LMFDB - Space of Modular Forms of Level 378, Weight 2, and Trivial Character

Defining parameters

Level:	$ N $	=	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	=	$ 2 $
Character orbit:	$[\chi]$	=	378.a (trivial)
Character field:			$\Q$
Newform subspaces:			$ 8 $
Sturm bound:			$144$
Trace bound:			$5$
Distinguishing $T_p$:			$5$, $17$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(378))$.

	Total	New	Old
Modular forms	84	8	76
Cusp forms	61	8	53
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.

$2$	$3$	$7$	Fricke	Dim.
$+$	$+$	$+$	$+$	$1$
$+$	$+$	$-$	$-$	$1$
$+$	$-$	$+$	$-$	$1$
$+$	$-$	$-$	$+$	$1$
$-$	$+$	$+$	$-$	$2$
$-$	$-$	$-$	$-$	$2$
Plus space			$+$	$2$
Minus space			$-$	$6$

Trace form

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(378))$ 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	7	$q$-expansion
378.2.a.a	$1$	$3.018$	$\Q$	None	$-1$	$0$	$-4$	$-1$	$+$	$-$	$+$	$q-q^{2}+q^{4}-4q^{5}-q^{7}-q^{8}+4q^{10}+\cdots$
378.2.a.b	$1$	$3.018$	$\Q$	None	$-1$	$0$	$-3$	$1$	$+$	$-$	$-$	$q-q^{2}+q^{4}-3q^{5}+q^{7}-q^{8}+3q^{10}+\cdots$
378.2.a.c	$1$	$3.018$	$\Q$	None	$-1$	$0$	$-1$	$-1$	$+$	$+$	$+$	$q-q^{2}+q^{4}-q^{5}-q^{7}-q^{8}+q^{10}+\cdots$
378.2.a.d	$1$	$3.018$	$\Q$	None	$-1$	$0$	$0$	$1$	$+$	$+$	$-$	$q-q^{2}+q^{4}+q^{7}-q^{8}+5q^{13}-q^{14}+\cdots$
378.2.a.e	$1$	$3.018$	$\Q$	None	$1$	$0$	$0$	$1$	$-$	$-$	$-$	$q+q^{2}+q^{4}+q^{7}+q^{8}+5q^{13}+q^{14}+\cdots$
378.2.a.f	$1$	$3.018$	$\Q$	None	$1$	$0$	$1$	$-1$	$-$	$+$	$+$	$q+q^{2}+q^{4}+q^{5}-q^{7}+q^{8}+q^{10}+\cdots$
378.2.a.g	$1$	$3.018$	$\Q$	None	$1$	$0$	$3$	$1$	$-$	$-$	$-$	$q+q^{2}+q^{4}+3q^{5}+q^{7}+q^{8}+3q^{10}+\cdots$
378.2.a.h	$1$	$3.018$	$\Q$	None	$1$	$0$	$4$	$-1$	$-$	$+$	$+$	$q+q^{2}+q^{4}+4q^{5}-q^{7}+q^{8}+4q^{10}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(378))$ into lower level spaces

$ S_{2}^{\mathrm{old}}(\Gamma_0(378)) \cong $ $S_{2}^{\mathrm{new}}(\Gamma_0(14))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(21))$$^{\oplus 6}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(27))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(42))$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(54))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(63))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(126))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(189))$$^{\oplus 2}$

Hecke Characteristic Polynomials

$p$	$F_p(T)$
$2$	($ 1 + T $)($ 1 + T $)($ 1 + T $)($ 1 + T $)($ 1 - T $)($ 1 - T $)($ 1 - T $)($ 1 - T $)
$3$	1
$5$	($ 1 + 4 T + 5 T^{2} $)($ 1 + 3 T + 5 T^{2} $)($ 1 + T + 5 T^{2} $)($ 1 + 5 T^{2} $)($ 1 + 5 T^{2} $)($ 1 - T + 5 T^{2} $)($ 1 - 3 T + 5 T^{2} $)($ 1 - 4 T + 5 T^{2} $)
$7$	($ 1 + T $)($ 1 - T $)($ 1 + T $)($ 1 - T $)($ 1 - T $)($ 1 + T $)($ 1 - T $)($ 1 + T $)
$11$	($ 1 - 4 T + 11 T^{2} $)($ 1 - 3 T + 11 T^{2} $)($ 1 + 5 T + 11 T^{2} $)($ 1 + 11 T^{2} $)($ 1 + 11 T^{2} $)($ 1 - 5 T + 11 T^{2} $)($ 1 + 3 T + 11 T^{2} $)($ 1 + 4 T + 11 T^{2} $)
$13$	($ 1 - 3 T + 13 T^{2} $)($ 1 + 4 T + 13 T^{2} $)($ 1 + 13 T^{2} $)($ 1 - 5 T + 13 T^{2} $)($ 1 - 5 T + 13 T^{2} $)($ 1 + 13 T^{2} $)($ 1 + 4 T + 13 T^{2} $)($ 1 - 3 T + 13 T^{2} $)
$17$	($ 1 - 7 T + 17 T^{2} $)($ 1 + 6 T + 17 T^{2} $)($ 1 + 2 T + 17 T^{2} $)($ 1 + 3 T + 17 T^{2} $)($ 1 - 3 T + 17 T^{2} $)($ 1 - 2 T + 17 T^{2} $)($ 1 - 6 T + 17 T^{2} $)($ 1 + 7 T + 17 T^{2} $)
$19$	($ 1 - 2 T + 19 T^{2} $)($ 1 + 7 T + 19 T^{2} $)($ 1 + T + 19 T^{2} $)($ 1 - 2 T + 19 T^{2} $)($ 1 - 2 T + 19 T^{2} $)($ 1 + T + 19 T^{2} $)($ 1 + 7 T + 19 T^{2} $)($ 1 - 2 T + 19 T^{2} $)
$23$	($ 1 - T + 23 T^{2} $)($ 1 + 3 T + 23 T^{2} $)($ 1 - T + 23 T^{2} $)($ 1 - 9 T + 23 T^{2} $)($ 1 + 9 T + 23 T^{2} $)($ 1 + T + 23 T^{2} $)($ 1 - 3 T + 23 T^{2} $)($ 1 + T + 23 T^{2} $)
$29$	($ 1 + T + 29 T^{2} $)($ 1 + 29 T^{2} $)($ 1 + 4 T + 29 T^{2} $)($ 1 - 3 T + 29 T^{2} $)($ 1 + 3 T + 29 T^{2} $)($ 1 - 4 T + 29 T^{2} $)($ 1 + 29 T^{2} $)($ 1 - T + 29 T^{2} $)
$31$	($ 1 + 9 T + 31 T^{2} $)($ 1 - 5 T + 31 T^{2} $)($ 1 + 9 T + 31 T^{2} $)($ 1 - 5 T + 31 T^{2} $)($ 1 - 5 T + 31 T^{2} $)($ 1 + 9 T + 31 T^{2} $)($ 1 - 5 T + 31 T^{2} $)($ 1 + 9 T + 31 T^{2} $)
$37$	($ 1 - 2 T + 37 T^{2} $)($ 1 + 7 T + 37 T^{2} $)($ 1 - 5 T + 37 T^{2} $)($ 1 - 2 T + 37 T^{2} $)($ 1 - 2 T + 37 T^{2} $)($ 1 - 5 T + 37 T^{2} $)($ 1 + 7 T + 37 T^{2} $)($ 1 - 2 T + 37 T^{2} $)
$41$	($ 1 + 6 T + 41 T^{2} $)($ 1 + 9 T + 41 T^{2} $)($ 1 - 9 T + 41 T^{2} $)($ 1 - 6 T + 41 T^{2} $)($ 1 + 6 T + 41 T^{2} $)($ 1 + 9 T + 41 T^{2} $)($ 1 - 9 T + 41 T^{2} $)($ 1 - 6 T + 41 T^{2} $)
$43$	($ 1 - 11 T + 43 T^{2} $)($ 1 + 10 T + 43 T^{2} $)($ 1 + 10 T + 43 T^{2} $)($ 1 + T + 43 T^{2} $)($ 1 + T + 43 T^{2} $)($ 1 + 10 T + 43 T^{2} $)($ 1 + 10 T + 43 T^{2} $)($ 1 - 11 T + 43 T^{2} $)
$47$	($ 1 - 6 T + 47 T^{2} $)($ 1 - 6 T + 47 T^{2} $)($ 1 + 6 T + 47 T^{2} $)($ 1 - 6 T + 47 T^{2} $)($ 1 + 6 T + 47 T^{2} $)($ 1 - 6 T + 47 T^{2} $)($ 1 + 6 T + 47 T^{2} $)($ 1 + 6 T + 47 T^{2} $)
$53$	($ 1 - 9 T + 53 T^{2} $)($ 1 - 12 T + 53 T^{2} $)($ 1 + 12 T + 53 T^{2} $)($ 1 + 3 T + 53 T^{2} $)($ 1 - 3 T + 53 T^{2} $)($ 1 - 12 T + 53 T^{2} $)($ 1 + 12 T + 53 T^{2} $)($ 1 + 9 T + 53 T^{2} $)
$59$	($ 1 - 5 T + 59 T^{2} $)($ 1 + 6 T + 59 T^{2} $)($ 1 - 14 T + 59 T^{2} $)($ 1 - 3 T + 59 T^{2} $)($ 1 + 3 T + 59 T^{2} $)($ 1 + 14 T + 59 T^{2} $)($ 1 - 6 T + 59 T^{2} $)($ 1 + 5 T + 59 T^{2} $)
$61$	($ 1 + 6 T + 61 T^{2} $)($ 1 - 8 T + 61 T^{2} $)($ 1 + 61 T^{2} $)($ 1 + 10 T + 61 T^{2} $)($ 1 + 10 T + 61 T^{2} $)($ 1 + 61 T^{2} $)($ 1 - 8 T + 61 T^{2} $)($ 1 + 6 T + 61 T^{2} $)
$67$	($ 1 - 7 T + 67 T^{2} $)($ 1 + 4 T + 67 T^{2} $)($ 1 + 8 T + 67 T^{2} $)($ 1 + 13 T + 67 T^{2} $)($ 1 + 13 T + 67 T^{2} $)($ 1 + 8 T + 67 T^{2} $)($ 1 + 4 T + 67 T^{2} $)($ 1 - 7 T + 67 T^{2} $)
$71$	($ 1 - 7 T + 71 T^{2} $)($ 1 - 9 T + 71 T^{2} $)($ 1 - 13 T + 71 T^{2} $)($ 1 + 9 T + 71 T^{2} $)($ 1 - 9 T + 71 T^{2} $)($ 1 + 13 T + 71 T^{2} $)($ 1 + 9 T + 71 T^{2} $)($ 1 + 7 T + 71 T^{2} $)
$73$	($ 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 - 2 T + 73 T^{2} $)($ 1 + 2 T + 73 T^{2} $)($ 1 - 2 T + 73 T^{2} $)($ 1 + 14 T + 73 T^{2} $)
$79$	($ 1 + 6 T + 79 T^{2} $)($ 1 + 10 T + 79 T^{2} $)($ 1 - 6 T + 79 T^{2} $)($ 1 + 10 T + 79 T^{2} $)($ 1 + 10 T + 79 T^{2} $)($ 1 - 6 T + 79 T^{2} $)($ 1 + 10 T + 79 T^{2} $)($ 1 + 6 T + 79 T^{2} $)
$83$	($ 1 - 4 T + 83 T^{2} $)($ 1 + 83 T^{2} $)($ 1 - 4 T + 83 T^{2} $)($ 1 - 12 T + 83 T^{2} $)($ 1 + 12 T + 83 T^{2} $)($ 1 + 4 T + 83 T^{2} $)($ 1 + 83 T^{2} $)($ 1 + 4 T + 83 T^{2} $)
$89$	($ 1 - 3 T + 89 T^{2} $)($ 1 - 15 T + 89 T^{2} $)($ 1 - 9 T + 89 T^{2} $)($ 1 + 15 T + 89 T^{2} $)($ 1 - 15 T + 89 T^{2} $)($ 1 + 9 T + 89 T^{2} $)($ 1 + 15 T + 89 T^{2} $)($ 1 + 3 T + 89 T^{2} $)
$97$	($ 1 + 8 T + 97 T^{2} $)($ 1 - 8 T + 97 T^{2} $)($ 1 - 16 T + 97 T^{2} $)($ 1 - 8 T + 97 T^{2} $)($ 1 - 8 T + 97 T^{2} $)($ 1 - 16 T + 97 T^{2} $)($ 1 - 8 T + 97 T^{2} $)($ 1 + 8 T + 97 T^{2} $)
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Introduction	Features
Universe	News

Level:	\( N \)	=	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	=	\( 2 \)
Character orbit:	\([\chi]\)	=	378.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(144\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\), \(17\)

\(2\)	\(3\)	\(7\)	Fricke	Dim.
\(+\)	\(+\)	\(+\)	\(+\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)	\(2\)
Plus space			\(+\)	\(2\)
Minus space			\(-\)	\(6\)

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	7	$q$-expansion
378.2.a.a	\(1\)	\(3.018\)	\(\Q\)	None	\(-1\)	\(0\)	\(-4\)	\(-1\)	\(+\)	\(-\)	\(+\)	\(q-q^{2}+q^{4}-4q^{5}-q^{7}-q^{8}+4q^{10}+\cdots\)
378.2.a.b	\(1\)	\(3.018\)	\(\Q\)	None	\(-1\)	\(0\)	\(-3\)	\(1\)	\(+\)	\(-\)	\(-\)	\(q-q^{2}+q^{4}-3q^{5}+q^{7}-q^{8}+3q^{10}+\cdots\)
378.2.a.c	\(1\)	\(3.018\)	\(\Q\)	None	\(-1\)	\(0\)	\(-1\)	\(-1\)	\(+\)	\(+\)	\(+\)	\(q-q^{2}+q^{4}-q^{5}-q^{7}-q^{8}+q^{10}+\cdots\)
378.2.a.d	\(1\)	\(3.018\)	\(\Q\)	None	\(-1\)	\(0\)	\(0\)	\(1\)	\(+\)	\(+\)	\(-\)	\(q-q^{2}+q^{4}+q^{7}-q^{8}+5q^{13}-q^{14}+\cdots\)
378.2.a.e	\(1\)	\(3.018\)	\(\Q\)	None	\(1\)	\(0\)	\(0\)	\(1\)	\(-\)	\(-\)	\(-\)	\(q+q^{2}+q^{4}+q^{7}+q^{8}+5q^{13}+q^{14}+\cdots\)
378.2.a.f	\(1\)	\(3.018\)	\(\Q\)	None	\(1\)	\(0\)	\(1\)	\(-1\)	\(-\)	\(+\)	\(+\)	\(q+q^{2}+q^{4}+q^{5}-q^{7}+q^{8}+q^{10}+\cdots\)
378.2.a.g	\(1\)	\(3.018\)	\(\Q\)	None	\(1\)	\(0\)	\(3\)	\(1\)	\(-\)	\(-\)	\(-\)	\(q+q^{2}+q^{4}+3q^{5}+q^{7}+q^{8}+3q^{10}+\cdots\)
378.2.a.h	\(1\)	\(3.018\)	\(\Q\)	None	\(1\)	\(0\)	\(4\)	\(-1\)	\(-\)	\(+\)	\(+\)	\(q+q^{2}+q^{4}+4q^{5}-q^{7}+q^{8}+4q^{10}+\cdots\)

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(378))\) into newform subspaces

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(378))\) 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	378.2.a

Level	378
Weight	2
Character orbit	a
Rep. character	\(\chi_{378}(1,\cdot)\)
Character field	\(\Q\)
Dimension	8
Newform subspaces	8
Sturm bound	144
Trace bound	5

$p$	$F_p(T)$
$2$	(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))
$3$	1
$5$	(\( 1 + 4 T + 5 T^{2} \))(\( 1 + 3 T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - T + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))
$7$	(\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + T \))
$11$	(\( 1 - 4 T + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 + 5 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 5 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))
$13$	(\( 1 - 3 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 - 5 T + 13 T^{2} \))(\( 1 - 5 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 3 T + 13 T^{2} \))
$17$	(\( 1 - 7 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 7 T + 17 T^{2} \))
$19$	(\( 1 - 2 T + 19 T^{2} \))(\( 1 + 7 T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 + 7 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))
$23$	(\( 1 - T + 23 T^{2} \))(\( 1 + 3 T + 23 T^{2} \))(\( 1 - T + 23 T^{2} \))(\( 1 - 9 T + 23 T^{2} \))(\( 1 + 9 T + 23 T^{2} \))(\( 1 + T + 23 T^{2} \))(\( 1 - 3 T + 23 T^{2} \))(\( 1 + T + 23 T^{2} \))
$29$	(\( 1 + T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 4 T + 29 T^{2} \))(\( 1 - 3 T + 29 T^{2} \))(\( 1 + 3 T + 29 T^{2} \))(\( 1 - 4 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 - T + 29 T^{2} \))
$31$	(\( 1 + 9 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 + 9 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 + 9 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 + 9 T + 31 T^{2} \))
$37$	(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 7 T + 37 T^{2} \))(\( 1 - 5 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 - 5 T + 37 T^{2} \))(\( 1 + 7 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))
$41$	(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 9 T + 41 T^{2} \))(\( 1 - 9 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 9 T + 41 T^{2} \))(\( 1 - 9 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))
$43$	(\( 1 - 11 T + 43 T^{2} \))(\( 1 + 10 T + 43 T^{2} \))(\( 1 + 10 T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 + 10 T + 43 T^{2} \))(\( 1 + 10 T + 43 T^{2} \))(\( 1 - 11 T + 43 T^{2} \))
$47$	(\( 1 - 6 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))
$53$	(\( 1 - 9 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 + 12 T + 53 T^{2} \))(\( 1 + 3 T + 53 T^{2} \))(\( 1 - 3 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 + 12 T + 53 T^{2} \))(\( 1 + 9 T + 53 T^{2} \))
$59$	(\( 1 - 5 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 - 14 T + 59 T^{2} \))(\( 1 - 3 T + 59 T^{2} \))(\( 1 + 3 T + 59 T^{2} \))(\( 1 + 14 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 + 5 T + 59 T^{2} \))
$61$	(\( 1 + 6 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))
$67$	(\( 1 - 7 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 + 13 T + 67 T^{2} \))(\( 1 + 13 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 7 T + 67 T^{2} \))
$71$	(\( 1 - 7 T + 71 T^{2} \))(\( 1 - 9 T + 71 T^{2} \))(\( 1 - 13 T + 71 T^{2} \))(\( 1 + 9 T + 71 T^{2} \))(\( 1 - 9 T + 71 T^{2} \))(\( 1 + 13 T + 71 T^{2} \))(\( 1 + 9 T + 71 T^{2} \))(\( 1 + 7 T + 71 T^{2} \))
$73$	(\( 1 + 14 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))
$79$	(\( 1 + 6 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 - 6 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 - 6 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 + 6 T + 79 T^{2} \))
$83$	(\( 1 - 4 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))
$89$	(\( 1 - 3 T + 89 T^{2} \))(\( 1 - 15 T + 89 T^{2} \))(\( 1 - 9 T + 89 T^{2} \))(\( 1 + 15 T + 89 T^{2} \))(\( 1 - 15 T + 89 T^{2} \))(\( 1 + 9 T + 89 T^{2} \))(\( 1 + 15 T + 89 T^{2} \))(\( 1 + 3 T + 89 T^{2} \))
$97$	(\( 1 + 8 T + 97 T^{2} \))(\( 1 - 8 T + 97 T^{2} \))(\( 1 - 16 T + 97 T^{2} \))(\( 1 - 8 T + 97 T^{2} \))(\( 1 - 8 T + 97 T^{2} \))(\( 1 - 16 T + 97 T^{2} \))(\( 1 - 8 T + 97 T^{2} \))(\( 1 + 8 T + 97 T^{2} \))
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