Properties

Label 1890.2.d
Level 1890
Weight 2
Character orbit d
Rep. character \(\chi_{1890}(1889,\cdot)\)
Character field \(\Q\)
Dimension 64
Newform subspaces 6
Sturm bound 864
Trace bound 23

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Defining parameters

Level: \( N \) = \( 1890 = 2 \cdot 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1890.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 105 \)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(864\)
Trace bound: \(23\)
Distinguishing \(T_p\): \(11\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1890, [\chi])\).

Total New Old
Modular forms 456 64 392
Cusp forms 408 64 344
Eisenstein series 48 0 48

Trace form

\( 64q + 64q^{4} + O(q^{10}) \) \( 64q + 64q^{4} + 64q^{16} - 8q^{46} + 68q^{49} + 64q^{64} - 2q^{70} + 64q^{79} - 32q^{85} - 16q^{91} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1890, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1890.2.d.a \(8\) \(15.092\) 8.0.303595776.1 None \(-8\) \(0\) \(0\) \(0\) \(q-q^{2}+q^{4}+(-\beta _{3}-\beta _{5}+\beta _{7})q^{5}+\cdots\)
1890.2.d.b \(8\) \(15.092\) 8.0.\(\cdots\).2 None \(-8\) \(0\) \(0\) \(0\) \(q-q^{2}+q^{4}+(-\beta _{1}-\beta _{3})q^{5}+(-\beta _{4}+\cdots)q^{7}+\cdots\)
1890.2.d.c \(8\) \(15.092\) 8.0.\(\cdots\).2 None \(8\) \(0\) \(0\) \(0\) \(q+q^{2}+q^{4}+(\beta _{1}-\beta _{3})q^{5}-\beta _{4}q^{7}+\cdots\)
1890.2.d.d \(8\) \(15.092\) 8.0.303595776.1 None \(8\) \(0\) \(0\) \(0\) \(q+q^{2}+q^{4}+(\beta _{1}+\beta _{3}-\beta _{5})q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
1890.2.d.e \(16\) \(15.092\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(-16\) \(0\) \(0\) \(0\) \(q-q^{2}+q^{4}-\beta _{15}q^{5}-\beta _{6}q^{7}-q^{8}+\cdots\)
1890.2.d.f \(16\) \(15.092\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(16\) \(0\) \(0\) \(0\) \(q+q^{2}+q^{4}-\beta _{8}q^{5}-\beta _{11}q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1890, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1890, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(315, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(630, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(945, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + T )^{8} \))(\( ( 1 + T )^{8} \))(\( ( 1 - T )^{8} \))(\( ( 1 - T )^{8} \))(\( ( 1 + T )^{16} \))(\( ( 1 - T )^{16} \))
$3$ 1
$5$ (\( 1 + T^{2} - 24 T^{4} + 25 T^{6} + 625 T^{8} \))(\( ( 1 - 2 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 2 T^{2} + 25 T^{4} )^{2} \))(\( 1 + T^{2} - 24 T^{4} + 25 T^{6} + 625 T^{8} \))(\( 1 + 3 T^{2} - 10 T^{4} + 33 T^{6} + 474 T^{8} + 825 T^{10} - 6250 T^{12} + 46875 T^{14} + 390625 T^{16} \))(\( 1 + 3 T^{2} - 10 T^{4} + 33 T^{6} + 474 T^{8} + 825 T^{10} - 6250 T^{12} + 46875 T^{14} + 390625 T^{16} \))
$7$ (\( 1 - 34 T^{4} + 2401 T^{8} \))(\( ( 1 - 8 T^{2} + 49 T^{4} )^{2} \))(\( ( 1 - 8 T^{2} + 49 T^{4} )^{2} \))(\( 1 - 34 T^{4} + 2401 T^{8} \))(\( 1 - T^{2} - 22 T^{4} - 15 T^{6} + 554 T^{8} - 735 T^{10} - 52822 T^{12} - 117649 T^{14} + 5764801 T^{16} \))(\( 1 - T^{2} - 22 T^{4} - 15 T^{6} + 554 T^{8} - 735 T^{10} - 52822 T^{12} - 117649 T^{14} + 5764801 T^{16} \))
$11$ (\( ( 1 - 37 T^{2} + 576 T^{4} - 4477 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 - 10 T^{2} + 3 T^{4} - 1210 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 - 10 T^{2} + 3 T^{4} - 1210 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 - 37 T^{2} + 576 T^{4} - 4477 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 - 38 T^{2} + 913 T^{4} - 15590 T^{6} + 194008 T^{8} - 1886390 T^{10} + 13367233 T^{12} - 67319318 T^{14} + 214358881 T^{16} )^{2} \))(\( ( 1 - 38 T^{2} + 913 T^{4} - 15590 T^{6} + 194008 T^{8} - 1886390 T^{10} + 13367233 T^{12} - 67319318 T^{14} + 214358881 T^{16} )^{2} \))
$13$ (\( ( 1 + 24 T^{2} + 350 T^{4} + 4056 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 + 2 T^{2} + 75 T^{4} + 338 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 + 2 T^{2} + 75 T^{4} + 338 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 + 24 T^{2} + 350 T^{4} + 4056 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 + 53 T^{2} + 1421 T^{4} + 26118 T^{6} + 376130 T^{8} + 4413942 T^{10} + 40585181 T^{12} + 255820877 T^{14} + 815730721 T^{16} )^{2} \))(\( ( 1 + 53 T^{2} + 1421 T^{4} + 26118 T^{6} + 376130 T^{8} + 4413942 T^{10} + 40585181 T^{12} + 255820877 T^{14} + 815730721 T^{16} )^{2} \))
$17$ (\( ( 1 - 8 T + 17 T^{2} )^{4}( 1 + 8 T + 17 T^{2} )^{4} \))(\( ( 1 + 2 T^{2} + 315 T^{4} + 578 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 + 2 T^{2} + 315 T^{4} + 578 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 - 8 T + 17 T^{2} )^{4}( 1 + 8 T + 17 T^{2} )^{4} \))(\( ( 1 - 58 T^{2} + 2009 T^{4} - 49530 T^{6} + 951284 T^{8} - 14314170 T^{10} + 167793689 T^{12} - 1399979002 T^{14} + 6975757441 T^{16} )^{2} \))(\( ( 1 - 58 T^{2} + 2009 T^{4} - 49530 T^{6} + 951284 T^{8} - 14314170 T^{10} + 167793689 T^{12} - 1399979002 T^{14} + 6975757441 T^{16} )^{2} \))
$19$ (\( ( 1 + T^{2} - 276 T^{4} + 361 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 20 T^{2} + 361 T^{4} )^{4} \))(\( ( 1 - 20 T^{2} + 361 T^{4} )^{4} \))(\( ( 1 + T^{2} - 276 T^{4} + 361 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 81 T^{2} + 3350 T^{4} - 98331 T^{6} + 2169690 T^{8} - 35497491 T^{10} + 436575350 T^{12} - 3810716361 T^{14} + 16983563041 T^{16} )^{2} \))(\( ( 1 - 81 T^{2} + 3350 T^{4} - 98331 T^{6} + 2169690 T^{8} - 35497491 T^{10} + 436575350 T^{12} - 3810716361 T^{14} + 16983563041 T^{16} )^{2} \))
$23$ (\( ( 1 + 7 T + 50 T^{2} + 161 T^{3} + 529 T^{4} )^{4} \))(\( ( 1 - 3 T + 23 T^{2} )^{8} \))(\( ( 1 + 3 T + 23 T^{2} )^{8} \))(\( ( 1 - 7 T + 50 T^{2} - 161 T^{3} + 529 T^{4} )^{4} \))(\( ( 1 - 2 T + 33 T^{2} + 820 T^{4} + 17457 T^{6} - 24334 T^{7} + 279841 T^{8} )^{4} \))(\( ( 1 + 2 T + 33 T^{2} + 820 T^{4} + 17457 T^{6} + 24334 T^{7} + 279841 T^{8} )^{4} \))
$29$ (\( ( 1 - 37 T^{2} + 1620 T^{4} - 31117 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 47 T^{2} + 841 T^{4} )^{4} \))(\( ( 1 - 47 T^{2} + 841 T^{4} )^{4} \))(\( ( 1 - 37 T^{2} + 1620 T^{4} - 31117 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 134 T^{2} + 9253 T^{4} - 425582 T^{6} + 14258284 T^{8} - 357914462 T^{10} + 6544471093 T^{12} - 79706325014 T^{14} + 500246412961 T^{16} )^{2} \))(\( ( 1 - 134 T^{2} + 9253 T^{4} - 425582 T^{6} + 14258284 T^{8} - 357914462 T^{10} + 6544471093 T^{12} - 79706325014 T^{14} + 500246412961 T^{16} )^{2} \))
$31$ (\( ( 1 - 37 T^{2} + 961 T^{4} )^{4} \))(\( ( 1 - 22 T^{2} - 333 T^{4} - 21142 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 - 22 T^{2} - 333 T^{4} - 21142 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 - 37 T^{2} + 961 T^{4} )^{4} \))(\( ( 1 - 159 T^{2} + 12581 T^{4} - 641190 T^{6} + 23260110 T^{8} - 616183590 T^{10} + 11618817701 T^{12} - 141113085279 T^{14} + 852891037441 T^{16} )^{2} \))(\( ( 1 - 159 T^{2} + 12581 T^{4} - 641190 T^{6} + 23260110 T^{8} - 616183590 T^{10} + 11618817701 T^{12} - 141113085279 T^{14} + 852891037441 T^{16} )^{2} \))
$37$ (\( ( 1 - 102 T^{2} + 4811 T^{4} - 139638 T^{6} + 1874161 T^{8} )^{2} \))(\( ( 1 - 50 T^{2} + 1369 T^{4} )^{4} \))(\( ( 1 - 50 T^{2} + 1369 T^{4} )^{4} \))(\( ( 1 - 102 T^{2} + 4811 T^{4} - 139638 T^{6} + 1874161 T^{8} )^{2} \))(\( ( 1 - 196 T^{2} + 19640 T^{4} - 1252476 T^{6} + 55246958 T^{8} - 1714639644 T^{10} + 36808522040 T^{12} - 502882376164 T^{14} + 3512479453921 T^{16} )^{2} \))(\( ( 1 - 196 T^{2} + 19640 T^{4} - 1252476 T^{6} + 55246958 T^{8} - 1714639644 T^{10} + 36808522040 T^{12} - 502882376164 T^{14} + 3512479453921 T^{16} )^{2} \))
$41$ (\( ( 1 + 70 T^{2} + 4059 T^{4} + 117670 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 + 70 T^{2} + 1681 T^{4} )^{4} \))(\( ( 1 + 70 T^{2} + 1681 T^{4} )^{4} \))(\( ( 1 + 70 T^{2} + 4059 T^{4} + 117670 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 + 159 T^{2} + 13778 T^{4} + 840417 T^{6} + 39217914 T^{8} + 1412740977 T^{10} + 38933335058 T^{12} + 755266574319 T^{14} + 7984925229121 T^{16} )^{2} \))(\( ( 1 + 159 T^{2} + 13778 T^{4} + 840417 T^{6} + 39217914 T^{8} + 1412740977 T^{10} + 38933335058 T^{12} + 755266574319 T^{14} + 7984925229121 T^{16} )^{2} \))
$43$ (\( ( 1 - 96 T^{2} + 4814 T^{4} - 177504 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 + 13 T^{2} + 1849 T^{4} )^{4} \))(\( ( 1 + 13 T^{2} + 1849 T^{4} )^{4} \))(\( ( 1 - 96 T^{2} + 4814 T^{4} - 177504 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 - 295 T^{2} + 39917 T^{4} - 3224142 T^{6} + 169742774 T^{8} - 5961438558 T^{10} + 136468279517 T^{12} - 1864802099455 T^{14} + 11688200277601 T^{16} )^{2} \))(\( ( 1 - 295 T^{2} + 39917 T^{4} - 3224142 T^{6} + 169742774 T^{8} - 5961438558 T^{10} + 136468279517 T^{12} - 1864802099455 T^{14} + 11688200277601 T^{16} )^{2} \))
$47$ (\( ( 1 - 167 T^{2} + 11316 T^{4} - 368903 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 - 106 T^{2} + 6171 T^{4} - 234154 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 - 106 T^{2} + 6171 T^{4} - 234154 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 - 167 T^{2} + 11316 T^{4} - 368903 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 + 15 T^{2} + 2378 T^{4} - 71535 T^{6} + 4137834 T^{8} - 158020815 T^{10} + 11603881418 T^{12} + 161688229935 T^{14} + 23811286661761 T^{16} )^{2} \))(\( ( 1 + 15 T^{2} + 2378 T^{4} - 71535 T^{6} + 4137834 T^{8} - 158020815 T^{10} + 11603881418 T^{12} + 161688229935 T^{14} + 23811286661761 T^{16} )^{2} \))
$53$ (\( ( 1 - 4 T + 53 T^{2} )^{8} \))(\( ( 1 + 12 T + 76 T^{2} + 636 T^{3} + 2809 T^{4} )^{4} \))(\( ( 1 - 12 T + 76 T^{2} - 636 T^{3} + 2809 T^{4} )^{4} \))(\( ( 1 + 4 T + 53 T^{2} )^{8} \))(\( ( 1 + 4 T + 72 T^{2} - 96 T^{3} + 3025 T^{4} - 5088 T^{5} + 202248 T^{6} + 595508 T^{7} + 7890481 T^{8} )^{4} \))(\( ( 1 - 4 T + 72 T^{2} + 96 T^{3} + 3025 T^{4} + 5088 T^{5} + 202248 T^{6} - 595508 T^{7} + 7890481 T^{8} )^{4} \))
$59$ (\( ( 1 + 85 T^{2} + 6912 T^{4} + 295885 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 + 106 T^{2} + 3481 T^{4} )^{4} \))(\( ( 1 + 106 T^{2} + 3481 T^{4} )^{4} \))(\( ( 1 + 85 T^{2} + 6912 T^{4} + 295885 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 + 198 T^{2} + 24269 T^{4} + 2110662 T^{6} + 143664636 T^{8} + 7347214422 T^{10} + 294076234109 T^{12} + 8351745660918 T^{14} + 146830437604321 T^{16} )^{2} \))(\( ( 1 + 198 T^{2} + 24269 T^{4} + 2110662 T^{6} + 143664636 T^{8} + 7347214422 T^{10} + 294076234109 T^{12} + 8351745660918 T^{14} + 146830437604321 T^{16} )^{2} \))
$61$ (\( ( 1 - 91 T^{2} + 8844 T^{4} - 338611 T^{6} + 13845841 T^{8} )^{2} \))(\( ( 1 - 104 T^{2} + 3721 T^{4} )^{4} \))(\( ( 1 - 104 T^{2} + 3721 T^{4} )^{4} \))(\( ( 1 - 91 T^{2} + 8844 T^{4} - 338611 T^{6} + 13845841 T^{8} )^{2} \))(\( ( 1 - 61 T^{2} )^{16} \))(\( ( 1 - 61 T^{2} )^{16} \))
$67$ (\( ( 1 - 96 T^{2} + 7982 T^{4} - 430944 T^{6} + 20151121 T^{8} )^{2} \))(\( ( 1 - 110 T^{2} + 4489 T^{4} )^{4} \))(\( ( 1 - 110 T^{2} + 4489 T^{4} )^{4} \))(\( ( 1 - 96 T^{2} + 7982 T^{4} - 430944 T^{6} + 20151121 T^{8} )^{2} \))(\( ( 1 - 79 T^{2} + 5069 T^{4} - 215790 T^{6} - 4579210 T^{8} - 968681310 T^{10} + 102146032349 T^{12} - 7146212191351 T^{14} + 406067677556641 T^{16} )^{2} \))(\( ( 1 - 79 T^{2} + 5069 T^{4} - 215790 T^{6} - 4579210 T^{8} - 968681310 T^{10} + 102146032349 T^{12} - 7146212191351 T^{14} + 406067677556641 T^{16} )^{2} \))
$71$ (\( ( 1 - 166 T^{2} + 14859 T^{4} - 836806 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 - 98 T^{2} + 5041 T^{4} )^{4} \))(\( ( 1 - 98 T^{2} + 5041 T^{4} )^{4} \))(\( ( 1 - 166 T^{2} + 14859 T^{4} - 836806 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 - 386 T^{2} + 71989 T^{4} - 8520110 T^{6} + 710279188 T^{8} - 42949874510 T^{10} + 1829361503509 T^{12} - 49446709593506 T^{14} + 645753531245761 T^{16} )^{2} \))(\( ( 1 - 386 T^{2} + 71989 T^{4} - 8520110 T^{6} + 710279188 T^{8} - 42949874510 T^{10} + 1829361503509 T^{12} - 49446709593506 T^{14} + 645753531245761 T^{16} )^{2} \))
$73$ (\( ( 1 + 229 T^{2} + 23100 T^{4} + 1220341 T^{6} + 28398241 T^{8} )^{2} \))(\( ( 1 + 124 T^{2} + 5329 T^{4} )^{4} \))(\( ( 1 + 124 T^{2} + 5329 T^{4} )^{4} \))(\( ( 1 + 229 T^{2} + 23100 T^{4} + 1220341 T^{6} + 28398241 T^{8} )^{2} \))(\( ( 1 + 306 T^{2} + 55085 T^{4} + 6477858 T^{6} + 556796076 T^{8} + 34520505282 T^{10} + 1564317105485 T^{12} + 46308273244434 T^{14} + 806460091894081 T^{16} )^{2} \))(\( ( 1 + 306 T^{2} + 55085 T^{4} + 6477858 T^{6} + 556796076 T^{8} + 34520505282 T^{10} + 1564317105485 T^{12} + 46308273244434 T^{14} + 806460091894081 T^{16} )^{2} \))
$79$ (\( ( 1 - 6 T + 79 T^{2} )^{8} \))(\( ( 1 + 2 T + 93 T^{2} + 158 T^{3} + 6241 T^{4} )^{4} \))(\( ( 1 + 2 T + 93 T^{2} + 158 T^{3} + 6241 T^{4} )^{4} \))(\( ( 1 - 6 T + 79 T^{2} )^{8} \))(\( ( 1 + 2 T + 110 T^{2} - 510 T^{3} + 3746 T^{4} - 40290 T^{5} + 686510 T^{6} + 986078 T^{7} + 38950081 T^{8} )^{4} \))(\( ( 1 + 2 T + 110 T^{2} - 510 T^{3} + 3746 T^{4} - 40290 T^{5} + 686510 T^{6} + 986078 T^{7} + 38950081 T^{8} )^{4} \))
$83$ (\( ( 1 - 36 T^{2} + 5654 T^{4} - 248004 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 - 116 T^{2} + 6889 T^{4} )^{4} \))(\( ( 1 - 116 T^{2} + 6889 T^{4} )^{4} \))(\( ( 1 - 36 T^{2} + 5654 T^{4} - 248004 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 - 586 T^{2} + 154913 T^{4} - 24245394 T^{6} + 2465561876 T^{8} - 167026519266 T^{10} + 7351910881073 T^{12} - 191587058794234 T^{14} + 2252292232139041 T^{16} )^{2} \))(\( ( 1 - 586 T^{2} + 154913 T^{4} - 24245394 T^{6} + 2465561876 T^{8} - 167026519266 T^{10} + 7351910881073 T^{12} - 191587058794234 T^{14} + 2252292232139041 T^{16} )^{2} \))
$89$ (\( ( 1 + 337 T^{2} + 44160 T^{4} + 2669377 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 + 166 T^{2} + 7921 T^{4} )^{4} \))(\( ( 1 + 166 T^{2} + 7921 T^{4} )^{4} \))(\( ( 1 + 337 T^{2} + 44160 T^{4} + 2669377 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 + 189 T^{2} + 9461 T^{4} + 745146 T^{6} + 124208310 T^{8} + 5902301466 T^{10} + 593604342101 T^{12} + 93929463991629 T^{14} + 3936588805702081 T^{16} )^{2} \))(\( ( 1 + 189 T^{2} + 9461 T^{4} + 745146 T^{6} + 124208310 T^{8} + 5902301466 T^{10} + 593604342101 T^{12} + 93929463991629 T^{14} + 3936588805702081 T^{16} )^{2} \))
$97$ (\( ( 1 + 146 T^{2} + 9409 T^{4} )^{4} \))(\( ( 1 + 248 T^{2} + 29970 T^{4} + 2333432 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 + 248 T^{2} + 29970 T^{4} + 2333432 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 + 146 T^{2} + 9409 T^{4} )^{4} \))(\( ( 1 + 507 T^{2} + 129614 T^{4} + 21362481 T^{6} + 2454936522 T^{8} + 200999583729 T^{10} + 11474634227534 T^{12} + 422316806499003 T^{14} + 7837433594376961 T^{16} )^{2} \))(\( ( 1 + 507 T^{2} + 129614 T^{4} + 21362481 T^{6} + 2454936522 T^{8} + 200999583729 T^{10} + 11474634227534 T^{12} + 422316806499003 T^{14} + 7837433594376961 T^{16} )^{2} \))
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