Properties

Label 1584.2.a
Level $1584$
Weight $2$
Character orbit 1584.a
Rep. character $\chi_{1584}(1,\cdot)$
Character field $\Q$
Dimension $25$
Newform subspaces $22$
Sturm bound $576$
Trace bound $13$

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) \(=\) \( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1584.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 22 \)
Sturm bound: \(576\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(5\), \(7\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1584))\).

Total New Old
Modular forms 312 25 287
Cusp forms 265 25 240
Eisenstein series 47 0 47

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

\(2\)\(3\)\(11\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(3\)
\(+\)\(+\)\(-\)\(-\)\(3\)
\(+\)\(-\)\(+\)\(-\)\(5\)
\(+\)\(-\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(4\)
\(-\)\(-\)\(-\)\(-\)\(4\)
Plus space\(+\)\(11\)
Minus space\(-\)\(14\)

Trace form

\( 25q - 2q^{5} - 4q^{7} + O(q^{10}) \) \( 25q - 2q^{5} - 4q^{7} - 3q^{11} - 2q^{13} + 2q^{17} + 4q^{19} + 6q^{23} + 27q^{25} + 6q^{29} + 10q^{31} - 12q^{35} - 2q^{37} + 2q^{41} + 8q^{43} - 8q^{47} + 17q^{49} - 2q^{53} - 4q^{55} - 14q^{59} - 18q^{61} + 28q^{65} - 10q^{67} + 18q^{71} + 2q^{73} - 20q^{79} + 32q^{83} - 20q^{85} - 2q^{89} - 16q^{91} - 24q^{95} - 10q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1584))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 11
1584.2.a.a \(1\) \(12.648\) \(\Q\) None \(0\) \(0\) \(-4\) \(2\) \(+\) \(-\) \(+\) \(q-4q^{5}+2q^{7}-q^{11}+6q^{17}-4q^{19}+\cdots\)
1584.2.a.b \(1\) \(12.648\) \(\Q\) None \(0\) \(0\) \(-4\) \(2\) \(-\) \(+\) \(-\) \(q-4q^{5}+2q^{7}+q^{11}-2q^{13}+2q^{17}+\cdots\)
1584.2.a.c \(1\) \(12.648\) \(\Q\) None \(0\) \(0\) \(-2\) \(-2\) \(-\) \(-\) \(+\) \(q-2q^{5}-2q^{7}-q^{11}+6q^{13}+4q^{17}+\cdots\)
1584.2.a.d \(1\) \(12.648\) \(\Q\) None \(0\) \(0\) \(-2\) \(0\) \(+\) \(-\) \(-\) \(q-2q^{5}+q^{11}+2q^{13}-6q^{17}+4q^{23}+\cdots\)
1584.2.a.e \(1\) \(12.648\) \(\Q\) None \(0\) \(0\) \(-2\) \(2\) \(-\) \(-\) \(-\) \(q-2q^{5}+2q^{7}+q^{11}-2q^{13}-4q^{17}+\cdots\)
1584.2.a.f \(1\) \(12.648\) \(\Q\) None \(0\) \(0\) \(-2\) \(4\) \(-\) \(-\) \(+\) \(q-2q^{5}+4q^{7}-q^{11}-6q^{13}-2q^{17}+\cdots\)
1584.2.a.g \(1\) \(12.648\) \(\Q\) None \(0\) \(0\) \(-1\) \(2\) \(-\) \(-\) \(-\) \(q-q^{5}+2q^{7}+q^{11}+4q^{13}+2q^{17}+\cdots\)
1584.2.a.h \(1\) \(12.648\) \(\Q\) None \(0\) \(0\) \(0\) \(-2\) \(-\) \(-\) \(+\) \(q-2q^{7}-q^{11}-4q^{13}+6q^{17}+4q^{19}+\cdots\)
1584.2.a.i \(1\) \(12.648\) \(\Q\) None \(0\) \(0\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(q-2q^{7}-q^{11}+2q^{13}+6q^{17}-2q^{19}+\cdots\)
1584.2.a.j \(1\) \(12.648\) \(\Q\) None \(0\) \(0\) \(0\) \(-2\) \(+\) \(-\) \(-\) \(q-2q^{7}+q^{11}+2q^{17}-8q^{19}-2q^{23}+\cdots\)
1584.2.a.k \(1\) \(12.648\) \(\Q\) None \(0\) \(0\) \(0\) \(-2\) \(-\) \(+\) \(-\) \(q-2q^{7}+q^{11}+2q^{13}-6q^{17}-2q^{19}+\cdots\)
1584.2.a.l \(1\) \(12.648\) \(\Q\) None \(0\) \(0\) \(0\) \(2\) \(+\) \(+\) \(+\) \(q+2q^{7}-q^{11}-6q^{13}-6q^{17}+2q^{19}+\cdots\)
1584.2.a.m \(1\) \(12.648\) \(\Q\) None \(0\) \(0\) \(0\) \(2\) \(+\) \(+\) \(-\) \(q+2q^{7}+q^{11}-6q^{13}+6q^{17}+2q^{19}+\cdots\)
1584.2.a.n \(1\) \(12.648\) \(\Q\) None \(0\) \(0\) \(2\) \(-4\) \(+\) \(-\) \(+\) \(q+2q^{5}-4q^{7}-q^{11}+6q^{13}-6q^{17}+\cdots\)
1584.2.a.o \(1\) \(12.648\) \(\Q\) None \(0\) \(0\) \(2\) \(-4\) \(-\) \(-\) \(-\) \(q+2q^{5}-4q^{7}+q^{11}-2q^{13}+2q^{17}+\cdots\)
1584.2.a.p \(1\) \(12.648\) \(\Q\) None \(0\) \(0\) \(3\) \(-2\) \(-\) \(-\) \(+\) \(q+3q^{5}-2q^{7}-q^{11}-4q^{13}-6q^{17}+\cdots\)
1584.2.a.q \(1\) \(12.648\) \(\Q\) None \(0\) \(0\) \(3\) \(2\) \(+\) \(-\) \(+\) \(q+3q^{5}+2q^{7}-q^{11}+6q^{17}-4q^{19}+\cdots\)
1584.2.a.r \(1\) \(12.648\) \(\Q\) None \(0\) \(0\) \(4\) \(2\) \(-\) \(+\) \(+\) \(q+4q^{5}+2q^{7}-q^{11}-2q^{13}-2q^{17}+\cdots\)
1584.2.a.s \(1\) \(12.648\) \(\Q\) None \(0\) \(0\) \(4\) \(2\) \(-\) \(-\) \(-\) \(q+4q^{5}+2q^{7}+q^{11}+4q^{13}+2q^{17}+\cdots\)
1584.2.a.t \(2\) \(12.648\) \(\Q(\sqrt{17}) \) None \(0\) \(0\) \(-3\) \(2\) \(+\) \(-\) \(+\) \(q+(-1-\beta )q^{5}+(2-2\beta )q^{7}-q^{11}+\cdots\)
1584.2.a.u \(2\) \(12.648\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(-4\) \(+\) \(+\) \(+\) \(q+\beta q^{5}+(-2-\beta )q^{7}-q^{11}+(2-\beta )q^{13}+\cdots\)
1584.2.a.v \(2\) \(12.648\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(-4\) \(+\) \(+\) \(-\) \(q+\beta q^{5}+(-2+\beta )q^{7}+q^{11}+(2+\beta )q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1584))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1584)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 15}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(132))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(176))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(198))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(264))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(396))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(528))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(792))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( 1 + 4 T + 5 T^{2} \))(\( 1 + 4 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 + 2 T^{2} + 25 T^{4} \))(\( 1 + 2 T^{2} + 25 T^{4} \))
$7$ (\( 1 - 2 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 - 2 T - 2 T^{2} - 14 T^{3} + 49 T^{4} \))(\( 1 + 4 T + 10 T^{2} + 28 T^{3} + 49 T^{4} \))(\( 1 + 4 T + 10 T^{2} + 28 T^{3} + 49 T^{4} \))
$11$ (\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))
$13$ (\( 1 + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 2 T + 10 T^{2} + 26 T^{3} + 169 T^{4} \))(\( 1 - 4 T + 22 T^{2} - 52 T^{3} + 169 T^{4} \))(\( 1 - 4 T + 22 T^{2} - 52 T^{3} + 169 T^{4} \))
$17$ (\( 1 - 6 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 - 4 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 - 2 T + 17 T^{2} )^{2} \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 - 8 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( ( 1 - 4 T + 19 T^{2} )^{2} \))(\( ( 1 + 2 T + 19 T^{2} )^{2} \))(\( ( 1 + 2 T + 19 T^{2} )^{2} \))
$23$ (\( 1 + 6 T + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 + T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 2 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( 1 + 3 T + 23 T^{2} \))(\( 1 - T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 - 9 T + 62 T^{2} - 207 T^{3} + 529 T^{4} \))(\( 1 + 38 T^{2} + 529 T^{4} \))(\( 1 + 38 T^{2} + 529 T^{4} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 - 8 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 - 8 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 10 T + 29 T^{2} \))(\( 1 - 2 T + 42 T^{2} - 58 T^{3} + 841 T^{4} \))(\( 1 + 4 T + 30 T^{2} + 116 T^{3} + 841 T^{4} \))(\( 1 - 4 T + 30 T^{2} - 116 T^{3} + 841 T^{4} \))
$31$ (\( 1 + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 7 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 5 T + 31 T^{2} \))(\( 1 - 7 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 - 7 T + 70 T^{2} - 217 T^{3} + 961 T^{4} \))(\( 1 + 8 T + 46 T^{2} + 248 T^{3} + 961 T^{4} \))(\( 1 + 8 T + 46 T^{2} + 248 T^{3} + 961 T^{4} \))
$37$ (\( 1 - 6 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 + 10 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 - 3 T + 37 T^{2} \))(\( 1 + 10 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 + T + 37 T^{2} \))(\( 1 + T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 11 T + 100 T^{2} + 407 T^{3} + 1369 T^{4} \))(\( 1 - 4 T + 46 T^{2} - 148 T^{3} + 1369 T^{4} \))(\( 1 - 4 T + 46 T^{2} - 148 T^{3} + 1369 T^{4} \))
$41$ (\( 1 - 10 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 8 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 - 8 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 4 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 + 6 T + 74 T^{2} + 246 T^{3} + 1681 T^{4} \))(\( 1 + 12 T + 86 T^{2} + 492 T^{3} + 1681 T^{4} \))(\( 1 - 12 T + 86 T^{2} - 492 T^{3} + 1681 T^{4} \))
$43$ (\( 1 - 8 T + 43 T^{2} \))(\( 1 + 6 T + 43 T^{2} \))(\( 1 + 10 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 - 2 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 6 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 - 10 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 10 T + 43 T^{2} \))(\( 1 - 6 T + 43 T^{2} \))(\( 1 - 6 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 + 43 T^{2} \))(\( 1 - 10 T + 43 T^{2} \))(\( 1 + 6 T + 43 T^{2} \))(\( 1 + 6 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 6 T + 78 T^{2} - 258 T^{3} + 1849 T^{4} \))(\( 1 + 12 T + 90 T^{2} + 516 T^{3} + 1849 T^{4} \))(\( 1 + 12 T + 90 T^{2} + 516 T^{3} + 1849 T^{4} \))
$47$ (\( 1 - 6 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( 1 - 4 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 + 2 T + 47 T^{2} \))(\( ( 1 - 8 T + 47 T^{2} )^{2} \))(\( 1 + 8 T + 38 T^{2} + 376 T^{3} + 2209 T^{4} \))(\( 1 - 8 T + 38 T^{2} - 376 T^{3} + 2209 T^{4} \))
$53$ (\( 1 - 12 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 + 14 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 2 T + 53 T^{2} \))(\( 1 + 2 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 - 8 T + 53 T^{2} \))(\( 1 + 12 T + 53 T^{2} \))(\( 1 - 4 T + 53 T^{2} \))(\( 1 + 4 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 2 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 + 4 T + 53 T^{2} \))(\( 1 + 8 T + 54 T^{2} + 424 T^{3} + 2809 T^{4} \))(\( 1 + 8 T + 114 T^{2} + 424 T^{3} + 2809 T^{4} \))(\( 1 - 8 T + 114 T^{2} - 424 T^{3} + 2809 T^{4} \))
$59$ (\( 1 + 8 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 - 5 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 8 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 - 3 T + 59 T^{2} \))(\( 1 + T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 5 T + 18 T^{2} + 295 T^{3} + 3481 T^{4} \))(\( 1 + 8 T + 102 T^{2} + 472 T^{3} + 3481 T^{4} \))(\( 1 - 8 T + 102 T^{2} - 472 T^{3} + 3481 T^{4} \))
$61$ (\( 1 - 4 T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))(\( 1 + 14 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 + 14 T + 61 T^{2} \))(\( 1 - 12 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 + 4 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 - 6 T + 61 T^{2} \))(\( 1 + 4 T + 61 T^{2} \))(\( 1 - 4 T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))(\( 1 + 8 T + 61 T^{2} \))(\( 1 + 6 T + 114 T^{2} + 366 T^{3} + 3721 T^{4} \))(\( 1 - 12 T + 150 T^{2} - 732 T^{3} + 3721 T^{4} \))(\( 1 - 12 T + 150 T^{2} - 732 T^{3} + 3721 T^{4} \))
$67$ (\( 1 - 12 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 7 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 - T + 67 T^{2} \))(\( 1 - 5 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 + 15 T + 186 T^{2} + 1005 T^{3} + 4489 T^{4} \))(\( 1 + 6 T^{2} + 4489 T^{4} \))(\( 1 + 6 T^{2} + 4489 T^{4} \))
$71$ (\( 1 - 10 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 + 3 T + 71 T^{2} \))(\( 1 - 6 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 + 10 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 15 T + 71 T^{2} \))(\( 1 - 3 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 2 T + 71 T^{2} \))(\( 1 + 5 T + 110 T^{2} + 355 T^{3} + 5041 T^{4} \))(\( 1 + 8 T + 86 T^{2} + 568 T^{3} + 5041 T^{4} \))(\( 1 - 8 T + 86 T^{2} - 568 T^{3} + 5041 T^{4} \))
$73$ (\( 1 - 2 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 - 4 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 - 14 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 - 14 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))(\( 1 + 4 T + 73 T^{2} \))(\( 1 - 16 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 - 2 T + 130 T^{2} - 146 T^{3} + 5329 T^{4} \))(\( 1 + 4 T + 22 T^{2} + 292 T^{3} + 5329 T^{4} \))(\( 1 + 4 T + 22 T^{2} + 292 T^{3} + 5329 T^{4} \))
$79$ (\( 1 + 2 T + 79 T^{2} \))(\( 1 - 10 T + 79 T^{2} \))(\( 1 + 2 T + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( 1 - 2 T + 79 T^{2} \))(\( 1 - 4 T + 79 T^{2} \))(\( 1 - 10 T + 79 T^{2} \))(\( 1 + 14 T + 79 T^{2} \))(\( 1 + 2 T + 79 T^{2} \))(\( 1 - 10 T + 79 T^{2} \))(\( 1 + 2 T + 79 T^{2} \))(\( 1 + 14 T + 79 T^{2} \))(\( 1 + 14 T + 79 T^{2} \))(\( 1 - 4 T + 79 T^{2} \))(\( 1 - 4 T + 79 T^{2} \))(\( 1 + 2 T + 79 T^{2} \))(\( 1 + 2 T + 79 T^{2} \))(\( 1 - 10 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 - 14 T + 190 T^{2} - 1106 T^{3} + 6241 T^{4} \))(\( 1 + 4 T + 154 T^{2} + 316 T^{3} + 6241 T^{4} \))(\( 1 + 4 T + 154 T^{2} + 316 T^{3} + 6241 T^{4} \))
$83$ (\( 1 - 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 16 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 16 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 + 2 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 - 10 T + 174 T^{2} - 830 T^{3} + 6889 T^{4} \))(\( 1 + 8 T + 150 T^{2} + 664 T^{3} + 6889 T^{4} \))(\( 1 - 8 T + 150 T^{2} - 664 T^{3} + 6889 T^{4} \))
$89$ (\( 1 - 6 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 - 14 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 - 14 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 + 15 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 - 9 T + 89 T^{2} \))(\( 1 + 15 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 - 7 T + 152 T^{2} - 623 T^{3} + 7921 T^{4} \))(\( ( 1 + 16 T + 89 T^{2} )^{2} \))(\( ( 1 - 16 T + 89 T^{2} )^{2} \))
$97$ (\( 1 - 14 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 14 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 14 T + 97 T^{2} \))(\( 1 + 7 T + 97 T^{2} \))(\( 1 - 14 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 + 7 T + 97 T^{2} \))(\( 1 + 7 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 27 T + 372 T^{2} - 2619 T^{3} + 9409 T^{4} \))(\( 1 + 12 T + 102 T^{2} + 1164 T^{3} + 9409 T^{4} \))(\( 1 + 12 T + 102 T^{2} + 1164 T^{3} + 9409 T^{4} \))
show more
show less