LMFDB - Space of Modular Forms of Level 1200, Weight 2, and Trivial Character

Defining parameters

Level:	$ N $	=	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	=	$ 2 $
Character orbit:	$[\chi]$	=	1200.a (trivial)
Character field:			$\Q$
Newform subspaces:			$ 19 $
Sturm bound:			$480$
Trace bound:			$13$
Distinguishing $T_p$:			$7$, $11$, $13$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(1200))$.

	Total	New	Old
Modular forms	276	19	257
Cusp forms	205	19	186
Eisenstein series	71	0	71

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

$2$	$3$	$5$	Fricke	Dim.
$+$	$+$	$+$	$+$	$1$
$+$	$+$	$-$	$-$	$3$
$+$	$-$	$+$	$-$	$4$
$+$	$-$	$-$	$+$	$1$
$-$	$+$	$+$	$-$	$2$
$-$	$+$	$-$	$+$	$3$
$-$	$-$	$+$	$+$	$2$
$-$	$-$	$-$	$-$	$3$
Plus space			$+$	$7$
Minus space			$-$	$12$

Trace form

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(1200))$ into newform subspaces

Label	Dim.	$A$	Field	CM	Traces				A-L signs			$q$-expansion
Label	Dim.	$A$	Field	CM	$a_2$	$a_3$	$a_5$	$a_7$	2	3	5	$q$-expansion
1200.2.a.a	$1$	$9.582$	$\Q$	None	$0$	$-1$	$0$	$-4$	$-$	$+$	$-$	$q-q^{3}-4q^{7}+q^{9}+4q^{11}+4q^{17}+\cdots$
1200.2.a.b	$1$	$9.582$	$\Q$	None	$0$	$-1$	$0$	$-3$	$+$	$+$	$-$	$q-q^{3}-3q^{7}+q^{9}-2q^{11}-3q^{13}+\cdots$
1200.2.a.c	$1$	$9.582$	$\Q$	None	$0$	$-1$	$0$	$-3$	$-$	$+$	$+$	$q-q^{3}-3q^{7}+q^{9}-2q^{11}-q^{13}+\cdots$
1200.2.a.d	$1$	$9.582$	$\Q$	None	$0$	$-1$	$0$	$0$	$+$	$+$	$+$	$q-q^{3}+q^{9}-4q^{11}+2q^{13}-2q^{17}+\cdots$
1200.2.a.e	$1$	$9.582$	$\Q$	None	$0$	$-1$	$0$	$0$	$-$	$+$	$+$	$q-q^{3}+q^{9}+4q^{11}+2q^{13}-2q^{17}+\cdots$
1200.2.a.f	$1$	$9.582$	$\Q$	None	$0$	$-1$	$0$	$1$	$-$	$+$	$-$	$q-q^{3}+q^{7}+q^{9}-6q^{11}+5q^{13}+\cdots$
1200.2.a.g	$1$	$9.582$	$\Q$	None	$0$	$-1$	$0$	$2$	$-$	$+$	$-$	$q-q^{3}+2q^{7}+q^{9}-2q^{11}-6q^{13}+\cdots$
1200.2.a.h	$1$	$9.582$	$\Q$	None	$0$	$-1$	$0$	$2$	$+$	$+$	$-$	$q-q^{3}+2q^{7}+q^{9}-2q^{11}+2q^{13}+\cdots$
1200.2.a.i	$1$	$9.582$	$\Q$	None	$0$	$-1$	$0$	$5$	$+$	$+$	$-$	$q-q^{3}+5q^{7}+q^{9}+6q^{11}-3q^{13}+\cdots$
1200.2.a.j	$1$	$9.582$	$\Q$	None	$0$	$1$	$0$	$-5$	$+$	$-$	$+$	$q+q^{3}-5q^{7}+q^{9}+6q^{11}+3q^{13}+\cdots$
1200.2.a.k	$1$	$9.582$	$\Q$	None	$0$	$1$	$0$	$-4$	$-$	$-$	$+$	$q+q^{3}-4q^{7}+q^{9}-2q^{13}-6q^{17}+\cdots$
1200.2.a.l	$1$	$9.582$	$\Q$	None	$0$	$1$	$0$	$-2$	$+$	$-$	$-$	$q+q^{3}-2q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots$
1200.2.a.m	$1$	$9.582$	$\Q$	None	$0$	$1$	$0$	$-2$	$-$	$-$	$-$	$q+q^{3}-2q^{7}+q^{9}-2q^{11}+6q^{13}+\cdots$
1200.2.a.n	$1$	$9.582$	$\Q$	None	$0$	$1$	$0$	$-1$	$-$	$-$	$+$	$q+q^{3}-q^{7}+q^{9}-6q^{11}-5q^{13}+\cdots$
1200.2.a.o	$1$	$9.582$	$\Q$	None	$0$	$1$	$0$	$0$	$+$	$-$	$+$	$q+q^{3}+q^{9}+4q^{11}-6q^{13}+6q^{17}+\cdots$
1200.2.a.p	$1$	$9.582$	$\Q$	None	$0$	$1$	$0$	$3$	$-$	$-$	$-$	$q+q^{3}+3q^{7}+q^{9}-2q^{11}+q^{13}+\cdots$
1200.2.a.q	$1$	$9.582$	$\Q$	None	$0$	$1$	$0$	$3$	$+$	$-$	$+$	$q+q^{3}+3q^{7}+q^{9}-2q^{11}+3q^{13}+\cdots$
1200.2.a.r	$1$	$9.582$	$\Q$	None	$0$	$1$	$0$	$4$	$+$	$-$	$+$	$q+q^{3}+4q^{7}+q^{9}+6q^{13}+2q^{17}+\cdots$
1200.2.a.s	$1$	$9.582$	$\Q$	None	$0$	$1$	$0$	$4$	$-$	$-$	$-$	$q+q^{3}+4q^{7}+q^{9}+4q^{11}-4q^{17}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(1200))$ into lower level spaces

Hecke Characteristic Polynomials

$p$	$F_p(T)$
$2$	()()()()()()()()()()()()()()()()()()()
$3$	($ 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 $)
$5$	()()()()()()()()()()()()()()()()()()()
$7$	($ 1 + 4 T + 7 T^{2} $)($ 1 + 3 T + 7 T^{2} $)($ 1 + 3 T + 7 T^{2} $)($ 1 + 7 T^{2} $)($ 1 + 7 T^{2} $)($ 1 - T + 7 T^{2} $)($ 1 - 2 T + 7 T^{2} $)($ 1 - 2 T + 7 T^{2} $)($ 1 - 5 T + 7 T^{2} $)($ 1 + 5 T + 7 T^{2} $)($ 1 + 4 T + 7 T^{2} $)($ 1 + 2 T + 7 T^{2} $)($ 1 + 2 T + 7 T^{2} $)($ 1 + T + 7 T^{2} $)($ 1 + 7 T^{2} $)($ 1 - 3 T + 7 T^{2} $)($ 1 - 3 T + 7 T^{2} $)($ 1 - 4 T + 7 T^{2} $)($ 1 - 4 T + 7 T^{2} $)
$11$	($ 1 - 4 T + 11 T^{2} $)($ 1 + 2 T + 11 T^{2} $)($ 1 + 2 T + 11 T^{2} $)($ 1 + 4 T + 11 T^{2} $)($ 1 - 4 T + 11 T^{2} $)($ 1 + 6 T + 11 T^{2} $)($ 1 + 2 T + 11 T^{2} $)($ 1 + 2 T + 11 T^{2} $)($ 1 - 6 T + 11 T^{2} $)($ 1 - 6 T + 11 T^{2} $)($ 1 + 11 T^{2} $)($ 1 + 2 T + 11 T^{2} $)($ 1 + 2 T + 11 T^{2} $)($ 1 + 6 T + 11 T^{2} $)($ 1 - 4 T + 11 T^{2} $)($ 1 + 2 T + 11 T^{2} $)($ 1 + 2 T + 11 T^{2} $)($ 1 + 11 T^{2} $)($ 1 - 4 T + 11 T^{2} $)
$13$	($ 1 + 13 T^{2} $)($ 1 + 3 T + 13 T^{2} $)($ 1 + T + 13 T^{2} $)($ 1 - 2 T + 13 T^{2} $)($ 1 - 2 T + 13 T^{2} $)($ 1 - 5 T + 13 T^{2} $)($ 1 + 6 T + 13 T^{2} $)($ 1 - 2 T + 13 T^{2} $)($ 1 + 3 T + 13 T^{2} $)($ 1 - 3 T + 13 T^{2} $)($ 1 + 2 T + 13 T^{2} $)($ 1 + 2 T + 13 T^{2} $)($ 1 - 6 T + 13 T^{2} $)($ 1 + 5 T + 13 T^{2} $)($ 1 + 6 T + 13 T^{2} $)($ 1 - T + 13 T^{2} $)($ 1 - 3 T + 13 T^{2} $)($ 1 - 6 T + 13 T^{2} $)($ 1 + 13 T^{2} $)
$17$	($ 1 - 4 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} $)($ 1 + 6 T + 17 T^{2} $)($ 1 + 2 T + 17 T^{2} $)($ 1 - 6 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 - 2 T + 17 T^{2} $)($ 1 + 6 T + 17 T^{2} $)($ 1 - 2 T + 17 T^{2} $)($ 1 + 4 T + 17 T^{2} $)
$19$	($ 1 + 19 T^{2} $)($ 1 - 7 T + 19 T^{2} $)($ 1 - 5 T + 19 T^{2} $)($ 1 - 4 T + 19 T^{2} $)($ 1 + 4 T + 19 T^{2} $)($ 1 + 5 T + 19 T^{2} $)($ 1 + 19 T^{2} $)($ 1 + 8 T + 19 T^{2} $)($ 1 + T + 19 T^{2} $)($ 1 + T + 19 T^{2} $)($ 1 - 4 T + 19 T^{2} $)($ 1 + 8 T + 19 T^{2} $)($ 1 + 19 T^{2} $)($ 1 + 5 T + 19 T^{2} $)($ 1 - 4 T + 19 T^{2} $)($ 1 - 5 T + 19 T^{2} $)($ 1 - 7 T + 19 T^{2} $)($ 1 + 4 T + 19 T^{2} $)($ 1 + 19 T^{2} $)
$23$	($ 1 + 4 T + 23 T^{2} $)($ 1 + 6 T + 23 T^{2} $)($ 1 - 6 T + 23 T^{2} $)($ 1 + 8 T + 23 T^{2} $)($ 1 + 23 T^{2} $)($ 1 - 6 T + 23 T^{2} $)($ 1 + 4 T + 23 T^{2} $)($ 1 - 4 T + 23 T^{2} $)($ 1 - 2 T + 23 T^{2} $)($ 1 + 2 T + 23 T^{2} $)($ 1 + 23 T^{2} $)($ 1 + 4 T + 23 T^{2} $)($ 1 - 4 T + 23 T^{2} $)($ 1 + 6 T + 23 T^{2} $)($ 1 + 23 T^{2} $)($ 1 + 6 T + 23 T^{2} $)($ 1 - 6 T + 23 T^{2} $)($ 1 + 8 T + 23 T^{2} $)($ 1 - 4 T + 23 T^{2} $)
$29$	($ 1 + 6 T + 29 T^{2} $)($ 1 + 2 T + 29 T^{2} $)($ 1 - 10 T + 29 T^{2} $)($ 1 - 6 T + 29 T^{2} $)($ 1 + 2 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 - 6 T + 29 T^{2} $)($ 1 + 6 T + 29 T^{2} $)($ 1 - 8 T + 29 T^{2} $)($ 1 + 29 T^{2} $)($ 1 + 6 T + 29 T^{2} $)($ 1 + 2 T + 29 T^{2} $)($ 1 - 10 T + 29 T^{2} $)($ 1 + 2 T + 29 T^{2} $)($ 1 + 6 T + 29 T^{2} $)($ 1 + 6 T + 29 T^{2} $)
$31$	($ 1 + 4 T + 31 T^{2} $)($ 1 - 5 T + 31 T^{2} $)($ 1 - 3 T + 31 T^{2} $)($ 1 + 8 T + 31 T^{2} $)($ 1 + 31 T^{2} $)($ 1 - T + 31 T^{2} $)($ 1 - 8 T + 31 T^{2} $)($ 1 + 31 T^{2} $)($ 1 + 3 T + 31 T^{2} $)($ 1 + 3 T + 31 T^{2} $)($ 1 + 8 T + 31 T^{2} $)($ 1 + 31 T^{2} $)($ 1 - 8 T + 31 T^{2} $)($ 1 - T + 31 T^{2} $)($ 1 - 8 T + 31 T^{2} $)($ 1 - 3 T + 31 T^{2} $)($ 1 - 5 T + 31 T^{2} $)($ 1 + 31 T^{2} $)($ 1 + 4 T + 31 T^{2} $)
$37$	($ 1 + 8 T + 37 T^{2} $)($ 1 - 10 T + 37 T^{2} $)($ 1 + 2 T + 37 T^{2} $)($ 1 + 6 T + 37 T^{2} $)($ 1 - 10 T + 37 T^{2} $)($ 1 - 2 T + 37 T^{2} $)($ 1 + 2 T + 37 T^{2} $)($ 1 + 10 T + 37 T^{2} $)($ 1 + 6 T + 37 T^{2} $)($ 1 - 6 T + 37 T^{2} $)($ 1 + 2 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 + 10 T + 37 T^{2} $)($ 1 - 6 T + 37 T^{2} $)($ 1 - 8 T + 37 T^{2} $)
$41$	($ 1 + 10 T + 41 T^{2} $)($ 1 - 12 T + 41 T^{2} $)($ 1 + 8 T + 41 T^{2} $)($ 1 + 6 T + 41 T^{2} $)($ 1 - 10 T + 41 T^{2} $)($ 1 + 41 T^{2} $)($ 1 - 2 T + 41 T^{2} $)($ 1 - 2 T + 41 T^{2} $)($ 1 - 4 T + 41 T^{2} $)($ 1 - 4 T + 41 T^{2} $)($ 1 + 6 T + 41 T^{2} $)($ 1 - 2 T + 41 T^{2} $)($ 1 - 2 T + 41 T^{2} $)($ 1 + 41 T^{2} $)($ 1 + 6 T + 41 T^{2} $)($ 1 + 8 T + 41 T^{2} $)($ 1 - 12 T + 41 T^{2} $)($ 1 - 10 T + 41 T^{2} $)($ 1 + 10 T + 41 T^{2} $)
$43$	($ 1 + 4 T + 43 T^{2} $)($ 1 + 3 T + 43 T^{2} $)($ 1 - T + 43 T^{2} $)($ 1 - 4 T + 43 T^{2} $)($ 1 - 4 T + 43 T^{2} $)($ 1 - T + 43 T^{2} $)($ 1 + 4 T + 43 T^{2} $)($ 1 - 12 T + 43 T^{2} $)($ 1 + 11 T + 43 T^{2} $)($ 1 - 11 T + 43 T^{2} $)($ 1 + 4 T + 43 T^{2} $)($ 1 + 12 T + 43 T^{2} $)($ 1 - 4 T + 43 T^{2} $)($ 1 + T + 43 T^{2} $)($ 1 - 12 T + 43 T^{2} $)($ 1 + T + 43 T^{2} $)($ 1 - 3 T + 43 T^{2} $)($ 1 + 4 T + 43 T^{2} $)($ 1 - 4 T + 43 T^{2} $)
$47$	($ 1 - 4 T + 47 T^{2} $)($ 1 - 10 T + 47 T^{2} $)($ 1 - 2 T + 47 T^{2} $)($ 1 + 47 T^{2} $)($ 1 - 8 T + 47 T^{2} $)($ 1 + 6 T + 47 T^{2} $)($ 1 + 8 T + 47 T^{2} $)($ 1 + 47 T^{2} $)($ 1 - 10 T + 47 T^{2} $)($ 1 + 10 T + 47 T^{2} $)($ 1 + 47 T^{2} $)($ 1 + 47 T^{2} $)($ 1 - 8 T + 47 T^{2} $)($ 1 - 6 T + 47 T^{2} $)($ 1 - 8 T + 47 T^{2} $)($ 1 + 2 T + 47 T^{2} $)($ 1 + 10 T + 47 T^{2} $)($ 1 - 8 T + 47 T^{2} $)($ 1 + 4 T + 47 T^{2} $)
$53$	($ 1 + 12 T + 53 T^{2} $)($ 1 + 53 T^{2} $)($ 1 - 4 T + 53 T^{2} $)($ 1 - 2 T + 53 T^{2} $)($ 1 - 10 T + 53 T^{2} $)($ 1 + 12 T + 53 T^{2} $)($ 1 + 6 T + 53 T^{2} $)($ 1 - 10 T + 53 T^{2} $)($ 1 + 8 T + 53 T^{2} $)($ 1 - 8 T + 53 T^{2} $)($ 1 - 6 T + 53 T^{2} $)($ 1 + 10 T + 53 T^{2} $)($ 1 - 6 T + 53 T^{2} $)($ 1 - 12 T + 53 T^{2} $)($ 1 + 6 T + 53 T^{2} $)($ 1 + 4 T + 53 T^{2} $)($ 1 + 53 T^{2} $)($ 1 + 10 T + 53 T^{2} $)($ 1 - 12 T + 53 T^{2} $)
$59$	($ 1 + 4 T + 59 T^{2} $)($ 1 - 6 T + 59 T^{2} $)($ 1 - 10 T + 59 T^{2} $)($ 1 + 4 T + 59 T^{2} $)($ 1 - 4 T + 59 T^{2} $)($ 1 - 6 T + 59 T^{2} $)($ 1 + 10 T + 59 T^{2} $)($ 1 - 6 T + 59 T^{2} $)($ 1 - 6 T + 59 T^{2} $)($ 1 - 6 T + 59 T^{2} $)($ 1 + 59 T^{2} $)($ 1 - 6 T + 59 T^{2} $)($ 1 + 10 T + 59 T^{2} $)($ 1 - 6 T + 59 T^{2} $)($ 1 + 12 T + 59 T^{2} $)($ 1 - 10 T + 59 T^{2} $)($ 1 - 6 T + 59 T^{2} $)($ 1 + 59 T^{2} $)($ 1 + 4 T + 59 T^{2} $)
$61$	($ 1 - 2 T + 61 T^{2} $)($ 1 + 13 T + 61 T^{2} $)($ 1 - 7 T + 61 T^{2} $)($ 1 + 2 T + 61 T^{2} $)($ 1 + 2 T + 61 T^{2} $)($ 1 + 13 T + 61 T^{2} $)($ 1 - 2 T + 61 T^{2} $)($ 1 - 2 T + 61 T^{2} $)($ 1 - 3 T + 61 T^{2} $)($ 1 - 3 T + 61 T^{2} $)($ 1 + 10 T + 61 T^{2} $)($ 1 - 2 T + 61 T^{2} $)($ 1 - 2 T + 61 T^{2} $)($ 1 + 13 T + 61 T^{2} $)($ 1 - 14 T + 61 T^{2} $)($ 1 - 7 T + 61 T^{2} $)($ 1 + 13 T + 61 T^{2} $)($ 1 - 6 T + 61 T^{2} $)($ 1 - 2 T + 61 T^{2} $)
$67$	($ 1 - 4 T + 67 T^{2} $)($ 1 + 7 T + 67 T^{2} $)($ 1 + 3 T + 67 T^{2} $)($ 1 + 4 T + 67 T^{2} $)($ 1 - 12 T + 67 T^{2} $)($ 1 + 11 T + 67 T^{2} $)($ 1 + 8 T + 67 T^{2} $)($ 1 - 8 T + 67 T^{2} $)($ 1 - T + 67 T^{2} $)($ 1 + T + 67 T^{2} $)($ 1 + 4 T + 67 T^{2} $)($ 1 + 8 T + 67 T^{2} $)($ 1 - 8 T + 67 T^{2} $)($ 1 - 11 T + 67 T^{2} $)($ 1 - 4 T + 67 T^{2} $)($ 1 - 3 T + 67 T^{2} $)($ 1 - 7 T + 67 T^{2} $)($ 1 + 4 T + 67 T^{2} $)($ 1 + 4 T + 67 T^{2} $)
$71$	($ 1 + 71 T^{2} $)($ 1 - 4 T + 71 T^{2} $)($ 1 - 8 T + 71 T^{2} $)($ 1 + 8 T + 71 T^{2} $)($ 1 - 8 T + 71 T^{2} $)($ 1 + 71 T^{2} $)($ 1 + 12 T + 71 T^{2} $)($ 1 - 4 T + 71 T^{2} $)($ 1 - 12 T + 71 T^{2} $)($ 1 - 12 T + 71 T^{2} $)($ 1 + 71 T^{2} $)($ 1 - 4 T + 71 T^{2} $)($ 1 + 12 T + 71 T^{2} $)($ 1 + 71 T^{2} $)($ 1 + 8 T + 71 T^{2} $)($ 1 - 8 T + 71 T^{2} $)($ 1 - 4 T + 71 T^{2} $)($ 1 + 71 T^{2} $)($ 1 + 71 T^{2} $)
$73$	($ 1 + 8 T + 73 T^{2} $)($ 1 + 6 T + 73 T^{2} $)($ 1 - 14 T + 73 T^{2} $)($ 1 + 10 T + 73 T^{2} $)($ 1 + 10 T + 73 T^{2} $)($ 1 - 2 T + 73 T^{2} $)($ 1 - 4 T + 73 T^{2} $)($ 1 - 4 T + 73 T^{2} $)($ 1 - 10 T + 73 T^{2} $)($ 1 + 10 T + 73 T^{2} $)($ 1 + 2 T + 73 T^{2} $)($ 1 + 4 T + 73 T^{2} $)($ 1 + 4 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 - 14 T + 73 T^{2} $)($ 1 - 8 T + 73 T^{2} $)
$79$	($ 1 - 12 T + 79 T^{2} $)($ 1 - 8 T + 79 T^{2} $)($ 1 + 79 T^{2} $)($ 1 - 8 T + 79 T^{2} $)($ 1 + 79 T^{2} $)($ 1 + 8 T + 79 T^{2} $)($ 1 + 79 T^{2} $)($ 1 - 8 T + 79 T^{2} $)($ 1 - 8 T + 79 T^{2} $)($ 1 - 8 T + 79 T^{2} $)($ 1 + 8 T + 79 T^{2} $)($ 1 - 8 T + 79 T^{2} $)($ 1 + 79 T^{2} $)($ 1 + 8 T + 79 T^{2} $)($ 1 - 8 T + 79 T^{2} $)($ 1 + 79 T^{2} $)($ 1 - 8 T + 79 T^{2} $)($ 1 + 16 T + 79 T^{2} $)($ 1 - 12 T + 79 T^{2} $)
$83$	($ 1 + 4 T + 83 T^{2} $)($ 1 - 6 T + 83 T^{2} $)($ 1 - 6 T + 83 T^{2} $)($ 1 + 4 T + 83 T^{2} $)($ 1 - 12 T + 83 T^{2} $)($ 1 - 6 T + 83 T^{2} $)($ 1 + 4 T + 83 T^{2} $)($ 1 + 4 T + 83 T^{2} $)($ 1 - 6 T + 83 T^{2} $)($ 1 + 6 T + 83 T^{2} $)($ 1 - 12 T + 83 T^{2} $)($ 1 - 4 T + 83 T^{2} $)($ 1 - 4 T + 83 T^{2} $)($ 1 + 6 T + 83 T^{2} $)($ 1 + 12 T + 83 T^{2} $)($ 1 + 6 T + 83 T^{2} $)($ 1 + 6 T + 83 T^{2} $)($ 1 - 12 T + 83 T^{2} $)($ 1 - 4 T + 83 T^{2} $)
$89$	($ 1 + 10 T + 89 T^{2} $)($ 1 - 16 T + 89 T^{2} $)($ 1 + 89 T^{2} $)($ 1 + 6 T + 89 T^{2} $)($ 1 + 6 T + 89 T^{2} $)($ 1 + 89 T^{2} $)($ 1 + 10 T + 89 T^{2} $)($ 1 - 6 T + 89 T^{2} $)($ 1 + 16 T + 89 T^{2} $)($ 1 + 16 T + 89 T^{2} $)($ 1 - 18 T + 89 T^{2} $)($ 1 - 6 T + 89 T^{2} $)($ 1 + 10 T + 89 T^{2} $)($ 1 + 89 T^{2} $)($ 1 - 10 T + 89 T^{2} $)($ 1 + 89 T^{2} $)($ 1 - 16 T + 89 T^{2} $)($ 1 - 2 T + 89 T^{2} $)($ 1 + 10 T + 89 T^{2} $)
$97$	($ 1 - 8 T + 97 T^{2} $)($ 1 + 7 T + 97 T^{2} $)($ 1 + 17 T + 97 T^{2} $)($ 1 + 2 T + 97 T^{2} $)($ 1 + 2 T + 97 T^{2} $)($ 1 + 7 T + 97 T^{2} $)($ 1 - 8 T + 97 T^{2} $)($ 1 - 8 T + 97 T^{2} $)($ 1 + 7 T + 97 T^{2} $)($ 1 - 7 T + 97 T^{2} $)($ 1 + 2 T + 97 T^{2} $)($ 1 + 8 T + 97 T^{2} $)($ 1 + 8 T + 97 T^{2} $)($ 1 - 7 T + 97 T^{2} $)($ 1 + 2 T + 97 T^{2} $)($ 1 - 17 T + 97 T^{2} $)($ 1 - 7 T + 97 T^{2} $)($ 1 + 2 T + 97 T^{2} $)($ 1 + 8 T + 97 T^{2} $)
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Introduction	Features
Universe	News

Level:	\( N \)	=	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	=	\( 2 \)
Character orbit:	\([\chi]\)	=	1200.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 19 \)
Sturm bound:			\(480\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\)

\(2\)	\(3\)	\(5\)	Fricke	Dim.
\(+\)	\(+\)	\(+\)	\(+\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)	\(3\)
Plus space			\(+\)	\(7\)
Minus space			\(-\)	\(12\)

Label	Dim.	\(A\)	Field	CM	Traces				A-L signs			$q$-expansion
Label	Dim.	\(A\)	Field	CM	\(a_2\)	\(a_3\)	\(a_5\)	\(a_7\)	2	3	5	$q$-expansion
1200.2.a.a	\(1\)	\(9.582\)	\(\Q\)	None	\(0\)	\(-1\)	\(0\)	\(-4\)	\(-\)	\(+\)	\(-\)	\(q-q^{3}-4q^{7}+q^{9}+4q^{11}+4q^{17}+\cdots\)
1200.2.a.b	\(1\)	\(9.582\)	\(\Q\)	None	\(0\)	\(-1\)	\(0\)	\(-3\)	\(+\)	\(+\)	\(-\)	\(q-q^{3}-3q^{7}+q^{9}-2q^{11}-3q^{13}+\cdots\)
1200.2.a.c	\(1\)	\(9.582\)	\(\Q\)	None	\(0\)	\(-1\)	\(0\)	\(-3\)	\(-\)	\(+\)	\(+\)	\(q-q^{3}-3q^{7}+q^{9}-2q^{11}-q^{13}+\cdots\)
1200.2.a.d	\(1\)	\(9.582\)	\(\Q\)	None	\(0\)	\(-1\)	\(0\)	\(0\)	\(+\)	\(+\)	\(+\)	\(q-q^{3}+q^{9}-4q^{11}+2q^{13}-2q^{17}+\cdots\)
1200.2.a.e	\(1\)	\(9.582\)	\(\Q\)	None	\(0\)	\(-1\)	\(0\)	\(0\)	\(-\)	\(+\)	\(+\)	\(q-q^{3}+q^{9}+4q^{11}+2q^{13}-2q^{17}+\cdots\)
1200.2.a.f	\(1\)	\(9.582\)	\(\Q\)	None	\(0\)	\(-1\)	\(0\)	\(1\)	\(-\)	\(+\)	\(-\)	\(q-q^{3}+q^{7}+q^{9}-6q^{11}+5q^{13}+\cdots\)
1200.2.a.g	\(1\)	\(9.582\)	\(\Q\)	None	\(0\)	\(-1\)	\(0\)	\(2\)	\(-\)	\(+\)	\(-\)	\(q-q^{3}+2q^{7}+q^{9}-2q^{11}-6q^{13}+\cdots\)
1200.2.a.h	\(1\)	\(9.582\)	\(\Q\)	None	\(0\)	\(-1\)	\(0\)	\(2\)	\(+\)	\(+\)	\(-\)	\(q-q^{3}+2q^{7}+q^{9}-2q^{11}+2q^{13}+\cdots\)
1200.2.a.i	\(1\)	\(9.582\)	\(\Q\)	None	\(0\)	\(-1\)	\(0\)	\(5\)	\(+\)	\(+\)	\(-\)	\(q-q^{3}+5q^{7}+q^{9}+6q^{11}-3q^{13}+\cdots\)
1200.2.a.j	\(1\)	\(9.582\)	\(\Q\)	None	\(0\)	\(1\)	\(0\)	\(-5\)	\(+\)	\(-\)	\(+\)	\(q+q^{3}-5q^{7}+q^{9}+6q^{11}+3q^{13}+\cdots\)
1200.2.a.k	\(1\)	\(9.582\)	\(\Q\)	None	\(0\)	\(1\)	\(0\)	\(-4\)	\(-\)	\(-\)	\(+\)	\(q+q^{3}-4q^{7}+q^{9}-2q^{13}-6q^{17}+\cdots\)
1200.2.a.l	\(1\)	\(9.582\)	\(\Q\)	None	\(0\)	\(1\)	\(0\)	\(-2\)	\(+\)	\(-\)	\(-\)	\(q+q^{3}-2q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
1200.2.a.m	\(1\)	\(9.582\)	\(\Q\)	None	\(0\)	\(1\)	\(0\)	\(-2\)	\(-\)	\(-\)	\(-\)	\(q+q^{3}-2q^{7}+q^{9}-2q^{11}+6q^{13}+\cdots\)
1200.2.a.n	\(1\)	\(9.582\)	\(\Q\)	None	\(0\)	\(1\)	\(0\)	\(-1\)	\(-\)	\(-\)	\(+\)	\(q+q^{3}-q^{7}+q^{9}-6q^{11}-5q^{13}+\cdots\)
1200.2.a.o	\(1\)	\(9.582\)	\(\Q\)	None	\(0\)	\(1\)	\(0\)	\(0\)	\(+\)	\(-\)	\(+\)	\(q+q^{3}+q^{9}+4q^{11}-6q^{13}+6q^{17}+\cdots\)
1200.2.a.p	\(1\)	\(9.582\)	\(\Q\)	None	\(0\)	\(1\)	\(0\)	\(3\)	\(-\)	\(-\)	\(-\)	\(q+q^{3}+3q^{7}+q^{9}-2q^{11}+q^{13}+\cdots\)
1200.2.a.q	\(1\)	\(9.582\)	\(\Q\)	None	\(0\)	\(1\)	\(0\)	\(3\)	\(+\)	\(-\)	\(+\)	\(q+q^{3}+3q^{7}+q^{9}-2q^{11}+3q^{13}+\cdots\)
1200.2.a.r	\(1\)	\(9.582\)	\(\Q\)	None	\(0\)	\(1\)	\(0\)	\(4\)	\(+\)	\(-\)	\(+\)	\(q+q^{3}+4q^{7}+q^{9}+6q^{13}+2q^{17}+\cdots\)
1200.2.a.s	\(1\)	\(9.582\)	\(\Q\)	None	\(0\)	\(1\)	\(0\)	\(4\)	\(-\)	\(-\)	\(-\)	\(q+q^{3}+4q^{7}+q^{9}+4q^{11}-4q^{17}+\cdots\)

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1200))\) into newform subspaces

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1200))\) 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	1200.2.a

Level	1200
Weight	2
Character orbit	a
Rep. character	\(\chi_{1200}(1,\cdot)\)
Character field	\(\Q\)
Dimension	19
Newform subspaces	19
Sturm bound	480
Trace bound	13

$p$	$F_p(T)$
$2$	(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$3$	(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))
$5$	(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$7$	(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 3 T + 7 T^{2} \))(\( 1 + 3 T + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 - 3 T + 7 T^{2} \))(\( 1 - 3 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))
$11$	(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 6 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 - 6 T + 11 T^{2} \))(\( 1 - 6 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + 6 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))
$13$	(\( 1 + 13 T^{2} \))(\( 1 + 3 T + 13 T^{2} \))(\( 1 + T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - 5 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 3 T + 13 T^{2} \))(\( 1 - 3 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 + 5 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 - 3 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))
$17$	(\( 1 - 4 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))
$19$	(\( 1 + 19 T^{2} \))(\( 1 - 7 T + 19 T^{2} \))(\( 1 - 5 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 5 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 5 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 - 5 T + 19 T^{2} \))(\( 1 - 7 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))
$23$	(\( 1 + 4 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 - 2 T + 23 T^{2} \))(\( 1 + 2 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))
$29$	(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 - 10 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 - 8 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 8 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 - 10 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))
$31$	(\( 1 + 4 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 - 3 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 - T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 3 T + 31 T^{2} \))(\( 1 + 3 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 - T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 - 3 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))
$37$	(\( 1 + 8 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 10 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 10 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 - 8 T + 37 T^{2} \))
$41$	(\( 1 + 10 T + 41 T^{2} \))(\( 1 - 12 T + 41 T^{2} \))(\( 1 + 8 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 - 4 T + 41 T^{2} \))(\( 1 - 4 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 8 T + 41 T^{2} \))(\( 1 - 12 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))
$43$	(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 3 T + 43 T^{2} \))(\( 1 - T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 12 T + 43 T^{2} \))(\( 1 + 11 T + 43 T^{2} \))(\( 1 - 11 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 12 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 - 12 T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 - 3 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))
$47$	(\( 1 - 4 T + 47 T^{2} \))(\( 1 - 10 T + 47 T^{2} \))(\( 1 - 2 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 10 T + 47 T^{2} \))(\( 1 + 10 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 2 T + 47 T^{2} \))(\( 1 + 10 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))
$53$	(\( 1 + 12 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 - 4 T + 53 T^{2} \))(\( 1 - 2 T + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))(\( 1 + 12 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))(\( 1 + 8 T + 53 T^{2} \))(\( 1 - 8 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 10 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 4 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 + 10 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))
$59$	(\( 1 + 4 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 - 10 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 + 10 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 + 10 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 - 10 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))
$61$	(\( 1 - 2 T + 61 T^{2} \))(\( 1 + 13 T + 61 T^{2} \))(\( 1 - 7 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 + 13 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( 1 - 3 T + 61 T^{2} \))(\( 1 - 3 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( 1 + 13 T + 61 T^{2} \))(\( 1 - 14 T + 61 T^{2} \))(\( 1 - 7 T + 61 T^{2} \))(\( 1 + 13 T + 61 T^{2} \))(\( 1 - 6 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))
$67$	(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 7 T + 67 T^{2} \))(\( 1 + 3 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 + 11 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 - T + 67 T^{2} \))(\( 1 + T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 - 11 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 - 3 T + 67 T^{2} \))(\( 1 - 7 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))
$71$	(\( 1 + 71 T^{2} \))(\( 1 - 4 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 - 4 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 4 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 - 4 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))
$73$	(\( 1 + 8 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 - 14 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 - 4 T + 73 T^{2} \))(\( 1 - 4 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 + 4 T + 73 T^{2} \))(\( 1 + 4 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 - 14 T + 73 T^{2} \))(\( 1 - 8 T + 73 T^{2} \))
$79$	(\( 1 - 12 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( 1 - 12 T + 79 T^{2} \))
$83$	(\( 1 + 4 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))
$89$	(\( 1 + 10 T + 89 T^{2} \))(\( 1 - 16 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 16 T + 89 T^{2} \))(\( 1 + 16 T + 89 T^{2} \))(\( 1 - 18 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 - 16 T + 89 T^{2} \))(\( 1 - 2 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))
$97$	(\( 1 - 8 T + 97 T^{2} \))(\( 1 + 7 T + 97 T^{2} \))(\( 1 + 17 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 7 T + 97 T^{2} \))(\( 1 - 8 T + 97 T^{2} \))(\( 1 - 8 T + 97 T^{2} \))(\( 1 + 7 T + 97 T^{2} \))(\( 1 - 7 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 8 T + 97 T^{2} \))(\( 1 + 8 T + 97 T^{2} \))(\( 1 - 7 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 17 T + 97 T^{2} \))(\( 1 - 7 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 8 T + 97 T^{2} \))
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