Properties

Label 441.2.e
Level $441$
Weight $2$
Character orbit 441.e
Rep. character $\chi_{441}(226,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $30$
Newform subspaces $10$
Sturm bound $112$
Trace bound $10$

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Defining parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 10 \)
Sturm bound: \(112\)
Trace bound: \(10\)
Distinguishing \(T_p\): \(2\), \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(441, [\chi])\).

Total New Old
Modular forms 144 38 106
Cusp forms 80 30 50
Eisenstein series 64 8 56

Trace form

\( 30q - 3q^{2} - 15q^{4} + 2q^{5} + 18q^{8} + O(q^{10}) \) \( 30q - 3q^{2} - 15q^{4} + 2q^{5} + 18q^{8} - 4q^{10} + 2q^{11} + 12q^{13} - 11q^{16} - 6q^{19} - 8q^{20} - 8q^{22} - 9q^{25} - 2q^{26} + 28q^{29} + 2q^{31} - q^{32} - 12q^{37} - 2q^{38} - 20q^{41} - 4q^{43} - 24q^{44} + 12q^{46} + 6q^{47} - 46q^{50} + 16q^{52} + 6q^{53} - 8q^{55} + 42q^{58} + 12q^{59} + 24q^{61} + 36q^{62} + 14q^{64} + 30q^{65} - 10q^{67} - 4q^{71} - 10q^{73} + 4q^{74} - 32q^{76} - 42q^{79} - 8q^{80} - 20q^{82} + 12q^{83} - 64q^{85} + 38q^{86} + 72q^{88} - 16q^{89} - 96q^{92} - 12q^{94} - 34q^{95} - 16q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(441, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
441.2.e.a \(2\) \(3.521\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(-2\) \(0\) \(q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{4}-2\zeta_{6}q^{5}-3q^{8}+\cdots\)
441.2.e.b \(2\) \(3.521\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(2\) \(0\) \(q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{4}+2\zeta_{6}q^{5}-3q^{8}+\cdots\)
441.2.e.c \(2\) \(3.521\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q+(2-2\zeta_{6})q^{4}+7q^{13}-4\zeta_{6}q^{16}+\cdots\)
441.2.e.d \(2\) \(3.521\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-7}) \) \(1\) \(0\) \(0\) \(0\) \(q+\zeta_{6}q^{2}+(1-\zeta_{6})q^{4}+3q^{8}+(4-4\zeta_{6})q^{11}+\cdots\)
441.2.e.e \(2\) \(3.521\) \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(2\) \(0\) \(q+2\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{4}+2\zeta_{6}q^{5}+\cdots\)
441.2.e.f \(4\) \(3.521\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(-2\) \(0\) \(-4\) \(0\) \(q+(-1+\beta _{1}-\beta _{2})q^{2}+(-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
441.2.e.g \(4\) \(3.521\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(-2\) \(0\) \(4\) \(0\) \(q+(-1+\beta _{1}-\beta _{2})q^{2}+(-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
441.2.e.h \(4\) \(3.521\) \(\Q(\sqrt{-3}, \sqrt{7})\) \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+5\beta _{2}q^{4}+3\beta _{3}q^{8}+(2\beta _{1}+\cdots)q^{11}+\cdots\)
441.2.e.i \(4\) \(3.521\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{12}^{2}q^{2}+(-1+\zeta_{12})q^{4}-2\zeta_{12}^{2}q^{5}+\cdots\)
441.2.e.j \(4\) \(3.521\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{12}^{2}q^{2}+(-1+\zeta_{12})q^{4}+2\zeta_{12}^{2}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(441, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(441, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T - T^{2} + 2 T^{3} + 4 T^{4} \))(\( 1 + T - T^{2} + 2 T^{3} + 4 T^{4} \))(\( 1 - 2 T^{2} + 4 T^{4} \))(\( 1 - T - T^{2} - 2 T^{3} + 4 T^{4} \))(\( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} \))(\( 1 + 2 T + T^{2} - 2 T^{3} - 3 T^{4} - 4 T^{5} + 4 T^{6} + 16 T^{7} + 16 T^{8} \))(\( 1 + 2 T + T^{2} - 2 T^{3} - 3 T^{4} - 4 T^{5} + 4 T^{6} + 16 T^{7} + 16 T^{8} \))(\( 1 + 3 T^{2} + 5 T^{4} + 12 T^{6} + 16 T^{8} \))(\( 1 - T^{2} - 3 T^{4} - 4 T^{6} + 16 T^{8} \))(\( 1 - T^{2} - 3 T^{4} - 4 T^{6} + 16 T^{8} \))
$3$ 1
$5$ (\( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 + 4 T + 4 T^{2} + 8 T^{3} + 39 T^{4} + 40 T^{5} + 100 T^{6} + 500 T^{7} + 625 T^{8} \))(\( 1 - 4 T + 4 T^{2} - 8 T^{3} + 39 T^{4} - 40 T^{5} + 100 T^{6} - 500 T^{7} + 625 T^{8} \))(\( ( 1 - 5 T^{2} + 25 T^{4} )^{2} \))(\( 1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8} \))(\( 1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8} \))
$7$ 1
$11$ (\( 1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4} \))(\( 1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4} \))(\( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} \))(\( ( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} )^{2} \))(\( 1 + 6 T^{2} - 85 T^{4} + 726 T^{6} + 14641 T^{8} \))(\( 1 - 10 T^{2} - 21 T^{4} - 1210 T^{6} + 14641 T^{8} \))(\( 1 - 10 T^{2} - 21 T^{4} - 1210 T^{6} + 14641 T^{8} \))
$13$ (\( ( 1 + 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 7 T + 13 T^{2} )^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 + T + 13 T^{2} )^{2} \))(\( ( 1 + 8 T + 40 T^{2} + 104 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 - 8 T + 40 T^{2} - 104 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 13 T^{2} )^{4} \))(\( ( 1 + 2 T + 13 T^{2} )^{4} \))(\( ( 1 - 2 T + 13 T^{2} )^{4} \))
$17$ (\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 + 4 T - 4 T^{2} - 56 T^{3} - 161 T^{4} - 952 T^{5} - 1156 T^{6} + 19652 T^{7} + 83521 T^{8} \))(\( 1 - 4 T - 4 T^{2} + 56 T^{3} - 161 T^{4} + 952 T^{5} - 1156 T^{6} - 19652 T^{7} + 83521 T^{8} \))(\( ( 1 - 17 T^{2} + 289 T^{4} )^{2} \))(\( 1 - 22 T^{2} + 195 T^{4} - 6358 T^{6} + 83521 T^{8} \))(\( 1 - 22 T^{2} + 195 T^{4} - 6358 T^{6} + 83521 T^{8} \))
$19$ (\( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} \))(\( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \))(\( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 - 19 T^{2} + 361 T^{4} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( 1 - 30 T^{2} + 539 T^{4} - 10830 T^{6} + 130321 T^{8} \))(\( 1 - 30 T^{2} + 539 T^{4} - 10830 T^{6} + 130321 T^{8} \))(\( ( 1 - 19 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} )^{2} \))
$23$ (\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 8 T + 41 T^{2} - 184 T^{3} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 + 4 T - 2 T^{2} - 112 T^{3} - 573 T^{4} - 2576 T^{5} - 1058 T^{6} + 48668 T^{7} + 279841 T^{8} \))(\( 1 + 4 T - 2 T^{2} - 112 T^{3} - 573 T^{4} - 2576 T^{5} - 1058 T^{6} + 48668 T^{7} + 279841 T^{8} \))(\( 1 - 18 T^{2} - 205 T^{4} - 9522 T^{6} + 279841 T^{8} \))(\( 1 - 34 T^{2} + 627 T^{4} - 17986 T^{6} + 279841 T^{8} \))(\( 1 - 34 T^{2} + 627 T^{4} - 17986 T^{6} + 279841 T^{8} \))
$29$ (\( ( 1 - 2 T + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 2 T + 29 T^{2} )^{2} \))(\( ( 1 + 4 T + 29 T^{2} )^{2} \))(\( ( 1 - 8 T + 66 T^{2} - 232 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 - 8 T + 66 T^{2} - 232 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 - 54 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 29 T^{2} )^{4} \))(\( ( 1 + 29 T^{2} )^{4} \))
$31$ (\( 1 - 31 T^{2} + 961 T^{4} \))(\( 1 - 31 T^{2} + 961 T^{4} \))(\( ( 1 - 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( 1 - 31 T^{2} + 961 T^{4} \))(\( 1 - 9 T + 50 T^{2} - 279 T^{3} + 961 T^{4} \))(\( 1 + 8 T - 6 T^{2} + 64 T^{3} + 1955 T^{4} + 1984 T^{5} - 5766 T^{6} + 238328 T^{7} + 923521 T^{8} \))(\( 1 - 8 T - 6 T^{2} - 64 T^{3} + 1955 T^{4} - 1984 T^{5} - 5766 T^{6} - 238328 T^{7} + 923521 T^{8} \))(\( ( 1 - 31 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 7 T + 31 T^{2} )^{2}( 1 + 11 T + 31 T^{2} )^{2} \))(\( ( 1 - 11 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \))
$37$ (\( 1 + 6 T - T^{2} + 222 T^{3} + 1369 T^{4} \))(\( 1 + 6 T - T^{2} + 222 T^{3} + 1369 T^{4} \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 - 6 T - T^{2} - 222 T^{3} + 1369 T^{4} \))(\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( ( 1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 6 T - T^{2} + 222 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} )^{2} \))
$41$ (\( ( 1 + 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 10 T + 41 T^{2} )^{2} \))(\( ( 1 - 4 T + 68 T^{2} - 164 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 4 T + 68 T^{2} + 164 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 - 26 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 26 T^{2} + 1681 T^{4} )^{2} \))
$43$ (\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 - 5 T + 43 T^{2} )^{2} \))(\( ( 1 + 12 T + 43 T^{2} )^{2} \))(\( ( 1 - 5 T + 43 T^{2} )^{2} \))(\( ( 1 + 54 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 54 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 12 T + 43 T^{2} )^{4} \))(\( ( 1 + 4 T + 43 T^{2} )^{4} \))(\( ( 1 + 4 T + 43 T^{2} )^{4} \))
$47$ (\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 - 86 T^{2} + 5187 T^{4} - 189974 T^{6} + 4879681 T^{8} \))(\( 1 - 86 T^{2} + 5187 T^{4} - 189974 T^{6} + 4879681 T^{8} \))(\( ( 1 - 47 T^{2} + 2209 T^{4} )^{2} \))(\( 1 - 46 T^{2} - 93 T^{4} - 101614 T^{6} + 4879681 T^{8} \))(\( 1 - 46 T^{2} - 93 T^{4} - 101614 T^{6} + 4879681 T^{8} \))
$53$ (\( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} \))(\( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} \))(\( 1 - 53 T^{2} + 2809 T^{4} \))(\( 1 + 10 T + 47 T^{2} + 530 T^{3} + 2809 T^{4} \))(\( 1 - 12 T + 91 T^{2} - 636 T^{3} + 2809 T^{4} \))(\( ( 1 + 2 T - 49 T^{2} + 106 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 + 2 T - 49 T^{2} + 106 T^{3} + 2809 T^{4} )^{2} \))(\( 1 + 6 T^{2} - 2773 T^{4} + 16854 T^{6} + 7890481 T^{8} \))(\( 1 - 58 T^{2} + 555 T^{4} - 162922 T^{6} + 7890481 T^{8} \))(\( 1 - 58 T^{2} + 555 T^{4} - 162922 T^{6} + 7890481 T^{8} \))
$59$ (\( 1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4} \))(\( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4} \))(\( 1 - 8 T - 62 T^{2} - 64 T^{3} + 8619 T^{4} - 3776 T^{5} - 215822 T^{6} - 1643032 T^{7} + 12117361 T^{8} \))(\( 1 + 8 T - 62 T^{2} + 64 T^{3} + 8619 T^{4} + 3776 T^{5} - 215822 T^{6} + 1643032 T^{7} + 12117361 T^{8} \))(\( ( 1 - 59 T^{2} + 3481 T^{4} )^{2} \))(\( 1 - 70 T^{2} + 1419 T^{4} - 243670 T^{6} + 12117361 T^{8} \))(\( 1 - 70 T^{2} + 1419 T^{4} - 243670 T^{6} + 12117361 T^{8} \))
$61$ (\( 1 - 2 T - 57 T^{2} - 122 T^{3} + 3721 T^{4} \))(\( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \))(\( ( 1 - 13 T + 61 T^{2} )( 1 - T + 61 T^{2} ) \))(\( 1 - 61 T^{2} + 3721 T^{4} \))(\( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} \))(\( 1 - 16 T + 88 T^{2} - 736 T^{3} + 8887 T^{4} - 44896 T^{5} + 327448 T^{6} - 3631696 T^{7} + 13845841 T^{8} \))(\( 1 + 16 T + 88 T^{2} + 736 T^{3} + 8887 T^{4} + 44896 T^{5} + 327448 T^{6} + 3631696 T^{7} + 13845841 T^{8} \))(\( ( 1 - 61 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} )^{2} \))
$67$ (\( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( ( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \))(\( 1 - 102 T^{2} + 5915 T^{4} - 457878 T^{6} + 20151121 T^{8} \))(\( 1 - 102 T^{2} + 5915 T^{4} - 457878 T^{6} + 20151121 T^{8} \))(\( ( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} )^{2} \))
$71$ (\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 16 T + 71 T^{2} )^{2} \))(\( ( 1 - 6 T + 71 T^{2} )^{2} \))(\( ( 1 - 4 T + 18 T^{2} - 284 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 4 T + 18 T^{2} - 284 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 114 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 34 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 34 T^{2} + 5041 T^{4} )^{2} \))
$73$ (\( 1 - 6 T - 37 T^{2} - 438 T^{3} + 5329 T^{4} \))(\( 1 + 6 T - 37 T^{2} + 438 T^{3} + 5329 T^{4} \))(\( ( 1 - 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( 1 - 73 T^{2} + 5329 T^{4} \))(\( 1 + 3 T - 64 T^{2} + 219 T^{3} + 5329 T^{4} \))(\( 1 - 8 T + 656 T^{3} - 5905 T^{4} + 47888 T^{5} - 3112136 T^{7} + 28398241 T^{8} \))(\( 1 + 8 T - 656 T^{3} - 5905 T^{4} - 47888 T^{5} + 3112136 T^{7} + 28398241 T^{8} \))(\( ( 1 - 73 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 14 T + 123 T^{2} + 1022 T^{3} + 5329 T^{4} )^{2} \))
$79$ (\( 1 - 16 T + 177 T^{2} - 1264 T^{3} + 6241 T^{4} \))(\( 1 - 16 T + 177 T^{2} - 1264 T^{3} + 6241 T^{4} \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \))(\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( 1 + 16 T + 66 T^{2} + 512 T^{3} + 9635 T^{4} + 40448 T^{5} + 411906 T^{6} + 7888624 T^{7} + 38950081 T^{8} \))(\( 1 + 16 T + 66 T^{2} + 512 T^{3} + 9635 T^{4} + 40448 T^{5} + 411906 T^{6} + 7888624 T^{7} + 38950081 T^{8} \))(\( ( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} )^{2} \))
$83$ (\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 8 T + 54 T^{2} + 664 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 - 8 T + 54 T^{2} - 664 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 83 T^{2} )^{4} \))(\( ( 1 + 83 T^{2} )^{4} \))(\( ( 1 + 83 T^{2} )^{4} \))
$89$ (\( 1 + 14 T + 107 T^{2} + 1246 T^{3} + 7921 T^{4} \))(\( 1 - 14 T + 107 T^{2} - 1246 T^{3} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 + 16 T + 167 T^{2} + 1424 T^{3} + 7921 T^{4} \))(\( 1 - 20 T + 140 T^{2} - 1640 T^{3} + 24079 T^{4} - 145960 T^{5} + 1108940 T^{6} - 14099380 T^{7} + 62742241 T^{8} \))(\( 1 + 20 T + 140 T^{2} + 1640 T^{3} + 24079 T^{4} + 145960 T^{5} + 1108940 T^{6} + 14099380 T^{7} + 62742241 T^{8} \))(\( ( 1 - 89 T^{2} + 7921 T^{4} )^{2} \))(\( 1 - 166 T^{2} + 19635 T^{4} - 1314886 T^{6} + 62742241 T^{8} \))(\( 1 - 166 T^{2} + 19635 T^{4} - 1314886 T^{6} + 62742241 T^{8} \))
$97$ (\( ( 1 - 18 T + 97 T^{2} )^{2} \))(\( ( 1 + 18 T + 97 T^{2} )^{2} \))(\( ( 1 + 14 T + 97 T^{2} )^{2} \))(\( ( 1 + 97 T^{2} )^{2} \))(\( ( 1 - 6 T + 97 T^{2} )^{2} \))(\( ( 1 + 8 T + 208 T^{2} + 776 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 8 T + 208 T^{2} - 776 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 97 T^{2} )^{4} \))(\( ( 1 + 14 T + 97 T^{2} )^{4} \))(\( ( 1 - 14 T + 97 T^{2} )^{4} \))
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