# Properties

 Label 144.2.i Level 144 Weight 2 Character orbit i Rep. character $$\chi_{144}(49,\cdot)$$ Character field $$\Q(\zeta_{3})$$ Dimension 10 Newform subspaces 4 Sturm bound 48 Trace bound 5

# Related objects

## Defining parameters

 Level: $$N$$ = $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 144.i (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$9$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$4$$ Sturm bound: $$48$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(144, [\chi])$$.

Total New Old
Modular forms 60 14 46
Cusp forms 36 10 26
Eisenstein series 24 4 20

## Trace form

 $$10q + 2q^{3} - q^{5} + q^{7} - 4q^{9} + O(q^{10})$$ $$10q + 2q^{3} - q^{5} + q^{7} - 4q^{9} + 7q^{11} - q^{13} + 5q^{15} - 8q^{17} + 4q^{19} - 3q^{21} - 5q^{23} - 2q^{25} - 16q^{27} + 3q^{29} + 7q^{31} - 5q^{33} - 30q^{35} - 4q^{37} - 31q^{39} - 3q^{41} + 7q^{43} + 11q^{45} - 15q^{47} + 10q^{51} - 4q^{53} - 6q^{55} - 10q^{57} + 25q^{59} - q^{61} + 33q^{63} + 17q^{65} + q^{67} + 13q^{69} + 56q^{71} - 16q^{73} + 46q^{75} + 21q^{77} + q^{79} + 20q^{81} + 29q^{83} - 6q^{85} + 15q^{87} + 12q^{89} - 22q^{91} + 5q^{93} - 28q^{95} + 5q^{97} - 47q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(144, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
144.2.i.a $$2$$ $$1.150$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-3$$ $$-1$$ $$q+(1-2\zeta_{6})q^{3}-3\zeta_{6}q^{5}+(-1+\zeta_{6})q^{7}+\cdots$$
144.2.i.b $$2$$ $$1.150$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$1$$ $$-3$$ $$q+(-1+2\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-3+3\zeta_{6})q^{7}+\cdots$$
144.2.i.c $$2$$ $$1.150$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$0$$ $$2$$ $$q+(1+\zeta_{6})q^{3}+(2-2\zeta_{6})q^{7}+3\zeta_{6}q^{9}+\cdots$$
144.2.i.d $$4$$ $$1.150$$ $$\Q(\sqrt{-3}, \sqrt{-11})$$ None $$0$$ $$-1$$ $$1$$ $$3$$ $$q-\beta _{1}q^{3}+(\beta _{1}-\beta _{2}-2\beta _{3})q^{5}+(-1+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(144, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(144, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(18, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(36, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(72, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ()()()()
$3$ ($$1 + 3 T^{2}$$)($$1 + 3 T^{2}$$)($$1 - 3 T + 3 T^{2}$$)($$1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4}$$)
$5$ ($$1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4}$$)($$1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4}$$)($$1 - 5 T^{2} + 25 T^{4}$$)($$1 - T - T^{2} + 8 T^{3} - 26 T^{4} + 40 T^{5} - 25 T^{6} - 125 T^{7} + 625 T^{8}$$)
$7$ ($$( 1 - 4 T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )$$)($$1 + 3 T + 2 T^{2} + 21 T^{3} + 49 T^{4}$$)($$1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4}$$)($$1 - 3 T + T^{2} + 18 T^{3} - 48 T^{4} + 126 T^{5} + 49 T^{6} - 1029 T^{7} + 2401 T^{8}$$)
$11$ ($$1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}$$)($$1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4}$$)($$1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4}$$)($$( 1 - T - 10 T^{2} - 11 T^{3} + 121 T^{4} )^{2}$$)
$13$ ($$1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4}$$)($$( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$)($$( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$)($$1 + 5 T + T^{2} - 10 T^{3} + 82 T^{4} - 130 T^{5} + 169 T^{6} + 10985 T^{7} + 28561 T^{8}$$)
$17$ ($$( 1 - 6 T + 17 T^{2} )^{2}$$)($$( 1 + 2 T + 17 T^{2} )^{2}$$)($$( 1 + 3 T + 17 T^{2} )^{2}$$)($$( 1 + 5 T + 32 T^{2} + 85 T^{3} + 289 T^{4} )^{2}$$)
$19$ ($$( 1 - 4 T + 19 T^{2} )^{2}$$)($$( 1 - 4 T + 19 T^{2} )^{2}$$)($$( 1 - T + 19 T^{2} )^{2}$$)($$( 1 + 7 T + 42 T^{2} + 133 T^{3} + 361 T^{4} )^{2}$$)
$23$ ($$1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4}$$)($$1 + T - 22 T^{2} + 23 T^{3} + 529 T^{4}$$)($$1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4}$$)($$1 - 5 T - 19 T^{2} + 10 T^{3} + 832 T^{4} + 230 T^{5} - 10051 T^{6} - 60835 T^{7} + 279841 T^{8}$$)
$29$ ($$1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4}$$)($$1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4}$$)($$1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4}$$)($$1 - 3 T - 43 T^{2} + 18 T^{3} + 1602 T^{4} + 522 T^{5} - 36163 T^{6} - 73167 T^{7} + 707281 T^{8}$$)
$31$ ($$1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4}$$)($$1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4}$$)($$( 1 - 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} )$$)($$1 - 7 T - 17 T^{2} - 28 T^{3} + 1876 T^{4} - 868 T^{5} - 16337 T^{6} - 208537 T^{7} + 923521 T^{8}$$)
$37$ ($$( 1 - 2 T + 37 T^{2} )^{2}$$)($$( 1 + 6 T + 37 T^{2} )^{2}$$)($$( 1 + 4 T + 37 T^{2} )^{2}$$)($$( 1 - 6 T + 50 T^{2} - 222 T^{3} + 1369 T^{4} )^{2}$$)
$41$ ($$1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4}$$)($$1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4}$$)($$1 + 9 T + 40 T^{2} + 369 T^{3} + 1681 T^{4}$$)($$1 - 12 T + 59 T^{2} - 36 T^{3} - 360 T^{4} - 1476 T^{5} + 99179 T^{6} - 827052 T^{7} + 2825761 T^{8}$$)
$43$ ($$1 + T - 42 T^{2} + 43 T^{3} + 1849 T^{4}$$)($$1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4}$$)($$1 + T - 42 T^{2} + 43 T^{3} + 1849 T^{4}$$)($$1 - 8 T - 5 T^{2} + 136 T^{3} + 160 T^{4} + 5848 T^{5} - 9245 T^{6} - 636056 T^{7} + 3418801 T^{8}$$)
$47$ ($$1 + 9 T + 34 T^{2} + 423 T^{3} + 2209 T^{4}$$)($$1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4}$$)($$1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4}$$)($$1 - 3 T - 79 T^{2} + 18 T^{3} + 5112 T^{4} + 846 T^{5} - 174511 T^{6} - 311469 T^{7} + 4879681 T^{8}$$)
$53$ ($$( 1 + 6 T + 53 T^{2} )^{2}$$)($$( 1 - 2 T + 53 T^{2} )^{2}$$)($$( 1 - 12 T + 53 T^{2} )^{2}$$)($$( 1 + 10 T + 98 T^{2} + 530 T^{3} + 2809 T^{4} )^{2}$$)
$59$ ($$1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4}$$)($$1 - 11 T + 62 T^{2} - 649 T^{3} + 3481 T^{4}$$)($$1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4}$$)($$( 1 - 7 T - 10 T^{2} - 413 T^{3} + 3481 T^{4} )^{2}$$)
$61$ ($$( 1 - 14 T + 61 T^{2} )( 1 + T + 61 T^{2} )$$)($$1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4}$$)($$1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4}$$)($$1 - T - 113 T^{2} + 8 T^{3} + 9214 T^{4} + 488 T^{5} - 420473 T^{6} - 226981 T^{7} + 13845841 T^{8}$$)
$67$ ($$1 + 7 T - 18 T^{2} + 469 T^{3} + 4489 T^{4}$$)($$1 + T - 66 T^{2} + 67 T^{3} + 4489 T^{4}$$)($$( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} )$$)($$1 - 4 T - 89 T^{2} + 116 T^{3} + 5464 T^{4} + 7772 T^{5} - 399521 T^{6} - 1203052 T^{7} + 20151121 T^{8}$$)
$71$ ($$( 1 - 12 T + 71 T^{2} )^{2}$$)($$( 1 + 4 T + 71 T^{2} )^{2}$$)($$( 1 - 12 T + 71 T^{2} )^{2}$$)($$( 1 - 4 T + 71 T^{2} )^{4}$$)
$73$ ($$( 1 + 10 T + 73 T^{2} )^{2}$$)($$( 1 + 2 T + 73 T^{2} )^{2}$$)($$( 1 - 11 T + 73 T^{2} )^{2}$$)($$( 1 + 7 T + 84 T^{2} + 511 T^{3} + 5329 T^{4} )^{2}$$)
$79$ ($$1 - 11 T + 42 T^{2} - 869 T^{3} + 6241 T^{4}$$)($$1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4}$$)($$( 1 - 13 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} )$$)($$1 + 7 T - 113 T^{2} + 28 T^{3} + 16132 T^{4} + 2212 T^{5} - 705233 T^{6} + 3451273 T^{7} + 38950081 T^{8}$$)
$83$ ($$1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4}$$)($$1 - T - 82 T^{2} - 83 T^{3} + 6889 T^{4}$$)($$1 - 12 T + 61 T^{2} - 996 T^{3} + 6889 T^{4}$$)($$1 - 25 T + 311 T^{2} - 3700 T^{3} + 39832 T^{4} - 307100 T^{5} + 2142479 T^{6} - 14294675 T^{7} + 47458321 T^{8}$$)
$89$ ($$( 1 - 6 T + 89 T^{2} )^{2}$$)($$( 1 + 18 T + 89 T^{2} )^{2}$$)($$( 1 - 6 T + 89 T^{2} )^{2}$$)($$( 1 - 6 T + 89 T^{2} )^{4}$$)
$97$ ($$1 + 11 T + 24 T^{2} + 1067 T^{3} + 9409 T^{4}$$)($$1 - 13 T + 72 T^{2} - 1261 T^{3} + 9409 T^{4}$$)($$( 1 - 14 T + 97 T^{2} )( 1 + 19 T + 97 T^{2} )$$)($$1 - 8 T - 113 T^{2} + 136 T^{3} + 15712 T^{4} + 13192 T^{5} - 1063217 T^{6} - 7301384 T^{7} + 88529281 T^{8}$$)