LMFDB - Space of modular forms of level 108, weight 3, and Character 108.d

Defining parameters

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	108.d (of order $2$ and degree $1$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$ 4 $
Character field:			$\Q$
Newform subspaces:			$ 4 $
Sturm bound:			$54$
Trace bound:			$2$
Distinguishing $T_p$:			$5$

Dimensions

The following table gives the dimensions of various subspaces of $M_{3}(108, [\chi])$.

	Total	New	Old
Modular forms	42	16	26
Cusp forms	30	16	14
Eisenstein series	12	0	12

Trace form

Decomposition of $S_{3}^{\mathrm{new}}(108, [\chi])$ 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
108.3.d.a	$2$	$2.943$	$\Q(\sqrt{-3}) $	None	$-2$	$0$	$14$	$0$	$q+(-1-\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{4}+7q^{5}+\cdots$
108.3.d.b	$2$	$2.943$	$\Q(\sqrt{-3}) $	None	$2$	$0$	$-14$	$0$	$q+(1+\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{4}-7q^{5}+\cdots$
108.3.d.c	$4$	$2.943$	$\Q(\sqrt{-3}, \sqrt{13})$	None	$0$	$0$	$0$	$0$	$q+(-\beta _{1}-\beta _{2})q^{2}+(3+\beta _{3})q^{4}+(-2\beta _{1}+\cdots)q^{5}+\cdots$
108.3.d.d	$8$	$2.943$	8.0.207360000.1	None	$0$	$0$	$0$	$0$	$q-\beta _{2}q^{2}+(-1+\beta _{5})q^{4}+(-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots$

Decomposition of $S_{3}^{\mathrm{old}}(108, [\chi])$ into lower level spaces

$ S_{3}^{\mathrm{old}}(108, [\chi]) \cong $ $S_{3}^{\mathrm{new}}(12, [\chi])$$^{\oplus 3}$$\oplus$$S_{3}^{\mathrm{new}}(36, [\chi])$$^{\oplus 2}$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	($ 1 + 2 T + 4 T^{2} $)($ 1 - 2 T + 4 T^{2} $)($ 1 - 5 T^{2} + 16 T^{4} $)($ 1 + 2 T^{2} - 12 T^{4} + 32 T^{6} + 256 T^{8} $)
$3$	1
$5$	($ ( 1 - 7 T + 25 T^{2} )^{2} $)($ ( 1 + 7 T + 25 T^{2} )^{2} $)($ ( 1 + 37 T^{2} + 625 T^{4} )^{2} $)($ ( 1 + 44 T^{2} + 1014 T^{4} + 27500 T^{6} + 390625 T^{8} )^{2} $)
$7$	($ ( 1 - 11 T + 49 T^{2} )( 1 + 11 T + 49 T^{2} ) $)($ ( 1 - 11 T + 49 T^{2} )( 1 + 11 T + 49 T^{2} ) $)($ ( 1 - 59 T^{2} + 2401 T^{4} )^{2} $)($ ( 1 - 70 T^{2} + 5307 T^{4} - 168070 T^{6} + 5764801 T^{8} )^{2} $)
$11$	($ 1 - 167 T^{2} + 14641 T^{4} $)($ 1 - 167 T^{2} + 14641 T^{4} $)($ ( 1 - 95 T^{2} + 14641 T^{4} )^{2} $)($ ( 1 - 124 T^{2} + 15126 T^{4} - 1815484 T^{6} + 214358881 T^{8} )^{2} $)
$13$	($ ( 1 - 20 T + 169 T^{2} )^{2} $)($ ( 1 - 20 T + 169 T^{2} )^{2} $)($ ( 1 + 16 T + 169 T^{2} )^{4} $)($ ( 1 + 2 T + 159 T^{2} + 338 T^{3} + 28561 T^{4} )^{4} $)
$17$	($ ( 1 + 8 T + 289 T^{2} )^{2} $)($ ( 1 - 8 T + 289 T^{2} )^{2} $)($ ( 1 + 370 T^{2} + 83521 T^{4} )^{2} $)($ ( 1 + 140 T^{2} + 136662 T^{4} + 11692940 T^{6} + 6975757441 T^{8} )^{2} $)
$19$	($ 1 - 614 T^{2} + 130321 T^{4} $)($ 1 - 614 T^{2} + 130321 T^{4} $)($ ( 1 + 682 T^{2} + 130321 T^{4} )^{2} $)($ ( 1 - 695 T^{2} + 130321 T^{4} )^{4} $)
$23$	($ 1 - 1046 T^{2} + 279841 T^{4} $)($ 1 - 1046 T^{2} + 279841 T^{4} $)($ ( 1 - 758 T^{2} + 279841 T^{4} )^{2} $)($ ( 1 - 604 T^{2} + 632886 T^{4} - 169023964 T^{6} + 78310985281 T^{8} )^{2} $)
$29$	($ ( 1 - 10 T + 841 T^{2} )^{2} $)($ ( 1 + 10 T + 841 T^{2} )^{2} $)($ ( 1 - 866 T^{2} + 707281 T^{4} )^{2} $)($ ( 1 + 2468 T^{2} + 2752998 T^{4} + 1745569508 T^{6} + 500246412961 T^{8} )^{2} $)
$31$	($ ( 1 - 31 T + 961 T^{2} )( 1 + 31 T + 961 T^{2} ) $)($ ( 1 - 31 T + 961 T^{2} )( 1 + 31 T + 961 T^{2} ) $)($ ( 1 - 1883 T^{2} + 923521 T^{4} )^{2} $)($ ( 1 - 1540 T^{2} + 1702662 T^{4} - 1422222340 T^{6} + 852891037441 T^{8} )^{2} $)
$37$	($ ( 1 + 10 T + 1369 T^{2} )^{2} $)($ ( 1 + 10 T + 1369 T^{2} )^{2} $)($ ( 1 - 26 T + 1369 T^{2} )^{4} $)($ ( 1 + 14 T + 2607 T^{2} + 19166 T^{3} + 1874161 T^{4} )^{4} $)
$41$	($ ( 1 + 50 T + 1681 T^{2} )^{2} $)($ ( 1 - 50 T + 1681 T^{2} )^{2} $)($ ( 1 + 3310 T^{2} + 2825761 T^{4} )^{2} $)($ ( 1 + 3140 T^{2} + 5167302 T^{4} + 8872889540 T^{6} + 7984925229121 T^{8} )^{2} $)
$43$	($ 1 - 3398 T^{2} + 3418801 T^{4} $)($ 1 - 3398 T^{2} + 3418801 T^{4} $)($ ( 1 - 3542 T^{2} + 3418801 T^{4} )^{2} $)($ ( 1 - 4516 T^{2} + 10784166 T^{4} - 15439305316 T^{6} + 11688200277601 T^{8} )^{2} $)
$47$	($ 1 + 3082 T^{2} + 4879681 T^{4} $)($ 1 + 3082 T^{2} + 4879681 T^{4} $)($ ( 1 - 4406 T^{2} + 4879681 T^{4} )^{2} $)($ ( 1 - 4444 T^{2} + 14436726 T^{4} - 21685302364 T^{6} + 23811286661761 T^{8} )^{2} $)
$53$	($ ( 1 + 47 T + 2809 T^{2} )^{2} $)($ ( 1 - 47 T + 2809 T^{2} )^{2} $)($ ( 1 + 925 T^{2} + 7890481 T^{4} )^{2} $)($ ( 1 - 508 T^{2} + 14912358 T^{4} - 4008364348 T^{6} + 62259690411361 T^{8} )^{2} $)
$59$	($ 1 - 5762 T^{2} + 12117361 T^{4} $)($ 1 - 5762 T^{2} + 12117361 T^{4} $)($ ( 1 - 1154 T^{2} + 12117361 T^{4} )^{2} $)($ ( 1 - 6076 T^{2} + 23942166 T^{4} - 73625085436 T^{6} + 146830437604321 T^{8} )^{2} $)
$61$	($ ( 1 + 64 T + 3721 T^{2} )^{2} $)($ ( 1 + 64 T + 3721 T^{2} )^{2} $)($ ( 1 - 8 T + 3721 T^{2} )^{4} $)($ ( 1 - 10 T + 7287 T^{2} - 37210 T^{3} + 13845841 T^{4} )^{4} $)
$67$	($ 1 - 1478 T^{2} + 20151121 T^{4} $)($ 1 - 1478 T^{2} + 20151121 T^{4} $)($ ( 1 - 5078 T^{2} + 20151121 T^{4} )^{2} $)($ ( 1 - 8110 T^{2} + 40110387 T^{4} - 163425591310 T^{6} + 406067677556641 T^{8} )^{2} $)
$71$	($ ( 1 - 71 T )^{2}( 1 + 71 T )^{2} $)($ ( 1 - 71 T )^{2}( 1 + 71 T )^{2} $)($ ( 1 - 6194 T^{2} + 25411681 T^{4} )^{2} $)($ ( 1 - 292 T^{2} + 42446598 T^{4} - 7420210852 T^{6} + 645753531245761 T^{8} )^{2} $)
$73$	($ ( 1 + 55 T + 5329 T^{2} )^{2} $)($ ( 1 + 55 T + 5329 T^{2} )^{2} $)($ ( 1 + 19 T + 5329 T^{2} )^{4} $)($ ( 1 - 58 T + 10779 T^{2} - 309082 T^{3} + 28398241 T^{4} )^{4} $)
$79$	($ 1 - 12434 T^{2} + 38950081 T^{4} $)($ 1 - 12434 T^{2} + 38950081 T^{4} $)($ ( 1 - 9986 T^{2} + 38950081 T^{4} )^{2} $)($ ( 1 - 934 T^{2} - 28603749 T^{4} - 36379375654 T^{6} + 1517108809906561 T^{8} )^{2} $)
$83$	($ 1 - 12911 T^{2} + 47458321 T^{4} $)($ 1 - 12911 T^{2} + 47458321 T^{4} $)($ ( 1 - 311 T^{2} + 47458321 T^{4} )^{2} $)($ ( 1 - 26116 T^{2} + 265140006 T^{4} - 1239421511236 T^{6} + 2252292232139041 T^{8} )^{2} $)
$89$	($ ( 1 - 10 T + 7921 T^{2} )^{2} $)($ ( 1 + 10 T + 7921 T^{2} )^{2} $)($ ( 1 + 9550 T^{2} + 62742241 T^{4} )^{2} $)($ ( 1 + 30668 T^{2} + 360580758 T^{4} + 1924179046988 T^{6} + 3936588805702081 T^{8} )^{2} $)
$97$	($ ( 1 + 25 T + 9409 T^{2} )^{2} $)($ ( 1 + 25 T + 9409 T^{2} )^{2} $)($ ( 1 - 119 T + 9409 T^{2} )^{4} $)($ ( 1 + 62 T + 8259 T^{2} + 583358 T^{3} + 88529281 T^{4} )^{4} $)
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Introduction	Features
Universe	News

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	108.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(54\)
Trace bound:			\(2\)
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
108.3.d.a	\(2\)	\(2.943\)	\(\Q(\sqrt{-3}) \)	None	\(-2\)	\(0\)	\(14\)	\(0\)	\(q+(-1-\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{4}+7q^{5}+\cdots\)
108.3.d.b	\(2\)	\(2.943\)	\(\Q(\sqrt{-3}) \)	None	\(2\)	\(0\)	\(-14\)	\(0\)	\(q+(1+\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{4}-7q^{5}+\cdots\)
108.3.d.c	\(4\)	\(2.943\)	\(\Q(\sqrt{-3}, \sqrt{13})\)	None	\(0\)	\(0\)	\(0\)	\(0\)	\(q+(-\beta _{1}-\beta _{2})q^{2}+(3+\beta _{3})q^{4}+(-2\beta _{1}+\cdots)q^{5}+\cdots\)
108.3.d.d	\(8\)	\(2.943\)	8.0.207360000.1	None	\(0\)	\(0\)	\(0\)	\(0\)	\(q-\beta _{2}q^{2}+(-1+\beta _{5})q^{4}+(-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{3}^{\mathrm{new}}(108, [\chi])\) into newform subspaces

Decomposition of \(S_{3}^{\mathrm{old}}(108, [\chi])\) 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	108.3.d

Level	108
Weight	3
Character orbit	d
Rep. character	\(\chi_{108}(55,\cdot)\)
Character field	\(\Q\)
Dimension	16
Newform subspaces	4
Sturm bound	54
Trace bound	2

$p$	$F_p(T)$
$2$	(\( 1 + 2 T + 4 T^{2} \))(\( 1 - 2 T + 4 T^{2} \))(\( 1 - 5 T^{2} + 16 T^{4} \))(\( 1 + 2 T^{2} - 12 T^{4} + 32 T^{6} + 256 T^{8} \))
$3$	1
$5$	(\( ( 1 - 7 T + 25 T^{2} )^{2} \))(\( ( 1 + 7 T + 25 T^{2} )^{2} \))(\( ( 1 + 37 T^{2} + 625 T^{4} )^{2} \))(\( ( 1 + 44 T^{2} + 1014 T^{4} + 27500 T^{6} + 390625 T^{8} )^{2} \))
$7$	(\( ( 1 - 11 T + 49 T^{2} )( 1 + 11 T + 49 T^{2} ) \))(\( ( 1 - 11 T + 49 T^{2} )( 1 + 11 T + 49 T^{2} ) \))(\( ( 1 - 59 T^{2} + 2401 T^{4} )^{2} \))(\( ( 1 - 70 T^{2} + 5307 T^{4} - 168070 T^{6} + 5764801 T^{8} )^{2} \))
$11$	(\( 1 - 167 T^{2} + 14641 T^{4} \))(\( 1 - 167 T^{2} + 14641 T^{4} \))(\( ( 1 - 95 T^{2} + 14641 T^{4} )^{2} \))(\( ( 1 - 124 T^{2} + 15126 T^{4} - 1815484 T^{6} + 214358881 T^{8} )^{2} \))
$13$	(\( ( 1 - 20 T + 169 T^{2} )^{2} \))(\( ( 1 - 20 T + 169 T^{2} )^{2} \))(\( ( 1 + 16 T + 169 T^{2} )^{4} \))(\( ( 1 + 2 T + 159 T^{2} + 338 T^{3} + 28561 T^{4} )^{4} \))
$17$	(\( ( 1 + 8 T + 289 T^{2} )^{2} \))(\( ( 1 - 8 T + 289 T^{2} )^{2} \))(\( ( 1 + 370 T^{2} + 83521 T^{4} )^{2} \))(\( ( 1 + 140 T^{2} + 136662 T^{4} + 11692940 T^{6} + 6975757441 T^{8} )^{2} \))
$19$	(\( 1 - 614 T^{2} + 130321 T^{4} \))(\( 1 - 614 T^{2} + 130321 T^{4} \))(\( ( 1 + 682 T^{2} + 130321 T^{4} )^{2} \))(\( ( 1 - 695 T^{2} + 130321 T^{4} )^{4} \))
$23$	(\( 1 - 1046 T^{2} + 279841 T^{4} \))(\( 1 - 1046 T^{2} + 279841 T^{4} \))(\( ( 1 - 758 T^{2} + 279841 T^{4} )^{2} \))(\( ( 1 - 604 T^{2} + 632886 T^{4} - 169023964 T^{6} + 78310985281 T^{8} )^{2} \))
$29$	(\( ( 1 - 10 T + 841 T^{2} )^{2} \))(\( ( 1 + 10 T + 841 T^{2} )^{2} \))(\( ( 1 - 866 T^{2} + 707281 T^{4} )^{2} \))(\( ( 1 + 2468 T^{2} + 2752998 T^{4} + 1745569508 T^{6} + 500246412961 T^{8} )^{2} \))
$31$	(\( ( 1 - 31 T + 961 T^{2} )( 1 + 31 T + 961 T^{2} ) \))(\( ( 1 - 31 T + 961 T^{2} )( 1 + 31 T + 961 T^{2} ) \))(\( ( 1 - 1883 T^{2} + 923521 T^{4} )^{2} \))(\( ( 1 - 1540 T^{2} + 1702662 T^{4} - 1422222340 T^{6} + 852891037441 T^{8} )^{2} \))
$37$	(\( ( 1 + 10 T + 1369 T^{2} )^{2} \))(\( ( 1 + 10 T + 1369 T^{2} )^{2} \))(\( ( 1 - 26 T + 1369 T^{2} )^{4} \))(\( ( 1 + 14 T + 2607 T^{2} + 19166 T^{3} + 1874161 T^{4} )^{4} \))
$41$	(\( ( 1 + 50 T + 1681 T^{2} )^{2} \))(\( ( 1 - 50 T + 1681 T^{2} )^{2} \))(\( ( 1 + 3310 T^{2} + 2825761 T^{4} )^{2} \))(\( ( 1 + 3140 T^{2} + 5167302 T^{4} + 8872889540 T^{6} + 7984925229121 T^{8} )^{2} \))
$43$	(\( 1 - 3398 T^{2} + 3418801 T^{4} \))(\( 1 - 3398 T^{2} + 3418801 T^{4} \))(\( ( 1 - 3542 T^{2} + 3418801 T^{4} )^{2} \))(\( ( 1 - 4516 T^{2} + 10784166 T^{4} - 15439305316 T^{6} + 11688200277601 T^{8} )^{2} \))
$47$	(\( 1 + 3082 T^{2} + 4879681 T^{4} \))(\( 1 + 3082 T^{2} + 4879681 T^{4} \))(\( ( 1 - 4406 T^{2} + 4879681 T^{4} )^{2} \))(\( ( 1 - 4444 T^{2} + 14436726 T^{4} - 21685302364 T^{6} + 23811286661761 T^{8} )^{2} \))
$53$	(\( ( 1 + 47 T + 2809 T^{2} )^{2} \))(\( ( 1 - 47 T + 2809 T^{2} )^{2} \))(\( ( 1 + 925 T^{2} + 7890481 T^{4} )^{2} \))(\( ( 1 - 508 T^{2} + 14912358 T^{4} - 4008364348 T^{6} + 62259690411361 T^{8} )^{2} \))
$59$	(\( 1 - 5762 T^{2} + 12117361 T^{4} \))(\( 1 - 5762 T^{2} + 12117361 T^{4} \))(\( ( 1 - 1154 T^{2} + 12117361 T^{4} )^{2} \))(\( ( 1 - 6076 T^{2} + 23942166 T^{4} - 73625085436 T^{6} + 146830437604321 T^{8} )^{2} \))
$61$	(\( ( 1 + 64 T + 3721 T^{2} )^{2} \))(\( ( 1 + 64 T + 3721 T^{2} )^{2} \))(\( ( 1 - 8 T + 3721 T^{2} )^{4} \))(\( ( 1 - 10 T + 7287 T^{2} - 37210 T^{3} + 13845841 T^{4} )^{4} \))
$67$	(\( 1 - 1478 T^{2} + 20151121 T^{4} \))(\( 1 - 1478 T^{2} + 20151121 T^{4} \))(\( ( 1 - 5078 T^{2} + 20151121 T^{4} )^{2} \))(\( ( 1 - 8110 T^{2} + 40110387 T^{4} - 163425591310 T^{6} + 406067677556641 T^{8} )^{2} \))
$71$	(\( ( 1 - 71 T )^{2}( 1 + 71 T )^{2} \))(\( ( 1 - 71 T )^{2}( 1 + 71 T )^{2} \))(\( ( 1 - 6194 T^{2} + 25411681 T^{4} )^{2} \))(\( ( 1 - 292 T^{2} + 42446598 T^{4} - 7420210852 T^{6} + 645753531245761 T^{8} )^{2} \))
$73$	(\( ( 1 + 55 T + 5329 T^{2} )^{2} \))(\( ( 1 + 55 T + 5329 T^{2} )^{2} \))(\( ( 1 + 19 T + 5329 T^{2} )^{4} \))(\( ( 1 - 58 T + 10779 T^{2} - 309082 T^{3} + 28398241 T^{4} )^{4} \))
$79$	(\( 1 - 12434 T^{2} + 38950081 T^{4} \))(\( 1 - 12434 T^{2} + 38950081 T^{4} \))(\( ( 1 - 9986 T^{2} + 38950081 T^{4} )^{2} \))(\( ( 1 - 934 T^{2} - 28603749 T^{4} - 36379375654 T^{6} + 1517108809906561 T^{8} )^{2} \))
$83$	(\( 1 - 12911 T^{2} + 47458321 T^{4} \))(\( 1 - 12911 T^{2} + 47458321 T^{4} \))(\( ( 1 - 311 T^{2} + 47458321 T^{4} )^{2} \))(\( ( 1 - 26116 T^{2} + 265140006 T^{4} - 1239421511236 T^{6} + 2252292232139041 T^{8} )^{2} \))
$89$	(\( ( 1 - 10 T + 7921 T^{2} )^{2} \))(\( ( 1 + 10 T + 7921 T^{2} )^{2} \))(\( ( 1 + 9550 T^{2} + 62742241 T^{4} )^{2} \))(\( ( 1 + 30668 T^{2} + 360580758 T^{4} + 1924179046988 T^{6} + 3936588805702081 T^{8} )^{2} \))
$97$	(\( ( 1 + 25 T + 9409 T^{2} )^{2} \))(\( ( 1 + 25 T + 9409 T^{2} )^{2} \))(\( ( 1 - 119 T + 9409 T^{2} )^{4} \))(\( ( 1 + 62 T + 8259 T^{2} + 583358 T^{3} + 88529281 T^{4} )^{4} \))
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