# Properties

 Label 252.2.j Level 252 Weight 2 Character orbit j Rep. character $$\chi_{252}(85,\cdot)$$ Character field $$\Q(\zeta_{3})$$ Dimension 12 Newform subspaces 2 Sturm bound 96 Trace bound 3

# Related objects

## Defining parameters

 Level: $$N$$ = $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 252.j (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$9$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$2$$ Sturm bound: $$96$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(252, [\chi])$$.

Total New Old
Modular forms 108 12 96
Cusp forms 84 12 72
Eisenstein series 24 0 24

## Trace form

 $$12q + 2q^{5} + 4q^{9} + O(q^{10})$$ $$12q + 2q^{5} + 4q^{9} + 4q^{11} - 2q^{15} + 4q^{17} + 12q^{19} - 2q^{21} - 8q^{23} + 14q^{29} + 6q^{31} + 8q^{33} - 8q^{35} - 12q^{37} - 20q^{39} + 6q^{41} - 6q^{43} - 10q^{45} - 6q^{47} - 6q^{49} - 22q^{51} - 24q^{53} - 12q^{55} - 30q^{57} - 28q^{59} - 18q^{65} + 32q^{69} + 4q^{71} + 12q^{73} + 46q^{75} + 8q^{77} + 6q^{79} - 32q^{81} - 2q^{83} + 30q^{85} - 68q^{87} + 36q^{89} - 12q^{91} + 22q^{93} + 4q^{95} + 24q^{97} - 32q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(252, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
252.2.j.a $$6$$ $$2.012$$ 6.0.309123.1 None $$0$$ $$-2$$ $$-1$$ $$3$$ $$q+(\beta _{2}+\beta _{4})q^{3}+(-\beta _{1}+\beta _{5})q^{5}+(1+\cdots)q^{7}+\cdots$$
252.2.j.b $$6$$ $$2.012$$ 6.0.309123.1 None $$0$$ $$2$$ $$3$$ $$-3$$ $$q+(1+\beta _{3}+\beta _{4})q^{3}+(-\beta _{1}-\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(252, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(252, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(18, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(36, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(126, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$1 + 2 T - 2 T^{2} - 9 T^{3} - 6 T^{4} + 18 T^{5} + 27 T^{6}$$)($$1 - 2 T + 4 T^{2} - 3 T^{3} + 12 T^{4} - 18 T^{5} + 27 T^{6}$$)
$5$ ($$1 + T - 10 T^{2} - 7 T^{3} + 57 T^{4} + 14 T^{5} - 299 T^{6} + 70 T^{7} + 1425 T^{8} - 875 T^{9} - 6250 T^{10} + 3125 T^{11} + 15625 T^{12}$$)($$1 - 3 T + 15 T^{3} - 27 T^{4} + 6 T^{5} + 61 T^{6} + 30 T^{7} - 675 T^{8} + 1875 T^{9} - 9375 T^{11} + 15625 T^{12}$$)
$7$ ($$( 1 - T + T^{2} )^{3}$$)($$( 1 + T + T^{2} )^{3}$$)
$11$ ($$1 + 2 T - 4 T^{2} + 46 T^{3} + 6 T^{4} - 230 T^{5} + 1699 T^{6} - 2530 T^{7} + 726 T^{8} + 61226 T^{9} - 58564 T^{10} + 322102 T^{11} + 1771561 T^{12}$$)($$1 - 6 T + 30 T^{3} + 162 T^{4} - 402 T^{5} - 821 T^{6} - 4422 T^{7} + 19602 T^{8} + 39930 T^{9} - 966306 T^{11} + 1771561 T^{12}$$)
$13$ ($$1 + 3 T + 3 T^{2} - 84 T^{3} - 195 T^{4} + 345 T^{5} + 5006 T^{6} + 4485 T^{7} - 32955 T^{8} - 184548 T^{9} + 85683 T^{10} + 1113879 T^{11} + 4826809 T^{12}$$)($$( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} )^{3}$$)
$17$ ($$( 1 - 2 T + 32 T^{2} - 21 T^{3} + 544 T^{4} - 578 T^{5} + 4913 T^{6} )^{2}$$)($$( 1 + 18 T^{2} + 9 T^{3} + 306 T^{4} + 4913 T^{6} )^{2}$$)
$19$ ($$( 1 - 3 T + 33 T^{2} - 35 T^{3} + 627 T^{4} - 1083 T^{5} + 6859 T^{6} )^{2}$$)($$( 1 - 3 T + 21 T^{2} - 65 T^{3} + 399 T^{4} - 1083 T^{5} + 6859 T^{6} )^{2}$$)
$23$ ($$1 + 14 T + 74 T^{2} + 358 T^{3} + 2628 T^{4} + 11188 T^{5} + 33943 T^{6} + 257324 T^{7} + 1390212 T^{8} + 4355786 T^{9} + 20708234 T^{10} + 90108802 T^{11} + 148035889 T^{12}$$)($$1 - 6 T - 18 T^{2} + 30 T^{3} + 612 T^{4} + 2172 T^{5} - 30449 T^{6} + 49956 T^{7} + 323748 T^{8} + 365010 T^{9} - 5037138 T^{10} - 38618058 T^{11} + 148035889 T^{12}$$)
$29$ ($$1 + T - 46 T^{2} + 149 T^{3} + 897 T^{4} - 4282 T^{5} - 13523 T^{6} - 124178 T^{7} + 754377 T^{8} + 3633961 T^{9} - 32534926 T^{10} + 20511149 T^{11} + 594823321 T^{12}$$)($$1 - 15 T + 90 T^{2} - 411 T^{3} + 2205 T^{4} - 4110 T^{5} - 17723 T^{6} - 119190 T^{7} + 1854405 T^{8} - 10023879 T^{9} + 63655290 T^{10} - 307667235 T^{11} + 594823321 T^{12}$$)
$31$ ($$1 - 3 T - 48 T^{2} + 147 T^{3} + 1005 T^{4} - 1344 T^{5} - 24505 T^{6} - 41664 T^{7} + 965805 T^{8} + 4379277 T^{9} - 44329008 T^{10} - 85887453 T^{11} + 887503681 T^{12}$$)($$1 - 3 T - 6 T^{2} + 221 T^{3} - 639 T^{4} - 2088 T^{5} + 61647 T^{6} - 64728 T^{7} - 614079 T^{8} + 6583811 T^{9} - 5541126 T^{10} - 85887453 T^{11} + 887503681 T^{12}$$)
$37$ ($$( 1 + 3 T + 81 T^{2} + 199 T^{3} + 2997 T^{4} + 4107 T^{5} + 50653 T^{6} )^{2}$$)($$( 1 + 3 T + 33 T^{2} + 115 T^{3} + 1221 T^{4} + 4107 T^{5} + 50653 T^{6} )^{2}$$)
$41$ ($$1 - 90 T^{2} + 18 T^{3} + 4410 T^{4} - 810 T^{5} - 194177 T^{6} - 33210 T^{7} + 7413210 T^{8} + 1240578 T^{9} - 254318490 T^{10} + 4750104241 T^{12}$$)($$1 - 6 T - 90 T^{2} + 210 T^{3} + 7812 T^{4} - 8952 T^{5} - 340301 T^{6} - 367032 T^{7} + 13131972 T^{8} + 14473410 T^{9} - 254318490 T^{10} - 695137206 T^{11} + 4750104241 T^{12}$$)
$43$ ($$1 + 3 T - 24 T^{2} - 979 T^{3} - 1947 T^{4} + 14820 T^{5} + 386067 T^{6} + 637260 T^{7} - 3600003 T^{8} - 77837353 T^{9} - 82051224 T^{10} + 441025329 T^{11} + 6321363049 T^{12}$$)($$1 + 3 T - 12 T^{2} + 473 T^{3} + 153 T^{4} - 4176 T^{5} + 165435 T^{6} - 179568 T^{7} + 282897 T^{8} + 37606811 T^{9} - 41025612 T^{10} + 441025329 T^{11} + 6321363049 T^{12}$$)
$47$ ($$1 + 21 T + 180 T^{2} + 1119 T^{3} + 10053 T^{4} + 100416 T^{5} + 788551 T^{6} + 4719552 T^{7} + 22207077 T^{8} + 116177937 T^{9} + 878342580 T^{10} + 4816245147 T^{11} + 10779215329 T^{12}$$)($$1 - 15 T + 48 T^{2} + 3 T^{3} + 3075 T^{4} - 18798 T^{5} + 4399 T^{6} - 883506 T^{7} + 6792675 T^{8} + 311469 T^{9} + 234224688 T^{10} - 3440175105 T^{11} + 10779215329 T^{12}$$)
$53$ ($$( 1 - 6 T + 162 T^{2} - 627 T^{3} + 8586 T^{4} - 16854 T^{5} + 148877 T^{6} )^{2}$$)($$( 1 + 18 T + 198 T^{2} + 1521 T^{3} + 10494 T^{4} + 50562 T^{5} + 148877 T^{6} )^{2}$$)
$59$ ($$1 + 31 T + 476 T^{2} + 5741 T^{3} + 62553 T^{4} + 587576 T^{5} + 4781851 T^{6} + 34666984 T^{7} + 217746993 T^{8} + 1179080839 T^{9} + 5767863836 T^{10} + 22162653269 T^{11} + 42180533641 T^{12}$$)($$1 - 3 T - 114 T^{2} + 501 T^{3} + 6567 T^{4} - 20406 T^{5} - 323957 T^{6} - 1203954 T^{7} + 22859727 T^{8} + 102894879 T^{9} - 1381379154 T^{10} - 2144772897 T^{11} + 42180533641 T^{12}$$)
$61$ ($$1 + 6 T + 48 T^{2} + 642 T^{3} + 3018 T^{4} + 35394 T^{5} + 438671 T^{6} + 2159034 T^{7} + 11229978 T^{8} + 145721802 T^{9} + 664600368 T^{10} + 5067577806 T^{11} + 51520374361 T^{12}$$)($$1 - 6 T - 102 T^{2} + 698 T^{3} + 6048 T^{4} - 26604 T^{5} - 259509 T^{6} - 1622844 T^{7} + 22504608 T^{8} + 158432738 T^{9} - 1412275782 T^{10} - 5067577806 T^{11} + 51520374361 T^{12}$$)
$67$ ($$1 + 6 T - 150 T^{2} - 506 T^{3} + 17268 T^{4} + 28236 T^{5} - 1220289 T^{6} + 1891812 T^{7} + 77516052 T^{8} - 152186078 T^{9} - 3022668150 T^{10} + 8100750642 T^{11} + 90458382169 T^{12}$$)($$1 - 6 T - 138 T^{2} + 446 T^{3} + 14148 T^{4} - 16668 T^{5} - 1033545 T^{6} - 1116756 T^{7} + 63510372 T^{8} + 134140298 T^{9} - 2780854698 T^{10} - 8100750642 T^{11} + 90458382169 T^{12}$$)
$71$ ($$( 1 - 17 T + 119 T^{2} - 507 T^{3} + 8449 T^{4} - 85697 T^{5} + 357911 T^{6} )^{2}$$)($$( 1 + 15 T + 231 T^{2} + 1833 T^{3} + 16401 T^{4} + 75615 T^{5} + 357911 T^{6} )^{2}$$)
$73$ ($$( 1 + 3 T + 195 T^{2} + 359 T^{3} + 14235 T^{4} + 15987 T^{5} + 389017 T^{6} )^{2}$$)($$( 1 - 9 T + 207 T^{2} - 1235 T^{3} + 15111 T^{4} - 47961 T^{5} + 389017 T^{6} )^{2}$$)
$79$ ($$1 - 9 T - 114 T^{2} + 351 T^{3} + 13143 T^{4} + 15786 T^{5} - 1414609 T^{6} + 1247094 T^{7} + 82025463 T^{8} + 173056689 T^{9} - 4440309234 T^{10} - 27693507591 T^{11} + 243087455521 T^{12}$$)($$1 + 3 T + 6 T^{2} + 1787 T^{3} + 2439 T^{4} + 14634 T^{5} + 1719519 T^{6} + 1156086 T^{7} + 15221799 T^{8} + 881060693 T^{9} + 233700486 T^{10} + 9231169197 T^{11} + 243087455521 T^{12}$$)
$83$ ($$1 + 20 T + 38 T^{2} + 346 T^{3} + 32058 T^{4} + 183754 T^{5} - 606869 T^{6} + 15251582 T^{7} + 220847562 T^{8} + 197838302 T^{9} + 1803416198 T^{10} + 78780812860 T^{11} + 326940373369 T^{12}$$)($$1 - 18 T + 198 T^{3} + 23814 T^{4} - 132750 T^{5} - 711245 T^{6} - 11018250 T^{7} + 164054646 T^{8} + 113213826 T^{9} - 70902731574 T^{11} + 326940373369 T^{12}$$)
$89$ ($$( 1 - 12 T + 216 T^{2} - 1425 T^{3} + 19224 T^{4} - 95052 T^{5} + 704969 T^{6} )^{2}$$)($$( 1 - 6 T + 72 T^{2} + 21 T^{3} + 6408 T^{4} - 47526 T^{5} + 704969 T^{6} )^{2}$$)
$97$ ($$1 - 9 T - 66 T^{2} + 2023 T^{3} - 7707 T^{4} - 73950 T^{5} + 1766073 T^{6} - 7173150 T^{7} - 72515163 T^{8} + 1846337479 T^{9} - 5842932546 T^{10} - 77286062313 T^{11} + 832972004929 T^{12}$$)($$1 - 15 T - 84 T^{2} + 1139 T^{3} + 22203 T^{4} - 134028 T^{5} - 1218567 T^{6} - 13000716 T^{7} + 208908027 T^{8} + 1039534547 T^{9} - 7436459604 T^{10} - 128810103855 T^{11} + 832972004929 T^{12}$$)