Properties

Label 704.2.a
Level $704$
Weight $2$
Character orbit 704.a
Rep. character $\chi_{704}(1,\cdot)$
Character field $\Q$
Dimension $20$
Newform subspaces $16$
Sturm bound $192$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 704 = 2^{6} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 704.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 16 \)
Sturm bound: \(192\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(704))\).

Total New Old
Modular forms 108 20 88
Cusp forms 85 20 65
Eisenstein series 23 0 23

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(11\)FrickeDim.
\(+\)\(+\)\(+\)\(4\)
\(+\)\(-\)\(-\)\(7\)
\(-\)\(+\)\(-\)\(6\)
\(-\)\(-\)\(+\)\(3\)
Plus space\(+\)\(7\)
Minus space\(-\)\(13\)

Trace form

\( 20q + 20q^{9} + O(q^{10}) \) \( 20q + 20q^{9} + 16q^{13} - 8q^{17} + 16q^{21} + 12q^{25} + 16q^{37} - 8q^{41} + 24q^{45} + 20q^{49} + 24q^{53} - 16q^{61} - 8q^{69} + 8q^{73} - 12q^{81} - 32q^{85} - 24q^{89} - 8q^{93} - 8q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(704))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 11
704.2.a.a \(1\) \(5.621\) \(\Q\) None \(0\) \(-3\) \(-1\) \(0\) \(-\) \(-\) \(q-3q^{3}-q^{5}+6q^{9}+q^{11}+6q^{13}+\cdots\)
704.2.a.b \(1\) \(5.621\) \(\Q\) None \(0\) \(-3\) \(3\) \(2\) \(-\) \(+\) \(q-3q^{3}+3q^{5}+2q^{7}+6q^{9}-q^{11}+\cdots\)
704.2.a.c \(1\) \(5.621\) \(\Q\) None \(0\) \(-1\) \(-1\) \(2\) \(-\) \(-\) \(q-q^{3}-q^{5}+2q^{7}-2q^{9}+q^{11}-4q^{13}+\cdots\)
704.2.a.d \(1\) \(5.621\) \(\Q\) None \(0\) \(-1\) \(-1\) \(4\) \(-\) \(+\) \(q-q^{3}-q^{5}+4q^{7}-2q^{9}-q^{11}+2q^{13}+\cdots\)
704.2.a.e \(1\) \(5.621\) \(\Q\) None \(0\) \(-1\) \(3\) \(-4\) \(+\) \(+\) \(q-q^{3}+3q^{5}-4q^{7}-2q^{9}-q^{11}+\cdots\)
704.2.a.f \(1\) \(5.621\) \(\Q\) None \(0\) \(-1\) \(3\) \(2\) \(+\) \(-\) \(q-q^{3}+3q^{5}+2q^{7}-2q^{9}+q^{11}+\cdots\)
704.2.a.g \(1\) \(5.621\) \(\Q\) None \(0\) \(1\) \(-1\) \(-4\) \(-\) \(-\) \(q+q^{3}-q^{5}-4q^{7}-2q^{9}+q^{11}+2q^{13}+\cdots\)
704.2.a.h \(1\) \(5.621\) \(\Q\) None \(0\) \(1\) \(-1\) \(-2\) \(+\) \(+\) \(q+q^{3}-q^{5}-2q^{7}-2q^{9}-q^{11}-4q^{13}+\cdots\)
704.2.a.i \(1\) \(5.621\) \(\Q\) None \(0\) \(1\) \(3\) \(-2\) \(-\) \(+\) \(q+q^{3}+3q^{5}-2q^{7}-2q^{9}-q^{11}+\cdots\)
704.2.a.j \(1\) \(5.621\) \(\Q\) None \(0\) \(1\) \(3\) \(4\) \(+\) \(-\) \(q+q^{3}+3q^{5}+4q^{7}-2q^{9}+q^{11}+\cdots\)
704.2.a.k \(1\) \(5.621\) \(\Q\) None \(0\) \(3\) \(-1\) \(0\) \(-\) \(+\) \(q+3q^{3}-q^{5}+6q^{9}-q^{11}+6q^{13}+\cdots\)
704.2.a.l \(1\) \(5.621\) \(\Q\) None \(0\) \(3\) \(3\) \(-2\) \(+\) \(-\) \(q+3q^{3}+3q^{5}-2q^{7}+6q^{9}+q^{11}+\cdots\)
704.2.a.m \(2\) \(5.621\) \(\Q(\sqrt{17}) \) None \(0\) \(-1\) \(-3\) \(-2\) \(+\) \(-\) \(q-\beta q^{3}+(-2+\beta )q^{5}-2\beta q^{7}+(1+\beta )q^{9}+\cdots\)
704.2.a.n \(2\) \(5.621\) \(\Q(\sqrt{17}) \) None \(0\) \(-1\) \(-3\) \(0\) \(+\) \(+\) \(q-\beta q^{3}+(-2+\beta )q^{5}+(1+\beta )q^{9}-q^{11}+\cdots\)
704.2.a.o \(2\) \(5.621\) \(\Q(\sqrt{17}) \) None \(0\) \(1\) \(-3\) \(0\) \(+\) \(-\) \(q+\beta q^{3}+(-2+\beta )q^{5}+(1+\beta )q^{9}+q^{11}+\cdots\)
704.2.a.p \(2\) \(5.621\) \(\Q(\sqrt{17}) \) None \(0\) \(1\) \(-3\) \(2\) \(-\) \(+\) \(q+\beta q^{3}+(-2+\beta )q^{5}+2\beta q^{7}+(1+\beta )q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(704))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(704)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(176))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(352))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 3 T + 3 T^{2} \))(\( 1 + 3 T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 + T + 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 + T + 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 - T + 2 T^{2} - 3 T^{3} + 9 T^{4} \))(\( 1 - T + 2 T^{2} - 3 T^{3} + 9 T^{4} \))
$5$ (\( 1 + T + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4} \))
$7$ (\( 1 + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + 2 T - 2 T^{2} + 14 T^{3} + 49 T^{4} \))(\( ( 1 + 7 T^{2} )^{2} \))(\( ( 1 + 7 T^{2} )^{2} \))(\( 1 - 2 T - 2 T^{2} - 14 T^{3} + 49 T^{4} \))
$11$ (\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))
$13$ (\( 1 - 6 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 - 2 T + 10 T^{2} - 26 T^{3} + 169 T^{4} \))(\( ( 1 + 2 T + 13 T^{2} )^{2} \))(\( ( 1 + 2 T + 13 T^{2} )^{2} \))(\( 1 - 2 T + 10 T^{2} - 26 T^{3} + 169 T^{4} \))
$17$ (\( 1 + 4 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 8 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 8 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( ( 1 - 2 T + 17 T^{2} )^{2} \))(\( 1 - 6 T + 26 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 26 T^{2} - 102 T^{3} + 289 T^{4} \))(\( ( 1 - 2 T + 17 T^{2} )^{2} \))
$19$ (\( 1 + 6 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 - 8 T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( ( 1 - 4 T + 19 T^{2} )^{2} \))(\( 1 + 6 T + 30 T^{2} + 114 T^{3} + 361 T^{4} \))(\( 1 - 6 T + 30 T^{2} - 114 T^{3} + 361 T^{4} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))
$23$ (\( 1 - 3 T + 23 T^{2} \))(\( 1 + T + 23 T^{2} \))(\( 1 - T + 23 T^{2} \))(\( 1 - 9 T + 23 T^{2} \))(\( 1 - 5 T + 23 T^{2} \))(\( 1 + 3 T + 23 T^{2} \))(\( 1 + 9 T + 23 T^{2} \))(\( 1 + T + 23 T^{2} \))(\( 1 - 3 T + 23 T^{2} \))(\( 1 + 5 T + 23 T^{2} \))(\( 1 + 3 T + 23 T^{2} \))(\( 1 - T + 23 T^{2} \))(\( 1 - 9 T + 62 T^{2} - 207 T^{3} + 529 T^{4} \))(\( 1 + 3 T + 10 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 - 3 T + 10 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 + 9 T + 62 T^{2} + 207 T^{3} + 529 T^{4} \))
$29$ (\( 1 - 4 T + 29 T^{2} \))(\( 1 - 8 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 4 T + 29 T^{2} \))(\( 1 + 4 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 4 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 4 T + 29 T^{2} \))(\( 1 - 4 T + 29 T^{2} \))(\( 1 - 8 T + 29 T^{2} \))(\( 1 - 2 T + 42 T^{2} - 58 T^{3} + 841 T^{4} \))(\( 1 + 6 T + 50 T^{2} + 174 T^{3} + 841 T^{4} \))(\( 1 + 6 T + 50 T^{2} + 174 T^{3} + 841 T^{4} \))(\( 1 - 2 T + 42 T^{2} - 58 T^{3} + 841 T^{4} \))
$31$ (\( 1 + 9 T + 31 T^{2} \))(\( 1 - 7 T + 31 T^{2} \))(\( 1 + 7 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 - T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 + 5 T + 31 T^{2} \))(\( 1 - 7 T + 31 T^{2} \))(\( 1 + 5 T + 31 T^{2} \))(\( 1 + T + 31 T^{2} \))(\( 1 - 9 T + 31 T^{2} \))(\( 1 + 7 T + 31 T^{2} \))(\( 1 + 7 T + 70 T^{2} + 217 T^{3} + 961 T^{4} \))(\( 1 + 15 T + 114 T^{2} + 465 T^{3} + 961 T^{4} \))(\( 1 - 15 T + 114 T^{2} - 465 T^{3} + 961 T^{4} \))(\( 1 - 7 T + 70 T^{2} - 217 T^{3} + 961 T^{4} \))
$37$ (\( 1 + 7 T + 37 T^{2} \))(\( 1 - T + 37 T^{2} \))(\( 1 + 3 T + 37 T^{2} \))(\( 1 - 9 T + 37 T^{2} \))(\( 1 + 3 T + 37 T^{2} \))(\( 1 - T + 37 T^{2} \))(\( 1 - 9 T + 37 T^{2} \))(\( 1 + 3 T + 37 T^{2} \))(\( 1 - T + 37 T^{2} \))(\( 1 + 3 T + 37 T^{2} \))(\( 1 + 7 T + 37 T^{2} \))(\( 1 - T + 37 T^{2} \))(\( 1 - 11 T + 100 T^{2} - 407 T^{3} + 1369 T^{4} \))(\( 1 + T + 36 T^{2} + 37 T^{3} + 1369 T^{4} \))(\( 1 + T + 36 T^{2} + 37 T^{3} + 1369 T^{4} \))(\( 1 - 11 T + 100 T^{2} - 407 T^{3} + 1369 T^{4} \))
$41$ (\( 1 + 2 T + 41 T^{2} \))(\( 1 - 4 T + 41 T^{2} \))(\( 1 + 8 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 8 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 - 4 T + 41 T^{2} \))(\( 1 - 6 T + 74 T^{2} - 246 T^{3} + 1681 T^{4} \))(\( 1 + 14 T^{2} + 1681 T^{4} \))(\( 1 + 14 T^{2} + 1681 T^{4} \))(\( 1 - 6 T + 74 T^{2} - 246 T^{3} + 1681 T^{4} \))
$43$ (\( 1 + 6 T + 43 T^{2} \))(\( 1 - 6 T + 43 T^{2} \))(\( 1 + 6 T + 43 T^{2} \))(\( 1 - 6 T + 43 T^{2} \))(\( 1 - 6 T + 43 T^{2} \))(\( 1 - 10 T + 43 T^{2} \))(\( 1 + 6 T + 43 T^{2} \))(\( 1 - 6 T + 43 T^{2} \))(\( 1 + 10 T + 43 T^{2} \))(\( 1 + 6 T + 43 T^{2} \))(\( 1 - 6 T + 43 T^{2} \))(\( 1 + 6 T + 43 T^{2} \))(\( 1 - 6 T + 78 T^{2} - 258 T^{3} + 1849 T^{4} \))(\( 1 + 6 T + 78 T^{2} + 258 T^{3} + 1849 T^{4} \))(\( 1 - 6 T + 78 T^{2} - 258 T^{3} + 1849 T^{4} \))(\( 1 + 6 T + 78 T^{2} + 258 T^{3} + 1849 T^{4} \))
$47$ (\( 1 - 12 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 4 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( ( 1 - 8 T + 47 T^{2} )^{2} \))(\( ( 1 + 4 T + 47 T^{2} )^{2} \))(\( ( 1 - 4 T + 47 T^{2} )^{2} \))(\( ( 1 + 8 T + 47 T^{2} )^{2} \))
$53$ (\( 1 + 2 T + 53 T^{2} \))(\( 1 + 2 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 2 T + 53 T^{2} \))(\( 1 + 2 T + 53 T^{2} \))(\( 1 + 8 T + 54 T^{2} + 424 T^{3} + 2809 T^{4} \))(\( 1 + 38 T^{2} + 2809 T^{4} \))(\( 1 + 38 T^{2} + 2809 T^{4} \))(\( 1 + 8 T + 54 T^{2} + 424 T^{3} + 2809 T^{4} \))
$59$ (\( 1 + 9 T + 59 T^{2} \))(\( 1 + T + 59 T^{2} \))(\( 1 - 5 T + 59 T^{2} \))(\( 1 - 5 T + 59 T^{2} \))(\( 1 + 3 T + 59 T^{2} \))(\( 1 + 3 T + 59 T^{2} \))(\( 1 + 5 T + 59 T^{2} \))(\( 1 + 5 T + 59 T^{2} \))(\( 1 - 3 T + 59 T^{2} \))(\( 1 - 3 T + 59 T^{2} \))(\( 1 - 9 T + 59 T^{2} \))(\( 1 - T + 59 T^{2} \))(\( 1 - 5 T + 18 T^{2} - 295 T^{3} + 3481 T^{4} \))(\( 1 - 13 T + 122 T^{2} - 767 T^{3} + 3481 T^{4} \))(\( 1 + 13 T + 122 T^{2} + 767 T^{3} + 3481 T^{4} \))(\( 1 + 5 T + 18 T^{2} + 295 T^{3} + 3481 T^{4} \))
$61$ (\( 1 + 8 T + 61 T^{2} \))(\( 1 + 4 T + 61 T^{2} \))(\( 1 + 12 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 - 4 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 + 12 T + 61 T^{2} \))(\( 1 - 4 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 + 8 T + 61 T^{2} \))(\( 1 + 4 T + 61 T^{2} \))(\( 1 - 6 T + 114 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( 1 - 6 T - 22 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( 1 - 6 T - 22 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( 1 - 6 T + 114 T^{2} - 366 T^{3} + 3721 T^{4} \))
$67$ (\( 1 - 15 T + 67 T^{2} \))(\( 1 + 5 T + 67 T^{2} \))(\( 1 + 7 T + 67 T^{2} \))(\( 1 - 13 T + 67 T^{2} \))(\( 1 + 11 T + 67 T^{2} \))(\( 1 - T + 67 T^{2} \))(\( 1 + 13 T + 67 T^{2} \))(\( 1 - 7 T + 67 T^{2} \))(\( 1 + T + 67 T^{2} \))(\( 1 - 11 T + 67 T^{2} \))(\( 1 + 15 T + 67 T^{2} \))(\( 1 - 5 T + 67 T^{2} \))(\( 1 + 15 T + 186 T^{2} + 1005 T^{3} + 4489 T^{4} \))(\( 1 + 3 T + 98 T^{2} + 201 T^{3} + 4489 T^{4} \))(\( 1 - 3 T + 98 T^{2} - 201 T^{3} + 4489 T^{4} \))(\( 1 - 15 T + 186 T^{2} - 1005 T^{3} + 4489 T^{4} \))
$71$ (\( 1 + 3 T + 71 T^{2} \))(\( 1 + 3 T + 71 T^{2} \))(\( 1 - 3 T + 71 T^{2} \))(\( 1 + T + 71 T^{2} \))(\( 1 + 5 T + 71 T^{2} \))(\( 1 - 15 T + 71 T^{2} \))(\( 1 - T + 71 T^{2} \))(\( 1 + 3 T + 71 T^{2} \))(\( 1 + 15 T + 71 T^{2} \))(\( 1 - 5 T + 71 T^{2} \))(\( 1 - 3 T + 71 T^{2} \))(\( 1 - 3 T + 71 T^{2} \))(\( 1 + 5 T + 110 T^{2} + 355 T^{3} + 5041 T^{4} \))(\( 1 - 19 T + 194 T^{2} - 1349 T^{3} + 5041 T^{4} \))(\( 1 + 19 T + 194 T^{2} + 1349 T^{3} + 5041 T^{4} \))(\( 1 - 5 T + 110 T^{2} - 355 T^{3} + 5041 T^{4} \))
$73$ (\( 1 + 6 T + 73 T^{2} \))(\( 1 - 16 T + 73 T^{2} \))(\( 1 - 4 T + 73 T^{2} \))(\( 1 - 14 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 + 4 T + 73 T^{2} \))(\( 1 - 14 T + 73 T^{2} \))(\( 1 - 4 T + 73 T^{2} \))(\( 1 + 4 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 - 16 T + 73 T^{2} \))(\( 1 - 2 T + 130 T^{2} - 146 T^{3} + 5329 T^{4} \))(\( ( 1 + 6 T + 73 T^{2} )^{2} \))(\( ( 1 + 6 T + 73 T^{2} )^{2} \))(\( 1 - 2 T + 130 T^{2} - 146 T^{3} + 5329 T^{4} \))
$79$ (\( 1 + 6 T + 79 T^{2} \))(\( 1 + 2 T + 79 T^{2} \))(\( 1 - 10 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 + 2 T + 79 T^{2} \))(\( 1 - 2 T + 79 T^{2} \))(\( 1 - 10 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 + 2 T + 79 T^{2} \))(\( 1 - 2 T + 79 T^{2} \))(\( 1 - 6 T + 79 T^{2} \))(\( 1 - 2 T + 79 T^{2} \))(\( 1 + 14 T + 190 T^{2} + 1106 T^{3} + 6241 T^{4} \))(\( 1 + 18 T + 222 T^{2} + 1422 T^{3} + 6241 T^{4} \))(\( 1 - 18 T + 222 T^{2} - 1422 T^{3} + 6241 T^{4} \))(\( 1 - 14 T + 190 T^{2} - 1106 T^{3} + 6241 T^{4} \))
$83$ (\( 1 - 6 T + 83 T^{2} \))(\( 1 + 2 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 + 14 T + 83 T^{2} \))(\( 1 - 2 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 - 14 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 + 2 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 - 2 T + 83 T^{2} \))(\( 1 + 10 T + 174 T^{2} + 830 T^{3} + 6889 T^{4} \))(\( 1 + 2 T + 14 T^{2} + 166 T^{3} + 6889 T^{4} \))(\( 1 - 2 T + 14 T^{2} - 166 T^{3} + 6889 T^{4} \))(\( 1 - 10 T + 174 T^{2} - 830 T^{3} + 6889 T^{4} \))
$89$ (\( 1 + 5 T + 89 T^{2} \))(\( 1 - 15 T + 89 T^{2} \))(\( 1 - 15 T + 89 T^{2} \))(\( 1 + 13 T + 89 T^{2} \))(\( 1 + 5 T + 89 T^{2} \))(\( 1 + 9 T + 89 T^{2} \))(\( 1 + 13 T + 89 T^{2} \))(\( 1 - 15 T + 89 T^{2} \))(\( 1 + 9 T + 89 T^{2} \))(\( 1 + 5 T + 89 T^{2} \))(\( 1 + 5 T + 89 T^{2} \))(\( 1 - 15 T + 89 T^{2} \))(\( 1 + 7 T + 152 T^{2} + 623 T^{3} + 7921 T^{4} \))(\( 1 + 3 T + 176 T^{2} + 267 T^{3} + 7921 T^{4} \))(\( 1 + 3 T + 176 T^{2} + 267 T^{3} + 7921 T^{4} \))(\( 1 + 7 T + 152 T^{2} + 623 T^{3} + 7921 T^{4} \))
$97$ (\( 1 + 3 T + 97 T^{2} \))(\( 1 + 7 T + 97 T^{2} \))(\( 1 + 7 T + 97 T^{2} \))(\( 1 + 19 T + 97 T^{2} \))(\( 1 - 13 T + 97 T^{2} \))(\( 1 + 7 T + 97 T^{2} \))(\( 1 + 19 T + 97 T^{2} \))(\( 1 + 7 T + 97 T^{2} \))(\( 1 + 7 T + 97 T^{2} \))(\( 1 - 13 T + 97 T^{2} \))(\( 1 + 3 T + 97 T^{2} \))(\( 1 + 7 T + 97 T^{2} \))(\( 1 - 27 T + 372 T^{2} - 2619 T^{3} + 9409 T^{4} \))(\( 1 + T + 156 T^{2} + 97 T^{3} + 9409 T^{4} \))(\( 1 + T + 156 T^{2} + 97 T^{3} + 9409 T^{4} \))(\( 1 - 27 T + 372 T^{2} - 2619 T^{3} + 9409 T^{4} \))
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