# Properties

 Label 105.2.s Level 105 Weight 2 Character orbit s Rep. character $$\chi_{105}(26,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 20 Newform subspaces 4 Sturm bound 32 Trace bound 2

# Related objects

## Defining parameters

 Level: $$N$$ = $$105 = 3 \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 105.s (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$21$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$4$$ Sturm bound: $$32$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(105, [\chi])$$.

Total New Old
Modular forms 40 20 20
Cusp forms 24 20 4
Eisenstein series 16 0 16

## Trace form

 $$20q + 8q^{4} - 6q^{7} - 4q^{9} + O(q^{10})$$ $$20q + 8q^{4} - 6q^{7} - 4q^{9} - 30q^{12} + 4q^{15} + 12q^{16} - 2q^{18} - 6q^{19} + 6q^{21} - 56q^{22} + 18q^{24} - 10q^{25} + 4q^{28} + 2q^{30} - 54q^{31} + 24q^{33} + 12q^{36} + 6q^{37} + 6q^{39} + 30q^{42} + 36q^{43} - 18q^{45} - 20q^{46} + 14q^{49} - 12q^{51} + 48q^{52} + 24q^{54} + 36q^{57} - 32q^{58} - 6q^{60} + 12q^{61} - 34q^{63} + 72q^{64} - 12q^{66} + 34q^{67} - 10q^{72} + 42q^{73} - 48q^{78} - 26q^{79} - 4q^{81} + 72q^{82} - 48q^{84} - 24q^{85} - 6q^{87} - 16q^{88} + 6q^{91} + 42q^{93} - 48q^{94} - 18q^{96} - 104q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(105, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
105.2.s.a $$2$$ $$0.838$$ $$\Q(\sqrt{-3})$$ None $$-3$$ $$-3$$ $$-1$$ $$-5$$ $$q+(-2+\zeta_{6})q^{2}+(-2+\zeta_{6})q^{3}+(1+\cdots)q^{4}+\cdots$$
105.2.s.b $$2$$ $$0.838$$ $$\Q(\sqrt{-3})$$ None $$3$$ $$0$$ $$1$$ $$-5$$ $$q+(2-\zeta_{6})q^{2}+(1-2\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots$$
105.2.s.c $$8$$ $$0.838$$ 8.0.856615824.2 None $$-3$$ $$1$$ $$4$$ $$2$$ $$q+(\beta _{1}+\beta _{3})q^{2}+(1-\beta _{1}+\beta _{4}+\beta _{6}+\cdots)q^{3}+\cdots$$
105.2.s.d $$8$$ $$0.838$$ 8.0.856615824.2 None $$3$$ $$2$$ $$-4$$ $$2$$ $$q-\beta _{3}q^{2}+(1+\beta _{3}+\beta _{6})q^{3}+(-1+\beta _{2}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(105, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(105, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}$$)($$1 - 3 T + 5 T^{2} - 6 T^{3} + 4 T^{4}$$)($$1 + 3 T + 7 T^{2} + 12 T^{3} + 16 T^{4} + 12 T^{5} - 8 T^{6} - 36 T^{7} - 68 T^{8} - 72 T^{9} - 32 T^{10} + 96 T^{11} + 256 T^{12} + 384 T^{13} + 448 T^{14} + 384 T^{15} + 256 T^{16}$$)($$1 - 3 T + 7 T^{2} - 12 T^{3} + 16 T^{4} - 12 T^{5} - 8 T^{6} + 36 T^{7} - 68 T^{8} + 72 T^{9} - 32 T^{10} - 96 T^{11} + 256 T^{12} - 384 T^{13} + 448 T^{14} - 384 T^{15} + 256 T^{16}$$)
$3$ ($$1 + 3 T + 3 T^{2}$$)($$1 + 3 T^{2}$$)($$1 - T + 3 T^{2} - 4 T^{3} + 16 T^{4} - 12 T^{5} + 27 T^{6} - 27 T^{7} + 81 T^{8}$$)($$1 - 2 T + 4 T^{3} - 11 T^{4} + 12 T^{5} - 54 T^{7} + 81 T^{8}$$)
$5$ ($$1 + T + T^{2}$$)($$1 - T + T^{2}$$)($$( 1 - T + T^{2} )^{4}$$)($$( 1 + T + T^{2} )^{4}$$)
$7$ ($$1 + 5 T + 7 T^{2}$$)($$1 + 5 T + 7 T^{2}$$)($$1 - 2 T + 4 T^{2} + 10 T^{3} - 41 T^{4} + 70 T^{5} + 196 T^{6} - 686 T^{7} + 2401 T^{8}$$)($$1 - 2 T + 4 T^{2} + 10 T^{3} - 41 T^{4} + 70 T^{5} + 196 T^{6} - 686 T^{7} + 2401 T^{8}$$)
$11$ ($$1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4}$$)($$1 - 6 T + 23 T^{2} - 66 T^{3} + 121 T^{4}$$)($$1 + 16 T^{2} - 2 T^{4} + 30 T^{5} + 268 T^{6} + 1548 T^{7} + 21079 T^{8} + 17028 T^{9} + 32428 T^{10} + 39930 T^{11} - 29282 T^{12} + 28344976 T^{14} + 214358881 T^{16}$$)($$1 + 16 T^{2} - 2 T^{4} - 30 T^{5} + 268 T^{6} - 1548 T^{7} + 21079 T^{8} - 17028 T^{9} + 32428 T^{10} - 39930 T^{11} - 29282 T^{12} + 28344976 T^{14} + 214358881 T^{16}$$)
$13$ ($$1 - 14 T^{2} + 169 T^{4}$$)($$1 - 14 T^{2} + 169 T^{4}$$)($$1 - 83 T^{2} + 3217 T^{4} - 76058 T^{6} + 1197778 T^{8} - 12853802 T^{10} + 91880737 T^{12} - 400625147 T^{14} + 815730721 T^{16}$$)($$1 - 83 T^{2} + 3217 T^{4} - 76058 T^{6} + 1197778 T^{8} - 12853802 T^{10} + 91880737 T^{12} - 400625147 T^{14} + 815730721 T^{16}$$)
$17$ ($$1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4}$$)($$1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4}$$)($$1 + 12 T + 34 T^{2} + 12 T^{3} + 1078 T^{4} + 6882 T^{5} + 8740 T^{6} + 70272 T^{7} + 637627 T^{8} + 1194624 T^{9} + 2525860 T^{10} + 33811266 T^{11} + 90035638 T^{12} + 17038284 T^{13} + 820677346 T^{14} + 4924064076 T^{15} + 6975757441 T^{16}$$)($$1 - 12 T + 34 T^{2} - 12 T^{3} + 1078 T^{4} - 6882 T^{5} + 8740 T^{6} - 70272 T^{7} + 637627 T^{8} - 1194624 T^{9} + 2525860 T^{10} - 33811266 T^{11} + 90035638 T^{12} - 17038284 T^{13} + 820677346 T^{14} - 4924064076 T^{15} + 6975757441 T^{16}$$)
$19$ ($$1 + 12 T + 67 T^{2} + 228 T^{3} + 361 T^{4}$$)($$1 + 12 T + 67 T^{2} + 228 T^{3} + 361 T^{4}$$)($$1 - 9 T + 100 T^{2} - 657 T^{3} + 4723 T^{4} - 26244 T^{5} + 148996 T^{6} - 704196 T^{7} + 3331528 T^{8} - 13379724 T^{9} + 53787556 T^{10} - 180007596 T^{11} + 615506083 T^{12} - 1626797043 T^{13} + 4704588100 T^{14} - 8044845651 T^{15} + 16983563041 T^{16}$$)($$1 - 9 T + 100 T^{2} - 657 T^{3} + 4723 T^{4} - 26244 T^{5} + 148996 T^{6} - 704196 T^{7} + 3331528 T^{8} - 13379724 T^{9} + 53787556 T^{10} - 180007596 T^{11} + 615506083 T^{12} - 1626797043 T^{13} + 4704588100 T^{14} - 8044845651 T^{15} + 16983563041 T^{16}$$)
$23$ ($$1 - 3 T + 26 T^{2} - 69 T^{3} + 529 T^{4}$$)($$1 + 3 T + 26 T^{2} + 69 T^{3} + 529 T^{4}$$)($$1 - 27 T + 391 T^{2} - 3996 T^{3} + 31651 T^{4} - 205875 T^{5} + 1157938 T^{6} - 5917779 T^{7} + 28782226 T^{8} - 136108917 T^{9} + 612549202 T^{10} - 2504881125 T^{11} + 8857247491 T^{12} - 25719626628 T^{13} + 57882032599 T^{14} - 91930287069 T^{15} + 78310985281 T^{16}$$)($$1 + 27 T + 391 T^{2} + 3996 T^{3} + 31651 T^{4} + 205875 T^{5} + 1157938 T^{6} + 5917779 T^{7} + 28782226 T^{8} + 136108917 T^{9} + 612549202 T^{10} + 2504881125 T^{11} + 8857247491 T^{12} + 25719626628 T^{13} + 57882032599 T^{14} + 91930287069 T^{15} + 78310985281 T^{16}$$)
$29$ ($$1 - 55 T^{2} + 841 T^{4}$$)($$1 - 55 T^{2} + 841 T^{4}$$)($$1 - 53 T^{2} + 3250 T^{4} - 128951 T^{6} + 4063174 T^{8} - 108447791 T^{10} + 2298663250 T^{12} - 31525636013 T^{14} + 500246412961 T^{16}$$)($$1 - 53 T^{2} + 3250 T^{4} - 128951 T^{6} + 4063174 T^{8} - 108447791 T^{10} + 2298663250 T^{12} - 31525636013 T^{14} + 500246412961 T^{16}$$)
$31$ ($$1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4}$$)($$1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4}$$)($$1 + 21 T + 262 T^{2} + 2415 T^{3} + 17293 T^{4} + 101304 T^{5} + 505090 T^{6} + 2328618 T^{7} + 11769748 T^{8} + 72187158 T^{9} + 485391490 T^{10} + 3017947464 T^{11} + 15970448653 T^{12} + 69139399665 T^{13} + 232525964422 T^{14} + 577764896331 T^{15} + 852891037441 T^{16}$$)($$1 + 21 T + 262 T^{2} + 2415 T^{3} + 17293 T^{4} + 101304 T^{5} + 505090 T^{6} + 2328618 T^{7} + 11769748 T^{8} + 72187158 T^{9} + 485391490 T^{10} + 3017947464 T^{11} + 15970448653 T^{12} + 69139399665 T^{13} + 232525964422 T^{14} + 577764896331 T^{15} + 852891037441 T^{16}$$)
$37$ ($$1 + 4 T - 21 T^{2} + 148 T^{3} + 1369 T^{4}$$)($$1 + 4 T - 21 T^{2} + 148 T^{3} + 1369 T^{4}$$)($$1 - 7 T - 24 T^{2} + 493 T^{3} - 973 T^{4} - 16188 T^{5} + 118336 T^{6} + 258098 T^{7} - 5756772 T^{8} + 9549626 T^{9} + 162001984 T^{10} - 819970764 T^{11} - 1823558653 T^{12} + 34186570801 T^{13} - 61577433816 T^{14} - 664523139931 T^{15} + 3512479453921 T^{16}$$)($$1 - 7 T - 24 T^{2} + 493 T^{3} - 973 T^{4} - 16188 T^{5} + 118336 T^{6} + 258098 T^{7} - 5756772 T^{8} + 9549626 T^{9} + 162001984 T^{10} - 819970764 T^{11} - 1823558653 T^{12} + 34186570801 T^{13} - 61577433816 T^{14} - 664523139931 T^{15} + 3512479453921 T^{16}$$)
$41$ ($$( 1 + 3 T + 41 T^{2} )^{2}$$)($$( 1 - 3 T + 41 T^{2} )^{2}$$)($$( 1 + 15 T + 218 T^{2} + 1791 T^{3} + 14136 T^{4} + 73431 T^{5} + 366458 T^{6} + 1033815 T^{7} + 2825761 T^{8} )^{2}$$)($$( 1 - 15 T + 218 T^{2} - 1791 T^{3} + 14136 T^{4} - 73431 T^{5} + 366458 T^{6} - 1033815 T^{7} + 2825761 T^{8} )^{2}$$)
$43$ ($$( 1 - T + 43 T^{2} )^{2}$$)($$( 1 - T + 43 T^{2} )^{2}$$)($$( 1 - 8 T + 184 T^{2} - 1022 T^{3} + 12127 T^{4} - 43946 T^{5} + 340216 T^{6} - 636056 T^{7} + 3418801 T^{8} )^{2}$$)($$( 1 - 8 T + 184 T^{2} - 1022 T^{3} + 12127 T^{4} - 43946 T^{5} + 340216 T^{6} - 636056 T^{7} + 3418801 T^{8} )^{2}$$)
$47$ ($$1 - 47 T^{2} + 2209 T^{4}$$)($$1 - 47 T^{2} + 2209 T^{4}$$)($$1 + 6 T - 116 T^{2} - 252 T^{3} + 10126 T^{4} + 1986 T^{5} - 595736 T^{6} - 157218 T^{7} + 25623007 T^{8} - 7389246 T^{9} - 1315980824 T^{10} + 206192478 T^{11} + 49411649806 T^{12} - 57794941764 T^{13} - 1250388978164 T^{14} + 3039738722778 T^{15} + 23811286661761 T^{16}$$)($$1 - 6 T - 116 T^{2} + 252 T^{3} + 10126 T^{4} - 1986 T^{5} - 595736 T^{6} + 157218 T^{7} + 25623007 T^{8} + 7389246 T^{9} - 1315980824 T^{10} - 206192478 T^{11} + 49411649806 T^{12} + 57794941764 T^{13} - 1250388978164 T^{14} - 3039738722778 T^{15} + 23811286661761 T^{16}$$)
$53$ ($$1 + 53 T^{2} + 2809 T^{4}$$)($$1 + 53 T^{2} + 2809 T^{4}$$)($$1 - 24 T + 340 T^{2} - 3552 T^{3} + 29050 T^{4} - 180120 T^{5} + 750160 T^{6} - 1659096 T^{7} + 1273315 T^{8} - 87932088 T^{9} + 2107199440 T^{10} - 26815725240 T^{11} + 229218473050 T^{12} - 1485430391136 T^{13} + 7535882783860 T^{14} - 28193067356088 T^{15} + 62259690411361 T^{16}$$)($$1 + 24 T + 340 T^{2} + 3552 T^{3} + 29050 T^{4} + 180120 T^{5} + 750160 T^{6} + 1659096 T^{7} + 1273315 T^{8} + 87932088 T^{9} + 2107199440 T^{10} + 26815725240 T^{11} + 229218473050 T^{12} + 1485430391136 T^{13} + 7535882783860 T^{14} + 28193067356088 T^{15} + 62259690411361 T^{16}$$)
$59$ ($$1 - 59 T^{2} + 3481 T^{4}$$)($$1 - 59 T^{2} + 3481 T^{4}$$)($$1 + 12 T - 80 T^{2} - 1164 T^{3} + 7690 T^{4} + 80082 T^{5} - 434420 T^{6} - 1772232 T^{7} + 28861927 T^{8} - 104561688 T^{9} - 1512216020 T^{10} + 16447161078 T^{11} + 93182506090 T^{12} - 832171884036 T^{13} - 3374442691280 T^{14} + 29863817817828 T^{15} + 146830437604321 T^{16}$$)($$1 - 12 T - 80 T^{2} + 1164 T^{3} + 7690 T^{4} - 80082 T^{5} - 434420 T^{6} + 1772232 T^{7} + 28861927 T^{8} + 104561688 T^{9} - 1512216020 T^{10} - 16447161078 T^{11} + 93182506090 T^{12} + 832171884036 T^{13} - 3374442691280 T^{14} - 29863817817828 T^{15} + 146830437604321 T^{16}$$)
$61$ ($$1 + 9 T + 88 T^{2} + 549 T^{3} + 3721 T^{4}$$)($$1 + 9 T + 88 T^{2} + 549 T^{3} + 3721 T^{4}$$)($$1 - 15 T + 223 T^{2} - 2220 T^{3} + 19711 T^{4} - 141723 T^{5} + 816310 T^{6} - 5175267 T^{7} + 31433836 T^{8} - 315691287 T^{9} + 3037489510 T^{10} - 32168428263 T^{11} + 272915371951 T^{12} - 1875003788220 T^{13} + 11489043482503 T^{14} - 47141142540315 T^{15} + 191707312997281 T^{16}$$)($$1 - 15 T + 223 T^{2} - 2220 T^{3} + 19711 T^{4} - 141723 T^{5} + 816310 T^{6} - 5175267 T^{7} + 31433836 T^{8} - 315691287 T^{9} + 3037489510 T^{10} - 32168428263 T^{11} + 272915371951 T^{12} - 1875003788220 T^{13} + 11489043482503 T^{14} - 47141142540315 T^{15} + 191707312997281 T^{16}$$)
$67$ ($$1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4}$$)($$1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4}$$)($$1 - 4 T - 234 T^{2} + 412 T^{3} + 35255 T^{4} - 28434 T^{5} - 3551522 T^{6} + 717722 T^{7} + 271900824 T^{8} + 48087374 T^{9} - 15942782258 T^{10} - 8551895142 T^{11} + 710427770855 T^{12} + 556251544084 T^{13} - 21167261427546 T^{14} - 24242846421292 T^{15} + 406067677556641 T^{16}$$)($$1 - 4 T - 234 T^{2} + 412 T^{3} + 35255 T^{4} - 28434 T^{5} - 3551522 T^{6} + 717722 T^{7} + 271900824 T^{8} + 48087374 T^{9} - 15942782258 T^{10} - 8551895142 T^{11} + 710427770855 T^{12} + 556251544084 T^{13} - 21167261427546 T^{14} - 24242846421292 T^{15} + 406067677556641 T^{16}$$)
$71$ ($$1 - 94 T^{2} + 5041 T^{4}$$)($$1 - 94 T^{2} + 5041 T^{4}$$)($$1 - 464 T^{2} + 99532 T^{4} - 12936548 T^{6} + 1114829374 T^{8} - 65213138468 T^{10} + 2529275433292 T^{12} - 59438531739344 T^{14} + 645753531245761 T^{16}$$)($$1 - 464 T^{2} + 99532 T^{4} - 12936548 T^{6} + 1114829374 T^{8} - 65213138468 T^{10} + 2529275433292 T^{12} - 59438531739344 T^{14} + 645753531245761 T^{16}$$)
$73$ ($$1 - 6 T + 85 T^{2} - 438 T^{3} + 5329 T^{4}$$)($$1 - 6 T + 85 T^{2} - 438 T^{3} + 5329 T^{4}$$)($$1 - 15 T + 280 T^{2} - 3075 T^{3} + 38779 T^{4} - 422928 T^{5} + 4017052 T^{6} - 39506334 T^{7} + 310273396 T^{8} - 2883962382 T^{9} + 21406870108 T^{10} - 164526181776 T^{11} + 1101255387739 T^{12} - 6374695148475 T^{13} + 42373583360920 T^{14} - 165710977786455 T^{15} + 806460091894081 T^{16}$$)($$1 - 15 T + 280 T^{2} - 3075 T^{3} + 38779 T^{4} - 422928 T^{5} + 4017052 T^{6} - 39506334 T^{7} + 310273396 T^{8} - 2883962382 T^{9} + 21406870108 T^{10} - 164526181776 T^{11} + 1101255387739 T^{12} - 6374695148475 T^{13} + 42373583360920 T^{14} - 165710977786455 T^{15} + 806460091894081 T^{16}$$)
$79$ ($$1 - 16 T + 177 T^{2} - 1264 T^{3} + 6241 T^{4}$$)($$1 - 16 T + 177 T^{2} - 1264 T^{3} + 6241 T^{4}$$)($$1 + 29 T + 294 T^{2} + 1975 T^{3} + 27377 T^{4} + 260496 T^{5} + 598654 T^{6} + 2403434 T^{7} + 77714340 T^{8} + 189871286 T^{9} + 3736199614 T^{10} + 128434687344 T^{11} + 1066336367537 T^{12} + 6077186388025 T^{13} + 71467711923174 T^{14} + 556913360598611 T^{15} + 1517108809906561 T^{16}$$)($$1 + 29 T + 294 T^{2} + 1975 T^{3} + 27377 T^{4} + 260496 T^{5} + 598654 T^{6} + 2403434 T^{7} + 77714340 T^{8} + 189871286 T^{9} + 3736199614 T^{10} + 128434687344 T^{11} + 1066336367537 T^{12} + 6077186388025 T^{13} + 71467711923174 T^{14} + 556913360598611 T^{15} + 1517108809906561 T^{16}$$)
$83$ ($$( 1 + 9 T + 83 T^{2} )^{2}$$)($$( 1 - 9 T + 83 T^{2} )^{2}$$)($$( 1 - 15 T + 380 T^{2} - 3759 T^{3} + 49260 T^{4} - 311997 T^{5} + 2617820 T^{6} - 8576805 T^{7} + 47458321 T^{8} )^{2}$$)($$( 1 + 15 T + 380 T^{2} + 3759 T^{3} + 49260 T^{4} + 311997 T^{5} + 2617820 T^{6} + 8576805 T^{7} + 47458321 T^{8} )^{2}$$)
$89$ ($$1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4}$$)($$1 - 3 T - 80 T^{2} - 267 T^{3} + 7921 T^{4}$$)($$1 + 3 T - 53 T^{2} + 2820 T^{3} + 14227 T^{4} - 160275 T^{5} + 3467116 T^{6} + 26593569 T^{7} - 193500020 T^{8} + 2366827641 T^{9} + 27463025836 T^{10} - 112988906475 T^{11} + 892633862707 T^{12} + 15747047646180 T^{13} - 26340008420933 T^{14} + 132694004686587 T^{15} + 3936588805702081 T^{16}$$)($$1 - 3 T - 53 T^{2} - 2820 T^{3} + 14227 T^{4} + 160275 T^{5} + 3467116 T^{6} - 26593569 T^{7} - 193500020 T^{8} - 2366827641 T^{9} + 27463025836 T^{10} + 112988906475 T^{11} + 892633862707 T^{12} - 15747047646180 T^{13} - 26340008420933 T^{14} - 132694004686587 T^{15} + 3936588805702081 T^{16}$$)
$97$ ($$1 - 86 T^{2} + 9409 T^{4}$$)($$1 - 86 T^{2} + 9409 T^{4}$$)($$1 - 368 T^{2} + 81676 T^{4} - 12257504 T^{6} + 1385094598 T^{8} - 115330855136 T^{10} + 7230717554956 T^{12} - 306533697813872 T^{14} + 7837433594376961 T^{16}$$)($$1 - 368 T^{2} + 81676 T^{4} - 12257504 T^{6} + 1385094598 T^{8} - 115330855136 T^{10} + 7230717554956 T^{12} - 306533697813872 T^{14} + 7837433594376961 T^{16}$$)