LMFDB - Space of modular forms of level 48, weight 7, and Character 48.e

Defining parameters

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	48.e (of order $2$ and degree $1$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$ 3 $
Character field:			$\Q$
Newform subspaces:			$ 4 $
Sturm bound:			$56$
Trace bound:			$3$
Distinguishing $T_p$:			$5$

Dimensions

The following table gives the dimensions of various subspaces of $M_{7}(48, [\chi])$.

	Total	New	Old
Modular forms	54	13	41
Cusp forms	42	11	31
Eisenstein series	12	2	10

Trace form

Decomposition of $S_{7}^{\mathrm{new}}(48, [\chi])$ 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
48.7.e.a	$1$	$11.043$	$\Q$	$\Q(\sqrt{-3}) $	$0$	$27$	$0$	$286$	$q+3^{3}q^{3}+286q^{7}+3^{6}q^{9}+506q^{13}+\cdots$
48.7.e.b	$2$	$11.043$	$\Q(\sqrt{-2}) $	None	$0$	$-42$	$0$	$-4$	$q+(-21+\beta )q^{3}+10\beta q^{5}-2q^{7}+(153+\cdots)q^{9}+\cdots$
48.7.e.c	$2$	$11.043$	$\Q(\sqrt{-5}) $	None	$0$	$6$	$0$	$-484$	$q+(3+\beta )q^{3}-6\beta q^{5}-242q^{7}+(-711+\cdots)q^{9}+\cdots$
48.7.e.d	$6$	$11.043$	6.0.1173604352.2	None	$0$	$10$	$0$	$-156$	$q+(2-\beta _{1})q^{3}+(-\beta _{1}-\beta _{2})q^{5}+(-3^{3}+\cdots)q^{7}+\cdots$

Decomposition of $S_{7}^{\mathrm{old}}(48, [\chi])$ into lower level spaces

$ S_{7}^{\mathrm{old}}(48, [\chi]) \cong $ $S_{7}^{\mathrm{new}}(3, [\chi])$$^{\oplus 5}$$\oplus$$S_{7}^{\mathrm{new}}(6, [\chi])$$^{\oplus 4}$$\oplus$$S_{7}^{\mathrm{new}}(12, [\chi])$$^{\oplus 3}$$\oplus$$S_{7}^{\mathrm{new}}(24, [\chi])$$^{\oplus 2}$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	1
$3$	($ 1 - 27 T $)($ 1 + 42 T + 729 T^{2} $)($ 1 - 6 T + 729 T^{2} $)($ 1 - 10 T + 87 T^{2} + 11988 T^{3} + 63423 T^{4} - 5314410 T^{5} + 387420489 T^{6} $)
$5$	($ ( 1 - 125 T )( 1 + 125 T ) $)($ 1 - 2450 T^{2} + 244140625 T^{4} $)($ 1 - 5330 T^{2} + 244140625 T^{4} $)($ 1 - 57558 T^{2} + 1665299775 T^{4} - 31537483192500 T^{6} + 406567327880859375 T^{8} - $$34\!\cdots\!50$$ T^{10} + $$14\!\cdots\!25$$ T^{12} $)
$7$	($ 1 - 286 T + 117649 T^{2} $)($ ( 1 + 2 T + 117649 T^{2} )^{2} $)($ ( 1 + 242 T + 117649 T^{2} )^{2} $)($ ( 1 + 78 T + 50079 T^{2} + 1090308 T^{3} + 5891744271 T^{4} + 1079620401678 T^{5} + 1628413597910449 T^{6} )^{2} $)
$11$	($ ( 1 - 1331 T )( 1 + 1331 T ) $)($ 1 - 3541970 T^{2} + 3138428376721 T^{4} $)($ 1 - 406802 T^{2} + 3138428376721 T^{4} $)($ 1 - 3337110 T^{2} + 13030923525375 T^{4} - 22216923537947614260 T^{6} + $$40\!\cdots\!75$$ T^{8} - $$32\!\cdots\!10$$ T^{10} + $$30\!\cdots\!61$$ T^{12} $)
$13$	($ 1 - 506 T + 4826809 T^{2} $)($ ( 1 + 2950 T + 4826809 T^{2} )^{2} $)($ ( 1 - 2618 T + 4826809 T^{2} )^{2} $)($ ( 1 - 78 T + 8645655 T^{2} - 5741493604 T^{3} + 41730925364895 T^{4} - 1817250639553518 T^{5} + $$11\!\cdots\!29$$ T^{6} )^{2} $)
$17$	($ ( 1 - 4913 T )( 1 + 4913 T ) $)($ 1 - 28202690 T^{2} + 582622237229761 T^{4} $)($ 1 + 1905982 T^{2} + 582622237229761 T^{4} $)($ 1 - 63219270 T^{2} + 2233832285753679 T^{4} - $$57\!\cdots\!32$$ T^{6} + $$13\!\cdots\!19$$ T^{8} - $$21\!\cdots\!70$$ T^{10} + $$19\!\cdots\!81$$ T^{12} $)
$19$	($ 1 - 10582 T + 47045881 T^{2} $)($ ( 1 + 5258 T + 47045881 T^{2} )^{2} $)($ ( 1 + 5786 T + 47045881 T^{2} )^{2} $)($ ( 1 - 2250 T + 64193463 T^{2} - 323410900012 T^{3} + 3020038021275903 T^{4} - 4979958567898862250 T^{5} + $$10\!\cdots\!41$$ T^{6} )^{2} $)
$23$	($ ( 1 - 12167 T )( 1 + 12167 T ) $)($ 1 - 191004770 T^{2} + 21914624432020321 T^{4} $)($ 1 - 208876898 T^{2} + 21914624432020321 T^{4} $)($ 1 + 37480026 T^{2} + 40240367212774575 T^{4} + $$51\!\cdots\!12$$ T^{6} + $$88\!\cdots\!75$$ T^{8} + $$17\!\cdots\!66$$ T^{10} + $$10\!\cdots\!61$$ T^{12} $)
$29$	($ ( 1 - 24389 T )( 1 + 24389 T ) $)($ 1 - 1184779442 T^{2} + 353814783205469041 T^{4} $)($ 1 - 1035966962 T^{2} + 353814783205469041 T^{4} $)($ 1 - 1660966710 T^{2} + 1523344801209102879 T^{4} - $$10\!\cdots\!08$$ T^{6} + $$53\!\cdots\!39$$ T^{8} - $$20\!\cdots\!10$$ T^{10} + $$44\!\cdots\!21$$ T^{12} $)
$31$	($ 1 + 35282 T + 887503681 T^{2} $)($ ( 1 + 22898 T + 887503681 T^{2} )^{2} $)($ ( 1 - 20446 T + 887503681 T^{2} )^{2} $)($ ( 1 - 37122 T + 2847433071 T^{2} - 66134143254940 T^{3} + 2527107331913634351 T^{4} - $$29\!\cdots\!42$$ T^{5} + $$69\!\cdots\!41$$ T^{6} )^{2} $)
$37$	($ 1 + 89206 T + 2565726409 T^{2} $)($ ( 1 - 34058 T + 2565726409 T^{2} )^{2} $)($ ( 1 + 46774 T + 2565726409 T^{2} )^{2} $)($ ( 1 - 85566 T + 8033556999 T^{2} - 380834469871044 T^{3} + 20611909350541086591 T^{4} - $$56\!\cdots\!46$$ T^{5} + $$16\!\cdots\!29$$ T^{6} )^{2} $)
$41$	($ ( 1 - 68921 T )( 1 + 68921 T ) $)($ 1 - 9219079010 T^{2} + 22563490300366186081 T^{4} $)($ 1 - 9487663202 T^{2} + 22563490300366186081 T^{4} $)($ 1 - 7569104550 T^{2} + 1363608137405888943 T^{4} + $$15\!\cdots\!00$$ T^{6} + $$30\!\cdots\!83$$ T^{8} - $$38\!\cdots\!50$$ T^{10} + $$11\!\cdots\!41$$ T^{12} $)
$43$	($ 1 + 111386 T + 6321363049 T^{2} $)($ ( 1 - 6406 T + 6321363049 T^{2} )^{2} $)($ ( 1 + 68618 T + 6321363049 T^{2} )^{2} $)($ ( 1 - 145530 T + 21837177927 T^{2} - 1677260136965132 T^{3} + $$13\!\cdots\!23$$ T^{4} - $$58\!\cdots\!30$$ T^{5} + $$25\!\cdots\!49$$ T^{6} )^{2} $)
$47$	($ ( 1 - 103823 T )( 1 + 103823 T ) $)($ 1 + 10801249342 T^{2} + $$11\!\cdots\!41$$ T^{4} $)($ 1 - 21106800578 T^{2} + $$11\!\cdots\!41$$ T^{4} $)($ 1 - 52339293510 T^{2} + $$12\!\cdots\!23$$ T^{4} - $$17\!\cdots\!20$$ T^{6} + $$14\!\cdots\!43$$ T^{8} - $$70\!\cdots\!10$$ T^{10} + $$15\!\cdots\!21$$ T^{12} $)
$53$	($ ( 1 - 148877 T )( 1 + 148877 T ) $)($ 1 - 7253988050 T^{2} + $$49\!\cdots\!41$$ T^{4} $)($ 1 - 14819087378 T^{2} + $$49\!\cdots\!41$$ T^{4} $)($ 1 - 75107476374 T^{2} + $$29\!\cdots\!67$$ T^{4} - $$77\!\cdots\!24$$ T^{6} + $$14\!\cdots\!47$$ T^{8} - $$18\!\cdots\!94$$ T^{10} + $$11\!\cdots\!21$$ T^{12} $)
$59$	($ ( 1 - 205379 T )( 1 + 205379 T ) $)($ 1 + 22449655150 T^{2} + $$17\!\cdots\!81$$ T^{4} $)($ 1 - 61991044562 T^{2} + $$17\!\cdots\!81$$ T^{4} $)($ 1 - 180693503190 T^{2} + $$15\!\cdots\!31$$ T^{4} - $$77\!\cdots\!36$$ T^{6} + $$26\!\cdots\!11$$ T^{8} - $$57\!\cdots\!90$$ T^{10} + $$56\!\cdots\!41$$ T^{12} $)
$61$	($ 1 + 420838 T + 51520374361 T^{2} $)($ ( 1 + 62566 T + 51520374361 T^{2} )^{2} $)($ ( 1 - 24794 T + 51520374361 T^{2} )^{2} $)($ ( 1 - 296046 T + 42371190135 T^{2} - 4032498794872228 T^{3} + $$21\!\cdots\!35$$ T^{4} - $$78\!\cdots\!66$$ T^{5} + $$13\!\cdots\!81$$ T^{6} )^{2} $)
$67$	($ 1 + 172874 T + 90458382169 T^{2} $)($ ( 1 + 438698 T + 90458382169 T^{2} )^{2} $)($ ( 1 - 84358 T + 90458382169 T^{2} )^{2} $)($ ( 1 - 285450 T + 220735162839 T^{2} - 36790378201873708 T^{3} + $$19\!\cdots\!91$$ T^{4} - $$23\!\cdots\!50$$ T^{5} + $$74\!\cdots\!09$$ T^{6} )^{2} $)
$71$	($ ( 1 - 357911 T )( 1 + 357911 T ) $)($ 1 - 251546372642 T^{2} + $$16\!\cdots\!41$$ T^{4} $)($ 1 - 151063967522 T^{2} + $$16\!\cdots\!41$$ T^{4} $)($ 1 - 402983850726 T^{2} + $$77\!\cdots\!75$$ T^{4} - $$10\!\cdots\!48$$ T^{6} + $$12\!\cdots\!75$$ T^{8} - $$10\!\cdots\!06$$ T^{10} + $$44\!\cdots\!21$$ T^{12} $)
$73$	($ 1 - 638066 T + 151334226289 T^{2} $)($ ( 1 + 730510 T + 151334226289 T^{2} )^{2} $)($ ( 1 + 113806 T + 151334226289 T^{2} )^{2} $)($ ( 1 - 559830 T + 328149232287 T^{2} - 146928200928984628 T^{3} + $$49\!\cdots\!43$$ T^{4} - $$12\!\cdots\!30$$ T^{5} + $$34\!\cdots\!69$$ T^{6} )^{2} $)
$79$	($ 1 - 204622 T + 243087455521 T^{2} $)($ ( 1 + 340562 T + 243087455521 T^{2} )^{2} $)($ ( 1 - 159742 T + 243087455521 T^{2} )^{2} $)($ ( 1 - 526818 T + 705484689807 T^{2} - 230000056692908636 T^{3} + $$17\!\cdots\!47$$ T^{4} - $$31\!\cdots\!38$$ T^{5} + $$14\!\cdots\!61$$ T^{6} )^{2} $)
$83$	($ ( 1 - 571787 T )( 1 + 571787 T ) $)($ 1 - 407613512306 T^{2} + $$10\!\cdots\!61$$ T^{4} $)($ 1 - 388294032818 T^{2} + $$10\!\cdots\!61$$ T^{4} $)($ 1 - 1779363154038 T^{2} + $$13\!\cdots\!67$$ T^{4} - $$58\!\cdots\!36$$ T^{6} + $$14\!\cdots\!87$$ T^{8} - $$20\!\cdots\!98$$ T^{10} + $$12\!\cdots\!81$$ T^{12} $)
$89$	($ ( 1 - 704969 T )( 1 + 704969 T ) $)($ 1 - 844406214050 T^{2} + $$24\!\cdots\!21$$ T^{4} $)($ 1 + 583819025758 T^{2} + $$24\!\cdots\!21$$ T^{4} $)($ 1 - 1464277161318 T^{2} + $$11\!\cdots\!39$$ T^{4} - $$68\!\cdots\!36$$ T^{6} + $$28\!\cdots\!19$$ T^{8} - $$89\!\cdots\!38$$ T^{10} + $$15\!\cdots\!61$$ T^{12} $)
$97$	($ 1 + 56446 T + 832972004929 T^{2} $)($ ( 1 + 281086 T + 832972004929 T^{2} )^{2} $)($ ( 1 - 899522 T + 832972004929 T^{2} )^{2} $)($ ( 1 + 399258 T + 1108436400207 T^{2} + 1034642865778948780 T^{3} + $$92\!\cdots\!03$$ T^{4} + $$27\!\cdots\!78$$ T^{5} + $$57\!\cdots\!89$$ T^{6} )^{2} $)
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Degree 1	Degree 2
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$\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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	48.e (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 3 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(56\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(5\)

Introduction

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{7}^{\mathrm{new}}(48, [\chi])\) into newform subspaces

Decomposition of \(S_{7}^{\mathrm{old}}(48, [\chi])\) 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	48.7.e

Level	48
Weight	7
Character orbit	e
Rep. character	\(\chi_{48}(17,\cdot)\)
Character field	\(\Q\)
Dimension	11
Newform subspaces	4
Sturm bound	56
Trace bound	3

Label	Dim.	\(A\)	Field	CM	Traces				$q$-expansion
Label	Dim.	\(A\)	Field	CM	\(a_2\)	\(a_3\)	\(a_5\)	\(a_7\)	$q$-expansion
48.7.e.a	\(1\)	\(11.043\)	\(\Q\)	\(\Q(\sqrt{-3}) \)	\(0\)	\(27\)	\(0\)	\(286\)	\(q+3^{3}q^{3}+286q^{7}+3^{6}q^{9}+506q^{13}+\cdots\)
48.7.e.b	\(2\)	\(11.043\)	\(\Q(\sqrt{-2}) \)	None	\(0\)	\(-42\)	\(0\)	\(-4\)	\(q+(-21+\beta )q^{3}+10\beta q^{5}-2q^{7}+(153+\cdots)q^{9}+\cdots\)
48.7.e.c	\(2\)	\(11.043\)	\(\Q(\sqrt{-5}) \)	None	\(0\)	\(6\)	\(0\)	\(-484\)	\(q+(3+\beta )q^{3}-6\beta q^{5}-242q^{7}+(-711+\cdots)q^{9}+\cdots\)
48.7.e.d	\(6\)	\(11.043\)	6.0.1173604352.2	None	\(0\)	\(10\)	\(0\)	\(-156\)	\(q+(2-\beta _{1})q^{3}+(-\beta _{1}-\beta _{2})q^{5}+(-3^{3}+\cdots)q^{7}+\cdots\)

$p$	$F_p(T)$
$2$	1
$3$	(\( 1 - 27 T \))(\( 1 + 42 T + 729 T^{2} \))(\( 1 - 6 T + 729 T^{2} \))(\( 1 - 10 T + 87 T^{2} + 11988 T^{3} + 63423 T^{4} - 5314410 T^{5} + 387420489 T^{6} \))
$5$	(\( ( 1 - 125 T )( 1 + 125 T ) \))(\( 1 - 2450 T^{2} + 244140625 T^{4} \))(\( 1 - 5330 T^{2} + 244140625 T^{4} \))(\( 1 - 57558 T^{2} + 1665299775 T^{4} - 31537483192500 T^{6} + 406567327880859375 T^{8} - \)\(34\!\cdots\!50\)\( T^{10} + \)\(14\!\cdots\!25\)\( T^{12} \))
$7$	(\( 1 - 286 T + 117649 T^{2} \))(\( ( 1 + 2 T + 117649 T^{2} )^{2} \))(\( ( 1 + 242 T + 117649 T^{2} )^{2} \))(\( ( 1 + 78 T + 50079 T^{2} + 1090308 T^{3} + 5891744271 T^{4} + 1079620401678 T^{5} + 1628413597910449 T^{6} )^{2} \))
$11$	(\( ( 1 - 1331 T )( 1 + 1331 T ) \))(\( 1 - 3541970 T^{2} + 3138428376721 T^{4} \))(\( 1 - 406802 T^{2} + 3138428376721 T^{4} \))(\( 1 - 3337110 T^{2} + 13030923525375 T^{4} - 22216923537947614260 T^{6} + \)\(40\!\cdots\!75\)\( T^{8} - \)\(32\!\cdots\!10\)\( T^{10} + \)\(30\!\cdots\!61\)\( T^{12} \))
$13$	(\( 1 - 506 T + 4826809 T^{2} \))(\( ( 1 + 2950 T + 4826809 T^{2} )^{2} \))(\( ( 1 - 2618 T + 4826809 T^{2} )^{2} \))(\( ( 1 - 78 T + 8645655 T^{2} - 5741493604 T^{3} + 41730925364895 T^{4} - 1817250639553518 T^{5} + \)\(11\!\cdots\!29\)\( T^{6} )^{2} \))
$17$	(\( ( 1 - 4913 T )( 1 + 4913 T ) \))(\( 1 - 28202690 T^{2} + 582622237229761 T^{4} \))(\( 1 + 1905982 T^{2} + 582622237229761 T^{4} \))(\( 1 - 63219270 T^{2} + 2233832285753679 T^{4} - \)\(57\!\cdots\!32\)\( T^{6} + \)\(13\!\cdots\!19\)\( T^{8} - \)\(21\!\cdots\!70\)\( T^{10} + \)\(19\!\cdots\!81\)\( T^{12} \))
$19$	(\( 1 - 10582 T + 47045881 T^{2} \))(\( ( 1 + 5258 T + 47045881 T^{2} )^{2} \))(\( ( 1 + 5786 T + 47045881 T^{2} )^{2} \))(\( ( 1 - 2250 T + 64193463 T^{2} - 323410900012 T^{3} + 3020038021275903 T^{4} - 4979958567898862250 T^{5} + \)\(10\!\cdots\!41\)\( T^{6} )^{2} \))
$23$	(\( ( 1 - 12167 T )( 1 + 12167 T ) \))(\( 1 - 191004770 T^{2} + 21914624432020321 T^{4} \))(\( 1 - 208876898 T^{2} + 21914624432020321 T^{4} \))(\( 1 + 37480026 T^{2} + 40240367212774575 T^{4} + \)\(51\!\cdots\!12\)\( T^{6} + \)\(88\!\cdots\!75\)\( T^{8} + \)\(17\!\cdots\!66\)\( T^{10} + \)\(10\!\cdots\!61\)\( T^{12} \))
$29$	(\( ( 1 - 24389 T )( 1 + 24389 T ) \))(\( 1 - 1184779442 T^{2} + 353814783205469041 T^{4} \))(\( 1 - 1035966962 T^{2} + 353814783205469041 T^{4} \))(\( 1 - 1660966710 T^{2} + 1523344801209102879 T^{4} - \)\(10\!\cdots\!08\)\( T^{6} + \)\(53\!\cdots\!39\)\( T^{8} - \)\(20\!\cdots\!10\)\( T^{10} + \)\(44\!\cdots\!21\)\( T^{12} \))
$31$	(\( 1 + 35282 T + 887503681 T^{2} \))(\( ( 1 + 22898 T + 887503681 T^{2} )^{2} \))(\( ( 1 - 20446 T + 887503681 T^{2} )^{2} \))(\( ( 1 - 37122 T + 2847433071 T^{2} - 66134143254940 T^{3} + 2527107331913634351 T^{4} - \)\(29\!\cdots\!42\)\( T^{5} + \)\(69\!\cdots\!41\)\( T^{6} )^{2} \))
$37$	(\( 1 + 89206 T + 2565726409 T^{2} \))(\( ( 1 - 34058 T + 2565726409 T^{2} )^{2} \))(\( ( 1 + 46774 T + 2565726409 T^{2} )^{2} \))(\( ( 1 - 85566 T + 8033556999 T^{2} - 380834469871044 T^{3} + 20611909350541086591 T^{4} - \)\(56\!\cdots\!46\)\( T^{5} + \)\(16\!\cdots\!29\)\( T^{6} )^{2} \))
$41$	(\( ( 1 - 68921 T )( 1 + 68921 T ) \))(\( 1 - 9219079010 T^{2} + 22563490300366186081 T^{4} \))(\( 1 - 9487663202 T^{2} + 22563490300366186081 T^{4} \))(\( 1 - 7569104550 T^{2} + 1363608137405888943 T^{4} + \)\(15\!\cdots\!00\)\( T^{6} + \)\(30\!\cdots\!83\)\( T^{8} - \)\(38\!\cdots\!50\)\( T^{10} + \)\(11\!\cdots\!41\)\( T^{12} \))
$43$	(\( 1 + 111386 T + 6321363049 T^{2} \))(\( ( 1 - 6406 T + 6321363049 T^{2} )^{2} \))(\( ( 1 + 68618 T + 6321363049 T^{2} )^{2} \))(\( ( 1 - 145530 T + 21837177927 T^{2} - 1677260136965132 T^{3} + \)\(13\!\cdots\!23\)\( T^{4} - \)\(58\!\cdots\!30\)\( T^{5} + \)\(25\!\cdots\!49\)\( T^{6} )^{2} \))
$47$	(\( ( 1 - 103823 T )( 1 + 103823 T ) \))(\( 1 + 10801249342 T^{2} + \)\(11\!\cdots\!41\)\( T^{4} \))(\( 1 - 21106800578 T^{2} + \)\(11\!\cdots\!41\)\( T^{4} \))(\( 1 - 52339293510 T^{2} + \)\(12\!\cdots\!23\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{6} + \)\(14\!\cdots\!43\)\( T^{8} - \)\(70\!\cdots\!10\)\( T^{10} + \)\(15\!\cdots\!21\)\( T^{12} \))
$53$	(\( ( 1 - 148877 T )( 1 + 148877 T ) \))(\( 1 - 7253988050 T^{2} + \)\(49\!\cdots\!41\)\( T^{4} \))(\( 1 - 14819087378 T^{2} + \)\(49\!\cdots\!41\)\( T^{4} \))(\( 1 - 75107476374 T^{2} + \)\(29\!\cdots\!67\)\( T^{4} - \)\(77\!\cdots\!24\)\( T^{6} + \)\(14\!\cdots\!47\)\( T^{8} - \)\(18\!\cdots\!94\)\( T^{10} + \)\(11\!\cdots\!21\)\( T^{12} \))
$59$	(\( ( 1 - 205379 T )( 1 + 205379 T ) \))(\( 1 + 22449655150 T^{2} + \)\(17\!\cdots\!81\)\( T^{4} \))(\( 1 - 61991044562 T^{2} + \)\(17\!\cdots\!81\)\( T^{4} \))(\( 1 - 180693503190 T^{2} + \)\(15\!\cdots\!31\)\( T^{4} - \)\(77\!\cdots\!36\)\( T^{6} + \)\(26\!\cdots\!11\)\( T^{8} - \)\(57\!\cdots\!90\)\( T^{10} + \)\(56\!\cdots\!41\)\( T^{12} \))
$61$	(\( 1 + 420838 T + 51520374361 T^{2} \))(\( ( 1 + 62566 T + 51520374361 T^{2} )^{2} \))(\( ( 1 - 24794 T + 51520374361 T^{2} )^{2} \))(\( ( 1 - 296046 T + 42371190135 T^{2} - 4032498794872228 T^{3} + \)\(21\!\cdots\!35\)\( T^{4} - \)\(78\!\cdots\!66\)\( T^{5} + \)\(13\!\cdots\!81\)\( T^{6} )^{2} \))
$67$	(\( 1 + 172874 T + 90458382169 T^{2} \))(\( ( 1 + 438698 T + 90458382169 T^{2} )^{2} \))(\( ( 1 - 84358 T + 90458382169 T^{2} )^{2} \))(\( ( 1 - 285450 T + 220735162839 T^{2} - 36790378201873708 T^{3} + \)\(19\!\cdots\!91\)\( T^{4} - \)\(23\!\cdots\!50\)\( T^{5} + \)\(74\!\cdots\!09\)\( T^{6} )^{2} \))
$71$	(\( ( 1 - 357911 T )( 1 + 357911 T ) \))(\( 1 - 251546372642 T^{2} + \)\(16\!\cdots\!41\)\( T^{4} \))(\( 1 - 151063967522 T^{2} + \)\(16\!\cdots\!41\)\( T^{4} \))(\( 1 - 402983850726 T^{2} + \)\(77\!\cdots\!75\)\( T^{4} - \)\(10\!\cdots\!48\)\( T^{6} + \)\(12\!\cdots\!75\)\( T^{8} - \)\(10\!\cdots\!06\)\( T^{10} + \)\(44\!\cdots\!21\)\( T^{12} \))
$73$	(\( 1 - 638066 T + 151334226289 T^{2} \))(\( ( 1 + 730510 T + 151334226289 T^{2} )^{2} \))(\( ( 1 + 113806 T + 151334226289 T^{2} )^{2} \))(\( ( 1 - 559830 T + 328149232287 T^{2} - 146928200928984628 T^{3} + \)\(49\!\cdots\!43\)\( T^{4} - \)\(12\!\cdots\!30\)\( T^{5} + \)\(34\!\cdots\!69\)\( T^{6} )^{2} \))
$79$	(\( 1 - 204622 T + 243087455521 T^{2} \))(\( ( 1 + 340562 T + 243087455521 T^{2} )^{2} \))(\( ( 1 - 159742 T + 243087455521 T^{2} )^{2} \))(\( ( 1 - 526818 T + 705484689807 T^{2} - 230000056692908636 T^{3} + \)\(17\!\cdots\!47\)\( T^{4} - \)\(31\!\cdots\!38\)\( T^{5} + \)\(14\!\cdots\!61\)\( T^{6} )^{2} \))
$83$	(\( ( 1 - 571787 T )( 1 + 571787 T ) \))(\( 1 - 407613512306 T^{2} + \)\(10\!\cdots\!61\)\( T^{4} \))(\( 1 - 388294032818 T^{2} + \)\(10\!\cdots\!61\)\( T^{4} \))(\( 1 - 1779363154038 T^{2} + \)\(13\!\cdots\!67\)\( T^{4} - \)\(58\!\cdots\!36\)\( T^{6} + \)\(14\!\cdots\!87\)\( T^{8} - \)\(20\!\cdots\!98\)\( T^{10} + \)\(12\!\cdots\!81\)\( T^{12} \))
$89$	(\( ( 1 - 704969 T )( 1 + 704969 T ) \))(\( 1 - 844406214050 T^{2} + \)\(24\!\cdots\!21\)\( T^{4} \))(\( 1 + 583819025758 T^{2} + \)\(24\!\cdots\!21\)\( T^{4} \))(\( 1 - 1464277161318 T^{2} + \)\(11\!\cdots\!39\)\( T^{4} - \)\(68\!\cdots\!36\)\( T^{6} + \)\(28\!\cdots\!19\)\( T^{8} - \)\(89\!\cdots\!38\)\( T^{10} + \)\(15\!\cdots\!61\)\( T^{12} \))
$97$	(\( 1 + 56446 T + 832972004929 T^{2} \))(\( ( 1 + 281086 T + 832972004929 T^{2} )^{2} \))(\( ( 1 - 899522 T + 832972004929 T^{2} )^{2} \))(\( ( 1 + 399258 T + 1108436400207 T^{2} + 1034642865778948780 T^{3} + \)\(92\!\cdots\!03\)\( T^{4} + \)\(27\!\cdots\!78\)\( T^{5} + \)\(57\!\cdots\!89\)\( T^{6} )^{2} \))
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