LMFDB - Space of modular forms of level 3024, weight 2, and Character 3024.k

Defining parameters

Level:	$ N $	$=$	$ 3024 = 2^{4} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3024.k (of order $2$ and degree $1$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$ 21 $
Character field:			$\Q$
Newform subspaces:			$ 12 $
Sturm bound:			$1152$
Trace bound:			$25$
Distinguishing $T_p$:			$5$, $11$, $13$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(3024, [\chi])$.

	Total	New	Old
Modular forms	612	64	548
Cusp forms	540	64	476
Eisenstein series	72	0	72

Trace form

Decomposition of $S_{2}^{\mathrm{new}}(3024, [\chi])$ 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
3024.2.k.a	$2$	$24.147$	$\Q(\sqrt{-3}) $	None	$0$	$0$	$-6$	$-4$	$q-3q^{5}+(-2-\zeta_{6})q^{7}+3\zeta_{6}q^{11}+\cdots$
3024.2.k.b	$2$	$24.147$	$\Q(\sqrt{-3}) $	$\Q(\sqrt{-3}) $	$0$	$0$	$0$	$-1$	$q+(-1+\zeta_{6})q^{7}+(-1+2\zeta_{6})q^{13}+\cdots$
3024.2.k.c	$2$	$24.147$	$\Q(\sqrt{-3}) $	$\Q(\sqrt{-3}) $	$0$	$0$	$0$	$5$	$q+(2+\zeta_{6})q^{7}+(1-2\zeta_{6})q^{13}+(-5+\cdots)q^{19}+\cdots$
3024.2.k.d	$2$	$24.147$	$\Q(\sqrt{-3}) $	None	$0$	$0$	$6$	$-4$	$q+3q^{5}+(-2+\zeta_{6})q^{7}+3\zeta_{6}q^{11}+\cdots$
3024.2.k.e	$4$	$24.147$	$\Q(\zeta_{12})$	None	$0$	$0$	$0$	$-8$	$q-\zeta_{12}^{2}q^{5}+(-2+\zeta_{12})q^{7}+\zeta_{12}^{3}q^{11}+\cdots$
3024.2.k.f	$4$	$24.147$	$\Q(\zeta_{12})$	None	$0$	$0$	$0$	$-8$	$q-2\zeta_{12}^{2}q^{5}+(-2+\zeta_{12})q^{7}+2\zeta_{12}^{3}q^{11}+\cdots$
3024.2.k.g	$4$	$24.147$	$\Q(\sqrt{-2}, \sqrt{3})$	None	$0$	$0$	$0$	$-4$	$q-\beta _{2}q^{5}+(-1-\beta _{3})q^{7}-\beta _{1}q^{11}+\cdots$
3024.2.k.h	$4$	$24.147$	$\Q(\sqrt{-2}, \sqrt{3})$	None	$0$	$0$	$0$	$4$	$q-\beta _{2}q^{5}+(1+\beta _{1})q^{7}+\beta _{3}q^{11}+\beta _{1}q^{13}+\cdots$
3024.2.k.i	$4$	$24.147$	$\Q(\sqrt{-3}, \sqrt{-5})$	None	$0$	$0$	$0$	$8$	$q+\beta _{3}q^{5}+(2-\beta _{2})q^{7}-\beta _{1}q^{11}-2\beta _{2}q^{13}+\cdots$
3024.2.k.j	$4$	$24.147$	$\Q(\zeta_{12})$	None	$0$	$0$	$0$	$10$	$q-\zeta_{12}^{3}q^{5}+(3-\zeta_{12})q^{7}+(2\zeta_{12}^{2}+\cdots)q^{11}+\cdots$
3024.2.k.k	$16$	$24.147$	$\mathbb{Q}[x]/(x^{16} - \cdots)$	None	$0$	$0$	$0$	$2$	$q-\beta _{9}q^{5}+\beta _{1}q^{7}+\beta _{10}q^{11}-\beta _{5}q^{13}+\cdots$
3024.2.k.l	$16$	$24.147$	$\mathbb{Q}[x]/(x^{16} - \cdots)$	None	$0$	$0$	$0$	$2$	$q+\beta _{1}q^{5}+\beta _{12}q^{7}-\beta _{10}q^{11}+(\beta _{4}+\cdots)q^{13}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(3024, [\chi])$ into lower level spaces

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	1
$3$	1
$5$	($ ( 1 + 3 T + 5 T^{2} )^{2} $)($ ( 1 + 5 T^{2} )^{2} $)($ ( 1 + 5 T^{2} )^{2} $)($ ( 1 - 3 T + 5 T^{2} )^{2} $)($ ( 1 + 7 T^{2} + 25 T^{4} )^{2} $)($ ( 1 - 2 T^{2} + 25 T^{4} )^{2} $)($ ( 1 + 7 T^{2} + 25 T^{4} )^{2} $)($ ( 1 + 7 T^{2} + 25 T^{4} )^{2} $)($ ( 1 - 5 T^{2} + 25 T^{4} )^{2} $)($ ( 1 + 7 T^{2} + 25 T^{4} )^{2} $)($ ( 1 + 23 T^{2} + 273 T^{4} + 2194 T^{6} + 12806 T^{8} + 54850 T^{10} + 170625 T^{12} + 359375 T^{14} + 390625 T^{16} )^{2} $)($ ( 1 + 11 T^{2} + 45 T^{4} + 34 T^{6} - 214 T^{8} + 850 T^{10} + 28125 T^{12} + 171875 T^{14} + 390625 T^{16} )^{2} $)
$7$	($ 1 + 4 T + 7 T^{2} $)($ 1 + T + 7 T^{2} $)($ 1 - 5 T + 7 T^{2} $)($ 1 + 4 T + 7 T^{2} $)($ ( 1 + 4 T + 7 T^{2} )^{2} $)($ ( 1 + 4 T + 7 T^{2} )^{2} $)($ ( 1 + 2 T + 7 T^{2} )^{2} $)($ ( 1 - 2 T + 7 T^{2} )^{2} $)($ ( 1 - 4 T + 7 T^{2} )^{2} $)($ ( 1 - 5 T + 7 T^{2} )^{2} $)($ ( 1 - T - 5 T^{3} - 34 T^{4} - 35 T^{5} - 343 T^{7} + 2401 T^{8} )^{2} $)($ ( 1 - T + 7 T^{3} + 14 T^{4} + 49 T^{5} - 343 T^{7} + 2401 T^{8} )^{2} $)
$11$	($ 1 + 5 T^{2} + 121 T^{4} $)($ ( 1 - 11 T^{2} )^{2} $)($ ( 1 - 11 T^{2} )^{2} $)($ 1 + 5 T^{2} + 121 T^{4} $)($ ( 1 - 13 T^{2} + 121 T^{4} )^{2} $)($ ( 1 + 14 T^{2} + 121 T^{4} )^{2} $)($ ( 1 - 20 T^{2} + 121 T^{4} )^{2} $)($ ( 1 - 4 T^{2} + 121 T^{4} )^{2} $)($ ( 1 - 17 T^{2} + 121 T^{4} )^{2} $)($ ( 1 - 13 T^{2} + 121 T^{4} )^{2} $)($ ( 1 - 37 T^{2} + 801 T^{4} - 12446 T^{6} + 149966 T^{8} - 1505966 T^{10} + 11727441 T^{12} - 65547757 T^{14} + 214358881 T^{16} )^{2} $)($ ( 1 - 55 T^{2} + 1590 T^{4} - 29597 T^{6} + 386186 T^{8} - 3581237 T^{10} + 23279190 T^{12} - 97435855 T^{14} + 214358881 T^{16} )^{2} $)
$13$	($ 1 - 14 T^{2} + 169 T^{4} $)($ ( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) $)($ ( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) $)($ 1 - 14 T^{2} + 169 T^{4} $)($ ( 1 - 14 T^{2} + 169 T^{4} )^{2} $)($ ( 1 - 7 T + 13 T^{2} )^{2}( 1 + 7 T + 13 T^{2} )^{2} $)($ ( 1 - 20 T^{2} + 169 T^{4} )^{2} $)($ ( 1 - 20 T^{2} + 169 T^{4} )^{2} $)($ ( 1 - 14 T^{2} + 169 T^{4} )^{2} $)($ ( 1 - 2 T + 13 T^{2} )^{2}( 1 + 2 T + 13 T^{2} )^{2} $)($ ( 1 - 49 T^{2} + 1398 T^{4} - 28055 T^{6} + 418226 T^{8} - 4741295 T^{10} + 39928278 T^{12} - 236513641 T^{14} + 815730721 T^{16} )^{2} $)($ ( 1 - 49 T^{2} + 1362 T^{4} - 26411 T^{6} + 392258 T^{8} - 4463459 T^{10} + 38900082 T^{12} - 236513641 T^{14} + 815730721 T^{16} )^{2} $)
$17$	($ ( 1 - 6 T + 17 T^{2} )^{2} $)($ ( 1 + 17 T^{2} )^{2} $)($ ( 1 + 17 T^{2} )^{2} $)($ ( 1 + 6 T + 17 T^{2} )^{2} $)($ ( 1 - 14 T^{2} + 289 T^{4} )^{2} $)($ ( 1 + 31 T^{2} + 289 T^{4} )^{2} $)($ ( 1 + 7 T^{2} + 289 T^{4} )^{2} $)($ ( 1 + 31 T^{2} + 289 T^{4} )^{2} $)($ ( 1 + 17 T^{2} )^{4} $)($ ( 1 - 14 T^{2} + 289 T^{4} )^{2} $)($ ( 1 + 72 T^{2} + 2652 T^{4} + 65784 T^{6} + 1248326 T^{8} + 19011576 T^{10} + 221497692 T^{12} + 1737904968 T^{14} + 6975757441 T^{16} )^{2} $)($ ( 1 + 87 T^{2} + 3537 T^{4} + 92514 T^{6} + 1788926 T^{8} + 26736546 T^{10} + 295413777 T^{12} + 2099968503 T^{14} + 6975757441 T^{16} )^{2} $)
$19$	($ 1 - 35 T^{2} + 361 T^{4} $)($ ( 1 - 7 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) $)($ ( 1 - T + 19 T^{2} )( 1 + T + 19 T^{2} ) $)($ 1 - 35 T^{2} + 361 T^{4} $)($ ( 1 - 35 T^{2} + 361 T^{4} )^{2} $)($ ( 1 + 10 T^{2} + 361 T^{4} )^{2} $)($ ( 1 + 16 T^{2} + 361 T^{4} )^{2} $)($ ( 1 - 32 T^{2} + 361 T^{4} )^{2} $)($ ( 1 - 7 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} $)($ ( 1 - 8 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2} $)($ ( 1 - 100 T^{2} + 4914 T^{4} - 156608 T^{6} + 3515195 T^{8} - 56535488 T^{10} + 640397394 T^{12} - 4704588100 T^{14} + 16983563041 T^{16} )^{2} $)($ ( 1 - 112 T^{2} + 5952 T^{4} - 197456 T^{6} + 4482014 T^{8} - 71281616 T^{10} + 775670592 T^{12} - 5269138672 T^{14} + 16983563041 T^{16} )^{2} $)
$23$	($ 1 - 19 T^{2} + 529 T^{4} $)($ ( 1 - 23 T^{2} )^{2} $)($ ( 1 - 23 T^{2} )^{2} $)($ 1 - 19 T^{2} + 529 T^{4} $)($ ( 1 - 37 T^{2} + 529 T^{4} )^{2} $)($ ( 1 - 37 T^{2} + 529 T^{4} )^{2} $)($ ( 1 - 38 T^{2} + 529 T^{4} )^{2} $)($ ( 1 + 26 T^{2} + 529 T^{4} )^{2} $)($ ( 1 - 41 T^{2} + 529 T^{4} )^{2} $)($ ( 1 - 10 T^{2} + 529 T^{4} )^{2} $)($ ( 1 - 21 T^{2} + 697 T^{4} - 11718 T^{6} + 312438 T^{8} - 6198822 T^{10} + 195049177 T^{12} - 3108753669 T^{14} + 78310985281 T^{16} )^{2} $)($ ( 1 - 99 T^{2} + 5182 T^{4} - 187533 T^{6} + 5005890 T^{8} - 99204957 T^{10} + 1450136062 T^{12} - 14655553011 T^{14} + 78310985281 T^{16} )^{2} $)
$29$	($ 1 + 50 T^{2} + 841 T^{4} $)($ ( 1 - 29 T^{2} )^{2} $)($ ( 1 - 29 T^{2} )^{2} $)($ 1 + 50 T^{2} + 841 T^{4} $)($ ( 1 - 22 T^{2} + 841 T^{4} )^{2} $)($ ( 1 - 49 T^{2} + 841 T^{4} )^{2} $)($ ( 1 - 8 T^{2} + 841 T^{4} )^{2} $)($ ( 1 - 40 T^{2} + 841 T^{4} )^{2} $)($ ( 1 - 38 T^{2} + 841 T^{4} )^{2} $)($ ( 1 - 22 T^{2} + 841 T^{4} )^{2} $)($ ( 1 - 172 T^{2} + 14052 T^{4} - 714740 T^{6} + 24798806 T^{8} - 601096340 T^{10} + 9938712612 T^{12} - 102309611212 T^{14} + 500246412961 T^{16} )^{2} $)($ ( 1 - 103 T^{2} + 5226 T^{4} - 168029 T^{6} + 4755218 T^{8} - 141312389 T^{10} + 3696250506 T^{12} - 61266802063 T^{14} + 500246412961 T^{16} )^{2} $)
$31$	($ 1 - 35 T^{2} + 961 T^{4} $)($ ( 1 - 4 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) $)($ ( 1 - 4 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) $)($ 1 - 35 T^{2} + 961 T^{4} $)($ ( 1 - 35 T^{2} + 961 T^{4} )^{2} $)($ ( 1 - 35 T^{2} + 961 T^{4} )^{2} $)($ ( 1 - 56 T^{2} + 961 T^{4} )^{2} $)($ ( 1 - 8 T^{2} + 961 T^{4} )^{2} $)($ ( 1 - 11 T + 31 T^{2} )^{2}( 1 + 11 T + 31 T^{2} )^{2} $)($ ( 1 - 35 T^{2} + 961 T^{4} )^{2} $)($ ( 1 - 187 T^{2} + 16353 T^{4} - 883634 T^{6} + 32653478 T^{8} - 849172274 T^{10} + 15102338913 T^{12} - 165963188347 T^{14} + 852891037441 T^{16} )^{2} $)($ ( 1 - 94 T^{2} + 5349 T^{4} - 230570 T^{6} + 7832744 T^{8} - 221577770 T^{10} + 4939913829 T^{12} - 83425346014 T^{14} + 852891037441 T^{16} )^{2} $)
$37$	($ ( 1 - T + 37 T^{2} )^{2} $)($ ( 1 + 11 T + 37 T^{2} )^{2} $)($ ( 1 - T + 37 T^{2} )^{2} $)($ ( 1 - T + 37 T^{2} )^{2} $)($ ( 1 - 7 T + 37 T^{2} )^{4} $)($ ( 1 + 2 T + 37 T^{2} )^{4} $)($ ( 1 - 5 T + 37 T^{2} )^{4} $)($ ( 1 - T + 37 T^{2} )^{4} $)($ ( 1 + T + 37 T^{2} )^{4} $)($ ( 1 + 2 T + 37 T^{2} )^{4} $)($ ( 1 + 2 T + 52 T^{2} - 100 T^{3} + 949 T^{4} - 3700 T^{5} + 71188 T^{6} + 101306 T^{7} + 1874161 T^{8} )^{4} $)($ ( 1 + 2 T + 49 T^{2} + 410 T^{3} + 1084 T^{4} + 15170 T^{5} + 67081 T^{6} + 101306 T^{7} + 1874161 T^{8} )^{4} $)
$41$	($ ( 1 - 3 T + 41 T^{2} )^{2} $)($ ( 1 + 41 T^{2} )^{2} $)($ ( 1 + 41 T^{2} )^{2} $)($ ( 1 + 3 T + 41 T^{2} )^{2} $)($ ( 1 - 65 T^{2} + 1681 T^{4} )^{2} $)($ ( 1 + 34 T^{2} + 1681 T^{4} )^{2} $)($ ( 1 + 7 T^{2} + 1681 T^{4} )^{2} $)($ ( 1 + 79 T^{2} + 1681 T^{4} )^{2} $)($ ( 1 + 67 T^{2} + 1681 T^{4} )^{2} $)($ ( 1 + 70 T^{2} + 1681 T^{4} )^{2} $)($ ( 1 + 135 T^{2} + 11289 T^{4} + 646218 T^{6} + 29821310 T^{8} + 1086292458 T^{10} + 31900015929 T^{12} + 641264072535 T^{14} + 7984925229121 T^{16} )^{2} $)($ ( 1 + 102 T^{2} + 10017 T^{4} + 548706 T^{6} + 28065644 T^{8} + 922374786 T^{10} + 28305647937 T^{12} + 484510632582 T^{14} + 7984925229121 T^{16} )^{2} $)
$43$	($ ( 1 + 10 T + 43 T^{2} )^{2} $)($ ( 1 - 8 T + 43 T^{2} )^{2} $)($ ( 1 - 8 T + 43 T^{2} )^{2} $)($ ( 1 + 10 T + 43 T^{2} )^{2} $)($ ( 1 - 2 T + 43 T^{2} )^{4} $)($ ( 1 - 11 T + 43 T^{2} )^{4} $)($ ( 1 + 5 T + 43 T^{2} )^{4} $)($ ( 1 + 7 T + 43 T^{2} )^{4} $)($ ( 1 + 2 T + 43 T^{2} )^{4} $)($ ( 1 - 2 T + 43 T^{2} )^{4} $)($ ( 1 + 2 T + 40 T^{2} - 118 T^{3} - 194 T^{4} - 5074 T^{5} + 73960 T^{6} + 159014 T^{7} + 3418801 T^{8} )^{4} $)($ ( 1 + 5 T + 109 T^{2} + 248 T^{3} + 5122 T^{4} + 10664 T^{5} + 201541 T^{6} + 397535 T^{7} + 3418801 T^{8} )^{4} $)
$47$	($ ( 1 + 6 T + 47 T^{2} )^{2} $)($ ( 1 + 47 T^{2} )^{2} $)($ ( 1 + 47 T^{2} )^{2} $)($ ( 1 - 6 T + 47 T^{2} )^{2} $)($ ( 1 + 82 T^{2} + 2209 T^{4} )^{2} $)($ ( 1 + 46 T^{2} + 2209 T^{4} )^{2} $)($ ( 1 + 19 T^{2} + 2209 T^{4} )^{2} $)($ ( 1 - 53 T^{2} + 2209 T^{4} )^{2} $)($ ( 1 + 34 T^{2} + 2209 T^{4} )^{2} $)($ ( 1 + 82 T^{2} + 2209 T^{4} )^{2} $)($ ( 1 + 20 T^{2} + 4068 T^{4} + 4972 T^{6} + 7535798 T^{8} + 10983148 T^{10} + 19850542308 T^{12} + 215584306580 T^{14} + 23811286661761 T^{16} )^{2} $)($ ( 1 + 278 T^{2} + 35193 T^{4} + 2749546 T^{6} + 150940940 T^{8} + 6073747114 T^{10} + 171730613433 T^{12} + 2996621861462 T^{14} + 23811286661761 T^{16} )^{2} $)
$53$	($ ( 1 - 53 T^{2} )^{2} $)($ ( 1 - 53 T^{2} )^{2} $)($ ( 1 - 53 T^{2} )^{2} $)($ ( 1 - 53 T^{2} )^{2} $)($ ( 1 + 38 T^{2} + 2809 T^{4} )^{2} $)($ ( 1 - 97 T^{2} + 2809 T^{4} )^{2} $)($ ( 1 + 22 T^{2} + 2809 T^{4} )^{2} $)($ ( 1 - 53 T^{2} )^{4} $)($ ( 1 - 26 T^{2} + 2809 T^{4} )^{2} $)($ ( 1 - 97 T^{2} + 2809 T^{4} )^{2} $)($ ( 1 - 184 T^{2} + 17436 T^{4} - 1193288 T^{6} + 68587430 T^{8} - 3351945992 T^{10} + 137578426716 T^{12} - 4078242447736 T^{14} + 62259690411361 T^{16} )^{2} $)($ ( 1 - 142 T^{2} + 13833 T^{4} - 1028534 T^{6} + 58011620 T^{8} - 2889152006 T^{10} + 109149023673 T^{12} - 3147339280318 T^{14} + 62259690411361 T^{16} )^{2} $)
$59$	($ ( 1 - 6 T + 59 T^{2} )^{2} $)($ ( 1 + 59 T^{2} )^{2} $)($ ( 1 + 59 T^{2} )^{2} $)($ ( 1 + 6 T + 59 T^{2} )^{2} $)($ ( 1 + 106 T^{2} + 3481 T^{4} )^{2} $)($ ( 1 + 43 T^{2} + 3481 T^{4} )^{2} $)($ ( 1 + 43 T^{2} + 3481 T^{4} )^{2} $)($ ( 1 + 43 T^{2} + 3481 T^{4} )^{2} $)($ ( 1 + 58 T^{2} + 3481 T^{4} )^{2} $)($ ( 1 + 106 T^{2} + 3481 T^{4} )^{2} $)($ ( 1 + 148 T^{2} + 10276 T^{4} + 457612 T^{6} + 23078422 T^{8} + 1592947372 T^{10} + 124518001636 T^{12} + 6242718978868 T^{14} + 146830437604321 T^{16} )^{2} $)($ ( 1 + 283 T^{2} + 41785 T^{4} + 4061626 T^{6} + 280646974 T^{8} + 14138520106 T^{10} + 506323929385 T^{12} + 11937091020403 T^{14} + 146830437604321 T^{16} )^{2} $)
$61$	($ 1 + 70 T^{2} + 3721 T^{4} $)($ ( 1 - T + 61 T^{2} )( 1 + T + 61 T^{2} ) $)($ ( 1 - 13 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) $)($ 1 + 70 T^{2} + 3721 T^{4} $)($ ( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2} $)($ ( 1 + 70 T^{2} + 3721 T^{4} )^{2} $)($ ( 1 - 116 T^{2} + 3721 T^{4} )^{2} $)($ ( 1 - 116 T^{2} + 3721 T^{4} )^{2} $)($ ( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2} $)($ ( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2} $)($ ( 1 - 317 T^{2} + 50098 T^{4} - 5135915 T^{6} + 370298362 T^{8} - 19110739715 T^{10} + 693648942418 T^{12} - 16331958672437 T^{14} + 191707312997281 T^{16} )^{2} $)($ ( 1 - 236 T^{2} + 32440 T^{4} - 3088964 T^{6} + 216205582 T^{8} - 11494035044 T^{10} + 449159082040 T^{12} - 12158808349196 T^{14} + 191707312997281 T^{16} )^{2} $)
$67$	($ ( 1 + 2 T + 67 T^{2} )^{2} $)($ ( 1 + 5 T + 67 T^{2} )^{2} $)($ ( 1 + 11 T + 67 T^{2} )^{2} $)($ ( 1 + 2 T + 67 T^{2} )^{2} $)($ ( 1 + 2 T + 67 T^{2} )^{4} $)($ ( 1 - 7 T + 67 T^{2} )^{4} $)($ ( 1 + 2 T + 67 T^{2} )^{4} $)($ ( 1 - 10 T + 67 T^{2} )^{4} $)($ ( 1 - 10 T + 67 T^{2} )^{4} $)($ ( 1 + 2 T + 67 T^{2} )^{4} $)($ ( 1 - 7 T + 114 T^{2} - 35 T^{3} + 3554 T^{4} - 2345 T^{5} + 511746 T^{6} - 2105341 T^{7} + 20151121 T^{8} )^{4} $)($ ( 1 + 11 T + 198 T^{2} + 2071 T^{3} + 17498 T^{4} + 138757 T^{5} + 888822 T^{6} + 3308393 T^{7} + 20151121 T^{8} )^{4} $)
$71$	($ 1 - 115 T^{2} + 5041 T^{4} $)($ ( 1 - 71 T^{2} )^{2} $)($ ( 1 - 71 T^{2} )^{2} $)($ 1 - 115 T^{2} + 5041 T^{4} $)($ ( 1 - 133 T^{2} + 5041 T^{4} )^{2} $)($ ( 1 - 133 T^{2} + 5041 T^{4} )^{2} $)($ ( 1 + 58 T^{2} + 5041 T^{4} )^{2} $)($ ( 1 - 70 T^{2} + 5041 T^{4} )^{2} $)($ ( 1 - 17 T^{2} + 5041 T^{4} )^{2} $)($ ( 1 + 2 T^{2} + 5041 T^{4} )^{2} $)($ ( 1 - 341 T^{2} + 63193 T^{4} - 7537622 T^{6} + 634982182 T^{8} - 37997152502 T^{10} + 1605840357433 T^{12} - 43682196817061 T^{14} + 645753531245761 T^{16} )^{2} $)($ ( 1 - 167 T^{2} + 18034 T^{4} - 1223801 T^{6} + 87222970 T^{8} - 6169180841 T^{10} + 458274255154 T^{12} - 21392747414807 T^{14} + 645753531245761 T^{16} )^{2} $)
$73$	($ 1 - 134 T^{2} + 5329 T^{4} $)($ ( 1 - 7 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) $)($ ( 1 - 17 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) $)($ 1 - 134 T^{2} + 5329 T^{4} $)($ ( 1 - 134 T^{2} + 5329 T^{4} )^{2} $)($ ( 1 - 98 T^{2} + 5329 T^{4} )^{2} $)($ ( 1 - 73 T^{2} )^{4} $)($ ( 1 - 14 T + 73 T^{2} )^{2}( 1 + 14 T + 73 T^{2} )^{2} $)($ ( 1 - 38 T^{2} + 5329 T^{4} )^{2} $)($ ( 1 + T^{2} + 5329 T^{4} )^{2} $)($ ( 1 - 321 T^{2} + 57726 T^{4} - 6821535 T^{6} + 583944098 T^{8} - 36351960015 T^{10} + 1639316859966 T^{12} - 48578286638769 T^{14} + 806460091894081 T^{16} )^{2} $)($ ( 1 - 117 T^{2} + 15114 T^{4} - 1400907 T^{6} + 111599882 T^{8} - 7465433403 T^{10} + 429211014474 T^{12} - 17706104475813 T^{14} + 806460091894081 T^{16} )^{2} $)
$79$	($ ( 1 + 14 T + 79 T^{2} )^{2} $)($ ( 1 + 17 T + 79 T^{2} )^{2} $)($ ( 1 - 13 T + 79 T^{2} )^{2} $)($ ( 1 + 14 T + 79 T^{2} )^{2} $)($ ( 1 - 10 T + 79 T^{2} )^{4} $)($ ( 1 + 8 T + 79 T^{2} )^{4} $)($ ( 1 - 13 T + 79 T^{2} )^{4} $)($ ( 1 + 5 T + 79 T^{2} )^{4} $)($ ( 1 + 2 T + 79 T^{2} )^{4} $)($ ( 1 + 8 T + 79 T^{2} )^{4} $)($ ( 1 - 11 T + 246 T^{2} - 2515 T^{3} + 26570 T^{4} - 198685 T^{5} + 1535286 T^{6} - 5423429 T^{7} + 38950081 T^{8} )^{4} $)($ ( 1 + 10 T + 261 T^{2} + 1850 T^{3} + 28244 T^{4} + 146150 T^{5} + 1628901 T^{6} + 4930390 T^{7} + 38950081 T^{8} )^{4} $)
$83$	($ ( 1 - 6 T + 83 T^{2} )^{2} $)($ ( 1 + 83 T^{2} )^{2} $)($ ( 1 + 83 T^{2} )^{2} $)($ ( 1 + 6 T + 83 T^{2} )^{2} $)($ ( 1 - 134 T^{2} + 6889 T^{4} )^{2} $)($ ( 1 + 154 T^{2} + 6889 T^{4} )^{2} $)($ ( 1 + 163 T^{2} + 6889 T^{4} )^{2} $)($ ( 1 + 19 T^{2} + 6889 T^{4} )^{2} $)($ ( 1 + 106 T^{2} + 6889 T^{4} )^{2} $)($ ( 1 + 163 T^{2} + 6889 T^{4} )^{2} $)($ ( 1 + 132 T^{2} + 12996 T^{4} + 1459932 T^{6} + 170705174 T^{8} + 10057471548 T^{10} + 616768339716 T^{12} + 43156129284708 T^{14} + 2252292232139041 T^{16} )^{2} $)($ ( 1 + 339 T^{2} + 56745 T^{4} + 6361962 T^{6} + 567506462 T^{8} + 43827556218 T^{10} + 2693022425145 T^{12} + 110832786572091 T^{14} + 2252292232139041 T^{16} )^{2} $)
$89$	($ ( 1 + 9 T + 89 T^{2} )^{2} $)($ ( 1 + 89 T^{2} )^{2} $)($ ( 1 + 89 T^{2} )^{2} $)($ ( 1 - 9 T + 89 T^{2} )^{2} $)($ ( 1 + 151 T^{2} + 7921 T^{4} )^{2} $)($ ( 1 + 151 T^{2} + 7921 T^{4} )^{2} $)($ ( 1 + 70 T^{2} + 7921 T^{4} )^{2} $)($ ( 1 + 70 T^{2} + 7921 T^{4} )^{2} $)($ ( 1 + 43 T^{2} + 7921 T^{4} )^{2} $)($ ( 1 + 70 T^{2} + 7921 T^{4} )^{2} $)($ ( 1 + 351 T^{2} + 57993 T^{4} + 6040074 T^{6} + 534549758 T^{8} + 47843426154 T^{10} + 3638610782313 T^{12} + 174440433127311 T^{14} + 3936588805702081 T^{16} )^{2} $)($ ( 1 + 321 T^{2} + 55134 T^{4} + 6909375 T^{6} + 691071074 T^{8} + 54729159375 T^{10} + 3459230715294 T^{12} + 159530994398481 T^{14} + 3936588805702081 T^{16} )^{2} $)
$97$	($ 1 - 146 T^{2} + 9409 T^{4} $)($ ( 1 - 19 T + 97 T^{2} )( 1 + 19 T + 97 T^{2} ) $)($ ( 1 - 5 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} ) $)($ 1 - 146 T^{2} + 9409 T^{4} $)($ ( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2} $)($ ( 1 - 146 T^{2} + 9409 T^{4} )^{2} $)($ ( 1 + 100 T^{2} + 9409 T^{4} )^{2} $)($ ( 1 - 188 T^{2} + 9409 T^{4} )^{2} $)($ ( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2} $)($ ( 1 - 47 T^{2} + 9409 T^{4} )^{2} $)($ ( 1 - 285 T^{2} + 64986 T^{4} - 9080499 T^{6} + 1059084650 T^{8} - 85438415091 T^{10} + 5753163855066 T^{12} - 237397021404765 T^{14} + 7837433594376961 T^{16} )^{2} $)($ ( 1 - 537 T^{2} + 140634 T^{4} - 23455755 T^{6} + 2707648082 T^{8} - 220695198795 T^{10} + 12450226904154 T^{12} - 447305966646873 T^{14} + 7837433594376961 T^{16} )^{2} $)
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Overview	Random
Universe	Knowledge

Degree 1	Degree 2
Degree 3	Degree 4
$\zeta$ zeros

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3024.k (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 21 \)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(1152\)
Trace bound:			\(25\)
Distinguishing \(T_p\):			\(5\), \(11\), \(13\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

Related objects

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{2}^{\mathrm{new}}(3024, [\chi])\) into newform subspaces

Decomposition of \(S_{2}^{\mathrm{old}}(3024, [\chi])\) 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	3024.2.k

Level	$3024$
Weight	$2$
Character orbit	3024.k
Rep. character	$\chi_{3024}(1889,\cdot)$
Character field	$\Q$
Dimension	$64$
Newform subspaces	$12$
Sturm bound	$1152$
Trace bound	$25$

Label	Dim.	\(A\)	Field	CM	Traces				$q$-expansion
Label	Dim.	\(A\)	Field	CM	\(a_2\)	\(a_3\)	\(a_5\)	\(a_7\)	$q$-expansion
3024.2.k.a	\(2\)	\(24.147\)	\(\Q(\sqrt{-3}) \)	None	\(0\)	\(0\)	\(-6\)	\(-4\)	\(q-3q^{5}+(-2-\zeta_{6})q^{7}+3\zeta_{6}q^{11}+\cdots\)
3024.2.k.b	\(2\)	\(24.147\)	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)	\(0\)	\(0\)	\(0\)	\(-1\)	\(q+(-1+\zeta_{6})q^{7}+(-1+2\zeta_{6})q^{13}+\cdots\)
3024.2.k.c	\(2\)	\(24.147\)	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)	\(0\)	\(0\)	\(0\)	\(5\)	\(q+(2+\zeta_{6})q^{7}+(1-2\zeta_{6})q^{13}+(-5+\cdots)q^{19}+\cdots\)
3024.2.k.d	\(2\)	\(24.147\)	\(\Q(\sqrt{-3}) \)	None	\(0\)	\(0\)	\(6\)	\(-4\)	\(q+3q^{5}+(-2+\zeta_{6})q^{7}+3\zeta_{6}q^{11}+\cdots\)
3024.2.k.e	\(4\)	\(24.147\)	\(\Q(\zeta_{12})\)	None	\(0\)	\(0\)	\(0\)	\(-8\)	\(q-\zeta_{12}^{2}q^{5}+(-2+\zeta_{12})q^{7}+\zeta_{12}^{3}q^{11}+\cdots\)
3024.2.k.f	\(4\)	\(24.147\)	\(\Q(\zeta_{12})\)	None	\(0\)	\(0\)	\(0\)	\(-8\)	\(q-2\zeta_{12}^{2}q^{5}+(-2+\zeta_{12})q^{7}+2\zeta_{12}^{3}q^{11}+\cdots\)
3024.2.k.g	\(4\)	\(24.147\)	\(\Q(\sqrt{-2}, \sqrt{3})\)	None	\(0\)	\(0\)	\(0\)	\(-4\)	\(q-\beta _{2}q^{5}+(-1-\beta _{3})q^{7}-\beta _{1}q^{11}+\cdots\)
3024.2.k.h	\(4\)	\(24.147\)	\(\Q(\sqrt{-2}, \sqrt{3})\)	None	\(0\)	\(0\)	\(0\)	\(4\)	\(q-\beta _{2}q^{5}+(1+\beta _{1})q^{7}+\beta _{3}q^{11}+\beta _{1}q^{13}+\cdots\)
3024.2.k.i	\(4\)	\(24.147\)	\(\Q(\sqrt{-3}, \sqrt{-5})\)	None	\(0\)	\(0\)	\(0\)	\(8\)	\(q+\beta _{3}q^{5}+(2-\beta _{2})q^{7}-\beta _{1}q^{11}-2\beta _{2}q^{13}+\cdots\)
3024.2.k.j	\(4\)	\(24.147\)	\(\Q(\zeta_{12})\)	None	\(0\)	\(0\)	\(0\)	\(10\)	\(q-\zeta_{12}^{3}q^{5}+(3-\zeta_{12})q^{7}+(2\zeta_{12}^{2}+\cdots)q^{11}+\cdots\)
3024.2.k.k	\(16\)	\(24.147\)	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	\(0\)	\(0\)	\(0\)	\(2\)	\(q-\beta _{9}q^{5}+\beta _{1}q^{7}+\beta _{10}q^{11}-\beta _{5}q^{13}+\cdots\)
3024.2.k.l	\(16\)	\(24.147\)	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	\(0\)	\(0\)	\(0\)	\(2\)	\(q+\beta _{1}q^{5}+\beta _{12}q^{7}-\beta _{10}q^{11}+(\beta _{4}+\cdots)q^{13}+\cdots\)

$p$	$F_p(T)$
$2$	1
$3$	1
$5$	(\( ( 1 + 3 T + 5 T^{2} )^{2} \))(\( ( 1 + 5 T^{2} )^{2} \))(\( ( 1 + 5 T^{2} )^{2} \))(\( ( 1 - 3 T + 5 T^{2} )^{2} \))(\( ( 1 + 7 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 2 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 + 7 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 + 7 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 5 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 + 7 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 + 23 T^{2} + 273 T^{4} + 2194 T^{6} + 12806 T^{8} + 54850 T^{10} + 170625 T^{12} + 359375 T^{14} + 390625 T^{16} )^{2} \))(\( ( 1 + 11 T^{2} + 45 T^{4} + 34 T^{6} - 214 T^{8} + 850 T^{10} + 28125 T^{12} + 171875 T^{14} + 390625 T^{16} )^{2} \))
$7$	(\( 1 + 4 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( ( 1 + 4 T + 7 T^{2} )^{2} \))(\( ( 1 + 4 T + 7 T^{2} )^{2} \))(\( ( 1 + 2 T + 7 T^{2} )^{2} \))(\( ( 1 - 2 T + 7 T^{2} )^{2} \))(\( ( 1 - 4 T + 7 T^{2} )^{2} \))(\( ( 1 - 5 T + 7 T^{2} )^{2} \))(\( ( 1 - T - 5 T^{3} - 34 T^{4} - 35 T^{5} - 343 T^{7} + 2401 T^{8} )^{2} \))(\( ( 1 - T + 7 T^{3} + 14 T^{4} + 49 T^{5} - 343 T^{7} + 2401 T^{8} )^{2} \))
$11$	(\( 1 + 5 T^{2} + 121 T^{4} \))(\( ( 1 - 11 T^{2} )^{2} \))(\( ( 1 - 11 T^{2} )^{2} \))(\( 1 + 5 T^{2} + 121 T^{4} \))(\( ( 1 - 13 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 14 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 20 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 4 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 17 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 13 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 37 T^{2} + 801 T^{4} - 12446 T^{6} + 149966 T^{8} - 1505966 T^{10} + 11727441 T^{12} - 65547757 T^{14} + 214358881 T^{16} )^{2} \))(\( ( 1 - 55 T^{2} + 1590 T^{4} - 29597 T^{6} + 386186 T^{8} - 3581237 T^{10} + 23279190 T^{12} - 97435855 T^{14} + 214358881 T^{16} )^{2} \))
$13$	(\( 1 - 14 T^{2} + 169 T^{4} \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( 1 - 14 T^{2} + 169 T^{4} \))(\( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 7 T + 13 T^{2} )^{2}( 1 + 7 T + 13 T^{2} )^{2} \))(\( ( 1 - 20 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 20 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2}( 1 + 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 49 T^{2} + 1398 T^{4} - 28055 T^{6} + 418226 T^{8} - 4741295 T^{10} + 39928278 T^{12} - 236513641 T^{14} + 815730721 T^{16} )^{2} \))(\( ( 1 - 49 T^{2} + 1362 T^{4} - 26411 T^{6} + 392258 T^{8} - 4463459 T^{10} + 38900082 T^{12} - 236513641 T^{14} + 815730721 T^{16} )^{2} \))
$17$	(\( ( 1 - 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( ( 1 + 6 T + 17 T^{2} )^{2} \))(\( ( 1 - 14 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 31 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 7 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 31 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 17 T^{2} )^{4} \))(\( ( 1 - 14 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 72 T^{2} + 2652 T^{4} + 65784 T^{6} + 1248326 T^{8} + 19011576 T^{10} + 221497692 T^{12} + 1737904968 T^{14} + 6975757441 T^{16} )^{2} \))(\( ( 1 + 87 T^{2} + 3537 T^{4} + 92514 T^{6} + 1788926 T^{8} + 26736546 T^{10} + 295413777 T^{12} + 2099968503 T^{14} + 6975757441 T^{16} )^{2} \))
$19$	(\( 1 - 35 T^{2} + 361 T^{4} \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( ( 1 - T + 19 T^{2} )( 1 + T + 19 T^{2} ) \))(\( 1 - 35 T^{2} + 361 T^{4} \))(\( ( 1 - 35 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 + 10 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 + 16 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 32 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 7 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 - 8 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2} \))(\( ( 1 - 100 T^{2} + 4914 T^{4} - 156608 T^{6} + 3515195 T^{8} - 56535488 T^{10} + 640397394 T^{12} - 4704588100 T^{14} + 16983563041 T^{16} )^{2} \))(\( ( 1 - 112 T^{2} + 5952 T^{4} - 197456 T^{6} + 4482014 T^{8} - 71281616 T^{10} + 775670592 T^{12} - 5269138672 T^{14} + 16983563041 T^{16} )^{2} \))
$23$	(\( 1 - 19 T^{2} + 529 T^{4} \))(\( ( 1 - 23 T^{2} )^{2} \))(\( ( 1 - 23 T^{2} )^{2} \))(\( 1 - 19 T^{2} + 529 T^{4} \))(\( ( 1 - 37 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 37 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 38 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 26 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 41 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 21 T^{2} + 697 T^{4} - 11718 T^{6} + 312438 T^{8} - 6198822 T^{10} + 195049177 T^{12} - 3108753669 T^{14} + 78310985281 T^{16} )^{2} \))(\( ( 1 - 99 T^{2} + 5182 T^{4} - 187533 T^{6} + 5005890 T^{8} - 99204957 T^{10} + 1450136062 T^{12} - 14655553011 T^{14} + 78310985281 T^{16} )^{2} \))
$29$	(\( 1 + 50 T^{2} + 841 T^{4} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( 1 + 50 T^{2} + 841 T^{4} \))(\( ( 1 - 22 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 49 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 8 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 40 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 38 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 172 T^{2} + 14052 T^{4} - 714740 T^{6} + 24798806 T^{8} - 601096340 T^{10} + 9938712612 T^{12} - 102309611212 T^{14} + 500246412961 T^{16} )^{2} \))(\( ( 1 - 103 T^{2} + 5226 T^{4} - 168029 T^{6} + 4755218 T^{8} - 141312389 T^{10} + 3696250506 T^{12} - 61266802063 T^{14} + 500246412961 T^{16} )^{2} \))
$31$	(\( 1 - 35 T^{2} + 961 T^{4} \))(\( ( 1 - 4 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( ( 1 - 4 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( 1 - 35 T^{2} + 961 T^{4} \))(\( ( 1 - 35 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 35 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 56 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 8 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 11 T + 31 T^{2} )^{2}( 1 + 11 T + 31 T^{2} )^{2} \))(\( ( 1 - 35 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 187 T^{2} + 16353 T^{4} - 883634 T^{6} + 32653478 T^{8} - 849172274 T^{10} + 15102338913 T^{12} - 165963188347 T^{14} + 852891037441 T^{16} )^{2} \))(\( ( 1 - 94 T^{2} + 5349 T^{4} - 230570 T^{6} + 7832744 T^{8} - 221577770 T^{10} + 4939913829 T^{12} - 83425346014 T^{14} + 852891037441 T^{16} )^{2} \))
$37$	(\( ( 1 - T + 37 T^{2} )^{2} \))(\( ( 1 + 11 T + 37 T^{2} )^{2} \))(\( ( 1 - T + 37 T^{2} )^{2} \))(\( ( 1 - T + 37 T^{2} )^{2} \))(\( ( 1 - 7 T + 37 T^{2} )^{4} \))(\( ( 1 + 2 T + 37 T^{2} )^{4} \))(\( ( 1 - 5 T + 37 T^{2} )^{4} \))(\( ( 1 - T + 37 T^{2} )^{4} \))(\( ( 1 + T + 37 T^{2} )^{4} \))(\( ( 1 + 2 T + 37 T^{2} )^{4} \))(\( ( 1 + 2 T + 52 T^{2} - 100 T^{3} + 949 T^{4} - 3700 T^{5} + 71188 T^{6} + 101306 T^{7} + 1874161 T^{8} )^{4} \))(\( ( 1 + 2 T + 49 T^{2} + 410 T^{3} + 1084 T^{4} + 15170 T^{5} + 67081 T^{6} + 101306 T^{7} + 1874161 T^{8} )^{4} \))
$41$	(\( ( 1 - 3 T + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 3 T + 41 T^{2} )^{2} \))(\( ( 1 - 65 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 34 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 7 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 79 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 67 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 70 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 135 T^{2} + 11289 T^{4} + 646218 T^{6} + 29821310 T^{8} + 1086292458 T^{10} + 31900015929 T^{12} + 641264072535 T^{14} + 7984925229121 T^{16} )^{2} \))(\( ( 1 + 102 T^{2} + 10017 T^{4} + 548706 T^{6} + 28065644 T^{8} + 922374786 T^{10} + 28305647937 T^{12} + 484510632582 T^{14} + 7984925229121 T^{16} )^{2} \))
$43$	(\( ( 1 + 10 T + 43 T^{2} )^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 + 10 T + 43 T^{2} )^{2} \))(\( ( 1 - 2 T + 43 T^{2} )^{4} \))(\( ( 1 - 11 T + 43 T^{2} )^{4} \))(\( ( 1 + 5 T + 43 T^{2} )^{4} \))(\( ( 1 + 7 T + 43 T^{2} )^{4} \))(\( ( 1 + 2 T + 43 T^{2} )^{4} \))(\( ( 1 - 2 T + 43 T^{2} )^{4} \))(\( ( 1 + 2 T + 40 T^{2} - 118 T^{3} - 194 T^{4} - 5074 T^{5} + 73960 T^{6} + 159014 T^{7} + 3418801 T^{8} )^{4} \))(\( ( 1 + 5 T + 109 T^{2} + 248 T^{3} + 5122 T^{4} + 10664 T^{5} + 201541 T^{6} + 397535 T^{7} + 3418801 T^{8} )^{4} \))
$47$	(\( ( 1 + 6 T + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 - 6 T + 47 T^{2} )^{2} \))(\( ( 1 + 82 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 46 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 19 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 - 53 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 34 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 82 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 20 T^{2} + 4068 T^{4} + 4972 T^{6} + 7535798 T^{8} + 10983148 T^{10} + 19850542308 T^{12} + 215584306580 T^{14} + 23811286661761 T^{16} )^{2} \))(\( ( 1 + 278 T^{2} + 35193 T^{4} + 2749546 T^{6} + 150940940 T^{8} + 6073747114 T^{10} + 171730613433 T^{12} + 2996621861462 T^{14} + 23811286661761 T^{16} )^{2} \))
$53$	(\( ( 1 - 53 T^{2} )^{2} \))(\( ( 1 - 53 T^{2} )^{2} \))(\( ( 1 - 53 T^{2} )^{2} \))(\( ( 1 - 53 T^{2} )^{2} \))(\( ( 1 + 38 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 97 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 + 22 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 53 T^{2} )^{4} \))(\( ( 1 - 26 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 97 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 184 T^{2} + 17436 T^{4} - 1193288 T^{6} + 68587430 T^{8} - 3351945992 T^{10} + 137578426716 T^{12} - 4078242447736 T^{14} + 62259690411361 T^{16} )^{2} \))(\( ( 1 - 142 T^{2} + 13833 T^{4} - 1028534 T^{6} + 58011620 T^{8} - 2889152006 T^{10} + 109149023673 T^{12} - 3147339280318 T^{14} + 62259690411361 T^{16} )^{2} \))
$59$	(\( ( 1 - 6 T + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 6 T + 59 T^{2} )^{2} \))(\( ( 1 + 106 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 43 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 43 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 43 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 58 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 106 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 148 T^{2} + 10276 T^{4} + 457612 T^{6} + 23078422 T^{8} + 1592947372 T^{10} + 124518001636 T^{12} + 6242718978868 T^{14} + 146830437604321 T^{16} )^{2} \))(\( ( 1 + 283 T^{2} + 41785 T^{4} + 4061626 T^{6} + 280646974 T^{8} + 14138520106 T^{10} + 506323929385 T^{12} + 11937091020403 T^{14} + 146830437604321 T^{16} )^{2} \))
$61$	(\( 1 + 70 T^{2} + 3721 T^{4} \))(\( ( 1 - T + 61 T^{2} )( 1 + T + 61 T^{2} ) \))(\( ( 1 - 13 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 + 70 T^{2} + 3721 T^{4} \))(\( ( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2} \))(\( ( 1 + 70 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 - 116 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 - 116 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2} \))(\( ( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2} \))(\( ( 1 - 317 T^{2} + 50098 T^{4} - 5135915 T^{6} + 370298362 T^{8} - 19110739715 T^{10} + 693648942418 T^{12} - 16331958672437 T^{14} + 191707312997281 T^{16} )^{2} \))(\( ( 1 - 236 T^{2} + 32440 T^{4} - 3088964 T^{6} + 216205582 T^{8} - 11494035044 T^{10} + 449159082040 T^{12} - 12158808349196 T^{14} + 191707312997281 T^{16} )^{2} \))
$67$	(\( ( 1 + 2 T + 67 T^{2} )^{2} \))(\( ( 1 + 5 T + 67 T^{2} )^{2} \))(\( ( 1 + 11 T + 67 T^{2} )^{2} \))(\( ( 1 + 2 T + 67 T^{2} )^{2} \))(\( ( 1 + 2 T + 67 T^{2} )^{4} \))(\( ( 1 - 7 T + 67 T^{2} )^{4} \))(\( ( 1 + 2 T + 67 T^{2} )^{4} \))(\( ( 1 - 10 T + 67 T^{2} )^{4} \))(\( ( 1 - 10 T + 67 T^{2} )^{4} \))(\( ( 1 + 2 T + 67 T^{2} )^{4} \))(\( ( 1 - 7 T + 114 T^{2} - 35 T^{3} + 3554 T^{4} - 2345 T^{5} + 511746 T^{6} - 2105341 T^{7} + 20151121 T^{8} )^{4} \))(\( ( 1 + 11 T + 198 T^{2} + 2071 T^{3} + 17498 T^{4} + 138757 T^{5} + 888822 T^{6} + 3308393 T^{7} + 20151121 T^{8} )^{4} \))
$71$	(\( 1 - 115 T^{2} + 5041 T^{4} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( 1 - 115 T^{2} + 5041 T^{4} \))(\( ( 1 - 133 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 133 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 58 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 70 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 17 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 2 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 341 T^{2} + 63193 T^{4} - 7537622 T^{6} + 634982182 T^{8} - 37997152502 T^{10} + 1605840357433 T^{12} - 43682196817061 T^{14} + 645753531245761 T^{16} )^{2} \))(\( ( 1 - 167 T^{2} + 18034 T^{4} - 1223801 T^{6} + 87222970 T^{8} - 6169180841 T^{10} + 458274255154 T^{12} - 21392747414807 T^{14} + 645753531245761 T^{16} )^{2} \))
$73$	(\( 1 - 134 T^{2} + 5329 T^{4} \))(\( ( 1 - 7 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( 1 - 134 T^{2} + 5329 T^{4} \))(\( ( 1 - 134 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 98 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 73 T^{2} )^{4} \))(\( ( 1 - 14 T + 73 T^{2} )^{2}( 1 + 14 T + 73 T^{2} )^{2} \))(\( ( 1 - 38 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 + T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 321 T^{2} + 57726 T^{4} - 6821535 T^{6} + 583944098 T^{8} - 36351960015 T^{10} + 1639316859966 T^{12} - 48578286638769 T^{14} + 806460091894081 T^{16} )^{2} \))(\( ( 1 - 117 T^{2} + 15114 T^{4} - 1400907 T^{6} + 111599882 T^{8} - 7465433403 T^{10} + 429211014474 T^{12} - 17706104475813 T^{14} + 806460091894081 T^{16} )^{2} \))
$79$	(\( ( 1 + 14 T + 79 T^{2} )^{2} \))(\( ( 1 + 17 T + 79 T^{2} )^{2} \))(\( ( 1 - 13 T + 79 T^{2} )^{2} \))(\( ( 1 + 14 T + 79 T^{2} )^{2} \))(\( ( 1 - 10 T + 79 T^{2} )^{4} \))(\( ( 1 + 8 T + 79 T^{2} )^{4} \))(\( ( 1 - 13 T + 79 T^{2} )^{4} \))(\( ( 1 + 5 T + 79 T^{2} )^{4} \))(\( ( 1 + 2 T + 79 T^{2} )^{4} \))(\( ( 1 + 8 T + 79 T^{2} )^{4} \))(\( ( 1 - 11 T + 246 T^{2} - 2515 T^{3} + 26570 T^{4} - 198685 T^{5} + 1535286 T^{6} - 5423429 T^{7} + 38950081 T^{8} )^{4} \))(\( ( 1 + 10 T + 261 T^{2} + 1850 T^{3} + 28244 T^{4} + 146150 T^{5} + 1628901 T^{6} + 4930390 T^{7} + 38950081 T^{8} )^{4} \))
$83$	(\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 - 134 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 154 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 163 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 19 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 106 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 163 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 132 T^{2} + 12996 T^{4} + 1459932 T^{6} + 170705174 T^{8} + 10057471548 T^{10} + 616768339716 T^{12} + 43156129284708 T^{14} + 2252292232139041 T^{16} )^{2} \))(\( ( 1 + 339 T^{2} + 56745 T^{4} + 6361962 T^{6} + 567506462 T^{8} + 43827556218 T^{10} + 2693022425145 T^{12} + 110832786572091 T^{14} + 2252292232139041 T^{16} )^{2} \))
$89$	(\( ( 1 + 9 T + 89 T^{2} )^{2} \))(\( ( 1 + 89 T^{2} )^{2} \))(\( ( 1 + 89 T^{2} )^{2} \))(\( ( 1 - 9 T + 89 T^{2} )^{2} \))(\( ( 1 + 151 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 151 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 70 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 70 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 43 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 70 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 351 T^{2} + 57993 T^{4} + 6040074 T^{6} + 534549758 T^{8} + 47843426154 T^{10} + 3638610782313 T^{12} + 174440433127311 T^{14} + 3936588805702081 T^{16} )^{2} \))(\( ( 1 + 321 T^{2} + 55134 T^{4} + 6909375 T^{6} + 691071074 T^{8} + 54729159375 T^{10} + 3459230715294 T^{12} + 159530994398481 T^{14} + 3936588805702081 T^{16} )^{2} \))
$97$	(\( 1 - 146 T^{2} + 9409 T^{4} \))(\( ( 1 - 19 T + 97 T^{2} )( 1 + 19 T + 97 T^{2} ) \))(\( ( 1 - 5 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} ) \))(\( 1 - 146 T^{2} + 9409 T^{4} \))(\( ( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2} \))(\( ( 1 - 146 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 100 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 188 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2} \))(\( ( 1 - 47 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 285 T^{2} + 64986 T^{4} - 9080499 T^{6} + 1059084650 T^{8} - 85438415091 T^{10} + 5753163855066 T^{12} - 237397021404765 T^{14} + 7837433594376961 T^{16} )^{2} \))(\( ( 1 - 537 T^{2} + 140634 T^{4} - 23455755 T^{6} + 2707648082 T^{8} - 220695198795 T^{10} + 12450226904154 T^{12} - 447305966646873 T^{14} + 7837433594376961 T^{16} )^{2} \))
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