# Properties

 Label 105.2.i Level 105 Weight 2 Character orbit i Rep. character $$\chi_{105}(16,\cdot)$$ Character field $$\Q(\zeta_{3})$$ Dimension 12 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.i (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$7$$ Character field: $$\Q(\zeta_{3})$$ 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 12 28
Cusp forms 24 12 12
Eisenstein series 16 0 16

## Trace form

 $$12q + 4q^{2} + 2q^{3} - 4q^{4} + 2q^{7} - 24q^{8} - 6q^{9} + O(q^{10})$$ $$12q + 4q^{2} + 2q^{3} - 4q^{4} + 2q^{7} - 24q^{8} - 6q^{9} + 4q^{10} + 4q^{11} + 4q^{12} - 4q^{13} - 4q^{14} - 8q^{16} - 8q^{17} + 4q^{18} - 6q^{19} + 16q^{20} - 12q^{21} + 24q^{22} + 8q^{23} - 12q^{24} - 6q^{25} - 8q^{26} - 4q^{27} + 24q^{28} - 8q^{29} - 6q^{31} + 8q^{32} + 8q^{33} - 16q^{34} + 8q^{35} + 8q^{36} + 10q^{37} + 32q^{38} + 6q^{39} + 12q^{40} + 8q^{41} - 4q^{42} - 44q^{43} + 4q^{44} - 8q^{46} - 8q^{47} - 16q^{48} - 14q^{49} - 8q^{50} + 4q^{51} - 16q^{53} - 24q^{55} - 36q^{56} + 12q^{57} - 20q^{58} + 4q^{59} - 4q^{61} + 24q^{62} + 2q^{63} + 64q^{64} - 12q^{65} - 12q^{66} + 14q^{67} + 12q^{68} + 24q^{69} - 4q^{70} + 32q^{71} + 12q^{72} - 10q^{73} - 4q^{74} + 2q^{75} + 40q^{76} + 44q^{77} + 8q^{78} + 6q^{79} - 16q^{80} - 6q^{81} - 28q^{82} + 8q^{83} + 48q^{84} + 8q^{85} + 20q^{86} + 24q^{87} + 8q^{88} + 16q^{89} - 8q^{90} + 10q^{91} - 88q^{92} - 6q^{93} - 24q^{94} - 24q^{96} - 32q^{97} - 8q^{99} + O(q^{100})$$

## 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.i.a $$2$$ $$0.838$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$1$$ $$5$$ $$q+(-1+\zeta_{6})q^{3}+(2-2\zeta_{6})q^{4}+\zeta_{6}q^{5}+\cdots$$
105.2.i.b $$2$$ $$0.838$$ $$\Q(\sqrt{-3})$$ None $$2$$ $$-1$$ $$-1$$ $$-1$$ $$q+2\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-2+2\zeta_{6})q^{4}+\cdots$$
105.2.i.c $$4$$ $$0.838$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$2$$ $$2$$ $$-2$$ $$q+\beta _{1}q^{2}-\beta _{2}q^{3}+(1+\beta _{2})q^{5}-\beta _{3}q^{6}+\cdots$$
105.2.i.d $$4$$ $$0.838$$ $$\Q(\zeta_{12})$$ None $$2$$ $$2$$ $$-2$$ $$0$$ $$q+(-\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(1-\zeta_{12}^{2}+\cdots)q^{3}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(105, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 - 2 T^{2} + 4 T^{4}$$)($$1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4}$$)($$( 1 - 2 T^{2} )^{2}( 1 + 2 T^{2} + 4 T^{4} )$$)($$( 1 - 2 T + 2 T^{2} )^{2}( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} )$$)
$3$ ($$1 + T + T^{2}$$)($$1 + T + T^{2}$$)($$( 1 - T + T^{2} )^{2}$$)($$( 1 - T + T^{2} )^{2}$$)
$5$ ($$1 - T + T^{2}$$)($$1 + T + T^{2}$$)($$( 1 - T + T^{2} )^{2}$$)($$( 1 + T + T^{2} )^{2}$$)
$7$ ($$1 - 5 T + 7 T^{2}$$)($$1 + T + 7 T^{2}$$)($$1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4}$$)($$1 + 11 T^{2} + 49 T^{4}$$)
$11$ ($$1 - 11 T^{2} + 121 T^{4}$$)($$1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4}$$)($$1 + 4 T - 8 T^{2} + 8 T^{3} + 279 T^{4} + 88 T^{5} - 968 T^{6} + 5324 T^{7} + 14641 T^{8}$$)($$1 - 2 T - 16 T^{2} + 4 T^{3} + 235 T^{4} + 44 T^{5} - 1936 T^{6} - 2662 T^{7} + 14641 T^{8}$$)
$13$ ($$( 1 + T + 13 T^{2} )^{2}$$)($$( 1 + 3 T + 13 T^{2} )^{2}$$)($$( 1 + 6 T + 33 T^{2} + 78 T^{3} + 169 T^{4} )^{2}$$)($$( 1 - 8 T + 39 T^{2} - 104 T^{3} + 169 T^{4} )^{2}$$)
$17$ ($$1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4}$$)($$1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4}$$)($$1 - 4 T - 4 T^{2} + 56 T^{3} - 161 T^{4} + 952 T^{5} - 1156 T^{6} - 19652 T^{7} + 83521 T^{8}$$)($$1 + 10 T + 44 T^{2} + 220 T^{3} + 1147 T^{4} + 3740 T^{5} + 12716 T^{6} + 49130 T^{7} + 83521 T^{8}$$)
$19$ ($$1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4}$$)($$( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$)($$1 - 2 T - 3 T^{2} + 62 T^{3} - 388 T^{4} + 1178 T^{5} - 1083 T^{6} - 13718 T^{7} + 130321 T^{8}$$)($$1 + 2 T - 23 T^{2} - 22 T^{3} + 292 T^{4} - 418 T^{5} - 8303 T^{6} + 13718 T^{7} + 130321 T^{8}$$)
$23$ ($$1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4}$$)($$1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4}$$)($$1 - 4 T - 16 T^{2} + 56 T^{3} + 127 T^{4} + 1288 T^{5} - 8464 T^{6} - 48668 T^{7} + 279841 T^{8}$$)($$1 - 6 T - 16 T^{2} - 36 T^{3} + 1347 T^{4} - 828 T^{5} - 8464 T^{6} - 73002 T^{7} + 279841 T^{8}$$)
$29$ ($$( 1 + 6 T + 29 T^{2} )^{2}$$)($$( 1 + 8 T + 29 T^{2} )^{2}$$)($$( 1 - 8 T + 56 T^{2} - 232 T^{3} + 841 T^{4} )^{2}$$)($$( 1 - 2 T + 32 T^{2} - 58 T^{3} + 841 T^{4} )^{2}$$)
$31$ ($$1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4}$$)($$1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4}$$)($$1 - 6 T - 27 T^{2} - 6 T^{3} + 1892 T^{4} - 186 T^{5} - 25947 T^{6} - 178746 T^{7} + 923521 T^{8}$$)($$1 + 6 T - 23 T^{2} - 18 T^{3} + 1404 T^{4} - 558 T^{5} - 22103 T^{6} + 178746 T^{7} + 923521 T^{8}$$)
$37$ ($$1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4}$$)($$1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4}$$)($$1 - 14 T + 75 T^{2} - 658 T^{3} + 6020 T^{4} - 24346 T^{5} + 102675 T^{6} - 709142 T^{7} + 1874161 T^{8}$$)($$1 + 4 T - 35 T^{2} - 92 T^{3} + 640 T^{4} - 3404 T^{5} - 47915 T^{6} + 202612 T^{7} + 1874161 T^{8}$$)
$41$ ($$( 1 - 12 T + 41 T^{2} )^{2}$$)($$( 1 + 6 T + 41 T^{2} )^{2}$$)($$( 1 + 4 T + 68 T^{2} + 164 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 - 2 T + 80 T^{2} - 82 T^{3} + 1681 T^{4} )^{2}$$)
$43$ ($$( 1 + T + 43 T^{2} )^{2}$$)($$( 1 - T + 43 T^{2} )^{2}$$)($$( 1 + 18 T + 165 T^{2} + 774 T^{3} + 1849 T^{4} )^{2}$$)($$( 1 + 4 T + 63 T^{2} + 172 T^{3} + 1849 T^{4} )^{2}$$)
$47$ ($$1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4}$$)($$1 + 2 T - 43 T^{2} + 94 T^{3} + 2209 T^{4}$$)($$1 - 4 T + 46 T^{2} + 496 T^{3} - 2061 T^{4} + 23312 T^{5} + 101614 T^{6} - 415292 T^{7} + 4879681 T^{8}$$)($$( 1 + 2 T - 43 T^{2} + 94 T^{3} + 2209 T^{4} )^{2}$$)
$53$ ($$1 - 53 T^{2} + 2809 T^{4}$$)($$1 + 4 T - 37 T^{2} + 212 T^{3} + 2809 T^{4}$$)($$1 + 8 T - 50 T^{2} + 64 T^{3} + 6795 T^{4} + 3392 T^{5} - 140450 T^{6} + 1191016 T^{7} + 7890481 T^{8}$$)($$1 + 4 T + 14 T^{2} - 416 T^{3} - 3653 T^{4} - 22048 T^{5} + 39326 T^{6} + 595508 T^{7} + 7890481 T^{8}$$)
$59$ ($$1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4}$$)($$1 - 8 T + 5 T^{2} - 472 T^{3} + 3481 T^{4}$$)($$1 - 116 T^{2} + 9975 T^{4} - 403796 T^{6} + 12117361 T^{8}$$)($$1 + 10 T - 16 T^{2} - 20 T^{3} + 4075 T^{4} - 1180 T^{5} - 55696 T^{6} + 2053790 T^{7} + 12117361 T^{8}$$)
$61$ ($$1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4}$$)($$( 1 - 13 T + 61 T^{2} )( 1 - T + 61 T^{2} )$$)($$1 + 8 T - 2 T^{2} - 448 T^{3} - 3269 T^{4} - 27328 T^{5} - 7442 T^{6} + 1815848 T^{7} + 13845841 T^{8}$$)($$( 1 + 4 T - 45 T^{2} + 244 T^{3} + 3721 T^{4} )^{2}$$)
$67$ ($$1 - 7 T - 18 T^{2} - 469 T^{3} + 4489 T^{4}$$)($$1 + 7 T - 18 T^{2} + 469 T^{3} + 4489 T^{4}$$)($$1 - 2 T + 31 T^{2} + 322 T^{3} - 4028 T^{4} + 21574 T^{5} + 139159 T^{6} - 601526 T^{7} + 20151121 T^{8}$$)($$1 - 12 T + 49 T^{2} + 468 T^{3} - 5112 T^{4} + 31356 T^{5} + 219961 T^{6} - 3609156 T^{7} + 20151121 T^{8}$$)
$71$ ($$( 1 - 12 T + 71 T^{2} )^{2}$$)($$( 1 - 6 T + 71 T^{2} )^{2}$$)($$( 1 + 4 T + 144 T^{2} + 284 T^{3} + 5041 T^{4} )^{2}$$)($$( 1 - 2 T + 116 T^{2} - 142 T^{3} + 5041 T^{4} )^{2}$$)
$73$ ($$1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4}$$)($$1 + T - 72 T^{2} + 73 T^{3} + 5329 T^{4}$$)($$1 - 10 T - 21 T^{2} + 250 T^{3} + 2012 T^{4} + 18250 T^{5} - 111909 T^{6} - 3890170 T^{7} + 28398241 T^{8}$$)($$1 + 8 T - 23 T^{2} - 472 T^{3} - 2432 T^{4} - 34456 T^{5} - 122567 T^{6} + 3112136 T^{7} + 28398241 T^{8}$$)
$79$ ($$( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} )$$)($$1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4}$$)($$1 + 2 T - 123 T^{2} - 62 T^{3} + 9572 T^{4} - 4898 T^{5} - 767643 T^{6} + 986078 T^{7} + 38950081 T^{8}$$)($$1 + 6 T - 23 T^{2} - 594 T^{3} - 5604 T^{4} - 46926 T^{5} - 143543 T^{6} + 2958234 T^{7} + 38950081 T^{8}$$)
$83$ ($$( 1 + 12 T + 83 T^{2} )^{2}$$)($$( 1 - 2 T + 83 T^{2} )^{2}$$)($$( 1 - 8 T + 180 T^{2} - 664 T^{3} + 6889 T^{4} )^{2}$$)($$( 1 - 6 T + 28 T^{2} - 498 T^{3} + 6889 T^{4} )^{2}$$)
$89$ ($$1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4}$$)($$1 - 12 T + 55 T^{2} - 1068 T^{3} + 7921 T^{4}$$)($$1 - 16 T + 32 T^{2} - 736 T^{3} + 19471 T^{4} - 65504 T^{5} + 253472 T^{6} - 11279504 T^{7} + 62742241 T^{8}$$)($$1 + 6 T - 4 T^{2} - 828 T^{3} - 9525 T^{4} - 73692 T^{5} - 31684 T^{6} + 4229814 T^{7} + 62742241 T^{8}$$)
$97$ ($$( 1 + 10 T + 97 T^{2} )^{2}$$)($$( 1 + 6 T + 97 T^{2} )^{2}$$)($$( 1 + 16 T + 250 T^{2} + 1552 T^{3} + 9409 T^{4} )^{2}$$)($$( 1 - 16 T + 210 T^{2} - 1552 T^{3} + 9409 T^{4} )^{2}$$)