Properties

Label 189.2.e
Level 189
Weight 2
Character orbit e
Rep. character \(\chi_{189}(109,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 22
Newform subspaces 6
Sturm bound 48
Trace bound 2

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Defining parameters

Level: \( N \) = \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 189.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 6 \)
Sturm bound: \(48\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(189, [\chi])\).

Total New Old
Modular forms 60 22 38
Cusp forms 36 22 14
Eisenstein series 24 0 24

Trace form

\( 22q - 12q^{4} + 9q^{7} + O(q^{10}) \) \( 22q - 12q^{4} + 9q^{7} - 2q^{10} - 30q^{16} - 9q^{19} + 40q^{22} - 15q^{25} - 20q^{28} + 13q^{31} + 4q^{37} - 70q^{43} - 18q^{46} - 11q^{49} + 20q^{52} + 32q^{55} + 4q^{58} + 15q^{61} + 112q^{64} - 30q^{67} + 152q^{70} + 7q^{73} - 76q^{76} - 22q^{79} - 118q^{82} - 36q^{85} - 54q^{88} - 52q^{91} + 30q^{94} - 14q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(189, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
189.2.e.a \(2\) \(1.509\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(4\) \(4\) \(q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{4}+4\zeta_{6}q^{5}+(1+\cdots)q^{7}+\cdots\)
189.2.e.b \(2\) \(1.509\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-1\) \(q+(2-2\zeta_{6})q^{4}+(1-3\zeta_{6})q^{7}+2q^{13}+\cdots\)
189.2.e.c \(2\) \(1.509\) \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(-4\) \(4\) \(q+\zeta_{6}q^{2}+(1-\zeta_{6})q^{4}-4\zeta_{6}q^{5}+(1+\cdots)q^{7}+\cdots\)
189.2.e.d \(4\) \(1.509\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(-2\) \(q-\beta _{2}q^{2}+(-4+4\beta _{1})q^{4}-\beta _{2}q^{5}+\cdots\)
189.2.e.e \(6\) \(1.509\) 6.0.309123.1 None \(-2\) \(0\) \(-1\) \(2\) \(q+(-\beta _{1}-\beta _{4}+\beta _{5})q^{2}+(-2+\beta _{2}+\cdots)q^{4}+\cdots\)
189.2.e.f \(6\) \(1.509\) 6.0.309123.1 None \(2\) \(0\) \(1\) \(2\) \(q+(1-\beta _{4}+\beta _{5})q^{2}+(-\beta _{1}-\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(189, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(189, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T - T^{2} + 2 T^{3} + 4 T^{4} \))(\( 1 - 2 T^{2} + 4 T^{4} \))(\( 1 - T - T^{2} - 2 T^{3} + 4 T^{4} \))(\( ( 1 + 2 T^{2} )^{2}( 1 - 2 T^{2} + 4 T^{4} ) \))(\( 1 + 2 T + T^{2} - 4 T^{3} - 7 T^{4} - T^{5} + 7 T^{6} - 2 T^{7} - 28 T^{8} - 32 T^{9} + 16 T^{10} + 64 T^{11} + 64 T^{12} \))(\( 1 - 2 T + T^{2} + 4 T^{3} - 7 T^{4} + T^{5} + 7 T^{6} + 2 T^{7} - 28 T^{8} + 32 T^{9} + 16 T^{10} - 64 T^{11} + 64 T^{12} \))
$3$ 1
$5$ (\( 1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} \))(\( 1 - 4 T^{2} - 9 T^{4} - 100 T^{6} + 625 T^{8} \))(\( 1 + T - 8 T^{2} - 17 T^{3} + 23 T^{4} + 52 T^{5} - 11 T^{6} + 260 T^{7} + 575 T^{8} - 2125 T^{9} - 5000 T^{10} + 3125 T^{11} + 15625 T^{12} \))(\( 1 - T - 8 T^{2} + 17 T^{3} + 23 T^{4} - 52 T^{5} - 11 T^{6} - 260 T^{7} + 575 T^{8} + 2125 T^{9} - 5000 T^{10} - 3125 T^{11} + 15625 T^{12} \))
$7$ (\( 1 - 4 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( ( 1 + T + 7 T^{2} )^{2} \))(\( 1 - 2 T - 4 T^{2} + 31 T^{3} - 28 T^{4} - 98 T^{5} + 343 T^{6} \))(\( 1 - 2 T - 4 T^{2} + 31 T^{3} - 28 T^{4} - 98 T^{5} + 343 T^{6} \))
$11$ (\( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 + 2 T^{2} - 117 T^{4} + 242 T^{6} + 14641 T^{8} \))(\( 1 + 7 T + 4 T^{2} + T^{3} + 431 T^{4} + 982 T^{5} - 893 T^{6} + 10802 T^{7} + 52151 T^{8} + 1331 T^{9} + 58564 T^{10} + 1127357 T^{11} + 1771561 T^{12} \))(\( 1 - 7 T + 4 T^{2} - T^{3} + 431 T^{4} - 982 T^{5} - 893 T^{6} - 10802 T^{7} + 52151 T^{8} - 1331 T^{9} + 58564 T^{10} - 1127357 T^{11} + 1771561 T^{12} \))
$13$ (\( ( 1 - T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 - T + 13 T^{2} )^{2} \))(\( ( 1 + 4 T + 13 T^{2} )^{4} \))(\( ( 1 - 2 T + 20 T^{2} - 5 T^{3} + 260 T^{4} - 338 T^{5} + 2197 T^{6} )^{2} \))(\( ( 1 - 2 T + 20 T^{2} - 5 T^{3} + 260 T^{4} - 338 T^{5} + 2197 T^{6} )^{2} \))
$17$ (\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 28 T^{2} + 495 T^{4} - 8092 T^{6} + 83521 T^{8} \))(\( 1 - 18 T^{2} - 18 T^{3} + 18 T^{4} + 162 T^{5} + 4399 T^{6} + 2754 T^{7} + 5202 T^{8} - 88434 T^{9} - 1503378 T^{10} + 24137569 T^{12} \))(\( 1 - 18 T^{2} + 18 T^{3} + 18 T^{4} - 162 T^{5} + 4399 T^{6} - 2754 T^{7} + 5202 T^{8} + 88434 T^{9} - 1503378 T^{10} + 24137569 T^{12} \))
$19$ (\( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \))(\( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} \))(\( ( 1 - 8 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \))(\( 1 + 5 T - 28 T^{2} - 57 T^{3} + 997 T^{4} + 268 T^{5} - 22757 T^{6} + 5092 T^{7} + 359917 T^{8} - 390963 T^{9} - 3648988 T^{10} + 12380495 T^{11} + 47045881 T^{12} \))(\( 1 + 5 T - 28 T^{2} - 57 T^{3} + 997 T^{4} + 268 T^{5} - 22757 T^{6} + 5092 T^{7} + 359917 T^{8} - 390963 T^{9} - 3648988 T^{10} + 12380495 T^{11} + 47045881 T^{12} \))
$23$ (\( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 - 40 T^{2} + 1071 T^{4} - 21160 T^{6} + 279841 T^{8} \))(\( 1 + 6 T - 36 T^{2} - 102 T^{3} + 1926 T^{4} + 2526 T^{5} - 42653 T^{6} + 58098 T^{7} + 1018854 T^{8} - 1241034 T^{9} - 10074276 T^{10} + 38618058 T^{11} + 148035889 T^{12} \))(\( 1 - 6 T - 36 T^{2} + 102 T^{3} + 1926 T^{4} - 2526 T^{5} - 42653 T^{6} - 58098 T^{7} + 1018854 T^{8} + 1241034 T^{9} - 10074276 T^{10} - 38618058 T^{11} + 148035889 T^{12} \))
$29$ (\( ( 1 - 2 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 2 T + 29 T^{2} )^{2} \))(\( ( 1 + 4 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 13 T + 117 T^{2} - 763 T^{3} + 3393 T^{4} - 10933 T^{5} + 24389 T^{6} )^{2} \))(\( ( 1 + 13 T + 117 T^{2} + 763 T^{3} + 3393 T^{4} + 10933 T^{5} + 24389 T^{6} )^{2} \))
$31$ (\( 1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4} \))(\( ( 1 + 4 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( 1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4} \))(\( ( 1 - 11 T + 31 T^{2} )^{2}( 1 + 4 T + 31 T^{2} )^{2} \))(\( 1 - 8 T - 30 T^{2} + 102 T^{3} + 2506 T^{4} - 1202 T^{5} - 93509 T^{6} - 37262 T^{7} + 2408266 T^{8} + 3038682 T^{9} - 27705630 T^{10} - 229033208 T^{11} + 887503681 T^{12} \))(\( 1 - 8 T - 30 T^{2} + 102 T^{3} + 2506 T^{4} - 1202 T^{5} - 93509 T^{6} - 37262 T^{7} + 2408266 T^{8} + 3038682 T^{9} - 27705630 T^{10} - 229033208 T^{11} + 887503681 T^{12} \))
$37$ (\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + T + 37 T^{2} ) \))(\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( ( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} )^{2} \))(\( 1 - 8 T - 42 T^{2} + 150 T^{3} + 3322 T^{4} - 1094 T^{5} - 153041 T^{6} - 40478 T^{7} + 4547818 T^{8} + 7597950 T^{9} - 78714762 T^{10} - 554751656 T^{11} + 2565726409 T^{12} \))(\( 1 - 8 T - 42 T^{2} + 150 T^{3} + 3322 T^{4} - 1094 T^{5} - 153041 T^{6} - 40478 T^{7} + 4547818 T^{8} + 7597950 T^{9} - 78714762 T^{10} - 554751656 T^{11} + 2565726409 T^{12} \))
$41$ (\( ( 1 + 2 T + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 + 28 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 2 T + 18 T^{2} + 223 T^{3} + 738 T^{4} - 3362 T^{5} + 68921 T^{6} )^{2} \))(\( ( 1 + 2 T + 18 T^{2} - 223 T^{3} + 738 T^{4} + 3362 T^{5} + 68921 T^{6} )^{2} \))
$43$ (\( ( 1 + T + 43 T^{2} )^{2} \))(\( ( 1 + 13 T + 43 T^{2} )^{2} \))(\( ( 1 + T + 43 T^{2} )^{2} \))(\( ( 1 + T + 43 T^{2} )^{4} \))(\( ( 1 + 9 T + 117 T^{2} + 673 T^{3} + 5031 T^{4} + 16641 T^{5} + 79507 T^{6} )^{2} \))(\( ( 1 + 9 T + 117 T^{2} + 673 T^{3} + 5031 T^{4} + 16641 T^{5} + 79507 T^{6} )^{2} \))
$47$ (\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 - 88 T^{2} + 5535 T^{4} - 194392 T^{6} + 4879681 T^{8} \))(\( 1 + 9 T - 18 T^{2} - 819 T^{3} - 2547 T^{4} + 20772 T^{5} + 299095 T^{6} + 976284 T^{7} - 5626323 T^{8} - 85031037 T^{9} - 87834258 T^{10} + 2064105063 T^{11} + 10779215329 T^{12} \))(\( 1 - 9 T - 18 T^{2} + 819 T^{3} - 2547 T^{4} - 20772 T^{5} + 299095 T^{6} - 976284 T^{7} - 5626323 T^{8} + 85031037 T^{9} - 87834258 T^{10} - 2064105063 T^{11} + 10779215329 T^{12} \))
$53$ (\( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} \))(\( 1 - 53 T^{2} + 2809 T^{4} \))(\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 - 100 T^{2} + 7191 T^{4} - 280900 T^{6} + 7890481 T^{8} \))(\( 1 + 24 T + 252 T^{2} + 2202 T^{3} + 20916 T^{4} + 146148 T^{5} + 883411 T^{6} + 7745844 T^{7} + 58753044 T^{8} + 327827154 T^{9} + 1988401212 T^{10} + 10036691832 T^{11} + 22164361129 T^{12} \))(\( 1 - 24 T + 252 T^{2} - 2202 T^{3} + 20916 T^{4} - 146148 T^{5} + 883411 T^{6} - 7745844 T^{7} + 58753044 T^{8} - 327827154 T^{9} + 1988401212 T^{10} - 10036691832 T^{11} + 22164361129 T^{12} \))
$59$ (\( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \))(\( 1 - 22 T^{2} - 2997 T^{4} - 76582 T^{6} + 12117361 T^{8} \))(\( 1 - 15 T - 18 T^{2} + 57 T^{3} + 16947 T^{4} - 71898 T^{5} - 430157 T^{6} - 4241982 T^{7} + 58992507 T^{8} + 11706603 T^{9} - 218112498 T^{10} - 10723864485 T^{11} + 42180533641 T^{12} \))(\( 1 + 15 T - 18 T^{2} - 57 T^{3} + 16947 T^{4} + 71898 T^{5} - 430157 T^{6} + 4241982 T^{7} + 58992507 T^{8} - 11706603 T^{9} - 218112498 T^{10} + 10723864485 T^{11} + 42180533641 T^{12} \))
$61$ (\( 1 - 5 T - 36 T^{2} - 305 T^{3} + 3721 T^{4} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \))(\( 1 - 5 T - 36 T^{2} - 305 T^{3} + 3721 T^{4} \))(\( ( 1 + 5 T - 36 T^{2} + 305 T^{3} + 3721 T^{4} )^{2} \))(\( 1 - T - 133 T^{2} - 132 T^{3} + 9781 T^{4} + 12493 T^{5} - 645074 T^{6} + 762073 T^{7} + 36395101 T^{8} - 29961492 T^{9} - 1841496853 T^{10} - 844596301 T^{11} + 51520374361 T^{12} \))(\( 1 - T - 133 T^{2} - 132 T^{3} + 9781 T^{4} + 12493 T^{5} - 645074 T^{6} + 762073 T^{7} + 36395101 T^{8} - 29961492 T^{9} - 1841496853 T^{10} - 844596301 T^{11} + 51520374361 T^{12} \))
$67$ (\( 1 + 7 T - 18 T^{2} + 469 T^{3} + 4489 T^{4} \))(\( ( 1 - 11 T + 67 T^{2} )( 1 - 5 T + 67 T^{2} ) \))(\( 1 + 7 T - 18 T^{2} + 469 T^{3} + 4489 T^{4} \))(\( ( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} )^{2} \))(\( 1 + 14 T - 58 T^{2} - 258 T^{3} + 20800 T^{4} + 70720 T^{5} - 964241 T^{6} + 4738240 T^{7} + 93371200 T^{8} - 77596854 T^{9} - 1168765018 T^{10} + 18901751498 T^{11} + 90458382169 T^{12} \))(\( 1 + 14 T - 58 T^{2} - 258 T^{3} + 20800 T^{4} + 70720 T^{5} - 964241 T^{6} + 4738240 T^{7} + 93371200 T^{8} - 77596854 T^{9} - 1168765018 T^{10} + 18901751498 T^{11} + 90458382169 T^{12} \))
$71$ (\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{4} \))(\( ( 1 + 3 T + 105 T^{2} + 669 T^{3} + 7455 T^{4} + 15123 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 - 3 T + 105 T^{2} - 669 T^{3} + 7455 T^{4} - 15123 T^{5} + 357911 T^{6} )^{2} \))
$73$ (\( 1 - 6 T - 37 T^{2} - 438 T^{3} + 5329 T^{4} \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \))(\( 1 - 6 T - 37 T^{2} - 438 T^{3} + 5329 T^{4} \))(\( ( 1 - T - 72 T^{2} - 73 T^{3} + 5329 T^{4} )^{2} \))(\( 1 + 7 T - 36 T^{2} + 513 T^{3} + 733 T^{4} - 46082 T^{5} - 8831 T^{6} - 3363986 T^{7} + 3906157 T^{8} + 199565721 T^{9} - 1022336676 T^{10} + 14511501151 T^{11} + 151334226289 T^{12} \))(\( 1 + 7 T - 36 T^{2} + 513 T^{3} + 733 T^{4} - 46082 T^{5} - 8831 T^{6} - 3363986 T^{7} + 3906157 T^{8} + 199565721 T^{9} - 1022336676 T^{10} + 14511501151 T^{11} + 151334226289 T^{12} \))
$79$ (\( 1 + 11 T + 42 T^{2} + 869 T^{3} + 6241 T^{4} \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( 1 + 11 T + 42 T^{2} + 869 T^{3} + 6241 T^{4} \))(\( ( 1 - 17 T + 79 T^{2} )^{2}( 1 + 13 T + 79 T^{2} )^{2} \))(\( 1 + 6 T - 132 T^{2} - 634 T^{3} + 10026 T^{4} + 16110 T^{5} - 794253 T^{6} + 1272690 T^{7} + 62572266 T^{8} - 312586726 T^{9} - 5141410692 T^{10} + 18462338394 T^{11} + 243087455521 T^{12} \))(\( 1 + 6 T - 132 T^{2} - 634 T^{3} + 10026 T^{4} + 16110 T^{5} - 794253 T^{6} + 1272690 T^{7} + 62572266 T^{8} - 312586726 T^{9} - 5141410692 T^{10} + 18462338394 T^{11} + 243087455521 T^{12} \))
$83$ (\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 - 50 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 3 T + 69 T^{2} - 1227 T^{3} + 5727 T^{4} - 20667 T^{5} + 571787 T^{6} )^{2} \))(\( ( 1 + 3 T + 69 T^{2} + 1227 T^{3} + 5727 T^{4} + 20667 T^{5} + 571787 T^{6} )^{2} \))
$89$ (\( 1 - 4 T - 73 T^{2} - 356 T^{3} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 + 4 T - 73 T^{2} + 356 T^{3} + 7921 T^{4} \))(\( 1 - 172 T^{2} + 21663 T^{4} - 1362412 T^{6} + 62742241 T^{8} \))(\( 1 - 5 T - 149 T^{2} - 68 T^{3} + 12785 T^{4} + 45481 T^{5} - 1321850 T^{6} + 4047809 T^{7} + 101269985 T^{8} - 47937892 T^{9} - 9348593909 T^{10} - 27920297245 T^{11} + 496981290961 T^{12} \))(\( 1 + 5 T - 149 T^{2} + 68 T^{3} + 12785 T^{4} - 45481 T^{5} - 1321850 T^{6} - 4047809 T^{7} + 101269985 T^{8} + 47937892 T^{9} - 9348593909 T^{10} + 27920297245 T^{11} + 496981290961 T^{12} \))
$97$ (\( ( 1 - 9 T + 97 T^{2} )^{2} \))(\( ( 1 - 5 T + 97 T^{2} )^{2} \))(\( ( 1 - 9 T + 97 T^{2} )^{2} \))(\( ( 1 + T + 97 T^{2} )^{4} \))(\( ( 1 + 14 T + 307 T^{2} + 2692 T^{3} + 29779 T^{4} + 131726 T^{5} + 912673 T^{6} )^{2} \))(\( ( 1 + 14 T + 307 T^{2} + 2692 T^{3} + 29779 T^{4} + 131726 T^{5} + 912673 T^{6} )^{2} \))
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