# Properties

 Label 108.6.a Level 108 Weight 6 Character orbit a Rep. character $$\chi_{108}(1,\cdot)$$ Character field $$\Q$$ Dimension 7 Newform subspaces 4 Sturm bound 108 Trace bound 11

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 108.a (trivial) Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$108$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$5$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(\Gamma_0(108))$$.

Total New Old
Modular forms 99 7 92
Cusp forms 81 7 74
Eisenstein series 18 0 18

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

$$2$$$$3$$FrickeDim.
$$-$$$$+$$$$-$$$$4$$
$$-$$$$-$$$$+$$$$3$$
Plus space$$+$$$$3$$
Minus space$$-$$$$4$$

## Trace form

 $$7q - 31q^{7} + O(q^{10})$$ $$7q - 31q^{7} + 647q^{13} + 623q^{19} + 6961q^{25} + 6788q^{31} - 9007q^{37} + 46520q^{43} + 5238q^{49} - 90720q^{55} + 71141q^{61} + 18035q^{67} - 259009q^{73} + 248351q^{79} - 123120q^{85} - 455123q^{91} + 381443q^{97} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(\Gamma_0(108))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2 3
108.6.a.a $$1$$ $$17.321$$ $$\Q$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$-25$$ $$-$$ $$-$$ $$q-5^{2}q^{7}-427q^{13}-1711q^{19}-5^{5}q^{25}+\cdots$$
108.6.a.b $$2$$ $$17.321$$ $$\Q(\sqrt{41})$$ None $$0$$ $$0$$ $$0$$ $$-32$$ $$-$$ $$-$$ $$q-\beta q^{5}+(-2^{4}+3\beta )q^{7}+(-3^{5}+8\beta )q^{11}+\cdots$$
108.6.a.c $$2$$ $$17.321$$ $$\Q(\sqrt{41})$$ None $$0$$ $$0$$ $$0$$ $$-32$$ $$-$$ $$+$$ $$q-\beta q^{5}+(-2^{4}-3\beta )q^{7}+(3^{5}+8\beta )q^{11}+\cdots$$
108.6.a.d $$2$$ $$17.321$$ $$\Q(\sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$58$$ $$-$$ $$+$$ $$q+\beta q^{5}+29q^{7}+\beta q^{11}+329q^{13}+\cdots$$

## Decomposition of $$S_{6}^{\mathrm{old}}(\Gamma_0(108))$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(\Gamma_0(108)) \cong$$ $$S_{6}^{\mathrm{new}}(\Gamma_0(3))$$$$^{\oplus 9}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(4))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(6))$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(9))$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(18))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(27))$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(36))$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(54))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ ($$1 + 3125 T^{2}$$)($$1 + 2929 T^{2} + 9765625 T^{4}$$)($$1 + 2929 T^{2} + 9765625 T^{4}$$)($$1 - 1526 T^{2} + 9765625 T^{4}$$)
$7$ ($$1 + 25 T + 16807 T^{2}$$)($$1 + 32 T + 3981 T^{2} + 537824 T^{3} + 282475249 T^{4}$$)($$1 + 32 T + 3981 T^{2} + 537824 T^{3} + 282475249 T^{4}$$)($$( 1 - 29 T + 16807 T^{2} )^{2}$$)
$11$ ($$1 + 161051 T^{2}$$)($$1 + 486 T + 168607 T^{2} + 78270786 T^{3} + 25937424601 T^{4}$$)($$1 - 486 T + 168607 T^{2} - 78270786 T^{3} + 25937424601 T^{4}$$)($$1 + 314326 T^{2} + 25937424601 T^{4}$$)
$13$ ($$1 + 427 T + 371293 T^{2}$$)($$1 - 208 T + 275178 T^{2} - 77228944 T^{3} + 137858491849 T^{4}$$)($$1 - 208 T + 275178 T^{2} - 77228944 T^{3} + 137858491849 T^{4}$$)($$( 1 - 329 T + 371293 T^{2} )^{2}$$)
$17$ ($$1 + 1419857 T^{2}$$)($$1 + 1944 T + 2456098 T^{2} + 2760202008 T^{3} + 2015993900449 T^{4}$$)($$1 - 1944 T + 2456098 T^{2} - 2760202008 T^{3} + 2015993900449 T^{4}$$)($$1 - 2020286 T^{2} + 2015993900449 T^{4}$$)
$19$ ($$1 + 1711 T + 2476099 T^{2}$$)($$1 + 632 T + 4932498 T^{2} + 1564894568 T^{3} + 6131066257801 T^{4}$$)($$1 + 632 T + 4932498 T^{2} + 1564894568 T^{3} + 6131066257801 T^{4}$$)($$( 1 - 1799 T + 2476099 T^{2} )^{2}$$)
$23$ ($$1 + 6436343 T^{2}$$)($$1 + 6804 T + 24393154 T^{2} + 43792877772 T^{3} + 41426511213649 T^{4}$$)($$1 - 6804 T + 24393154 T^{2} - 43792877772 T^{3} + 41426511213649 T^{4}$$)($$1 - 198770 T^{2} + 41426511213649 T^{4}$$)
$29$ ($$1 + 20511149 T^{2}$$)($$1 + 11664 T + 70238998 T^{2} + 239242041936 T^{3} + 420707233300201 T^{4}$$)($$1 - 11664 T + 70238998 T^{2} - 239242041936 T^{3} + 420707233300201 T^{4}$$)($$1 + 39031642 T^{2} + 420707233300201 T^{4}$$)
$31$ ($$1 + 10324 T + 28629151 T^{2}$$)($$1 - 3328 T + 38238117 T^{2} - 95277814528 T^{3} + 819628286980801 T^{4}$$)($$1 - 3328 T + 38238117 T^{2} - 95277814528 T^{3} + 819628286980801 T^{4}$$)($$( 1 - 5228 T + 28629151 T^{2} )^{2}$$)
$37$ ($$1 + 6661 T + 69343957 T^{2}$$)($$1 + 9956 T + 151512798 T^{2} + 690388435892 T^{3} + 4808584372417849 T^{4}$$)($$1 + 9956 T + 151512798 T^{2} + 690388435892 T^{3} + 4808584372417849 T^{4}$$)($$( 1 - 8783 T + 69343957 T^{2} )^{2}$$)
$41$ ($$1 + 115856201 T^{2}$$)($$1 + 13608 T + 266834974 T^{2} + 1576571183208 T^{3} + 13422659310152401 T^{4}$$)($$1 - 13608 T + 266834974 T^{2} - 1576571183208 T^{3} + 13422659310152401 T^{4}$$)($$1 - 9156974 T^{2} + 13422659310152401 T^{4}$$)
$43$ ($$1 + 3352 T + 147008443 T^{2}$$)($$1 - 4960 T + 213010962 T^{2} - 729161877280 T^{3} + 21611482313284249 T^{4}$$)($$1 - 4960 T + 213010962 T^{2} - 729161877280 T^{3} + 21611482313284249 T^{4}$$)($$( 1 - 19976 T + 147008443 T^{2} )^{2}$$)
$47$ ($$1 + 229345007 T^{2}$$)($$1 + 18468 T + 312496354 T^{2} + 4235543589276 T^{3} + 52599132235830049 T^{4}$$)($$1 - 18468 T + 312496354 T^{2} - 4235543589276 T^{3} + 52599132235830049 T^{4}$$)($$1 + 341046910 T^{2} + 52599132235830049 T^{4}$$)
$53$ ($$1 + 418195493 T^{2}$$)($$1 - 11664 T + 484234009 T^{2} - 4877832230352 T^{3} + 174887470365513049 T^{4}$$)($$1 + 11664 T + 484234009 T^{2} + 4877832230352 T^{3} + 174887470365513049 T^{4}$$)($$1 - 31068470 T^{2} + 174887470365513049 T^{4}$$)
$59$ ($$1 + 714924299 T^{2}$$)($$1 + 1944 T + 961283686 T^{2} + 1389812837256 T^{3} + 511116753300641401 T^{4}$$)($$1 - 1944 T + 961283686 T^{2} - 1389812837256 T^{3} + 511116753300641401 T^{4}$$)($$1 + 1396994998 T^{2} + 511116753300641401 T^{4}$$)
$61$ ($$1 - 56927 T + 844596301 T^{2}$$)($$1 - 8176 T - 15702054 T^{2} - 6905419356976 T^{3} + 713342911662882601 T^{4}$$)($$1 - 8176 T - 15702054 T^{2} - 6905419356976 T^{3} + 713342911662882601 T^{4}$$)($$( 1 + 1069 T + 844596301 T^{2} )^{2}$$)
$67$ ($$1 + 37939 T + 1350125107 T^{2}$$)($$1 - 90064 T + 4055628738 T^{2} - 121597667636848 T^{3} + 1822837804551761449 T^{4}$$)($$1 - 90064 T + 4055628738 T^{2} - 121597667636848 T^{3} + 1822837804551761449 T^{4}$$)($$( 1 + 62077 T + 1350125107 T^{2} )^{2}$$)
$71$ ($$1 + 1804229351 T^{2}$$)($$1 - 44712 T + 2046094414 T^{2} - 80670702741912 T^{3} + 3255243551009881201 T^{4}$$)($$1 + 44712 T + 2046094414 T^{2} + 80670702741912 T^{3} + 3255243551009881201 T^{4}$$)($$1 + 1457026126 T^{2} + 3255243551009881201 T^{4}$$)
$73$ ($$1 - 79577 T + 2073071593 T^{2}$$)($$1 + 121214 T + 6975764499 T^{2} + 251285300073902 T^{3} + 4297625829703557649 T^{4}$$)($$1 + 121214 T + 6975764499 T^{2} + 251285300073902 T^{3} + 4297625829703557649 T^{4}$$)($$( 1 + 48079 T + 2073071593 T^{2} )^{2}$$)
$79$ ($$1 - 90857 T + 3077056399 T^{2}$$)($$1 - 28768 T + 4017714654 T^{2} - 88520758486432 T^{3} + 9468276082626847201 T^{4}$$)($$1 - 28768 T + 4017714654 T^{2} - 88520758486432 T^{3} + 9468276082626847201 T^{4}$$)($$( 1 - 49979 T + 3077056399 T^{2} )^{2}$$)
$83$ ($$1 + 3939040643 T^{2}$$)($$1 - 15066 T - 2258995409 T^{2} - 59345586327438 T^{3} + 15516041187205853449 T^{4}$$)($$1 + 15066 T - 2258995409 T^{2} + 59345586327438 T^{3} + 15516041187205853449 T^{4}$$)($$1 + 4552161670 T^{2} + 15516041187205853449 T^{4}$$)
$89$ ($$1 + 5584059449 T^{2}$$)($$1 - 178848 T + 18739138030 T^{2} - 998697864334752 T^{3} + 31181719929966183601 T^{4}$$)($$1 + 178848 T + 18739138030 T^{2} + 998697864334752 T^{3} + 31181719929966183601 T^{4}$$)($$1 + 3438704914 T^{2} + 31181719929966183601 T^{4}$$)
$97$ ($$1 - 177725 T + 8587340257 T^{2}$$)($$1 - 88942 T + 13779025491 T^{2} - 763775217138094 T^{3} + 73742412689492826049 T^{4}$$)($$1 - 88942 T + 13779025491 T^{2} - 763775217138094 T^{3} + 73742412689492826049 T^{4}$$)($$( 1 - 12917 T + 8587340257 T^{2} )^{2}$$)