# Properties

 Label 144.2.s Level 144 Weight 2 Character orbit s Rep. character $$\chi_{144}(47,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 12 Newform subspaces 5 Sturm bound 48 Trace bound 7

# Related objects

## Defining parameters

 Level: $$N$$ = $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 144.s (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$36$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$5$$ Sturm bound: $$48$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$5$$, $$7$$

## Dimensions

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

Total New Old
Modular forms 60 12 48
Cusp forms 36 12 24
Eisenstein series 24 0 24

## Trace form

 $$12q + 6q^{9} + O(q^{10})$$ $$12q + 6q^{9} - 12q^{21} + 6q^{25} - 36q^{29} - 18q^{33} - 18q^{41} - 36q^{45} + 6q^{49} + 30q^{57} + 72q^{65} + 72q^{69} - 36q^{73} + 72q^{77} + 54q^{81} - 18q^{97} + O(q^{100})$$

## 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.s.a $$2$$ $$1.150$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$-6$$ $$-6$$ $$q+(-1-\zeta_{6})q^{3}+(-4+2\zeta_{6})q^{5}+(-2+\cdots)q^{7}+\cdots$$
144.2.s.b $$2$$ $$1.150$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$3$$ $$-3$$ $$q+(1-2\zeta_{6})q^{3}+(2-\zeta_{6})q^{5}+(-1-\zeta_{6})q^{7}+\cdots$$
144.2.s.c $$2$$ $$1.150$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$3$$ $$3$$ $$q+(-1+2\zeta_{6})q^{3}+(2-\zeta_{6})q^{5}+(1+\zeta_{6})q^{7}+\cdots$$
144.2.s.d $$2$$ $$1.150$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$-6$$ $$6$$ $$q+(1+\zeta_{6})q^{3}+(-4+2\zeta_{6})q^{5}+(2+2\zeta_{6})q^{7}+\cdots$$
144.2.s.e $$4$$ $$1.150$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$6$$ $$0$$ $$q-\zeta_{12}^{3}q^{3}+(1+\zeta_{12})q^{5}+(-\zeta_{12}^{2}+\cdots)q^{7}+\cdots$$

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

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ()()()()()
$3$ ($$1 + 3 T + 3 T^{2}$$)($$1 + 3 T^{2}$$)($$1 + 3 T^{2}$$)($$1 - 3 T + 3 T^{2}$$)($$( 1 - 3 T^{2} )^{2}$$)
$5$ ($$1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4}$$)($$1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4}$$)($$1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4}$$)($$1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4}$$)($$( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} )^{2}$$)
$7$ ($$( 1 + T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )$$)($$( 1 - T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )$$)($$( 1 - 4 T + 7 T^{2} )( 1 + T + 7 T^{2} )$$)($$( 1 - 5 T + 7 T^{2} )( 1 - T + 7 T^{2} )$$)($$1 + 5 T^{2} - 24 T^{4} + 245 T^{6} + 2401 T^{8}$$)
$11$ ($$1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4}$$)($$1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}$$)($$1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4}$$)($$1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}$$)($$1 + 5 T^{2} - 96 T^{4} + 605 T^{6} + 14641 T^{8}$$)
$13$ ($$1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4}$$)($$( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$)($$( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$)($$1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4}$$)($$( 1 + T - 12 T^{2} + 13 T^{3} + 169 T^{4} )^{2}$$)
$17$ ($$1 - 31 T^{2} + 289 T^{4}$$)($$1 + 14 T^{2} + 289 T^{4}$$)($$1 + 14 T^{2} + 289 T^{4}$$)($$1 - 31 T^{2} + 289 T^{4}$$)($$( 1 - 22 T^{2} + 289 T^{4} )^{2}$$)
$19$ ($$1 - 35 T^{2} + 361 T^{4}$$)($$( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$)($$( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$)($$1 - 35 T^{2} + 361 T^{4}$$)($$( 1 - 2 T^{2} + 361 T^{4} )^{2}$$)
$23$ ($$1 - 23 T^{2} + 529 T^{4}$$)($$1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4}$$)($$1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4}$$)($$1 - 23 T^{2} + 529 T^{4}$$)($$1 - 19 T^{2} - 168 T^{4} - 10051 T^{6} + 279841 T^{8}$$)
$29$ ($$1 + 6 T + 41 T^{2} + 174 T^{3} + 841 T^{4}$$)($$1 - 3 T + 32 T^{2} - 87 T^{3} + 841 T^{4}$$)($$1 - 3 T + 32 T^{2} - 87 T^{3} + 841 T^{4}$$)($$1 + 6 T + 41 T^{2} + 174 T^{3} + 841 T^{4}$$)($$( 1 + 15 T + 104 T^{2} + 435 T^{3} + 841 T^{4} )^{2}$$)
$31$ ($$1 + 31 T^{2} + 961 T^{4}$$)($$1 + 9 T + 58 T^{2} + 279 T^{3} + 961 T^{4}$$)($$1 - 9 T + 58 T^{2} - 279 T^{3} + 961 T^{4}$$)($$1 + 31 T^{2} + 961 T^{4}$$)($$1 + 53 T^{2} + 1848 T^{4} + 50933 T^{6} + 923521 T^{8}$$)
$37$ ($$( 1 - 2 T + 37 T^{2} )^{2}$$)($$( 1 - 2 T + 37 T^{2} )^{2}$$)($$( 1 - 2 T + 37 T^{2} )^{2}$$)($$( 1 - 2 T + 37 T^{2} )^{2}$$)($$( 1 + 4 T + 37 T^{2} )^{4}$$)
$41$ ($$1 + 9 T + 68 T^{2} + 369 T^{3} + 1681 T^{4}$$)($$1 + 9 T + 68 T^{2} + 369 T^{3} + 1681 T^{4}$$)($$1 + 9 T + 68 T^{2} + 369 T^{3} + 1681 T^{4}$$)($$1 + 9 T + 68 T^{2} + 369 T^{3} + 1681 T^{4}$$)($$( 1 - 9 T + 68 T^{2} - 369 T^{3} + 1681 T^{4} )^{2}$$)
$43$ ($$1 - 9 T + 70 T^{2} - 387 T^{3} + 1849 T^{4}$$)($$1 - 9 T + 70 T^{2} - 387 T^{3} + 1849 T^{4}$$)($$1 + 9 T + 70 T^{2} + 387 T^{3} + 1849 T^{4}$$)($$1 + 9 T + 70 T^{2} + 387 T^{3} + 1849 T^{4}$$)($$1 + 77 T^{2} + 4080 T^{4} + 142373 T^{6} + 3418801 T^{8}$$)
$47$ ($$1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4}$$)($$1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4}$$)($$1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4}$$)($$1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4}$$)($$1 - 67 T^{2} + 2280 T^{4} - 148003 T^{6} + 4879681 T^{8}$$)
$53$ ($$( 1 - 53 T^{2} )^{2}$$)($$( 1 - 53 T^{2} )^{2}$$)($$( 1 - 53 T^{2} )^{2}$$)($$( 1 - 53 T^{2} )^{2}$$)($$( 1 + 2 T^{2} + 2809 T^{4} )^{2}$$)
$59$ ($$1 + 15 T + 166 T^{2} + 885 T^{3} + 3481 T^{4}$$)($$1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4}$$)($$1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4}$$)($$1 - 15 T + 166 T^{2} - 885 T^{3} + 3481 T^{4}$$)($$1 - 91 T^{2} + 4800 T^{4} - 316771 T^{6} + 12117361 T^{8}$$)
$61$ ($$1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4}$$)($$( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$)($$( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$)($$1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4}$$)($$( 1 - 7 T - 12 T^{2} - 427 T^{3} + 3721 T^{4} )^{2}$$)
$67$ ($$1 + 15 T + 142 T^{2} + 1005 T^{3} + 4489 T^{4}$$)($$1 - 15 T + 142 T^{2} - 1005 T^{3} + 4489 T^{4}$$)($$1 + 15 T + 142 T^{2} + 1005 T^{3} + 4489 T^{4}$$)($$1 - 15 T + 142 T^{2} - 1005 T^{3} + 4489 T^{4}$$)($$1 + 53 T^{2} - 1680 T^{4} + 237917 T^{6} + 20151121 T^{8}$$)
$71$ ($$( 1 + 6 T + 71 T^{2} )^{2}$$)($$( 1 + 12 T + 71 T^{2} )^{2}$$)($$( 1 - 12 T + 71 T^{2} )^{2}$$)($$( 1 - 6 T + 71 T^{2} )^{2}$$)($$( 1 + 34 T^{2} + 5041 T^{4} )^{2}$$)
$73$ ($$( 1 + 11 T + 73 T^{2} )^{2}$$)($$( 1 + 2 T + 73 T^{2} )^{2}$$)($$( 1 + 2 T + 73 T^{2} )^{2}$$)($$( 1 + 11 T + 73 T^{2} )^{2}$$)($$( 1 - 4 T + 73 T^{2} )^{4}$$)
$79$ ($$1 - 6 T + 91 T^{2} - 474 T^{3} + 6241 T^{4}$$)($$1 + 15 T + 154 T^{2} + 1185 T^{3} + 6241 T^{4}$$)($$1 - 15 T + 154 T^{2} - 1185 T^{3} + 6241 T^{4}$$)($$1 + 6 T + 91 T^{2} + 474 T^{3} + 6241 T^{4}$$)($$1 - 67 T^{2} - 1752 T^{4} - 418147 T^{6} + 38950081 T^{8}$$)
$83$ ($$1 - 12 T + 61 T^{2} - 996 T^{3} + 6889 T^{4}$$)($$1 - 15 T + 142 T^{2} - 1245 T^{3} + 6889 T^{4}$$)($$1 + 15 T + 142 T^{2} + 1245 T^{3} + 6889 T^{4}$$)($$1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4}$$)($$1 - 139 T^{2} + 12432 T^{4} - 957571 T^{6} + 47458321 T^{8}$$)
$89$ ($$1 + 14 T^{2} + 7921 T^{4}$$)($$1 - 130 T^{2} + 7921 T^{4}$$)($$1 - 130 T^{2} + 7921 T^{4}$$)($$1 + 14 T^{2} + 7921 T^{4}$$)($$( 1 - 166 T^{2} + 7921 T^{4} )^{2}$$)
$97$ ($$1 + 13 T + 72 T^{2} + 1261 T^{3} + 9409 T^{4}$$)($$( 1 - 19 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} )$$)($$( 1 - 19 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} )$$)($$1 + 13 T + 72 T^{2} + 1261 T^{3} + 9409 T^{4}$$)($$( 1 + T - 96 T^{2} + 97 T^{3} + 9409 T^{4} )^{2}$$)