LMFDB - Space of modular forms of level 338, weight 2, and Trivial character

Defining parameters

Level:	$ N $	$=$	$ 338 = 2 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	338.a (trivial)
Character field:			$\Q$
Newform subspaces:			$ 8 $
Sturm bound:			$91$
Trace bound:			$3$
Distinguishing $T_p$:			$3$, $5$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(338))$.

	Total	New	Old
Modular forms	59	12	47
Cusp forms	32	12	20
Eisenstein series	27	0	27

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

$2$	$13$	Fricke	Dim.
$+$	$+$	$+$	$2$
$+$	$-$	$-$	$4$
$-$	$+$	$-$	$5$
$-$	$-$	$+$	$1$
Plus space		$+$	$3$
Minus space		$-$	$9$

Trace form

$ 12q + 2q^{3} + 12q^{4} + 4q^{5} + 4q^{6} + 10q^{9} + O(q^{10}) $

$ 12q + 2q^{3} + 12q^{4} + 4q^{5} + 4q^{6} + 10q^{9} - 2q^{10} - 4q^{11} + 2q^{12} - 4q^{14} + 12q^{16} + 4q^{17} - 8q^{18} - 8q^{19} + 4q^{20} + 4q^{21} + 6q^{22} + 4q^{24} + 10q^{25} + 8q^{27} - 14q^{29} - 12q^{30} - 12q^{33} - 12q^{35} + 10q^{36} + 4q^{37} - 6q^{38} - 2q^{40} - 4q^{42} + 2q^{43} - 4q^{44} + 4q^{46} - 16q^{47} + 2q^{48} + 16q^{49} + 8q^{50} - 20q^{51} - 18q^{53} + 4q^{54} + 8q^{55} - 4q^{56} + 16q^{57} + 4q^{58} + 16q^{59} - 10q^{61} - 16q^{62} - 8q^{63} + 12q^{64} - 8q^{66} - 12q^{67} + 4q^{68} - 28q^{69} + 4q^{70} + 8q^{71} - 8q^{72} + 8q^{73} - 18q^{74} - 38q^{75} - 8q^{76} - 12q^{77} - 20q^{79} + 4q^{80} - 20q^{81} - 4q^{82} - 12q^{83} + 4q^{84} - 12q^{85} + 4q^{86} - 20q^{87} + 6q^{88} - 2q^{90} + 16q^{93} - 16q^{94} - 12q^{95} + 4q^{96} - 4q^{97} + 24q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(338))$ 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	13	$q$-expansion
338.2.a.a	$1$	$2.699$	$\Q$	None	$-1$	$-3$	$1$	$-1$	$+$	$+$	$q-q^{2}-3q^{3}+q^{4}+q^{5}+3q^{6}-q^{7}+\cdots$
338.2.a.b	$1$	$2.699$	$\Q$	None	$-1$	$-1$	$3$	$3$	$+$	$-$	$q-q^{2}-q^{3}+q^{4}+3q^{5}+q^{6}+3q^{7}+\cdots$
338.2.a.c	$1$	$2.699$	$\Q$	None	$-1$	$0$	$1$	$-4$	$+$	$+$	$q-q^{2}+q^{4}+q^{5}-4q^{7}-q^{8}-3q^{9}+\cdots$
338.2.a.d	$1$	$2.699$	$\Q$	None	$1$	$-1$	$-3$	$-3$	$-$	$-$	$q+q^{2}-q^{3}+q^{4}-3q^{5}-q^{6}-3q^{7}+\cdots$
338.2.a.e	$1$	$2.699$	$\Q$	None	$1$	$0$	$-1$	$4$	$-$	$+$	$q+q^{2}+q^{4}-q^{5}+4q^{7}+q^{8}-3q^{9}+\cdots$
338.2.a.f	$1$	$2.699$	$\Q$	None	$1$	$1$	$3$	$1$	$-$	$+$	$q+q^{2}+q^{3}+q^{4}+3q^{5}+q^{6}+q^{7}+\cdots$
338.2.a.g	$3$	$2.699$	$\Q(\zeta_{14})^+$	None	$-3$	$3$	$-2$	$4$	$+$	$-$	$q-q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+q^{4}-2\beta _{1}q^{5}+\cdots$
338.2.a.h	$3$	$2.699$	$\Q(\zeta_{14})^+$	None	$3$	$3$	$2$	$-4$	$-$	$+$	$q+q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+q^{4}+2\beta _{1}q^{5}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(338))$ into lower level spaces

$ S_{2}^{\mathrm{old}}(\Gamma_0(338)) \cong $ $S_{2}^{\mathrm{new}}(\Gamma_0(26))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(169))$$^{\oplus 2}$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	($ 1 + T $)($ 1 + T $)($ 1 + T $)($ 1 - T $)($ 1 - T $)($ 1 - T $)($ ( 1 + T )^{3} $)($ ( 1 - T )^{3} $)
$3$	($ 1 + 3 T + 3 T^{2} $)($ 1 + T + 3 T^{2} $)($ 1 + 3 T^{2} $)($ 1 + T + 3 T^{2} $)($ 1 + 3 T^{2} $)($ 1 - T + 3 T^{2} $)($ 1 - 3 T + 5 T^{2} - 5 T^{3} + 15 T^{4} - 27 T^{5} + 27 T^{6} $)($ 1 - 3 T + 5 T^{2} - 5 T^{3} + 15 T^{4} - 27 T^{5} + 27 T^{6} $)
$5$	($ 1 - T + 5 T^{2} $)($ 1 - 3 T + 5 T^{2} $)($ 1 - T + 5 T^{2} $)($ 1 + 3 T + 5 T^{2} $)($ 1 + T + 5 T^{2} $)($ 1 - 3 T + 5 T^{2} $)($ 1 + 2 T + 7 T^{2} + 12 T^{3} + 35 T^{4} + 50 T^{5} + 125 T^{6} $)($ 1 - 2 T + 7 T^{2} - 12 T^{3} + 35 T^{4} - 50 T^{5} + 125 T^{6} $)
$7$	($ 1 + T + 7 T^{2} $)($ 1 - 3 T + 7 T^{2} $)($ 1 + 4 T + 7 T^{2} $)($ 1 + 3 T + 7 T^{2} $)($ 1 - 4 T + 7 T^{2} $)($ 1 - T + 7 T^{2} $)($ 1 - 4 T + 17 T^{2} - 48 T^{3} + 119 T^{4} - 196 T^{5} + 343 T^{6} $)($ 1 + 4 T + 17 T^{2} + 48 T^{3} + 119 T^{4} + 196 T^{5} + 343 T^{6} $)
$11$	($ 1 - 2 T + 11 T^{2} $)($ 1 + 11 T^{2} $)($ 1 + 4 T + 11 T^{2} $)($ 1 + 11 T^{2} $)($ 1 - 4 T + 11 T^{2} $)($ 1 + 6 T + 11 T^{2} $)($ 1 + 3 T + 29 T^{2} + 53 T^{3} + 319 T^{4} + 363 T^{5} + 1331 T^{6} $)($ 1 - 3 T + 29 T^{2} - 53 T^{3} + 319 T^{4} - 363 T^{5} + 1331 T^{6} $)
$13$	1
$17$	($ 1 + 3 T + 17 T^{2} $)($ 1 + 3 T + 17 T^{2} $)($ 1 - 3 T + 17 T^{2} $)($ 1 + 3 T + 17 T^{2} $)($ 1 - 3 T + 17 T^{2} $)($ 1 + 3 T + 17 T^{2} $)($ 1 - 5 T + 29 T^{2} - 73 T^{3} + 493 T^{4} - 1445 T^{5} + 4913 T^{6} $)($ 1 - 5 T + 29 T^{2} - 73 T^{3} + 493 T^{4} - 1445 T^{5} + 4913 T^{6} $)
$19$	($ 1 + 6 T + 19 T^{2} $)($ 1 - 6 T + 19 T^{2} $)($ 1 + 19 T^{2} $)($ 1 + 6 T + 19 T^{2} $)($ 1 + 19 T^{2} $)($ 1 + 2 T + 19 T^{2} $)($ 1 + T + 41 T^{2} + 51 T^{3} + 779 T^{4} + 361 T^{5} + 6859 T^{6} $)($ 1 - T + 41 T^{2} - 51 T^{3} + 779 T^{4} - 361 T^{5} + 6859 T^{6} $)
$23$	($ 1 + 4 T + 23 T^{2} $)($ 1 - 6 T + 23 T^{2} $)($ 1 + 4 T + 23 T^{2} $)($ 1 - 6 T + 23 T^{2} $)($ 1 + 4 T + 23 T^{2} $)($ 1 + 23 T^{2} $)($ 1 + 41 T^{2} + 56 T^{3} + 943 T^{4} + 12167 T^{6} $)($ 1 + 41 T^{2} + 56 T^{3} + 943 T^{4} + 12167 T^{6} $)
$29$	($ 1 - 2 T + 29 T^{2} $)($ 1 + 29 T^{2} $)($ 1 + T + 29 T^{2} $)($ 1 + 29 T^{2} $)($ 1 + T + 29 T^{2} $)($ 1 - 6 T + 29 T^{2} $)($ 1 + 10 T + 83 T^{2} + 476 T^{3} + 2407 T^{4} + 8410 T^{5} + 24389 T^{6} $)($ 1 + 10 T + 83 T^{2} + 476 T^{3} + 2407 T^{4} + 8410 T^{5} + 24389 T^{6} $)
$31$	($ 1 + 4 T + 31 T^{2} $)($ 1 + 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 - 16 T + 169 T^{2} - 1096 T^{3} + 5239 T^{4} - 15376 T^{5} + 29791 T^{6} $)($ 1 + 16 T + 169 T^{2} + 1096 T^{3} + 5239 T^{4} + 15376 T^{5} + 29791 T^{6} $)
$37$	($ 1 + 3 T + 37 T^{2} $)($ 1 - 3 T + 37 T^{2} $)($ 1 + 3 T + 37 T^{2} $)($ 1 + 3 T + 37 T^{2} $)($ 1 - 3 T + 37 T^{2} $)($ 1 - 7 T + 37 T^{2} $)($ 1 - 14 T + 167 T^{2} - 1092 T^{3} + 6179 T^{4} - 19166 T^{5} + 50653 T^{6} $)($ 1 + 14 T + 167 T^{2} + 1092 T^{3} + 6179 T^{4} + 19166 T^{5} + 50653 T^{6} $)
$41$	($ 1 + 41 T^{2} $)($ 1 + 41 T^{2} $)($ 1 - 9 T + 41 T^{2} $)($ 1 + 41 T^{2} $)($ 1 + 9 T + 41 T^{2} $)($ 1 + 41 T^{2} $)($ 1 + 7 T + 109 T^{2} + 483 T^{3} + 4469 T^{4} + 11767 T^{5} + 68921 T^{6} $)($ 1 - 7 T + 109 T^{2} - 483 T^{3} + 4469 T^{4} - 11767 T^{5} + 68921 T^{6} $)
$43$	($ 1 + 5 T + 43 T^{2} $)($ 1 - T + 43 T^{2} $)($ 1 + 8 T + 43 T^{2} $)($ 1 - T + 43 T^{2} $)($ 1 + 8 T + 43 T^{2} $)($ 1 + T + 43 T^{2} $)($ 1 - 11 T + 125 T^{2} - 945 T^{3} + 5375 T^{4} - 20339 T^{5} + 79507 T^{6} $)($ 1 - 11 T + 125 T^{2} - 945 T^{3} + 5375 T^{4} - 20339 T^{5} + 79507 T^{6} $)
$47$	($ 1 + 13 T + 47 T^{2} $)($ 1 - 3 T + 47 T^{2} $)($ 1 - 8 T + 47 T^{2} $)($ 1 + 3 T + 47 T^{2} $)($ 1 + 8 T + 47 T^{2} $)($ 1 + 3 T + 47 T^{2} $)($ 1 - 2 T + 105 T^{2} - 180 T^{3} + 4935 T^{4} - 4418 T^{5} + 103823 T^{6} $)($ 1 + 2 T + 105 T^{2} + 180 T^{3} + 4935 T^{4} + 4418 T^{5} + 103823 T^{6} $)
$53$	($ 1 - 12 T + 53 T^{2} $)($ 1 + 6 T + 53 T^{2} $)($ 1 + 9 T + 53 T^{2} $)($ 1 + 6 T + 53 T^{2} $)($ 1 + 9 T + 53 T^{2} $)($ 1 + 53 T^{2} $)($ 1 + 131 T^{2} - 56 T^{3} + 6943 T^{4} + 148877 T^{6} $)($ 1 + 131 T^{2} - 56 T^{3} + 6943 T^{4} + 148877 T^{6} $)
$59$	($ 1 - 10 T + 59 T^{2} $)($ 1 + 6 T + 59 T^{2} $)($ 1 - 4 T + 59 T^{2} $)($ 1 - 6 T + 59 T^{2} $)($ 1 + 4 T + 59 T^{2} $)($ 1 - 6 T + 59 T^{2} $)($ 1 - 3 T + 117 T^{2} - 481 T^{3} + 6903 T^{4} - 10443 T^{5} + 205379 T^{6} $)($ 1 + 3 T + 117 T^{2} + 481 T^{3} + 6903 T^{4} + 10443 T^{5} + 205379 T^{6} $)
$61$	($ 1 + 8 T + 61 T^{2} $)($ 1 + 8 T + 61 T^{2} $)($ 1 - 7 T + 61 T^{2} $)($ 1 + 8 T + 61 T^{2} $)($ 1 - 7 T + 61 T^{2} $)($ 1 - 8 T + 61 T^{2} $)($ 1 + 4 T + 39 T^{2} + 424 T^{3} + 2379 T^{4} + 14884 T^{5} + 226981 T^{6} $)($ 1 + 4 T + 39 T^{2} + 424 T^{3} + 2379 T^{4} + 14884 T^{5} + 226981 T^{6} $)
$67$	($ 1 - 2 T + 67 T^{2} $)($ 1 - 12 T + 67 T^{2} $)($ 1 + 4 T + 67 T^{2} $)($ 1 + 12 T + 67 T^{2} $)($ 1 - 4 T + 67 T^{2} $)($ 1 + 14 T + 67 T^{2} $)($ 1 + 21 T + 285 T^{2} + 2527 T^{3} + 19095 T^{4} + 94269 T^{5} + 300763 T^{6} $)($ 1 - 21 T + 285 T^{2} - 2527 T^{3} + 19095 T^{4} - 94269 T^{5} + 300763 T^{6} $)
$71$	($ 1 - 5 T + 71 T^{2} $)($ 1 + 15 T + 71 T^{2} $)($ 1 - 8 T + 71 T^{2} $)($ 1 - 15 T + 71 T^{2} $)($ 1 + 8 T + 71 T^{2} $)($ 1 - 3 T + 71 T^{2} $)($ 1 - 12 T + 233 T^{2} - 1712 T^{3} + 16543 T^{4} - 60492 T^{5} + 357911 T^{6} $)($ 1 + 12 T + 233 T^{2} + 1712 T^{3} + 16543 T^{4} + 60492 T^{5} + 357911 T^{6} $)
$73$	($ 1 - 10 T + 73 T^{2} $)($ 1 - 6 T + 73 T^{2} $)($ 1 + 11 T + 73 T^{2} $)($ 1 + 6 T + 73 T^{2} $)($ 1 - 11 T + 73 T^{2} $)($ 1 + 2 T + 73 T^{2} $)($ 1 + T + 133 T^{2} + 397 T^{3} + 9709 T^{4} + 5329 T^{5} + 389017 T^{6} $)($ 1 - T + 133 T^{2} - 397 T^{3} + 9709 T^{4} - 5329 T^{5} + 389017 T^{6} $)
$79$	($ 1 + 4 T + 79 T^{2} $)($ 1 - 10 T + 79 T^{2} $)($ 1 + 4 T + 79 T^{2} $)($ 1 - 10 T + 79 T^{2} $)($ 1 + 4 T + 79 T^{2} $)($ 1 - 8 T + 79 T^{2} $)($ 1 + 18 T + 261 T^{2} + 2612 T^{3} + 20619 T^{4} + 112338 T^{5} + 493039 T^{6} $)($ 1 + 18 T + 261 T^{2} + 2612 T^{3} + 20619 T^{4} + 112338 T^{5} + 493039 T^{6} $)
$83$	($ 1 + 83 T^{2} $)($ 1 + 6 T + 83 T^{2} $)($ 1 + 83 T^{2} $)($ 1 - 6 T + 83 T^{2} $)($ 1 + 83 T^{2} $)($ 1 + 12 T + 83 T^{2} $)($ 1 - T + 233 T^{2} - 179 T^{3} + 19339 T^{4} - 6889 T^{5} + 571787 T^{6} $)($ 1 + T + 233 T^{2} + 179 T^{3} + 19339 T^{4} + 6889 T^{5} + 571787 T^{6} $)
$89$	($ 1 + 6 T + 89 T^{2} $)($ 1 + 6 T + 89 T^{2} $)($ 1 - 6 T + 89 T^{2} $)($ 1 - 6 T + 89 T^{2} $)($ 1 + 6 T + 89 T^{2} $)($ 1 - 6 T + 89 T^{2} $)($ 1 + 25 T + 361 T^{2} + 3693 T^{3} + 32129 T^{4} + 198025 T^{5} + 704969 T^{6} $)($ 1 - 25 T + 361 T^{2} - 3693 T^{3} + 32129 T^{4} - 198025 T^{5} + 704969 T^{6} $)
$97$	($ 1 + 14 T + 97 T^{2} $)($ 1 - 12 T + 97 T^{2} $)($ 1 + 2 T + 97 T^{2} $)($ 1 + 12 T + 97 T^{2} $)($ 1 - 2 T + 97 T^{2} $)($ 1 - 10 T + 97 T^{2} $)($ 1 + 23 T + 353 T^{2} + 3579 T^{3} + 34241 T^{4} + 216407 T^{5} + 912673 T^{6} $)($ 1 - 23 T + 353 T^{2} - 3579 T^{3} + 34241 T^{4} - 216407 T^{5} + 912673 T^{6} $)
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Introduction	Features
Universe	News

Level:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	338.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(91\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\), \(5\)

\(2\)	\(13\)	Fricke	Dim.
\(+\)	\(+\)	\(+\)	\(2\)
\(+\)	\(-\)	\(-\)	\(4\)
\(-\)	\(+\)	\(-\)	\(5\)
\(-\)	\(-\)	\(+\)	\(1\)
Plus space		\(+\)	\(3\)
Minus space		\(-\)	\(9\)

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	13	$q$-expansion
338.2.a.a	\(1\)	\(2.699\)	\(\Q\)	None	\(-1\)	\(-3\)	\(1\)	\(-1\)	\(+\)	\(+\)	\(q-q^{2}-3q^{3}+q^{4}+q^{5}+3q^{6}-q^{7}+\cdots\)
338.2.a.b	\(1\)	\(2.699\)	\(\Q\)	None	\(-1\)	\(-1\)	\(3\)	\(3\)	\(+\)	\(-\)	\(q-q^{2}-q^{3}+q^{4}+3q^{5}+q^{6}+3q^{7}+\cdots\)
338.2.a.c	\(1\)	\(2.699\)	\(\Q\)	None	\(-1\)	\(0\)	\(1\)	\(-4\)	\(+\)	\(+\)	\(q-q^{2}+q^{4}+q^{5}-4q^{7}-q^{8}-3q^{9}+\cdots\)
338.2.a.d	\(1\)	\(2.699\)	\(\Q\)	None	\(1\)	\(-1\)	\(-3\)	\(-3\)	\(-\)	\(-\)	\(q+q^{2}-q^{3}+q^{4}-3q^{5}-q^{6}-3q^{7}+\cdots\)
338.2.a.e	\(1\)	\(2.699\)	\(\Q\)	None	\(1\)	\(0\)	\(-1\)	\(4\)	\(-\)	\(+\)	\(q+q^{2}+q^{4}-q^{5}+4q^{7}+q^{8}-3q^{9}+\cdots\)
338.2.a.f	\(1\)	\(2.699\)	\(\Q\)	None	\(1\)	\(1\)	\(3\)	\(1\)	\(-\)	\(+\)	\(q+q^{2}+q^{3}+q^{4}+3q^{5}+q^{6}+q^{7}+\cdots\)
338.2.a.g	\(3\)	\(2.699\)	\(\Q(\zeta_{14})^+\)	None	\(-3\)	\(3\)	\(-2\)	\(4\)	\(+\)	\(-\)	\(q-q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+q^{4}-2\beta _{1}q^{5}+\cdots\)
338.2.a.h	\(3\)	\(2.699\)	\(\Q(\zeta_{14})^+\)	None	\(3\)	\(3\)	\(2\)	\(-4\)	\(-\)	\(+\)	\(q+q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+q^{4}+2\beta _{1}q^{5}+\cdots\)

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(338))\) into newform subspaces

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(338))\) 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	338.2.a

Level	338
Weight	2
Character orbit	a
Rep. character	\(\chi_{338}(1,\cdot)\)
Character field	\(\Q\)
Dimension	12
Newform subspaces	8
Sturm bound	91
Trace bound	3

$p$	$F_p(T)$
$2$	(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{3} \))
$3$	(\( 1 + 3 T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 - 3 T + 5 T^{2} - 5 T^{3} + 15 T^{4} - 27 T^{5} + 27 T^{6} \))(\( 1 - 3 T + 5 T^{2} - 5 T^{3} + 15 T^{4} - 27 T^{5} + 27 T^{6} \))
$5$	(\( 1 - T + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 - T + 5 T^{2} \))(\( 1 + 3 T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 + 2 T + 7 T^{2} + 12 T^{3} + 35 T^{4} + 50 T^{5} + 125 T^{6} \))(\( 1 - 2 T + 7 T^{2} - 12 T^{3} + 35 T^{4} - 50 T^{5} + 125 T^{6} \))
$7$	(\( 1 + T + 7 T^{2} \))(\( 1 - 3 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 3 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - 4 T + 17 T^{2} - 48 T^{3} + 119 T^{4} - 196 T^{5} + 343 T^{6} \))(\( 1 + 4 T + 17 T^{2} + 48 T^{3} + 119 T^{4} + 196 T^{5} + 343 T^{6} \))
$11$	(\( 1 - 2 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 6 T + 11 T^{2} \))(\( 1 + 3 T + 29 T^{2} + 53 T^{3} + 319 T^{4} + 363 T^{5} + 1331 T^{6} \))(\( 1 - 3 T + 29 T^{2} - 53 T^{3} + 319 T^{4} - 363 T^{5} + 1331 T^{6} \))
$13$	1
$17$	(\( 1 + 3 T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 - 5 T + 29 T^{2} - 73 T^{3} + 493 T^{4} - 1445 T^{5} + 4913 T^{6} \))(\( 1 - 5 T + 29 T^{2} - 73 T^{3} + 493 T^{4} - 1445 T^{5} + 4913 T^{6} \))
$19$	(\( 1 + 6 T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 + T + 41 T^{2} + 51 T^{3} + 779 T^{4} + 361 T^{5} + 6859 T^{6} \))(\( 1 - T + 41 T^{2} - 51 T^{3} + 779 T^{4} - 361 T^{5} + 6859 T^{6} \))
$23$	(\( 1 + 4 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 41 T^{2} + 56 T^{3} + 943 T^{4} + 12167 T^{6} \))(\( 1 + 41 T^{2} + 56 T^{3} + 943 T^{4} + 12167 T^{6} \))
$29$	(\( 1 - 2 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 10 T + 83 T^{2} + 476 T^{3} + 2407 T^{4} + 8410 T^{5} + 24389 T^{6} \))(\( 1 + 10 T + 83 T^{2} + 476 T^{3} + 2407 T^{4} + 8410 T^{5} + 24389 T^{6} \))
$31$	(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 - 16 T + 169 T^{2} - 1096 T^{3} + 5239 T^{4} - 15376 T^{5} + 29791 T^{6} \))(\( 1 + 16 T + 169 T^{2} + 1096 T^{3} + 5239 T^{4} + 15376 T^{5} + 29791 T^{6} \))
$37$	(\( 1 + 3 T + 37 T^{2} \))(\( 1 - 3 T + 37 T^{2} \))(\( 1 + 3 T + 37 T^{2} \))(\( 1 + 3 T + 37 T^{2} \))(\( 1 - 3 T + 37 T^{2} \))(\( 1 - 7 T + 37 T^{2} \))(\( 1 - 14 T + 167 T^{2} - 1092 T^{3} + 6179 T^{4} - 19166 T^{5} + 50653 T^{6} \))(\( 1 + 14 T + 167 T^{2} + 1092 T^{3} + 6179 T^{4} + 19166 T^{5} + 50653 T^{6} \))
$41$	(\( 1 + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 - 9 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 9 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 7 T + 109 T^{2} + 483 T^{3} + 4469 T^{4} + 11767 T^{5} + 68921 T^{6} \))(\( 1 - 7 T + 109 T^{2} - 483 T^{3} + 4469 T^{4} - 11767 T^{5} + 68921 T^{6} \))
$43$	(\( 1 + 5 T + 43 T^{2} \))(\( 1 - T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 - T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 - 11 T + 125 T^{2} - 945 T^{3} + 5375 T^{4} - 20339 T^{5} + 79507 T^{6} \))(\( 1 - 11 T + 125 T^{2} - 945 T^{3} + 5375 T^{4} - 20339 T^{5} + 79507 T^{6} \))
$47$	(\( 1 + 13 T + 47 T^{2} \))(\( 1 - 3 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 3 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 + 3 T + 47 T^{2} \))(\( 1 - 2 T + 105 T^{2} - 180 T^{3} + 4935 T^{4} - 4418 T^{5} + 103823 T^{6} \))(\( 1 + 2 T + 105 T^{2} + 180 T^{3} + 4935 T^{4} + 4418 T^{5} + 103823 T^{6} \))
$53$	(\( 1 - 12 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 9 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 9 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 + 131 T^{2} - 56 T^{3} + 6943 T^{4} + 148877 T^{6} \))(\( 1 + 131 T^{2} - 56 T^{3} + 6943 T^{4} + 148877 T^{6} \))
$59$	(\( 1 - 10 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 - 3 T + 117 T^{2} - 481 T^{3} + 6903 T^{4} - 10443 T^{5} + 205379 T^{6} \))(\( 1 + 3 T + 117 T^{2} + 481 T^{3} + 6903 T^{4} + 10443 T^{5} + 205379 T^{6} \))
$61$	(\( 1 + 8 T + 61 T^{2} \))(\( 1 + 8 T + 61 T^{2} \))(\( 1 - 7 T + 61 T^{2} \))(\( 1 + 8 T + 61 T^{2} \))(\( 1 - 7 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 4 T + 39 T^{2} + 424 T^{3} + 2379 T^{4} + 14884 T^{5} + 226981 T^{6} \))(\( 1 + 4 T + 39 T^{2} + 424 T^{3} + 2379 T^{4} + 14884 T^{5} + 226981 T^{6} \))
$67$	(\( 1 - 2 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 14 T + 67 T^{2} \))(\( 1 + 21 T + 285 T^{2} + 2527 T^{3} + 19095 T^{4} + 94269 T^{5} + 300763 T^{6} \))(\( 1 - 21 T + 285 T^{2} - 2527 T^{3} + 19095 T^{4} - 94269 T^{5} + 300763 T^{6} \))
$71$	(\( 1 - 5 T + 71 T^{2} \))(\( 1 + 15 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 - 15 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 - 3 T + 71 T^{2} \))(\( 1 - 12 T + 233 T^{2} - 1712 T^{3} + 16543 T^{4} - 60492 T^{5} + 357911 T^{6} \))(\( 1 + 12 T + 233 T^{2} + 1712 T^{3} + 16543 T^{4} + 60492 T^{5} + 357911 T^{6} \))
$73$	(\( 1 - 10 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 + 11 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 - 11 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 + T + 133 T^{2} + 397 T^{3} + 9709 T^{4} + 5329 T^{5} + 389017 T^{6} \))(\( 1 - T + 133 T^{2} - 397 T^{3} + 9709 T^{4} - 5329 T^{5} + 389017 T^{6} \))
$79$	(\( 1 + 4 T + 79 T^{2} \))(\( 1 - 10 T + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))(\( 1 - 10 T + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 18 T + 261 T^{2} + 2612 T^{3} + 20619 T^{4} + 112338 T^{5} + 493039 T^{6} \))(\( 1 + 18 T + 261 T^{2} + 2612 T^{3} + 20619 T^{4} + 112338 T^{5} + 493039 T^{6} \))
$83$	(\( 1 + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - T + 233 T^{2} - 179 T^{3} + 19339 T^{4} - 6889 T^{5} + 571787 T^{6} \))(\( 1 + T + 233 T^{2} + 179 T^{3} + 19339 T^{4} + 6889 T^{5} + 571787 T^{6} \))
$89$	(\( 1 + 6 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 25 T + 361 T^{2} + 3693 T^{3} + 32129 T^{4} + 198025 T^{5} + 704969 T^{6} \))(\( 1 - 25 T + 361 T^{2} - 3693 T^{3} + 32129 T^{4} - 198025 T^{5} + 704969 T^{6} \))
$97$	(\( 1 + 14 T + 97 T^{2} \))(\( 1 - 12 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 12 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 + 23 T + 353 T^{2} + 3579 T^{3} + 34241 T^{4} + 216407 T^{5} + 912673 T^{6} \))(\( 1 - 23 T + 353 T^{2} - 3579 T^{3} + 34241 T^{4} - 216407 T^{5} + 912673 T^{6} \))
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