LMFDB - Space of Modular Forms of Level 1008, Weight 2, and Character 1008.b

Defining parameters

Level:	$ N $	=	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	=	$ 2 $
Character orbit:	$[\chi]$	=	1008.b (of order $2$ and degree $1$)
Character conductor:	$\operatorname{cond}(\chi)$	=	$ 28 $
Character field:			$\Q$
Newform subspaces:			$ 9 $
Sturm bound:			$384$
Trace bound:			$19$
Distinguishing $T_p$:			$5$, $11$, $19$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(1008, [\chi])$.

	Total	New	Old
Modular forms	216	20	196
Cusp forms	168	20	148
Eisenstein series	48	0	48

Trace form

Decomposition of $S_{2}^{\mathrm{new}}(1008, [\chi])$ into newform subspaces

Label	Dim.	$A$	Field	CM	Traces				$q$-expansion
Label	Dim.	$A$	Field	CM	$a_2$	$a_3$	$a_5$	$a_7$	$q$-expansion
1008.2.b.a	$2$	$8.049$	$\Q(\sqrt{-3}) $	None	$0$	$0$	$0$	$-4$	$q+(-2+\zeta_{6})q^{7}-2\zeta_{6}q^{11}-4\zeta_{6}q^{17}+\cdots$
1008.2.b.b	$2$	$8.049$	$\Q(\sqrt{-3}) $	None	$0$	$0$	$0$	$-4$	$q-2\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+2\zeta_{6}q^{11}+\cdots$
1008.2.b.c	$2$	$8.049$	$\Q(\sqrt{-3}) $	$\Q(\sqrt{-3}) $	$0$	$0$	$0$	$-4$	$q+(-2+\zeta_{6})q^{7}-4\zeta_{6}q^{13}+8q^{19}+\cdots$
1008.2.b.d	$2$	$8.049$	$\Q(\sqrt{-6}) $	None	$0$	$0$	$0$	$-2$	$q+\beta q^{5}+(-1-\beta )q^{7}-\beta q^{11}-2\beta q^{13}+\cdots$
1008.2.b.e	$2$	$8.049$	$\Q(\sqrt{-6}) $	None	$0$	$0$	$0$	$2$	$q+\beta q^{5}+(1+\beta )q^{7}+\beta q^{11}-2\beta q^{13}+\cdots$
1008.2.b.f	$2$	$8.049$	$\Q(\sqrt{-3}) $	$\Q(\sqrt{-3}) $	$0$	$0$	$0$	$4$	$q+(2-\zeta_{6})q^{7}-4\zeta_{6}q^{13}-8q^{19}+5q^{25}+\cdots$
1008.2.b.g	$2$	$8.049$	$\Q(\sqrt{-3}) $	None	$0$	$0$	$0$	$4$	$q-2\zeta_{6}q^{5}+(2+\zeta_{6})q^{7}-2\zeta_{6}q^{11}+\cdots$
1008.2.b.h	$2$	$8.049$	$\Q(\sqrt{-3}) $	None	$0$	$0$	$0$	$4$	$q+(2+\zeta_{6})q^{7}-2\zeta_{6}q^{11}+4\zeta_{6}q^{17}+\cdots$
1008.2.b.i	$4$	$8.049$	$\Q(\sqrt{-6}, \sqrt{7})$	$\Q(\sqrt{-21}) $	$0$	$0$	$0$	$0$	$q-\beta _{1}q^{5}-\beta _{3}q^{7}-\beta _{2}q^{11}+3\beta _{1}q^{17}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(1008, [\chi])$ into lower level spaces

$ S_{2}^{\mathrm{old}}(1008, [\chi]) \cong $ $S_{2}^{\mathrm{new}}(28, [\chi])$$^{\oplus 9}$$\oplus$$S_{2}^{\mathrm{new}}(84, [\chi])$$^{\oplus 6}$$\oplus$$S_{2}^{\mathrm{new}}(112, [\chi])$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(252, [\chi])$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(336, [\chi])$$^{\oplus 2}$

Hecke Characteristic Polynomials

$p$	$F_p(T)$
$2$	1
$3$	1
$5$	($ ( 1 - 5 T^{2} )^{2} $)($ 1 + 2 T^{2} + 25 T^{4} $)($ ( 1 - 5 T^{2} )^{2} $)($ 1 - 4 T^{2} + 25 T^{4} $)($ 1 - 4 T^{2} + 25 T^{4} $)($ ( 1 - 5 T^{2} )^{2} $)($ 1 + 2 T^{2} + 25 T^{4} $)($ ( 1 - 5 T^{2} )^{2} $)($ ( 1 - 4 T^{2} + 25 T^{4} )^{2} $)
$7$	($ 1 + 4 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 - 4 T + 7 T^{2} $)($ 1 - 4 T + 7 T^{2} $)($ 1 - 4 T + 7 T^{2} $)($ ( 1 - 7 T^{2} )^{2} $)
$11$	($ 1 - 10 T^{2} + 121 T^{4} $)($ 1 - 10 T^{2} + 121 T^{4} $)($ ( 1 - 11 T^{2} )^{2} $)($ 1 - 16 T^{2} + 121 T^{4} $)($ 1 - 16 T^{2} + 121 T^{4} $)($ ( 1 - 11 T^{2} )^{2} $)($ 1 - 10 T^{2} + 121 T^{4} $)($ 1 - 10 T^{2} + 121 T^{4} $)($ ( 1 + 20 T^{2} + 121 T^{4} )^{2} $)
$13$	($ ( 1 - 13 T^{2} )^{2} $)($ 1 - 14 T^{2} + 169 T^{4} $)($ ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) $)($ 1 - 2 T^{2} + 169 T^{4} $)($ 1 - 2 T^{2} + 169 T^{4} $)($ ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) $)($ 1 - 14 T^{2} + 169 T^{4} $)($ ( 1 - 13 T^{2} )^{2} $)($ ( 1 - 13 T^{2} )^{4} $)
$17$	($ 1 + 14 T^{2} + 289 T^{4} $)($ ( 1 - 17 T^{2} )^{2} $)($ ( 1 - 17 T^{2} )^{2} $)($ 1 - 28 T^{2} + 289 T^{4} $)($ 1 - 28 T^{2} + 289 T^{4} $)($ ( 1 - 17 T^{2} )^{2} $)($ ( 1 - 17 T^{2} )^{2} $)($ 1 + 14 T^{2} + 289 T^{4} $)($ ( 1 + 20 T^{2} + 289 T^{4} )^{2} $)
$19$	($ ( 1 + 4 T + 19 T^{2} )^{2} $)($ ( 1 - 2 T + 19 T^{2} )^{2} $)($ ( 1 - 8 T + 19 T^{2} )^{2} $)($ ( 1 + 2 T + 19 T^{2} )^{2} $)($ ( 1 - 2 T + 19 T^{2} )^{2} $)($ ( 1 + 8 T + 19 T^{2} )^{2} $)($ ( 1 + 2 T + 19 T^{2} )^{2} $)($ ( 1 - 4 T + 19 T^{2} )^{2} $)($ ( 1 + 10 T^{2} + 361 T^{4} )^{2} $)
$23$	($ 1 - 34 T^{2} + 529 T^{4} $)($ 1 - 34 T^{2} + 529 T^{4} $)($ ( 1 - 23 T^{2} )^{2} $)($ 1 + 8 T^{2} + 529 T^{4} $)($ 1 + 8 T^{2} + 529 T^{4} $)($ ( 1 - 23 T^{2} )^{2} $)($ 1 - 34 T^{2} + 529 T^{4} $)($ 1 - 34 T^{2} + 529 T^{4} $)($ ( 1 - 4 T^{2} + 529 T^{4} )^{2} $)
$29$	($ ( 1 - 6 T + 29 T^{2} )^{2} $)($ ( 1 + 6 T + 29 T^{2} )^{2} $)($ ( 1 + 29 T^{2} )^{2} $)($ ( 1 + 6 T + 29 T^{2} )^{2} $)($ ( 1 + 6 T + 29 T^{2} )^{2} $)($ ( 1 + 29 T^{2} )^{2} $)($ ( 1 + 6 T + 29 T^{2} )^{2} $)($ ( 1 - 6 T + 29 T^{2} )^{2} $)($ ( 1 + 29 T^{2} )^{4} $)
$31$	($ ( 1 - 4 T + 31 T^{2} )^{2} $)($ ( 1 + 8 T + 31 T^{2} )^{2} $)($ ( 1 - 4 T + 31 T^{2} )^{2} $)($ ( 1 - 8 T + 31 T^{2} )^{2} $)($ ( 1 + 8 T + 31 T^{2} )^{2} $)($ ( 1 + 4 T + 31 T^{2} )^{2} $)($ ( 1 - 8 T + 31 T^{2} )^{2} $)($ ( 1 + 4 T + 31 T^{2} )^{2} $)($ ( 1 - 50 T^{2} + 961 T^{4} )^{2} $)
$37$	($ ( 1 + 2 T + 37 T^{2} )^{2} $)($ ( 1 + 2 T + 37 T^{2} )^{2} $)($ ( 1 - 10 T + 37 T^{2} )^{2} $)($ ( 1 - 4 T + 37 T^{2} )^{2} $)($ ( 1 - 4 T + 37 T^{2} )^{2} $)($ ( 1 - 10 T + 37 T^{2} )^{2} $)($ ( 1 + 2 T + 37 T^{2} )^{2} $)($ ( 1 + 2 T + 37 T^{2} )^{2} $)($ ( 1 + 8 T + 37 T^{2} )^{4} $)
$41$	($ 1 - 34 T^{2} + 1681 T^{4} $)($ 1 - 34 T^{2} + 1681 T^{4} $)($ ( 1 - 41 T^{2} )^{2} $)($ 1 - 28 T^{2} + 1681 T^{4} $)($ 1 - 28 T^{2} + 1681 T^{4} $)($ ( 1 - 41 T^{2} )^{2} $)($ 1 - 34 T^{2} + 1681 T^{4} $)($ 1 - 34 T^{2} + 1681 T^{4} $)($ ( 1 + 68 T^{2} + 1681 T^{4} )^{2} $)
$43$	($ 1 - 74 T^{2} + 1849 T^{4} $)($ ( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} ) $)($ ( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} ) $)($ 1 - 62 T^{2} + 1849 T^{4} $)($ 1 - 62 T^{2} + 1849 T^{4} $)($ ( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} ) $)($ ( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} ) $)($ 1 - 74 T^{2} + 1849 T^{4} $)($ ( 1 - 43 T^{2} )^{4} $)
$47$	($ ( 1 + 47 T^{2} )^{2} $)($ ( 1 + 47 T^{2} )^{2} $)($ ( 1 + 47 T^{2} )^{2} $)($ ( 1 - 12 T + 47 T^{2} )^{2} $)($ ( 1 + 12 T + 47 T^{2} )^{2} $)($ ( 1 + 47 T^{2} )^{2} $)($ ( 1 + 47 T^{2} )^{2} $)($ ( 1 + 47 T^{2} )^{2} $)($ ( 1 + 47 T^{2} )^{4} $)
$53$	($ ( 1 - 6 T + 53 T^{2} )^{2} $)($ ( 1 + 6 T + 53 T^{2} )^{2} $)($ ( 1 + 53 T^{2} )^{2} $)($ ( 1 - 6 T + 53 T^{2} )^{2} $)($ ( 1 - 6 T + 53 T^{2} )^{2} $)($ ( 1 + 53 T^{2} )^{2} $)($ ( 1 + 6 T + 53 T^{2} )^{2} $)($ ( 1 - 6 T + 53 T^{2} )^{2} $)($ ( 1 + 53 T^{2} )^{4} $)
$59$	($ ( 1 + 12 T + 59 T^{2} )^{2} $)($ ( 1 + 6 T + 59 T^{2} )^{2} $)($ ( 1 + 59 T^{2} )^{2} $)($ ( 1 + 12 T + 59 T^{2} )^{2} $)($ ( 1 - 12 T + 59 T^{2} )^{2} $)($ ( 1 + 59 T^{2} )^{2} $)($ ( 1 - 6 T + 59 T^{2} )^{2} $)($ ( 1 - 12 T + 59 T^{2} )^{2} $)($ ( 1 + 59 T^{2} )^{4} $)
$61$	($ 1 + 70 T^{2} + 3721 T^{4} $)($ 1 - 110 T^{2} + 3721 T^{4} $)($ ( 1 - 14 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) $)($ ( 1 - 61 T^{2} )^{2} $)($ ( 1 - 61 T^{2} )^{2} $)($ ( 1 - 14 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) $)($ 1 - 110 T^{2} + 3721 T^{4} $)($ 1 + 70 T^{2} + 3721 T^{4} $)($ ( 1 - 61 T^{2} )^{4} $)
$67$	($ ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) $)($ ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) $)($ ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) $)($ ( 1 - 67 T^{2} )^{2} $)($ ( 1 - 67 T^{2} )^{2} $)($ ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) $)($ ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) $)($ ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) $)($ ( 1 - 67 T^{2} )^{4} $)
$71$	($ 1 - 34 T^{2} + 5041 T^{4} $)($ 1 - 130 T^{2} + 5041 T^{4} $)($ ( 1 - 71 T^{2} )^{2} $)($ 1 + 8 T^{2} + 5041 T^{4} $)($ 1 + 8 T^{2} + 5041 T^{4} $)($ ( 1 - 71 T^{2} )^{2} $)($ 1 - 130 T^{2} + 5041 T^{4} $)($ 1 - 34 T^{2} + 5041 T^{4} $)($ ( 1 - 100 T^{2} + 5041 T^{4} )^{2} $)
$73$	($ ( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) $)($ 1 - 98 T^{2} + 5329 T^{4} $)($ ( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) $)($ 1 + 70 T^{2} + 5329 T^{4} $)($ 1 + 70 T^{2} + 5329 T^{4} $)($ ( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) $)($ 1 - 98 T^{2} + 5329 T^{4} $)($ ( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) $)($ ( 1 - 73 T^{2} )^{4} $)
$79$	($ 1 - 50 T^{2} + 6241 T^{4} $)($ 1 - 146 T^{2} + 6241 T^{4} $)($ ( 1 - 4 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) $)($ 1 - 62 T^{2} + 6241 T^{4} $)($ 1 - 62 T^{2} + 6241 T^{4} $)($ ( 1 - 4 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) $)($ 1 - 146 T^{2} + 6241 T^{4} $)($ 1 - 50 T^{2} + 6241 T^{4} $)($ ( 1 - 79 T^{2} )^{4} $)
$83$	($ ( 1 + 12 T + 83 T^{2} )^{2} $)($ ( 1 - 6 T + 83 T^{2} )^{2} $)($ ( 1 + 83 T^{2} )^{2} $)($ ( 1 + 83 T^{2} )^{2} $)($ ( 1 + 83 T^{2} )^{2} $)($ ( 1 + 83 T^{2} )^{2} $)($ ( 1 + 6 T + 83 T^{2} )^{2} $)($ ( 1 - 12 T + 83 T^{2} )^{2} $)($ ( 1 + 83 T^{2} )^{4} $)
$89$	($ 1 - 130 T^{2} + 7921 T^{4} $)($ 1 - 130 T^{2} + 7921 T^{4} $)($ ( 1 - 89 T^{2} )^{2} $)($ 1 - 28 T^{2} + 7921 T^{4} $)($ 1 - 28 T^{2} + 7921 T^{4} $)($ ( 1 - 89 T^{2} )^{2} $)($ 1 - 130 T^{2} + 7921 T^{4} $)($ 1 - 130 T^{2} + 7921 T^{4} $)($ ( 1 - 172 T^{2} + 7921 T^{4} )^{2} $)
$97$	($ ( 1 - 97 T^{2} )^{2} $)($ ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) $)($ ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) $)($ 1 - 170 T^{2} + 9409 T^{4} $)($ 1 - 170 T^{2} + 9409 T^{4} $)($ ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) $)($ ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) $)($ ( 1 - 97 T^{2} )^{2} $)($ ( 1 - 97 T^{2} )^{4} $)
show more
show less

Introduction	Features
Universe	News

Level:	\( N \)	=	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	=	\( 2 \)
Character orbit:	\([\chi]\)	=	1008.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	=	\( 28 \)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(384\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(5\), \(11\), \(19\)

Label	Dim.	\(A\)	Field	CM	Traces				$q$-expansion
Label	Dim.	\(A\)	Field	CM	\(a_2\)	\(a_3\)	\(a_5\)	\(a_7\)	$q$-expansion
1008.2.b.a	\(2\)	\(8.049\)	\(\Q(\sqrt{-3}) \)	None	\(0\)	\(0\)	\(0\)	\(-4\)	\(q+(-2+\zeta_{6})q^{7}-2\zeta_{6}q^{11}-4\zeta_{6}q^{17}+\cdots\)
1008.2.b.b	\(2\)	\(8.049\)	\(\Q(\sqrt{-3}) \)	None	\(0\)	\(0\)	\(0\)	\(-4\)	\(q-2\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+2\zeta_{6}q^{11}+\cdots\)
1008.2.b.c	\(2\)	\(8.049\)	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)	\(0\)	\(0\)	\(0\)	\(-4\)	\(q+(-2+\zeta_{6})q^{7}-4\zeta_{6}q^{13}+8q^{19}+\cdots\)
1008.2.b.d	\(2\)	\(8.049\)	\(\Q(\sqrt{-6}) \)	None	\(0\)	\(0\)	\(0\)	\(-2\)	\(q+\beta q^{5}+(-1-\beta )q^{7}-\beta q^{11}-2\beta q^{13}+\cdots\)
1008.2.b.e	\(2\)	\(8.049\)	\(\Q(\sqrt{-6}) \)	None	\(0\)	\(0\)	\(0\)	\(2\)	\(q+\beta q^{5}+(1+\beta )q^{7}+\beta q^{11}-2\beta q^{13}+\cdots\)
1008.2.b.f	\(2\)	\(8.049\)	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)	\(0\)	\(0\)	\(0\)	\(4\)	\(q+(2-\zeta_{6})q^{7}-4\zeta_{6}q^{13}-8q^{19}+5q^{25}+\cdots\)
1008.2.b.g	\(2\)	\(8.049\)	\(\Q(\sqrt{-3}) \)	None	\(0\)	\(0\)	\(0\)	\(4\)	\(q-2\zeta_{6}q^{5}+(2+\zeta_{6})q^{7}-2\zeta_{6}q^{11}+\cdots\)
1008.2.b.h	\(2\)	\(8.049\)	\(\Q(\sqrt{-3}) \)	None	\(0\)	\(0\)	\(0\)	\(4\)	\(q+(2+\zeta_{6})q^{7}-2\zeta_{6}q^{11}+4\zeta_{6}q^{17}+\cdots\)
1008.2.b.i	\(4\)	\(8.049\)	\(\Q(\sqrt{-6}, \sqrt{7})\)	\(\Q(\sqrt{-21}) \)	\(0\)	\(0\)	\(0\)	\(0\)	\(q-\beta _{1}q^{5}-\beta _{3}q^{7}-\beta _{2}q^{11}+3\beta _{1}q^{17}+\cdots\)

Introduction and more

L-functions

Modular Forms

Varieties

Fields

Representations

Groups

Knowledge

Properties

Related objects

Downloads

Learn more about

Defining parameters

Dimensions

Trace form

Decomposition of \(S_{2}^{\mathrm{new}}(1008, [\chi])\) into newform subspaces

Decomposition of \(S_{2}^{\mathrm{old}}(1008, [\chi])\) into lower level spaces

Hecke Characteristic Polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1008.2.b

Level	1008
Weight	2
Character orbit	b
Rep. character	\(\chi_{1008}(559,\cdot)\)
Character field	\(\Q\)
Dimension	20
Newform subspaces	9
Sturm bound	384
Trace bound	19

$p$	$F_p(T)$
$2$	1
$3$	1
$5$	(\( ( 1 - 5 T^{2} )^{2} \))(\( 1 + 2 T^{2} + 25 T^{4} \))(\( ( 1 - 5 T^{2} )^{2} \))(\( 1 - 4 T^{2} + 25 T^{4} \))(\( 1 - 4 T^{2} + 25 T^{4} \))(\( ( 1 - 5 T^{2} )^{2} \))(\( 1 + 2 T^{2} + 25 T^{4} \))(\( ( 1 - 5 T^{2} )^{2} \))(\( ( 1 - 4 T^{2} + 25 T^{4} )^{2} \))
$7$	(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( ( 1 - 7 T^{2} )^{2} \))
$11$	(\( 1 - 10 T^{2} + 121 T^{4} \))(\( 1 - 10 T^{2} + 121 T^{4} \))(\( ( 1 - 11 T^{2} )^{2} \))(\( 1 - 16 T^{2} + 121 T^{4} \))(\( 1 - 16 T^{2} + 121 T^{4} \))(\( ( 1 - 11 T^{2} )^{2} \))(\( 1 - 10 T^{2} + 121 T^{4} \))(\( 1 - 10 T^{2} + 121 T^{4} \))(\( ( 1 + 20 T^{2} + 121 T^{4} )^{2} \))
$13$	(\( ( 1 - 13 T^{2} )^{2} \))(\( 1 - 14 T^{2} + 169 T^{4} \))(\( ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( 1 - 2 T^{2} + 169 T^{4} \))(\( 1 - 2 T^{2} + 169 T^{4} \))(\( ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( 1 - 14 T^{2} + 169 T^{4} \))(\( ( 1 - 13 T^{2} )^{2} \))(\( ( 1 - 13 T^{2} )^{4} \))
$17$	(\( 1 + 14 T^{2} + 289 T^{4} \))(\( ( 1 - 17 T^{2} )^{2} \))(\( ( 1 - 17 T^{2} )^{2} \))(\( 1 - 28 T^{2} + 289 T^{4} \))(\( 1 - 28 T^{2} + 289 T^{4} \))(\( ( 1 - 17 T^{2} )^{2} \))(\( ( 1 - 17 T^{2} )^{2} \))(\( 1 + 14 T^{2} + 289 T^{4} \))(\( ( 1 + 20 T^{2} + 289 T^{4} )^{2} \))
$19$	(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 - 2 T + 19 T^{2} )^{2} \))(\( ( 1 - 8 T + 19 T^{2} )^{2} \))(\( ( 1 + 2 T + 19 T^{2} )^{2} \))(\( ( 1 - 2 T + 19 T^{2} )^{2} \))(\( ( 1 + 8 T + 19 T^{2} )^{2} \))(\( ( 1 + 2 T + 19 T^{2} )^{2} \))(\( ( 1 - 4 T + 19 T^{2} )^{2} \))(\( ( 1 + 10 T^{2} + 361 T^{4} )^{2} \))
$23$	(\( 1 - 34 T^{2} + 529 T^{4} \))(\( 1 - 34 T^{2} + 529 T^{4} \))(\( ( 1 - 23 T^{2} )^{2} \))(\( 1 + 8 T^{2} + 529 T^{4} \))(\( 1 + 8 T^{2} + 529 T^{4} \))(\( ( 1 - 23 T^{2} )^{2} \))(\( 1 - 34 T^{2} + 529 T^{4} \))(\( 1 - 34 T^{2} + 529 T^{4} \))(\( ( 1 - 4 T^{2} + 529 T^{4} )^{2} \))
$29$	(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{4} \))
$31$	(\( ( 1 - 4 T + 31 T^{2} )^{2} \))(\( ( 1 + 8 T + 31 T^{2} )^{2} \))(\( ( 1 - 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{2} \))(\( ( 1 + 8 T + 31 T^{2} )^{2} \))(\( ( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{2} \))(\( ( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 50 T^{2} + 961 T^{4} )^{2} \))
$37$	(\( ( 1 + 2 T + 37 T^{2} )^{2} \))(\( ( 1 + 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 10 T + 37 T^{2} )^{2} \))(\( ( 1 - 4 T + 37 T^{2} )^{2} \))(\( ( 1 - 4 T + 37 T^{2} )^{2} \))(\( ( 1 - 10 T + 37 T^{2} )^{2} \))(\( ( 1 + 2 T + 37 T^{2} )^{2} \))(\( ( 1 + 2 T + 37 T^{2} )^{2} \))(\( ( 1 + 8 T + 37 T^{2} )^{4} \))
$41$	(\( 1 - 34 T^{2} + 1681 T^{4} \))(\( 1 - 34 T^{2} + 1681 T^{4} \))(\( ( 1 - 41 T^{2} )^{2} \))(\( 1 - 28 T^{2} + 1681 T^{4} \))(\( 1 - 28 T^{2} + 1681 T^{4} \))(\( ( 1 - 41 T^{2} )^{2} \))(\( 1 - 34 T^{2} + 1681 T^{4} \))(\( 1 - 34 T^{2} + 1681 T^{4} \))(\( ( 1 + 68 T^{2} + 1681 T^{4} )^{2} \))
$43$	(\( 1 - 74 T^{2} + 1849 T^{4} \))(\( ( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} ) \))(\( ( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} ) \))(\( 1 - 62 T^{2} + 1849 T^{4} \))(\( 1 - 62 T^{2} + 1849 T^{4} \))(\( ( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} ) \))(\( ( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} ) \))(\( 1 - 74 T^{2} + 1849 T^{4} \))(\( ( 1 - 43 T^{2} )^{4} \))
$47$	(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 - 12 T + 47 T^{2} )^{2} \))(\( ( 1 + 12 T + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{4} \))
$53$	(\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( ( 1 + 53 T^{2} )^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 + 53 T^{2} )^{2} \))(\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 + 53 T^{2} )^{4} \))
$59$	(\( ( 1 + 12 T + 59 T^{2} )^{2} \))(\( ( 1 + 6 T + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 12 T + 59 T^{2} )^{2} \))(\( ( 1 - 12 T + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 - 6 T + 59 T^{2} )^{2} \))(\( ( 1 - 12 T + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{4} \))
$61$	(\( 1 + 70 T^{2} + 3721 T^{4} \))(\( 1 - 110 T^{2} + 3721 T^{4} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))(\( ( 1 - 61 T^{2} )^{2} \))(\( ( 1 - 61 T^{2} )^{2} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))(\( 1 - 110 T^{2} + 3721 T^{4} \))(\( 1 + 70 T^{2} + 3721 T^{4} \))(\( ( 1 - 61 T^{2} )^{4} \))
$67$	(\( ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 67 T^{2} )^{2} \))(\( ( 1 - 67 T^{2} )^{2} \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 67 T^{2} )^{4} \))
$71$	(\( 1 - 34 T^{2} + 5041 T^{4} \))(\( 1 - 130 T^{2} + 5041 T^{4} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( 1 + 8 T^{2} + 5041 T^{4} \))(\( 1 + 8 T^{2} + 5041 T^{4} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( 1 - 130 T^{2} + 5041 T^{4} \))(\( 1 - 34 T^{2} + 5041 T^{4} \))(\( ( 1 - 100 T^{2} + 5041 T^{4} )^{2} \))
$73$	(\( ( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \))(\( 1 - 98 T^{2} + 5329 T^{4} \))(\( ( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \))(\( 1 + 70 T^{2} + 5329 T^{4} \))(\( 1 + 70 T^{2} + 5329 T^{4} \))(\( ( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \))(\( 1 - 98 T^{2} + 5329 T^{4} \))(\( ( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \))(\( ( 1 - 73 T^{2} )^{4} \))
$79$	(\( 1 - 50 T^{2} + 6241 T^{4} \))(\( 1 - 146 T^{2} + 6241 T^{4} \))(\( ( 1 - 4 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \))(\( 1 - 62 T^{2} + 6241 T^{4} \))(\( 1 - 62 T^{2} + 6241 T^{4} \))(\( ( 1 - 4 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \))(\( 1 - 146 T^{2} + 6241 T^{4} \))(\( 1 - 50 T^{2} + 6241 T^{4} \))(\( ( 1 - 79 T^{2} )^{4} \))
$83$	(\( ( 1 + 12 T + 83 T^{2} )^{2} \))(\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{4} \))
$89$	(\( 1 - 130 T^{2} + 7921 T^{4} \))(\( 1 - 130 T^{2} + 7921 T^{4} \))(\( ( 1 - 89 T^{2} )^{2} \))(\( 1 - 28 T^{2} + 7921 T^{4} \))(\( 1 - 28 T^{2} + 7921 T^{4} \))(\( ( 1 - 89 T^{2} )^{2} \))(\( 1 - 130 T^{2} + 7921 T^{4} \))(\( 1 - 130 T^{2} + 7921 T^{4} \))(\( ( 1 - 172 T^{2} + 7921 T^{4} )^{2} \))
$97$	(\( ( 1 - 97 T^{2} )^{2} \))(\( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( 1 - 170 T^{2} + 9409 T^{4} \))(\( 1 - 170 T^{2} + 9409 T^{4} \))(\( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( ( 1 - 97 T^{2} )^{2} \))(\( ( 1 - 97 T^{2} )^{4} \))
show more
show less