Properties

Label 350.2.e
Level 350
Weight 2
Character orbit e
Rep. character \(\chi_{350}(51,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 24
Newform subspaces 12
Sturm bound 120
Trace bound 7

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Defining parameters

Level: \( N \) = \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 350.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 12 \)
Sturm bound: \(120\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(350, [\chi])\).

Total New Old
Modular forms 144 24 120
Cusp forms 96 24 72
Eisenstein series 48 0 48

Trace form

\( 24q - 12q^{4} + 8q^{7} - 8q^{9} + O(q^{10}) \) \( 24q - 12q^{4} + 8q^{7} - 8q^{9} - 4q^{14} - 12q^{16} - 4q^{17} + 8q^{18} + 4q^{19} + 4q^{21} - 8q^{22} + 12q^{23} + 8q^{26} - 24q^{27} - 4q^{28} - 8q^{29} - 20q^{31} - 24q^{33} - 16q^{34} + 16q^{36} - 4q^{37} - 8q^{38} + 8q^{39} - 24q^{41} - 4q^{42} + 40q^{43} - 8q^{46} + 20q^{47} - 12q^{51} + 4q^{53} + 12q^{54} - 4q^{56} + 24q^{57} + 4q^{59} + 32q^{61} + 48q^{62} - 36q^{63} + 24q^{64} - 4q^{68} + 48q^{69} - 24q^{71} + 8q^{72} - 28q^{73} + 20q^{74} - 8q^{76} + 8q^{77} + 16q^{78} + 12q^{79} - 20q^{81} - 16q^{82} + 32q^{83} + 28q^{84} - 8q^{86} + 24q^{87} + 4q^{88} - 8q^{89} + 28q^{91} - 24q^{92} - 12q^{93} + 4q^{94} - 64q^{97} - 24q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(350, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
350.2.e.a \(2\) \(2.795\) \(\Q(\sqrt{-3}) \) None \(-1\) \(-2\) \(0\) \(-5\) \(q-\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
350.2.e.b \(2\) \(2.795\) \(\Q(\sqrt{-3}) \) None \(-1\) \(-2\) \(0\) \(4\) \(q-\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
350.2.e.c \(2\) \(2.795\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(0\) \(-4\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(-3+2\zeta_{6})q^{7}+\cdots\)
350.2.e.d \(2\) \(2.795\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(0\) \(1\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(2-3\zeta_{6})q^{7}+\cdots\)
350.2.e.e \(2\) \(2.795\) \(\Q(\sqrt{-3}) \) None \(-1\) \(1\) \(0\) \(1\) \(q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
350.2.e.f \(2\) \(2.795\) \(\Q(\sqrt{-3}) \) None \(-1\) \(3\) \(0\) \(5\) \(q-\zeta_{6}q^{2}+(3-3\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
350.2.e.g \(2\) \(2.795\) \(\Q(\sqrt{-3}) \) None \(1\) \(-3\) \(0\) \(-5\) \(q+\zeta_{6}q^{2}+(-3+3\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
350.2.e.h \(2\) \(2.795\) \(\Q(\sqrt{-3}) \) None \(1\) \(-2\) \(0\) \(4\) \(q+\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
350.2.e.i \(2\) \(2.795\) \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(0\) \(-1\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(-2+3\zeta_{6})q^{7}+\cdots\)
350.2.e.j \(2\) \(2.795\) \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(0\) \(4\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(3-2\zeta_{6})q^{7}+\cdots\)
350.2.e.k \(2\) \(2.795\) \(\Q(\sqrt{-3}) \) None \(1\) \(2\) \(0\) \(5\) \(q+\zeta_{6}q^{2}+(2-2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
350.2.e.l \(2\) \(2.795\) \(\Q(\sqrt{-3}) \) None \(1\) \(3\) \(0\) \(-1\) \(q+\zeta_{6}q^{2}+(3-3\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(350, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(350, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))
$3$ (\( 1 + 2 T + T^{2} + 6 T^{3} + 9 T^{4} \))(\( 1 + 2 T + T^{2} + 6 T^{3} + 9 T^{4} \))(\( ( 1 - 3 T + 3 T^{2} )( 1 + 3 T + 3 T^{2} ) \))(\( ( 1 - 3 T + 3 T^{2} )( 1 + 3 T + 3 T^{2} ) \))(\( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4} \))(\( ( 1 - 3 T + 3 T^{2} )( 1 + 3 T^{2} ) \))(\( ( 1 + 3 T^{2} )( 1 + 3 T + 3 T^{2} ) \))(\( 1 + 2 T + T^{2} + 6 T^{3} + 9 T^{4} \))(\( ( 1 - 3 T + 3 T^{2} )( 1 + 3 T + 3 T^{2} ) \))(\( ( 1 - 3 T + 3 T^{2} )( 1 + 3 T + 3 T^{2} ) \))(\( 1 - 2 T + T^{2} - 6 T^{3} + 9 T^{4} \))(\( ( 1 - 3 T + 3 T^{2} )( 1 + 3 T^{2} ) \))
$5$ 1
$7$ (\( 1 + 5 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))
$11$ (\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \))
$13$ (\( ( 1 + 2 T + 13 T^{2} )^{2} \))(\( ( 1 - T + 13 T^{2} )^{2} \))(\( ( 1 + 5 T + 13 T^{2} )^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 - 4 T + 13 T^{2} )^{2} \))(\( ( 1 + 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 + 5 T + 13 T^{2} )^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 - 5 T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))
$17$ (\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 2 T - 13 T^{2} - 34 T^{3} + 289 T^{4} \))(\( 1 - 7 T + 32 T^{2} - 119 T^{3} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 2 T - 13 T^{2} - 34 T^{3} + 289 T^{4} \))(\( 1 + 2 T - 13 T^{2} + 34 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 + 7 T + 32 T^{2} + 119 T^{3} + 289 T^{4} \))(\( 1 + 2 T - 13 T^{2} + 34 T^{3} + 289 T^{4} \))(\( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \))(\( 1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4} \))
$19$ (\( ( 1 + T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} \))(\( 1 - 19 T^{2} + 361 T^{4} \))(\( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 - 2 T - 15 T^{2} - 38 T^{3} + 361 T^{4} \))(\( 1 - 2 T - 15 T^{2} - 38 T^{3} + 361 T^{4} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( 1 - 19 T^{2} + 361 T^{4} \))(\( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} \))(\( ( 1 + T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( 1 - 6 T + 17 T^{2} - 114 T^{3} + 361 T^{4} \))
$23$ (\( 1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4} \))(\( 1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4} \))(\( 1 - 7 T + 26 T^{2} - 161 T^{3} + 529 T^{4} \))(\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4} \))(\( 1 + T - 22 T^{2} + 23 T^{3} + 529 T^{4} \))(\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 + 7 T + 26 T^{2} + 161 T^{3} + 529 T^{4} \))(\( 1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4} \))(\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \))
$29$ (\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 4 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 3 T + 29 T^{2} )^{2} \))(\( ( 1 + T + 29 T^{2} )^{2} \))(\( ( 1 + T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 4 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 9 T + 29 T^{2} )^{2} \))
$31$ (\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \))(\( 1 - 2 T - 27 T^{2} - 62 T^{3} + 961 T^{4} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \))(\( 1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4} \))(\( 1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( 1 - 2 T - 27 T^{2} - 62 T^{3} + 961 T^{4} \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))
$37$ (\( 1 - 8 T + 27 T^{2} - 296 T^{3} + 1369 T^{4} \))(\( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} \))(\( ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( 1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4} \))(\( 1 + 4 T - 21 T^{2} + 148 T^{3} + 1369 T^{4} \))(\( 1 - 8 T + 27 T^{2} - 296 T^{3} + 1369 T^{4} \))(\( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} \))(\( ( 1 - 10 T + 37 T^{2} )( 1 - T + 37 T^{2} ) \))(\( 1 + 4 T - 21 T^{2} + 148 T^{3} + 1369 T^{4} \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} \))(\( 1 + 4 T - 21 T^{2} + 148 T^{3} + 1369 T^{4} \))
$41$ (\( ( 1 + 3 T + 41 T^{2} )^{2} \))(\( ( 1 - 3 T + 41 T^{2} )^{2} \))(\( ( 1 - 3 T + 41 T^{2} )^{2} \))(\( ( 1 + 7 T + 41 T^{2} )^{2} \))(\( ( 1 - 9 T + 41 T^{2} )^{2} \))(\( ( 1 + 3 T + 41 T^{2} )^{2} \))(\( ( 1 + 3 T + 41 T^{2} )^{2} \))(\( ( 1 - 3 T + 41 T^{2} )^{2} \))(\( ( 1 + 7 T + 41 T^{2} )^{2} \))(\( ( 1 - 3 T + 41 T^{2} )^{2} \))(\( ( 1 + 3 T + 41 T^{2} )^{2} \))(\( ( 1 + 7 T + 41 T^{2} )^{2} \))
$43$ (\( ( 1 - 10 T + 43 T^{2} )^{2} \))(\( ( 1 + 2 T + 43 T^{2} )^{2} \))(\( ( 1 - 2 T + 43 T^{2} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 7 T + 43 T^{2} )^{2} \))(\( ( 1 - 5 T + 43 T^{2} )^{2} \))(\( ( 1 + 5 T + 43 T^{2} )^{2} \))(\( ( 1 - 10 T + 43 T^{2} )^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 + 2 T + 43 T^{2} )^{2} \))(\( ( 1 + 10 T + 43 T^{2} )^{2} \))(\( ( 1 - 5 T + 43 T^{2} )^{2} \))
$47$ (\( 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} \))(\( 1 - 9 T + 34 T^{2} - 423 T^{3} + 2209 T^{4} \))(\( 1 - 7 T + 2 T^{2} - 329 T^{3} + 2209 T^{4} \))(\( 1 - 7 T + 2 T^{2} - 329 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 + 8 T + 17 T^{2} + 376 T^{3} + 2209 T^{4} \))(\( 1 - 8 T + 17 T^{2} - 376 T^{3} + 2209 T^{4} \))(\( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \))(\( 1 + 7 T + 2 T^{2} + 329 T^{3} + 2209 T^{4} \))(\( 1 + 7 T + 2 T^{2} + 329 T^{3} + 2209 T^{4} \))(\( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \))(\( 1 - 8 T + 17 T^{2} - 376 T^{3} + 2209 T^{4} \))
$53$ (\( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} \))(\( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \))(\( 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} \))(\( 1 + 4 T - 37 T^{2} + 212 T^{3} + 2809 T^{4} \))(\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} \))(\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4} \))(\( 1 - 4 T - 37 T^{2} - 212 T^{3} + 2809 T^{4} \))(\( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \))(\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 + 2 T - 49 T^{2} + 106 T^{3} + 2809 T^{4} \))
$59$ (\( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4} \))(\( 1 - 14 T + 137 T^{2} - 826 T^{3} + 3481 T^{4} \))(\( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \))(\( 1 + 2 T - 55 T^{2} + 118 T^{3} + 3481 T^{4} \))(\( 1 + 2 T - 55 T^{2} + 118 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 14 T + 137 T^{2} - 826 T^{3} + 3481 T^{4} \))(\( 1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4} \))(\( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 + 10 T + 41 T^{2} + 590 T^{3} + 3481 T^{4} \))
$61$ (\( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} \))(\( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( 1 + 6 T - 25 T^{2} + 366 T^{3} + 3721 T^{4} \))(\( ( 1 - 13 T + 61 T^{2} )( 1 - T + 61 T^{2} ) \))(\( 1 + 5 T - 36 T^{2} + 305 T^{3} + 3721 T^{4} \))(\( 1 - 9 T + 20 T^{2} - 549 T^{3} + 3721 T^{4} \))(\( 1 - 9 T + 20 T^{2} - 549 T^{3} + 3721 T^{4} \))(\( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} \))(\( ( 1 - 13 T + 61 T^{2} )( 1 - T + 61 T^{2} ) \))(\( 1 + 6 T - 25 T^{2} + 366 T^{3} + 3721 T^{4} \))(\( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} \))(\( ( 1 - 13 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))
$67$ (\( 1 - 2 T - 63 T^{2} - 134 T^{3} + 4489 T^{4} \))(\( 1 - 8 T - 3 T^{2} - 536 T^{3} + 4489 T^{4} \))(\( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} \))(\( 1 + 12 T + 77 T^{2} + 804 T^{3} + 4489 T^{4} \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \))(\( 1 - 7 T - 18 T^{2} - 469 T^{3} + 4489 T^{4} \))(\( 1 + 7 T - 18 T^{2} + 469 T^{3} + 4489 T^{4} \))(\( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 - 12 T + 77 T^{2} - 804 T^{3} + 4489 T^{4} \))(\( 1 - 2 T - 63 T^{2} - 134 T^{3} + 4489 T^{4} \))(\( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} \))(\( 1 + 9 T + 14 T^{2} + 603 T^{3} + 4489 T^{4} \))
$71$ (\( ( 1 + 9 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( ( 1 + T + 71 T^{2} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( ( 1 - 6 T + 71 T^{2} )^{2} \))(\( ( 1 - 6 T + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 + T + 71 T^{2} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 9 T + 71 T^{2} )^{2} \))(\( ( 1 - 2 T + 71 T^{2} )^{2} \))
$73$ (\( ( 1 - 7 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( 1 + 4 T - 57 T^{2} + 292 T^{3} + 5329 T^{4} \))(\( 1 - 16 T + 183 T^{2} - 1168 T^{3} + 5329 T^{4} \))(\( 1 + 14 T + 123 T^{2} + 1022 T^{3} + 5329 T^{4} \))(\( 1 + 16 T + 183 T^{2} + 1168 T^{3} + 5329 T^{4} \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( ( 1 - 7 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( 1 + 4 T - 57 T^{2} + 292 T^{3} + 5329 T^{4} \))(\( 1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4} \))(\( 1 + 16 T + 183 T^{2} + 1168 T^{3} + 5329 T^{4} \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( 1 + 4 T - 57 T^{2} + 292 T^{3} + 5329 T^{4} \))
$79$ (\( 1 + 5 T - 54 T^{2} + 395 T^{3} + 6241 T^{4} \))(\( 1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4} \))(\( 1 + 14 T + 117 T^{2} + 1106 T^{3} + 6241 T^{4} \))(\( 1 - 11 T + 42 T^{2} - 869 T^{3} + 6241 T^{4} \))(\( 1 + 2 T - 75 T^{2} + 158 T^{3} + 6241 T^{4} \))(\( 1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4} \))(\( 1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4} \))(\( 1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4} \))(\( 1 - 11 T + 42 T^{2} - 869 T^{3} + 6241 T^{4} \))(\( 1 + 14 T + 117 T^{2} + 1106 T^{3} + 6241 T^{4} \))(\( 1 + 5 T - 54 T^{2} + 395 T^{3} + 6241 T^{4} \))(\( 1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4} \))
$83$ (\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 - 14 T + 83 T^{2} )^{2} \))(\( ( 1 + 3 T + 83 T^{2} )^{2} \))(\( ( 1 + 9 T + 83 T^{2} )^{2} \))(\( ( 1 - 9 T + 83 T^{2} )^{2} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 14 T + 83 T^{2} )^{2} \))(\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 - 7 T + 83 T^{2} )^{2} \))
$89$ (\( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 + 2 T - 85 T^{2} + 178 T^{3} + 7921 T^{4} \))(\( 1 + 7 T - 40 T^{2} + 623 T^{3} + 7921 T^{4} \))(\( 1 - 15 T + 136 T^{2} - 1335 T^{3} + 7921 T^{4} \))(\( 1 - 7 T - 40 T^{2} - 623 T^{3} + 7921 T^{4} \))(\( 1 - 7 T - 40 T^{2} - 623 T^{3} + 7921 T^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 + 7 T - 40 T^{2} + 623 T^{3} + 7921 T^{4} \))(\( 1 + 2 T - 85 T^{2} + 178 T^{3} + 7921 T^{4} \))(\( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} \))(\( 1 + T - 88 T^{2} + 89 T^{3} + 7921 T^{4} \))
$97$ (\( ( 1 + 5 T + 97 T^{2} )^{2} \))(\( ( 1 - 10 T + 97 T^{2} )^{2} \))(\( ( 1 + 12 T + 97 T^{2} )^{2} \))(\( ( 1 + 7 T + 97 T^{2} )^{2} \))(\( ( 1 + 14 T + 97 T^{2} )^{2} \))(\( ( 1 + 97 T^{2} )^{2} \))(\( ( 1 + 97 T^{2} )^{2} \))(\( ( 1 + 14 T + 97 T^{2} )^{2} \))(\( ( 1 - 7 T + 97 T^{2} )^{2} \))(\( ( 1 - 12 T + 97 T^{2} )^{2} \))(\( ( 1 - 5 T + 97 T^{2} )^{2} \))(\( ( 1 + 14 T + 97 T^{2} )^{2} \))
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