta_{7} ) q^{57} + ( 37607 + 10498 \beta_{3} + 1181 \beta_{5} + 122 \beta_{7} ) q^{58} + ( -151147 \beta_{1} - 1460 \beta_{2} - 703 \beta_{4} + 377 \beta_{6} ) q^{59} + ( -106515 - 1242 \beta_{3} + 1215 \beta_{5} + 594 \beta_{7} ) q^{60} + ( -33528 \beta_{1} - 2144 \beta_{2} + 964 \beta_{4} + 52 \beta_{6} ) q^{61} + ( -25589 \beta_{1} - 6415 \beta_{2} + 2044 \beta_{4} - 704 \beta_{6} ) q^{62} + ( -176205 + 4266 \beta_{3} + 3193 \beta_{5} + 738 \beta_{7} ) q^{64} + ( 93648 + 19168 \beta_{3} + 1584 \beta_{5} + 1586 \beta_{7} ) q^{65} + ( 1521 \beta_{1} + 5130 \beta_{2} - 702 \beta_{4} + 99 \beta_{6} ) q^{66} + ( -101116 + 18853 \beta_{3} - 1851 \beta_{5} + 167 \beta_{7} ) q^{67} + ( -140802 \beta_{1} - 22668 \beta_{2} + 1644 \beta_{4} + 114 \beta_{6} ) q^{68} + ( -37296 \beta_{1} - 11790 \beta_{2} - 108 \beta_{4} + 486 \beta_{6} ) q^{69} + ( 28372 - 5388 \beta_{3} - 1332 \beta_{5} - 1706 \beta_{7} ) q^{71} + ( 14823 - 3888 \beta_{3} - 729 \beta_{5} + 486 \beta_{7} ) q^{72} + ( -77984 \beta_{1} - 10218 \beta_{2} - 195 \beta_{4} + 887 \beta_{6} ) q^{73} + ( -304420 + 32159 \beta_{3} + 3045 \beta_{5} - 1518 \beta_{7} ) q^{74} + ( 46782 \beta_{1} - 12582 \beta_{2} + 27 \beta_{4} + 729 \beta_{6} ) q^{75} + ( -58806 \beta_{1} + 34440 \beta_{2} - 3102 \beta_{4} - 1200 \beta_{6} ) q^{76} + ( -36882 - 28557 \beta_{3} - 2781 \beta_{5} - 954 \beta_{7} ) q^{78} + ( 127061 + 51202 \beta_{3} + 822 \beta_{5} - 484 \beta_{7} ) q^{79} + ( -70377 \beta_{1} + 6204 \beta_{2} + 4450 \beta_{4} + 691 \beta_{6} ) q^{80} + 59049 q^{81} + ( 35342 \beta_{1} + 20938 \beta_{2} - 4664 \beta_{4} + 112 \beta_{6} ) q^{82} + ( 73913 \beta_{1} - 25496 \beta_{2} - 1625 \beta_{4} + 4843 \beta_{6} ) q^{83} + ( 225666 + 75398 \beta_{3} - 1962 \beta_{5} + 1618 \beta_{7} ) q^{85} + ( 578108 - 25879 \beta_{3} - 7533 \beta_{5} - 4158 \beta_{7} ) q^{86} + ( -35541 \beta_{1} + 4194 \beta_{2} + 1647 \beta_{4} - 1647 \beta_{6} ) q^{87} + ( -83477 + 28310 \beta_{3} - 107 \beta_{5} - 302 \beta_{7} ) q^{88} + ( -148486 \beta_{1} - 3980 \beta_{2} + 78 \beta_{4} - 126 \beta_{6} ) q^{89} + ( 2187 \beta_{1} - 8262 \beta_{2} + 3402 \beta_{4} + 729 \beta_{6} ) q^{90} + ( 8028 - 91444 \beta_{3} - 8832 \beta_{5} + 3016 \beta_{7} ) q^{92} + ( -91827 - 2898 \beta_{3} - 7614 \beta_{5} - 396 \beta_{7} ) q^{93} + ( 153920 \beta_{1} - 6106 \beta_{2} - 6496 \beta_{4} - 2750 \beta_{6} ) q^{94} + ( -265014 - 78386 \beta_{3} + 9798 \beta_{5} + 5936 \beta_{7} ) q^{95} + ( 153657 \beta_{1} - 17280 \beta_{2} + 1566 \beta_{4} - 1647 \beta_{6} ) q^{96} + ( -135633 \beta_{1} + 3574 \beta_{2} + 6931 \beta_{4} - 1627 \beta_{6} ) q^{97} + ( -64395 + 243 \beta_{3} + 2187 \beta_{5} + 243 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 10q^{2} + 346q^{4} - 454q^{8} - 1944q^{9} + O(q^{10}) \) \( 8q + 10q^{2} + 346q^{4} - 454q^{8} - 1944q^{9} + 2140q^{11} - 756q^{15} - 7822q^{16} - 2430q^{18} - 78q^{22} + 30448q^{23} - 44548q^{25} + 32524q^{29} - 4698q^{30} - 140406q^{32} - 84078q^{36} + 91340q^{37} - 186732q^{39} - 445660q^{43} + 377658q^{44} - 1051608q^{46} - 1218884q^{50} - 129816q^{51} + 26068q^{53} - 442908q^{57} + 319002q^{58} - 859410q^{60} - 1410446q^{64} + 778008q^{65} - 768188q^{67} + 225688q^{71} + 110322q^{72} - 2371060q^{74} - 342792q^{78} + 1119184q^{79} + 472392q^{81} + 1953576q^{85} + 4604804q^{86} - 609774q^{88} - 113064q^{92} - 723600q^{93} - 2320224q^{95} - 520020q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} + 212 x^{6} - 787 x^{5} + 38792 x^{4} - 92833 x^{3} + 1563109 x^{2} + 3107772 x + 38787984\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(1585013359 \nu^{7} - 18232571539 \nu^{6} + 303603349712 \nu^{5} - 4245938445433 \nu^{4} + 65749585575908 \nu^{3} - 780366646056751 \nu^{2} + 2109023702500351 \nu - 10280963027029950\)\()/ 11901315886480818 \)
\(\beta_{2}\)\(=\)\((\)\(8019055 \nu^{7} + 15616321 \nu^{6} + 1444380025 \nu^{5} - 2053828205 \nu^{4} + 305021724904 \nu^{3} + 177516832405 \nu^{2} + 24523492689402 \nu + 29614389599556\)\()/ 11465622241311 \)
\(\beta_{3}\)\(=\)\((\)\(-8019055 \nu^{7} - 15616321 \nu^{6} - 1444380025 \nu^{5} + 2053828205 \nu^{4} - 305021724904 \nu^{3} - 177516832405 \nu^{2} - 1592248206780 \nu - 29614389599556\)\()/ 11465622241311 \)
\(\beta_{4}\)\(=\)\((\)\(27322708193 \nu^{7} + 1000103862583 \nu^{6} + 22563530233273 \nu^{5} + 219959885092027 \nu^{4} + 2723907868150765 \nu^{3} + 19609334703803473 \nu^{2} + 301119122619702170 \nu + 699314583065439780\)\()/ 11901315886480818 \)
\(\beta_{5}\)\(=\)\((\)\(39673486 \nu^{7} - 224426993 \nu^{6} + 7145928130 \nu^{5} - 10161113066 \nu^{4} + 1531993215028 \nu^{3} + 878247070906 \nu^{2} + 7877491417656 \nu + 963541759742958\)\()/ 11465622241311 \)
\(\beta_{6}\)\(=\)\((\)\(-80062726973 \nu^{7} + 817429199804 \nu^{6} - 15380773767166 \nu^{5} + 224024867280275 \nu^{4} - 3322848657824296 \nu^{3} + 29545401640254419 \nu^{2} - 134797977236992055 \nu + 352719998947020942\)\()/ 5950657943240409 \)
\(\beta_{7}\)\(=\)\((\)\(-313586723 \nu^{7} - 1214053715 \nu^{6} - 56482764965 \nu^{5} + 80315355913 \nu^{4} - 8060204009803 \nu^{3} - 6941830646033 \nu^{2} - 62265179296908 \nu - 487133839162452\)\()/ 7643748160874 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{6} + \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - 106 \beta_{1} - 106\)\()/2\)
\(\nu^{3}\)\(=\)\(2 \beta_{7} - 6 \beta_{5} - 147 \beta_{3} + 252\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{7} + 205 \beta_{6} + 205 \beta_{5} + 6 \beta_{4} + 774 \beta_{3} + 776 \beta_{2} + 15886 \beta_{1} - 15886\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-424 \beta_{7} - 1560 \beta_{6} + 1560 \beta_{5} + 1272 \beta_{4} + 25013 \beta_{3} - 24589 \beta_{2} - 90180 \beta_{1} - 90180\)\()/2\)
\(\nu^{6}\)\(=\)\(152 \beta_{7} - 38461 \beta_{5} - 199198 \beta_{3} + 2727346\)
\(\nu^{7}\)\(=\)\((\)\(-75858 \beta_{7} + 355626 \beta_{6} + 355626 \beta_{5} - 227574 \beta_{4} + 4549255 \beta_{3} + 4473397 \beta_{2} + 22658292 \beta_{1} - 22658292\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\) \(50\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
−7.08935 + 12.2791i
−7.08935 12.2791i
−2.30325 + 3.98935i
−2.30325 3.98935i
4.15432 7.19549i
4.15432 + 7.19549i
5.73828 9.93899i
5.73828 + 9.93899i
−13.1787 15.5885i 109.678 79.5668i 205.435i 0 −601.976 −243.000 1048.59i
97.2 −13.1787 15.5885i 109.678 79.5668i 205.435i 0 −601.976 −243.000 1048.59i
97.3 −3.60650 15.5885i −50.9932 83.0589i 56.2198i 0 414.723 −243.000 299.552i
97.4 −3.60650 15.5885i −50.9932 83.0589i 56.2198i 0 414.723 −243.000 299.552i
97.5 9.30863 15.5885i 22.6506 174.756i 145.107i 0 −384.906 −243.000 1626.74i
97.6 9.30863 15.5885i 22.6506 174.756i 145.107i 0 −384.906 −243.000 1626.74i
97.7 12.4766 15.5885i 91.6646 202.496i 194.490i 0 345.159 −243.000 2526.46i
97.8 12.4766 15.5885i 91.6646 202.496i 194.490i 0 345.159 −243.000 2526.46i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.7.d.a 8
3.b odd 2 1 441.7.d.d 8
7.b odd 2 1 inner 147.7.d.a 8
7.c even 3 1 21.7.f.b 8
7.c even 3 1 147.7.f.a 8
7.d odd 6 1 21.7.f.b 8
7.d odd 6 1 147.7.f.a 8
21.c even 2 1 441.7.d.d 8
21.g even 6 1 63.7.m.c 8
21.h odd 6 1 63.7.m.c 8
28.f even 6 1 336.7.bh.b 8
28.g odd 6 1 336.7.bh.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.7.f.b 8 7.c even 3 1
21.7.f.b 8 7.d odd 6 1
63.7.m.c 8 21.g even 6 1
63.7.m.c 8 21.h odd 6 1
147.7.d.a 8 1.a even 1 1 trivial
147.7.d.a 8 7.b odd 2 1 inner
147.7.f.a 8 7.c even 3 1
147.7.f.a 8 7.d odd 6 1
336.7.bh.b 8 28.f even 6 1
336.7.bh.b 8 28.g odd 6 1
441.7.d.d 8 3.b odd 2 1
441.7.d.d 8 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 5 T_{2}^{3} - 202 T_{2}^{2} + 914 T_{2} + 5520 \) acting on \(S_{7}^{\mathrm{new}}(147, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 5520 + 914 T - 202 T^{2} - 5 T^{3} + T^{4} )^{2} \)
$3$ \( ( 243 + T^{2} )^{4} \)
$5$ \( 54693259182810000 + 19691805407400 T^{2} + 2242450449 T^{4} + 84774 T^{6} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( ( 59090060580 + 312885260 T - 264211 T^{2} - 1070 T^{3} + T^{4} )^{2} \)
$13$ \( \)\(34\!\cdots\!00\)\( + 53257929335845404768 T^{2} + 105787465152777 T^{4} + 22315686 T^{6} + T^{8} \)
$17$ \( \)\(22\!\cdots\!00\)\( + \)\(26\!\cdots\!28\)\( T^{2} + 10270113626486736 T^{4} + 169210728 T^{6} + T^{8} \)
$19$ \( \)\(12\!\cdots\!96\)\( + \)\(27\!\cdots\!48\)\( T^{2} + 14949101230916409 T^{4} + 256355982 T^{6} + T^{8} \)
$23$ \( ( 25130113528867200 + 2827786801280 T - 288968680 T^{2} - 15224 T^{3} + T^{4} )^{2} \)
$29$ \( ( -39242852020022400 + 9430137809600 T - 442580539 T^{2} - 16262 T^{3} + T^{4} )^{2} \)
$31$ \( \)\(42\!\cdots\!21\)\( + \)\(24\!\cdots\!12\)\( T^{2} + 5004381354236358894 T^{4} + 4028119908 T^{6} + T^{8} \)
$37$ \( ( 503660415922686500 + 23949968583100 T - 3089311755 T^{2} - 45670 T^{3} + T^{4} )^{2} \)
$41$ \( \)\(53\!\cdots\!16\)\( + \)\(87\!\cdots\!08\)\( T^{2} + 50479416076831845984 T^{4} + 11994986352 T^{6} + T^{8} \)
$43$ \( ( -\)\(14\!\cdots\!84\)\( - 2845100102192540 T - 1577056827 T^{2} + 222830 T^{3} + T^{4} )^{2} \)
$47$ \( \)\(30\!\cdots\!00\)\( + \)\(12\!\cdots\!00\)\( T^{2} + \)\(11\!\cdots\!60\)\( T^{4} + 27041579112 T^{6} + T^{8} \)
$53$ \( ( \)\(48\!\cdots\!00\)\( + 802901081989208 T - 50576547643 T^{2} - 13034 T^{3} + T^{4} )^{2} \)
$59$ \( \)\(13\!\cdots\!36\)\( + \)\(11\!\cdots\!60\)\( T^{2} + \)\(29\!\cdots\!97\)\( T^{4} + 302515596390 T^{6} + T^{8} \)
$61$ \( \)\(78\!\cdots\!00\)\( + \)\(58\!\cdots\!72\)\( T^{2} + \)\(10\!\cdots\!96\)\( T^{4} + 58965042912 T^{6} + T^{8} \)
$67$ \( ( \)\(96\!\cdots\!80\)\( - 14714428944980048 T - 44897227815 T^{2} + 384094 T^{3} + T^{4} )^{2} \)
$71$ \( ( -\)\(25\!\cdots\!12\)\( + 44439430271275520 T - 187952640388 T^{2} - 112844 T^{3} + T^{4} )^{2} \)
$73$ \( \)\(12\!\cdots\!24\)\( + \)\(15\!\cdots\!36\)\( T^{2} + \)\(13\!\cdots\!25\)\( T^{4} + 231622808886 T^{6} + T^{8} \)
$79$ \( ( -\)\(76\!\cdots\!35\)\( + 247069108400054368 T - 466345189254 T^{2} - 559592 T^{3} + T^{4} )^{2} \)
$83$ \( \)\(36\!\cdots\!24\)\( + \)\(24\!\cdots\!96\)\( T^{2} + \)\(12\!\cdots\!25\)\( T^{4} + 2038066317246 T^{6} + T^{8} \)
$89$ \( \)\(13\!\cdots\!44\)\( + \)\(10\!\cdots\!04\)\( T^{2} + \)\(26\!\cdots\!60\)\( T^{4} + 282963720024 T^{6} + T^{8} \)
$97$ \( \)\(22\!\cdots\!00\)\( + \)\(33\!\cdots\!12\)\( T^{2} + \)\(14\!\cdots\!01\)\( T^{4} + 2348711138742 T^{6} + T^{8} \)
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