LMFDB - Space of modular forms of level 108, weight 4, and Character 108.h

Defining parameters

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	108.h (of order $6$ and degree $2$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$ 36 $
Character field:			$\Q(\zeta_{6})$
Newform subspaces:			$ 2 $
Sturm bound:			$72$
Trace bound:			$1$
Distinguishing $T_p$:			$5$

Dimensions

The following table gives the dimensions of various subspaces of $M_{4}(108, [\chi])$.

	Total	New	Old
Modular forms	120	40	80
Cusp forms	96	32	64
Eisenstein series	24	8	16

Trace form

Decomposition of $S_{4}^{\mathrm{new}}(108, [\chi])$ 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
108.4.h.a	$8$	$6.372$	8.0.$\cdots$.1	None	$3$	$0$	$-66$	$0$	$q+\beta _{7}q^{2}+(2\beta _{2}-2\beta _{3}-2\beta _{4}+\beta _{6}+\cdots)q^{4}+\cdots$
108.4.h.b	$24$	$6.372$		None	$0$	$0$	$72$	$0$

Decomposition of $S_{4}^{\mathrm{old}}(108, [\chi])$ into lower level spaces

$ S_{4}^{\mathrm{old}}(108, [\chi]) \cong $ $S_{4}^{\mathrm{new}}(36, [\chi])$$^{\oplus 2}$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	($ 1 - 3 T - T^{2} + 12 T^{3} - 24 T^{4} + 96 T^{5} - 64 T^{6} - 1536 T^{7} + 4096 T^{8} $)
$3$	1
$5$	($ ( 1 + 33 T + 683 T^{2} + 10560 T^{3} + 132150 T^{4} + 1320000 T^{5} + 10671875 T^{6} + 64453125 T^{7} + 244140625 T^{8} )^{2} $)
$7$	($ 1 + 763 T^{2} + 286549 T^{4} + 46025686 T^{6} + 5058296518 T^{8} + 5414875932214 T^{10} + 3966207006159349 T^{12} + 1242479575205672587 T^{14} + $$19\!\cdots\!01$$ T^{16} $)
$11$	($ 1 - 3248 T^{2} + 5235775 T^{4} - 5750931536 T^{6} + 6408197652976 T^{8} - 10188126022847696 T^{10} + 16432104834126393775 T^{12} - $$18\!\cdots\!88$$ T^{14} + $$98\!\cdots\!41$$ T^{16} $)
$13$	($ ( 1 - 107 T + 4255 T^{2} - 299600 T^{3} + 21773374 T^{4} - 658221200 T^{5} + 20538072295 T^{6} - 1134681432911 T^{7} + 23298085122481 T^{8} )^{2} $)
$17$	($ ( 1 - 9325 T^{2} + 46774812 T^{4} - 225082830925 T^{6} + 582622237229761 T^{8} )^{2} $)
$19$	($ ( 1 + 239 T^{2} + 51737136 T^{4} + 11243965559 T^{6} + 2213314919066161 T^{8} )^{2} $)
$23$	($ 1 - 37877 T^{2} + 781967941 T^{4} - 13507976408570 T^{6} + 191719381723246342 T^{8} - $$19\!\cdots\!30$$ T^{10} + $$17\!\cdots\!61$$ T^{12} - $$12\!\cdots\!13$$ T^{14} + $$48\!\cdots\!41$$ T^{16} $)
$29$	($ ( 1 - 249 T + 63635 T^{2} - 10699032 T^{3} + 1755473166 T^{4} - 260938691448 T^{5} + 37851582031835 T^{6} - 3612279347991381 T^{7} + 353814783205469041 T^{8} )^{2} $)
$31$	($ 1 + 72991 T^{2} + 2479562881 T^{4} + 78327798131458 T^{6} + 2595801970489726078 T^{8} + $$69\!\cdots\!98$$ T^{10} + $$19\!\cdots\!41$$ T^{12} + $$51\!\cdots\!31$$ T^{14} + $$62\!\cdots\!21$$ T^{16} $)
$37$	($ ( 1 + 314 T + 125706 T^{2} + 15905042 T^{3} + 2565726409 T^{4} )^{4} $)
$41$	($ ( 1 - 636 T + 254507 T^{2} - 76113300 T^{3} + 18864757656 T^{4} - 5245804749300 T^{5} + 1208934780064187 T^{6} - 208214910274559196 T^{7} + 22563490300366186081 T^{8} )^{2} $)
$43$	($ 1 + 261520 T^{2} + 39141940543 T^{4} + 4343335603853680 T^{6} + $$38\!\cdots\!48$$ T^{8} + $$27\!\cdots\!20$$ T^{10} + $$15\!\cdots\!43$$ T^{12} + $$66\!\cdots\!80$$ T^{14} + $$15\!\cdots\!01$$ T^{16} $)
$47$	($ 1 - 58493 T^{2} - 17900011739 T^{4} + 13862131479910 T^{6} + $$30\!\cdots\!82$$ T^{8} + $$14\!\cdots\!90$$ T^{10} - $$20\!\cdots\!99$$ T^{12} - $$73\!\cdots\!77$$ T^{14} + $$13\!\cdots\!81$$ T^{16} $)
$53$	($ ( 1 - 363880 T^{2} + 76436876862 T^{4} - 8065167727620520 T^{6} + $$49\!\cdots\!41$$ T^{8} )^{2} $)
$59$	($ 1 - 717032 T^{2} + 302479462375 T^{4} - 91274129085638744 T^{6} + $$21\!\cdots\!36$$ T^{8} - $$38\!\cdots\!04$$ T^{10} + $$53\!\cdots\!75$$ T^{12} - $$53\!\cdots\!72$$ T^{14} + $$31\!\cdots\!61$$ T^{16} $)
$61$	($ ( 1 - 131 T - 127289 T^{2} + 40546072 T^{3} - 34549792802 T^{4} + 9203187968632 T^{5} - 6557976932037329 T^{6} - 1531933138161272471 T^{7} + $$26\!\cdots\!21$$ T^{8} )^{2} $)
$67$	($ 1 + 1070656 T^{2} + 683126750095 T^{4} + 302204171872082368 T^{6} + $$10\!\cdots\!36$$ T^{8} + $$27\!\cdots\!92$$ T^{10} + $$55\!\cdots\!95$$ T^{12} + $$79\!\cdots\!04$$ T^{14} + $$66\!\cdots\!21$$ T^{16} $)
$71$	($ ( 1 + 1260428 T^{2} + 653321166774 T^{4} + 161461184661978188 T^{6} + $$16\!\cdots\!41$$ T^{8} )^{2} $)
$73$	($ ( 1 - 985 T + 570834 T^{2} - 383181745 T^{3} + 151334226289 T^{4} )^{4} $)
$79$	($ 1 + 259483 T^{2} - 289013191643 T^{4} - 33688753687579130 T^{6} + $$51\!\cdots\!14$$ T^{8} - $$81\!\cdots\!30$$ T^{10} - $$17\!\cdots\!63$$ T^{12} + $$37\!\cdots\!63$$ T^{14} + $$34\!\cdots\!81$$ T^{16} $)
$83$	($ 1 - 1120157 T^{2} + 290685528061 T^{4} - 347456380544486450 T^{6} + $$41\!\cdots\!02$$ T^{8} - $$11\!\cdots\!50$$ T^{10} + $$31\!\cdots\!21$$ T^{12} - $$39\!\cdots\!13$$ T^{14} + $$11\!\cdots\!21$$ T^{16} $)
$89$	($ ( 1 - 2694616 T^{2} + 2805913145742 T^{4} - 1339173738324165976 T^{6} + $$24\!\cdots\!21$$ T^{8} )^{2} $)
$97$	($ ( 1 + 286 T - 1237115 T^{2} - 144840410 T^{3} + 831901220284 T^{4} - 132191931515930 T^{5} - 1030482161877739835 T^{6} + $$21\!\cdots\!62$$ T^{7} + $$69\!\cdots\!41$$ T^{8} )^{2} $)
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Introduction	Features
Universe	News

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	108.h (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 36 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(72\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(5\)

Label	Dim.	\(A\)	Field	CM	Traces				$q$-expansion
Label	Dim.	\(A\)	Field	CM	\(a_2\)	\(a_3\)	\(a_5\)	\(a_7\)	$q$-expansion
108.4.h.a	\(8\)	\(6.372\)	8.0.\(\cdots\).1	None	\(3\)	\(0\)	\(-66\)	\(0\)	\(q+\beta _{7}q^{2}+(2\beta _{2}-2\beta _{3}-2\beta _{4}+\beta _{6}+\cdots)q^{4}+\cdots\)
108.4.h.b	\(24\)	\(6.372\)		None	\(0\)	\(0\)	\(72\)	\(0\)

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{4}^{\mathrm{new}}(108, [\chi])\) into newform subspaces

Decomposition of \(S_{4}^{\mathrm{old}}(108, [\chi])\) 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	108.4.h

Level	108
Weight	4
Character orbit	h
Rep. character	\(\chi_{108}(35,\cdot)\)
Character field	\(\Q(\zeta_{6})\)
Dimension	32
Newform subspaces	2
Sturm bound	72
Trace bound	1

$p$	$F_p(T)$
$2$	(\( 1 - 3 T - T^{2} + 12 T^{3} - 24 T^{4} + 96 T^{5} - 64 T^{6} - 1536 T^{7} + 4096 T^{8} \))
$3$	1
$5$	(\( ( 1 + 33 T + 683 T^{2} + 10560 T^{3} + 132150 T^{4} + 1320000 T^{5} + 10671875 T^{6} + 64453125 T^{7} + 244140625 T^{8} )^{2} \))
$7$	(\( 1 + 763 T^{2} + 286549 T^{4} + 46025686 T^{6} + 5058296518 T^{8} + 5414875932214 T^{10} + 3966207006159349 T^{12} + 1242479575205672587 T^{14} + \)\(19\!\cdots\!01\)\( T^{16} \))
$11$	(\( 1 - 3248 T^{2} + 5235775 T^{4} - 5750931536 T^{6} + 6408197652976 T^{8} - 10188126022847696 T^{10} + 16432104834126393775 T^{12} - \)\(18\!\cdots\!88\)\( T^{14} + \)\(98\!\cdots\!41\)\( T^{16} \))
$13$	(\( ( 1 - 107 T + 4255 T^{2} - 299600 T^{3} + 21773374 T^{4} - 658221200 T^{5} + 20538072295 T^{6} - 1134681432911 T^{7} + 23298085122481 T^{8} )^{2} \))
$17$	(\( ( 1 - 9325 T^{2} + 46774812 T^{4} - 225082830925 T^{6} + 582622237229761 T^{8} )^{2} \))
$19$	(\( ( 1 + 239 T^{2} + 51737136 T^{4} + 11243965559 T^{6} + 2213314919066161 T^{8} )^{2} \))
$23$	(\( 1 - 37877 T^{2} + 781967941 T^{4} - 13507976408570 T^{6} + 191719381723246342 T^{8} - \)\(19\!\cdots\!30\)\( T^{10} + \)\(17\!\cdots\!61\)\( T^{12} - \)\(12\!\cdots\!13\)\( T^{14} + \)\(48\!\cdots\!41\)\( T^{16} \))
$29$	(\( ( 1 - 249 T + 63635 T^{2} - 10699032 T^{3} + 1755473166 T^{4} - 260938691448 T^{5} + 37851582031835 T^{6} - 3612279347991381 T^{7} + 353814783205469041 T^{8} )^{2} \))
$31$	(\( 1 + 72991 T^{2} + 2479562881 T^{4} + 78327798131458 T^{6} + 2595801970489726078 T^{8} + \)\(69\!\cdots\!98\)\( T^{10} + \)\(19\!\cdots\!41\)\( T^{12} + \)\(51\!\cdots\!31\)\( T^{14} + \)\(62\!\cdots\!21\)\( T^{16} \))
$37$	(\( ( 1 + 314 T + 125706 T^{2} + 15905042 T^{3} + 2565726409 T^{4} )^{4} \))
$41$	(\( ( 1 - 636 T + 254507 T^{2} - 76113300 T^{3} + 18864757656 T^{4} - 5245804749300 T^{5} + 1208934780064187 T^{6} - 208214910274559196 T^{7} + 22563490300366186081 T^{8} )^{2} \))
$43$	(\( 1 + 261520 T^{2} + 39141940543 T^{4} + 4343335603853680 T^{6} + \)\(38\!\cdots\!48\)\( T^{8} + \)\(27\!\cdots\!20\)\( T^{10} + \)\(15\!\cdots\!43\)\( T^{12} + \)\(66\!\cdots\!80\)\( T^{14} + \)\(15\!\cdots\!01\)\( T^{16} \))
$47$	(\( 1 - 58493 T^{2} - 17900011739 T^{4} + 13862131479910 T^{6} + \)\(30\!\cdots\!82\)\( T^{8} + \)\(14\!\cdots\!90\)\( T^{10} - \)\(20\!\cdots\!99\)\( T^{12} - \)\(73\!\cdots\!77\)\( T^{14} + \)\(13\!\cdots\!81\)\( T^{16} \))
$53$	(\( ( 1 - 363880 T^{2} + 76436876862 T^{4} - 8065167727620520 T^{6} + \)\(49\!\cdots\!41\)\( T^{8} )^{2} \))
$59$	(\( 1 - 717032 T^{2} + 302479462375 T^{4} - 91274129085638744 T^{6} + \)\(21\!\cdots\!36\)\( T^{8} - \)\(38\!\cdots\!04\)\( T^{10} + \)\(53\!\cdots\!75\)\( T^{12} - \)\(53\!\cdots\!72\)\( T^{14} + \)\(31\!\cdots\!61\)\( T^{16} \))
$61$	(\( ( 1 - 131 T - 127289 T^{2} + 40546072 T^{3} - 34549792802 T^{4} + 9203187968632 T^{5} - 6557976932037329 T^{6} - 1531933138161272471 T^{7} + \)\(26\!\cdots\!21\)\( T^{8} )^{2} \))
$67$	(\( 1 + 1070656 T^{2} + 683126750095 T^{4} + 302204171872082368 T^{6} + \)\(10\!\cdots\!36\)\( T^{8} + \)\(27\!\cdots\!92\)\( T^{10} + \)\(55\!\cdots\!95\)\( T^{12} + \)\(79\!\cdots\!04\)\( T^{14} + \)\(66\!\cdots\!21\)\( T^{16} \))
$71$	(\( ( 1 + 1260428 T^{2} + 653321166774 T^{4} + 161461184661978188 T^{6} + \)\(16\!\cdots\!41\)\( T^{8} )^{2} \))
$73$	(\( ( 1 - 985 T + 570834 T^{2} - 383181745 T^{3} + 151334226289 T^{4} )^{4} \))
$79$	(\( 1 + 259483 T^{2} - 289013191643 T^{4} - 33688753687579130 T^{6} + \)\(51\!\cdots\!14\)\( T^{8} - \)\(81\!\cdots\!30\)\( T^{10} - \)\(17\!\cdots\!63\)\( T^{12} + \)\(37\!\cdots\!63\)\( T^{14} + \)\(34\!\cdots\!81\)\( T^{16} \))
$83$	(\( 1 - 1120157 T^{2} + 290685528061 T^{4} - 347456380544486450 T^{6} + \)\(41\!\cdots\!02\)\( T^{8} - \)\(11\!\cdots\!50\)\( T^{10} + \)\(31\!\cdots\!21\)\( T^{12} - \)\(39\!\cdots\!13\)\( T^{14} + \)\(11\!\cdots\!21\)\( T^{16} \))
$89$	(\( ( 1 - 2694616 T^{2} + 2805913145742 T^{4} - 1339173738324165976 T^{6} + \)\(24\!\cdots\!21\)\( T^{8} )^{2} \))
$97$	(\( ( 1 + 286 T - 1237115 T^{2} - 144840410 T^{3} + 831901220284 T^{4} - 132191931515930 T^{5} - 1030482161877739835 T^{6} + \)\(21\!\cdots\!62\)\( T^{7} + \)\(69\!\cdots\!41\)\( T^{8} )^{2} \))
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