Properties

Label 336.2.bc
Level 336
Weight 2
Character orbit bc
Rep. character \(\chi_{336}(17,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 28
Newform subspaces 6
Sturm bound 128
Trace bound 5

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Defining parameters

Level: \( N \) = \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 336.bc (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 21 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 6 \)
Sturm bound: \(128\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(336, [\chi])\).

Total New Old
Modular forms 152 36 116
Cusp forms 104 28 76
Eisenstein series 48 8 40

Trace form

\( 28q + 3q^{3} + 2q^{7} - q^{9} + O(q^{10}) \) \( 28q + 3q^{3} + 2q^{7} - q^{9} - 2q^{15} + 12q^{19} - q^{21} - 12q^{25} + 48q^{31} - 3q^{33} - 2q^{37} + 10q^{39} - 20q^{43} - 15q^{45} - 4q^{49} - 27q^{51} - 26q^{57} - 6q^{61} - 7q^{63} + 16q^{67} - 18q^{73} - 66q^{75} - 28q^{79} + 3q^{81} + 28q^{85} - 60q^{87} - 54q^{91} + q^{93} + 34q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(336, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
336.2.bc.a \(2\) \(2.683\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(-3\) \(0\) \(5\) \(q+(-1-\zeta_{6})q^{3}+(2+\zeta_{6})q^{7}+3\zeta_{6}q^{9}+\cdots\)
336.2.bc.b \(2\) \(2.683\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-3\) \(-4\) \(q+(-1+2\zeta_{6})q^{3}-3\zeta_{6}q^{5}+(-1-2\zeta_{6})q^{7}+\cdots\)
336.2.bc.c \(2\) \(2.683\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(3\) \(0\) \(-1\) \(q+(1+\zeta_{6})q^{3}+(-2+3\zeta_{6})q^{7}+3\zeta_{6}q^{9}+\cdots\)
336.2.bc.d \(2\) \(2.683\) \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(3\) \(-4\) \(q+(2-\zeta_{6})q^{3}+3\zeta_{6}q^{5}+(-1-2\zeta_{6})q^{7}+\cdots\)
336.2.bc.e \(4\) \(2.683\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(10\) \(q-\zeta_{12}^{2}q^{3}+(-\zeta_{12}^{2}+\zeta_{12}^{3})q^{5}+\cdots\)
336.2.bc.f \(16\) \(2.683\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(-4\) \(q+(-\beta _{5}+\beta _{8})q^{3}-\beta _{14}q^{5}+\beta _{10}q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(336, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(336, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$3$ (\( 1 + 3 T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 + 3 T^{2} + 9 T^{4} \))(\( 1 - T^{2} - 7 T^{4} + 24 T^{5} + 22 T^{6} - 48 T^{7} + 10 T^{8} - 144 T^{9} + 198 T^{10} + 648 T^{11} - 567 T^{12} - 729 T^{14} + 6561 T^{16} \))
$5$ (\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 - 7 T^{2} + 24 T^{4} - 175 T^{6} + 625 T^{8} \))(\( ( 1 - 13 T^{2} + 93 T^{4} - 434 T^{6} + 1886 T^{8} - 10850 T^{10} + 58125 T^{12} - 203125 T^{14} + 390625 T^{16} )( 1 + 2 T^{2} - 39 T^{4} - 38 T^{6} + 836 T^{8} - 950 T^{10} - 24375 T^{12} + 31250 T^{14} + 390625 T^{16} ) \))
$7$ (\( 1 - 5 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( ( 1 - 5 T + 7 T^{2} )^{2} \))(\( ( 1 + 2 T + 9 T^{2} + 10 T^{3} + 44 T^{4} + 70 T^{5} + 441 T^{6} + 686 T^{7} + 2401 T^{8} )^{2} \))
$11$ (\( 1 + 11 T^{2} + 121 T^{4} \))(\( 1 + 9 T + 38 T^{2} + 99 T^{3} + 121 T^{4} \))(\( 1 + 11 T^{2} + 121 T^{4} \))(\( 1 - 9 T + 38 T^{2} - 99 T^{3} + 121 T^{4} \))(\( 1 + 13 T^{2} + 48 T^{4} + 1573 T^{6} + 14641 T^{8} \))(\( 1 + 49 T^{2} + 1300 T^{4} + 22265 T^{6} + 252641 T^{8} + 1395328 T^{10} - 12534274 T^{12} - 440430082 T^{14} - 6222779240 T^{16} - 53292039922 T^{18} - 183514305634 T^{20} + 2471908667008 T^{22} + 54155842054721 T^{24} + 577496758741265 T^{26} + 4079956889737300 T^{28} + 18607741845578809 T^{30} + 45949729863572161 T^{32} \))
$13$ (\( ( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( ( 1 - 13 T^{2} )^{2} \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( ( 1 - 13 T^{2} )^{2} \))(\( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 49 T^{2} + 1278 T^{4} - 23495 T^{6} + 341186 T^{8} - 3970655 T^{10} + 36500958 T^{12} - 236513641 T^{14} + 815730721 T^{16} )^{2} \))
$17$ (\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \))(\( 1 - 22 T^{2} + 195 T^{4} - 6358 T^{6} + 83521 T^{8} \))(\( 1 - 42 T^{2} + 1059 T^{4} - 6894 T^{6} - 204407 T^{8} + 7559412 T^{10} - 49293810 T^{12} - 1407253056 T^{14} + 49740922386 T^{16} - 406696133184 T^{18} - 4117068305010 T^{20} + 182465828749428 T^{22} - 1425893651242487 T^{24} - 13898261949695406 T^{26} + 616996949226316899 T^{28} - 7071868715494839018 T^{30} + 48661191875666868481 T^{32} \))
$19$ (\( ( 1 - 8 T + 19 T^{2} )( 1 - 7 T + 19 T^{2} ) \))(\( 1 + 3 T + 22 T^{2} + 57 T^{3} + 361 T^{4} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 - T + 19 T^{2} ) \))(\( 1 + 3 T + 22 T^{2} + 57 T^{3} + 361 T^{4} \))(\( ( 1 - T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 - 3 T + 62 T^{2} - 177 T^{3} + 2049 T^{4} - 6000 T^{5} + 55390 T^{6} - 152826 T^{7} + 1217108 T^{8} - 2903694 T^{9} + 19995790 T^{10} - 41154000 T^{11} + 267027729 T^{12} - 438269523 T^{13} + 2916844622 T^{14} - 2681615217 T^{15} + 16983563041 T^{16} )^{2} \))
$23$ (\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 - 9 T + 50 T^{2} - 207 T^{3} + 529 T^{4} \))(\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 + 9 T + 50 T^{2} + 207 T^{3} + 529 T^{4} \))(\( 1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8} \))(\( 1 + 102 T^{2} + 6323 T^{4} + 248898 T^{6} + 6664873 T^{8} + 89132724 T^{10} - 1162854034 T^{12} - 108042764448 T^{14} - 3248548802990 T^{16} - 57154622392992 T^{18} - 325414235728594 T^{20} + 13194842036331636 T^{22} + 521932771402734313 T^{24} + 10310975788054808802 T^{26} + \)\(13\!\cdots\!83\)\( T^{28} + \)\(11\!\cdots\!18\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 49 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 103 T^{2} + 6606 T^{4} - 293969 T^{6} + 9810626 T^{8} - 247227929 T^{10} + 4672298286 T^{12} - 61266802063 T^{14} + 500246412961 T^{16} )^{2} \))
$31$ (\( ( 1 - 7 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( ( 1 - 7 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( ( 1 + 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( ( 1 - 7 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( ( 1 - 7 T + 31 T^{2} )^{2}( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 24 T + 350 T^{2} - 3792 T^{3} + 33057 T^{4} - 243840 T^{5} + 1597150 T^{6} - 9593544 T^{7} + 54548420 T^{8} - 297399864 T^{9} + 1534861150 T^{10} - 7264237440 T^{11} + 30528833697 T^{12} - 108561740592 T^{13} + 310626288350 T^{14} - 660302738664 T^{15} + 852891037441 T^{16} )^{2} \))
$37$ (\( ( 1 - 10 T + 37 T^{2} )( 1 - T + 37 T^{2} ) \))(\( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} \))(\( ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} \))(\( ( 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + T - 60 T^{2} + 689 T^{3} + 2741 T^{4} - 33360 T^{5} + 198298 T^{6} + 1145242 T^{7} - 8840520 T^{8} + 42373954 T^{9} + 271469962 T^{10} - 1689784080 T^{11} + 5137075301 T^{12} + 47777986373 T^{13} - 153943584540 T^{14} + 94931877133 T^{15} + 3512479453921 T^{16} )^{2} \))
$41$ (\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 34 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 240 T^{2} + 27996 T^{4} + 2035728 T^{6} + 100303238 T^{8} + 3422058768 T^{10} + 79110004956 T^{12} + 1140025017840 T^{14} + 7984925229121 T^{16} )^{2} \))
$43$ (\( ( 1 + 13 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 - 5 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{4} \))(\( ( 1 + 5 T + 76 T^{2} + 341 T^{3} + 2710 T^{4} + 14663 T^{5} + 140524 T^{6} + 397535 T^{7} + 3418801 T^{8} )^{4} \))
$47$ (\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} \))(\( 1 - 46 T^{2} - 93 T^{4} - 101614 T^{6} + 4879681 T^{8} \))(\( 1 - 158 T^{2} + 13171 T^{4} - 396034 T^{6} - 16747399 T^{8} + 2338864468 T^{10} - 73391270338 T^{12} - 1850417430616 T^{14} + 239524986298546 T^{16} - 4087572104230744 T^{18} - 358125987434202178 T^{20} + 25211123725919029972 T^{22} - \)\(39\!\cdots\!39\)\( T^{24} - \)\(20\!\cdots\!66\)\( T^{26} + \)\(15\!\cdots\!11\)\( T^{28} - \)\(40\!\cdots\!02\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( 1 + 53 T^{2} + 2809 T^{4} \))(\( 1 + 9 T + 80 T^{2} + 477 T^{3} + 2809 T^{4} \))(\( 1 + 53 T^{2} + 2809 T^{4} \))(\( 1 - 9 T + 80 T^{2} - 477 T^{3} + 2809 T^{4} \))(\( 1 + 25 T^{2} - 2184 T^{4} + 70225 T^{6} + 7890481 T^{8} \))(\( 1 + 265 T^{2} + 32968 T^{4} + 3168029 T^{6} + 288393293 T^{8} + 22350648928 T^{10} + 1460664537746 T^{12} + 90342579430370 T^{14} + 5160533915433520 T^{16} + 253772305619909330 T^{18} + 11525345782458595826 T^{20} + \)\(49\!\cdots\!12\)\( T^{22} + \)\(17\!\cdots\!73\)\( T^{24} + \)\(55\!\cdots\!21\)\( T^{26} + \)\(16\!\cdots\!88\)\( T^{28} + \)\(36\!\cdots\!85\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} \))(\( 1 - 115 T^{2} + 9744 T^{4} - 400315 T^{6} + 12117361 T^{8} \))(\( 1 - 187 T^{2} + 11904 T^{4} - 267479 T^{6} + 18518765 T^{8} - 3131679552 T^{10} + 215589458578 T^{12} - 8824777678534 T^{14} + 393339242196864 T^{16} - 30719051098976854 T^{18} + 2612375297384172658 T^{20} - \)\(13\!\cdots\!32\)\( T^{22} + \)\(27\!\cdots\!65\)\( T^{24} - \)\(13\!\cdots\!79\)\( T^{26} + \)\(21\!\cdots\!24\)\( T^{28} - \)\(11\!\cdots\!07\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( ( 1 - T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 + 21 T + 208 T^{2} + 1281 T^{3} + 3721 T^{4} \))(\( ( 1 - 13 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \))(\( 1 + 21 T + 208 T^{2} + 1281 T^{3} + 3721 T^{4} \))(\( ( 1 + 61 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 - 18 T + 331 T^{2} - 4014 T^{3} + 46777 T^{4} - 446148 T^{5} + 4203790 T^{6} - 34615152 T^{7} + 285617722 T^{8} - 2111524272 T^{9} + 15642302590 T^{10} - 101267119188 T^{11} + 647666904457 T^{12} - 3390209552214 T^{13} + 17053243913491 T^{14} - 56569371048378 T^{15} + 191707312997281 T^{16} )^{2} \))
$67$ (\( ( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 5 T + 67 T^{2} ) \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \))(\( ( 1 - 2 T - 63 T^{2} - 134 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 7 T - 200 T^{2} - 865 T^{3} + 28259 T^{4} + 67468 T^{5} - 2804524 T^{6} - 1482940 T^{7} + 222922288 T^{8} - 99356980 T^{9} - 12589508236 T^{10} + 20291878084 T^{11} + 569450528339 T^{12} - 1167858217555 T^{13} - 18091676433800 T^{14} + 42424981237261 T^{15} + 406067677556641 T^{16} )^{2} \))
$71$ (\( ( 1 - 71 T^{2} )^{2} \))(\( 1 - 34 T^{2} + 5041 T^{4} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( 1 - 34 T^{2} + 5041 T^{4} \))(\( ( 1 + 2 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 224 T^{2} + 21244 T^{4} - 988448 T^{6} + 37865158 T^{8} - 4982766368 T^{10} + 539845751164 T^{12} - 28694463598304 T^{14} + 645753531245761 T^{16} )^{2} \))
$73$ (\( ( 1 - 7 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \))(\( 1 + 21 T + 220 T^{2} + 1533 T^{3} + 5329 T^{4} \))(\( ( 1 + 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( 1 + 21 T + 220 T^{2} + 1533 T^{3} + 5329 T^{4} \))(\( ( 1 - 12 T + 121 T^{2} - 876 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 15 T + 312 T^{2} - 3555 T^{3} + 48909 T^{4} - 547104 T^{5} + 5639826 T^{6} - 55083990 T^{7} + 457851344 T^{8} - 4021131270 T^{9} + 30054632754 T^{10} - 212832756768 T^{11} + 1388929569069 T^{12} - 7369769513115 T^{13} + 47216278602168 T^{14} - 165710977786455 T^{15} + 806460091894081 T^{16} )^{2} \))
$79$ (\( ( 1 - 13 T + 79 T^{2} )( 1 - 4 T + 79 T^{2} ) \))(\( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \))(\( ( 1 - 4 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \))(\( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \))(\( ( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 14 T - 152 T^{2} - 1448 T^{3} + 34763 T^{4} + 170492 T^{5} - 3816772 T^{6} - 3401582 T^{7} + 382142944 T^{8} - 268724978 T^{9} - 23820474052 T^{10} + 84059205188 T^{11} + 1354021665803 T^{12} - 4455577665752 T^{13} - 36949293239192 T^{14} + 268854725806226 T^{15} + 1517108809906561 T^{16} )^{2} \))
$83$ (\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 91 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 141 T^{2} + 22278 T^{4} + 1798779 T^{6} + 189218258 T^{8} + 12391788531 T^{10} + 1057276475238 T^{12} + 46098592645029 T^{14} + 2252292232139041 T^{16} )^{2} \))
$89$ (\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 + 9 T - 8 T^{2} + 801 T^{3} + 7921 T^{4} \))(\( 1 - 70 T^{2} - 3021 T^{4} - 554470 T^{6} + 62742241 T^{8} \))(\( 1 - 378 T^{2} + 69795 T^{4} - 8481534 T^{6} + 768803689 T^{8} - 53847736620 T^{10} + 2382276614382 T^{12} + 24847322779392 T^{14} - 11761219221673230 T^{16} + 196815643735564032 T^{18} + \)\(14\!\cdots\!62\)\( T^{20} - \)\(26\!\cdots\!20\)\( T^{22} + \)\(30\!\cdots\!09\)\( T^{24} - \)\(26\!\cdots\!34\)\( T^{26} + \)\(17\!\cdots\!95\)\( T^{28} - \)\(73\!\cdots\!98\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \))
$97$ (\( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( 1 - 146 T^{2} + 9409 T^{4} \))(\( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( 1 - 146 T^{2} + 9409 T^{4} \))(\( ( 1 - 19 T + 97 T^{2} )^{2}( 1 + 19 T + 97 T^{2} )^{2} \))(\( ( 1 - 429 T^{2} + 87258 T^{4} - 11450019 T^{6} + 1191212138 T^{8} - 107733228771 T^{10} + 7724888001498 T^{12} - 357344990114541 T^{14} + 7837433594376961 T^{16} )^{2} \))
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