# Properties

 Label 1350.2.j Level 1350 Weight 2 Character orbit j Rep. character $$\chi_{1350}(199,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 36 Newform subspaces 7 Sturm bound 540 Trace bound 19

# Related objects

## Defining parameters

 Level: $$N$$ = $$1350 = 2 \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1350.j (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$45$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$7$$ Sturm bound: $$540$$ Trace bound: $$19$$ Distinguishing $$T_p$$: $$7$$, $$11$$, $$19$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1350, [\chi])$$.

Total New Old
Modular forms 612 36 576
Cusp forms 468 36 432
Eisenstein series 144 0 144

## Trace form

 $$36q + 18q^{4} + O(q^{10})$$ $$36q + 18q^{4} + 12q^{11} - 12q^{14} - 18q^{16} - 24q^{26} + 12q^{31} + 36q^{41} + 24q^{44} + 24q^{46} + 30q^{49} + 12q^{56} + 30q^{59} - 12q^{61} - 36q^{64} + 48q^{71} - 36q^{74} + 36q^{79} + 6q^{86} + 96q^{89} + 48q^{91} + 12q^{94} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1350, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
1350.2.j.a $$4$$ $$10.780$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+2\zeta_{12}q^{7}-\zeta_{12}^{3}q^{8}+\cdots$$
1350.2.j.b $$4$$ $$10.780$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+2\zeta_{12}q^{7}-\zeta_{12}^{3}q^{8}+\cdots$$
1350.2.j.c $$4$$ $$10.780$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+4\zeta_{12}q^{7}-\zeta_{12}^{3}q^{8}+\cdots$$
1350.2.j.d $$4$$ $$10.780$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+4\zeta_{12}q^{7}-\zeta_{12}^{3}q^{8}+\cdots$$
1350.2.j.e $$4$$ $$10.780$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+\zeta_{12}q^{7}+\zeta_{12}^{3}q^{8}+\cdots$$
1350.2.j.f $$8$$ $$10.780$$ 8.0.303595776.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}-\beta _{3}q^{4}-\beta _{7}q^{7}+(\beta _{1}+\beta _{5}+\cdots)q^{8}+\cdots$$
1350.2.j.g $$8$$ $$10.780$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-\zeta_{24}+\zeta_{24}^{3})q^{2}+(1-\zeta_{24}^{2})q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1350, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1350, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(135, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(225, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(270, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(450, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(675, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 - T^{2} + T^{4}$$)($$1 - T^{2} + T^{4}$$)($$1 - T^{2} + T^{4}$$)($$1 - T^{2} + T^{4}$$)($$1 - T^{2} + T^{4}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)
$3$ ()()()()()()()
$5$ ()()()()()()()
$7$ ($$1 + 10 T^{2} + 51 T^{4} + 490 T^{6} + 2401 T^{8}$$)($$1 + 10 T^{2} + 51 T^{4} + 490 T^{6} + 2401 T^{8}$$)($$( 1 - 13 T^{2} + 49 T^{4} )( 1 + 11 T^{2} + 49 T^{4} )$$)($$( 1 - 13 T^{2} + 49 T^{4} )( 1 + 11 T^{2} + 49 T^{4} )$$)($$( 1 + 2 T^{2} + 49 T^{4} )( 1 + 11 T^{2} + 49 T^{4} )$$)($$1 + 11 T^{2} + T^{4} + 242 T^{6} + 6070 T^{8} + 11858 T^{10} + 2401 T^{12} + 1294139 T^{14} + 5764801 T^{16}$$)($$1 + 8 T^{2} + 46 T^{4} - 640 T^{6} - 5213 T^{8} - 31360 T^{10} + 110446 T^{12} + 941192 T^{14} + 5764801 T^{16}$$)
$11$ ($$( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} )^{2}$$)($$( 1 - 11 T^{2} + 121 T^{4} )^{2}$$)($$( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} )^{2}$$)($$( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} )^{2}$$)($$( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} )^{2}$$)($$( 1 + 3 T - 7 T^{2} - 18 T^{3} + 36 T^{4} - 198 T^{5} - 847 T^{6} + 3993 T^{7} + 14641 T^{8} )^{2}$$)($$( 1 + 2 T^{2} - 117 T^{4} + 242 T^{6} + 14641 T^{8} )^{2}$$)
$13$ ($$( 1 - T^{2} + 169 T^{4} )( 1 + 23 T^{2} + 169 T^{4} )$$)($$( 1 - 6 T + 23 T^{2} - 78 T^{3} + 169 T^{4} )( 1 + 6 T + 23 T^{2} + 78 T^{3} + 169 T^{4} )$$)($$( 1 - 6 T + 23 T^{2} - 78 T^{3} + 169 T^{4} )( 1 + 6 T + 23 T^{2} + 78 T^{3} + 169 T^{4} )$$)($$( 1 - 6 T + 23 T^{2} - 78 T^{3} + 169 T^{4} )( 1 + 6 T + 23 T^{2} + 78 T^{3} + 169 T^{4} )$$)($$( 1 - T^{2} + 169 T^{4} )( 1 + 23 T^{2} + 169 T^{4} )$$)($$1 - 16 T^{2} - 14 T^{4} + 1088 T^{6} + 1075 T^{8} + 183872 T^{10} - 399854 T^{12} - 77228944 T^{14} + 815730721 T^{16}$$)($$1 + 32 T^{2} + 526 T^{4} + 5120 T^{6} + 46387 T^{8} + 865280 T^{10} + 15023086 T^{12} + 154457888 T^{14} + 815730721 T^{16}$$)
$17$ ($$( 1 - 25 T^{2} + 289 T^{4} )^{2}$$)($$( 1 + 2 T^{2} + 289 T^{4} )^{2}$$)($$( 1 - 25 T^{2} + 289 T^{4} )^{2}$$)($$( 1 - 25 T^{2} + 289 T^{4} )^{2}$$)($$( 1 - 17 T^{2} )^{4}$$)($$( 1 - 11 T^{2} - 60 T^{4} - 3179 T^{6} + 83521 T^{8} )^{2}$$)($$( 1 - 10 T^{2} + 289 T^{4} )^{4}$$)
$19$ ($$( 1 - T + 19 T^{2} )^{4}$$)($$( 1 - 7 T + 19 T^{2} )^{4}$$)($$( 1 + 5 T + 19 T^{2} )^{4}$$)($$( 1 - 4 T + 19 T^{2} )^{4}$$)($$( 1 - 4 T + 19 T^{2} )^{4}$$)($$( 1 + T + 30 T^{2} + 19 T^{3} + 361 T^{4} )^{4}$$)($$( 1 + 10 T + 57 T^{2} + 190 T^{3} + 361 T^{4} )^{4}$$)
$23$ ($$1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8}$$)($$( 1 + 23 T^{2} + 529 T^{4} )^{2}$$)($$1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8}$$)($$1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8}$$)($$1 - 35 T^{2} + 696 T^{4} - 18515 T^{6} + 279841 T^{8}$$)($$1 + 71 T^{2} + 2797 T^{4} + 84206 T^{6} + 2089006 T^{8} + 44544974 T^{10} + 782715277 T^{12} + 10510548119 T^{14} + 78310985281 T^{16}$$)($$( 1 + 40 T^{2} + 1071 T^{4} + 21160 T^{6} + 279841 T^{8} )^{2}$$)
$29$ ($$( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} )^{2}$$)($$( 1 - 6 T + 7 T^{2} - 174 T^{3} + 841 T^{4} )^{2}$$)($$( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} )^{2}$$)($$( 1 - 6 T + 7 T^{2} - 174 T^{3} + 841 T^{4} )^{2}$$)($$( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4} )^{2}$$)($$( 1 - 3 T - 43 T^{2} + 18 T^{3} + 1602 T^{4} + 522 T^{5} - 36163 T^{6} - 73167 T^{7} + 707281 T^{8} )^{2}$$)($$( 1 - 52 T^{2} + 1863 T^{4} - 43732 T^{6} + 707281 T^{8} )^{2}$$)
$31$ ($$( 1 - 11 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2}$$)($$( 1 - 10 T + 69 T^{2} - 310 T^{3} + 961 T^{4} )^{2}$$)($$( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} )^{2}$$)($$( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} )^{2}$$)($$( 1 - 11 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2}$$)($$( 1 - 2 T - 26 T^{2} + 64 T^{3} - 185 T^{4} + 1984 T^{5} - 24986 T^{6} - 59582 T^{7} + 923521 T^{8} )^{2}$$)($$( 1 + 4 T - 44 T^{2} - 8 T^{3} + 2143 T^{4} - 248 T^{5} - 42284 T^{6} + 119164 T^{7} + 923521 T^{8} )^{2}$$)
$37$ ($$( 1 - 58 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2}$$)($$( 1 - 58 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 - 10 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 - 10 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 - 58 T^{2} + 1369 T^{4} )^{4}$$)($$( 1 - 8 T^{2} - 702 T^{4} - 10952 T^{6} + 1874161 T^{8} )^{2}$$)
$41$ ($$( 1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 + 6 T - 5 T^{2} + 246 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} )^{4}$$)($$( 1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4} )^{4}$$)
$43$ ($$1 + 85 T^{2} + 5376 T^{4} + 157165 T^{6} + 3418801 T^{8}$$)($$1 + 85 T^{2} + 5376 T^{4} + 157165 T^{6} + 3418801 T^{8}$$)($$1 - 35 T^{2} - 624 T^{4} - 64715 T^{6} + 3418801 T^{8}$$)($$1 + 85 T^{2} + 5376 T^{4} + 157165 T^{6} + 3418801 T^{8}$$)($$( 1 - 61 T^{2} + 1849 T^{4} )( 1 + 83 T^{2} + 1849 T^{4} )$$)($$1 + 11 T^{2} - 1223 T^{4} - 25894 T^{6} - 1836194 T^{8} - 47878006 T^{10} - 4181193623 T^{12} + 69534993539 T^{14} + 11688200277601 T^{16}$$)($$1 + 110 T^{2} + 5977 T^{4} + 266750 T^{6} + 11699428 T^{8} + 493220750 T^{10} + 20434173577 T^{12} + 695349935390 T^{14} + 11688200277601 T^{16}$$)
$47$ ($$1 + 58 T^{2} + 1155 T^{4} + 128122 T^{6} + 4879681 T^{8}$$)($$1 + 58 T^{2} + 1155 T^{4} + 128122 T^{6} + 4879681 T^{8}$$)($$( 1 + 47 T^{2} + 2209 T^{4} )^{2}$$)($$1 - 50 T^{2} + 291 T^{4} - 110450 T^{6} + 4879681 T^{8}$$)($$1 + 85 T^{2} + 5016 T^{4} + 187765 T^{6} + 4879681 T^{8}$$)($$1 + 131 T^{2} + 9121 T^{4} + 474482 T^{6} + 21853270 T^{8} + 1048130738 T^{10} + 44507570401 T^{12} + 1412077208099 T^{14} + 23811286661761 T^{16}$$)($$1 + 68 T^{2} + 2506 T^{4} - 156400 T^{6} - 10608173 T^{8} - 345487600 T^{10} + 12228480586 T^{12} + 732986642372 T^{14} + 23811286661761 T^{16}$$)
$53$ ($$( 1 + 38 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 + 38 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 70 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 53 T^{2} )^{4}$$)($$( 1 - 70 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 + 26 T^{2} + 2809 T^{4} )^{4}$$)($$( 1 - 128 T^{2} + 8850 T^{4} - 359552 T^{6} + 7890481 T^{8} )^{2}$$)
$59$ ($$( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} )^{2}$$)($$( 1 - 9 T + 22 T^{2} - 531 T^{3} + 3481 T^{4} )^{2}$$)($$( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} )^{2}$$)($$( 1 - 9 T + 22 T^{2} - 531 T^{3} + 3481 T^{4} )^{2}$$)($$( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} )^{2}$$)($$( 1 + 3 T - 103 T^{2} - 18 T^{3} + 8532 T^{4} - 1062 T^{5} - 358543 T^{6} + 616137 T^{7} + 12117361 T^{8} )^{2}$$)($$( 1 - 6 T - 85 T^{2} - 18 T^{3} + 9036 T^{4} - 1062 T^{5} - 295885 T^{6} - 1232274 T^{7} + 12117361 T^{8} )^{2}$$)
$61$ ($$( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} )^{2}$$)($$( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} )^{2}$$)($$( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} )^{2}$$)($$( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} )^{2}$$)($$( 1 - 14 T + 61 T^{2} )^{2}( 1 + T + 61 T^{2} )^{2}$$)($$( 1 + T - 47 T^{2} - 74 T^{3} - 1478 T^{4} - 4514 T^{5} - 174887 T^{6} + 226981 T^{7} + 13845841 T^{8} )^{2}$$)($$( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} )^{4}$$)
$67$ ($$( 1 - 13 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} )$$)($$1 - 35 T^{2} - 3264 T^{4} - 157115 T^{6} + 20151121 T^{8}$$)($$( 1 - 13 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} )$$)($$1 + 118 T^{2} + 9435 T^{4} + 529702 T^{6} + 20151121 T^{8}$$)($$1 - 35 T^{2} - 3264 T^{4} - 157115 T^{6} + 20151121 T^{8}$$)($$( 1 + 85 T^{2} + 2736 T^{4} + 381565 T^{6} + 20151121 T^{8} )^{2}$$)($$( 1 + 62 T^{2} + 4489 T^{4} )^{2}( 1 - 62 T^{2} - 645 T^{4} - 278318 T^{6} + 20151121 T^{8} )$$)
$71$ ($$( 1 - 12 T + 71 T^{2} )^{4}$$)($$( 1 + 6 T + 71 T^{2} )^{4}$$)($$( 1 + 6 T + 71 T^{2} )^{4}$$)($$( 1 - 6 T + 71 T^{2} )^{4}$$)($$( 1 - 6 T + 71 T^{2} )^{4}$$)($$( 1 - 6 T + 71 T^{2} )^{8}$$)($$( 1 + 12 T + 124 T^{2} + 852 T^{3} + 5041 T^{4} )^{4}$$)
$73$ ($$( 1 - 25 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 - 145 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 - 97 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 + 50 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 - 130 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 - 83 T^{2} + 3396 T^{4} - 442307 T^{6} + 28398241 T^{8} )^{2}$$)($$( 1 - 145 T^{2} + 5329 T^{4} )^{4}$$)
$79$ ($$( 1 - 13 T + 79 T^{2} )^{2}( 1 + 17 T + 79 T^{2} )^{2}$$)($$( 1 - 2 T - 75 T^{2} - 158 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 - 14 T + 117 T^{2} - 1106 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 - 2 T - 75 T^{2} - 158 T^{3} + 6241 T^{4} )^{4}$$)($$( 1 - 4 T + 70 T^{2} + 848 T^{3} - 4589 T^{4} + 66992 T^{5} + 436870 T^{6} - 1972156 T^{7} + 38950081 T^{8} )^{2}$$)
$83$ ($$1 + 22 T^{2} - 6405 T^{4} + 151558 T^{6} + 47458321 T^{8}$$)($$1 + 85 T^{2} + 336 T^{4} + 585565 T^{6} + 47458321 T^{8}$$)($$1 + 22 T^{2} - 6405 T^{4} + 151558 T^{6} + 47458321 T^{8}$$)($$1 + 85 T^{2} + 336 T^{4} + 585565 T^{6} + 47458321 T^{8}$$)($$1 + 85 T^{2} + 336 T^{4} + 585565 T^{6} + 47458321 T^{8}$$)($$1 + 275 T^{2} + 43609 T^{4} + 5015450 T^{6} + 456585310 T^{8} + 34551435050 T^{10} + 2069609920489 T^{12} + 89908602676475 T^{14} + 2252292232139041 T^{16}$$)($$1 + 302 T^{2} + 54841 T^{4} + 6820670 T^{6} + 646505092 T^{8} + 46987595630 T^{10} + 2602661781961 T^{12} + 98735992757438 T^{14} + 2252292232139041 T^{16}$$)
$89$ ($$( 1 - 6 T + 89 T^{2} )^{4}$$)($$( 1 - 15 T + 89 T^{2} )^{4}$$)($$( 1 - 6 T + 89 T^{2} )^{4}$$)($$( 1 + 9 T + 89 T^{2} )^{4}$$)($$( 1 - 9 T + 89 T^{2} )^{4}$$)($$( 1 - 15 T + 160 T^{2} - 1335 T^{3} + 7921 T^{4} )^{4}$$)($$( 1 + 9 T + 89 T^{2} )^{8}$$)
$97$ ($$( 1 + 2 T^{2} + 9409 T^{4} )( 1 + 167 T^{2} + 9409 T^{4} )$$)($$1 - 95 T^{2} - 384 T^{4} - 893855 T^{6} + 88529281 T^{8}$$)($$1 + 73 T^{2} - 4080 T^{4} + 686857 T^{6} + 88529281 T^{8}$$)($$1 + 145 T^{2} + 11616 T^{4} + 1364305 T^{6} + 88529281 T^{8}$$)($$1 + 190 T^{2} + 26691 T^{4} + 1787710 T^{6} + 88529281 T^{8}$$)($$1 + 311 T^{2} + 54721 T^{4} + 7209602 T^{6} + 765422830 T^{8} + 67835145218 T^{10} + 4844410785601 T^{12} + 259054293532919 T^{14} + 7837433594376961 T^{16}$$)($$1 + 194 T^{2} + 9793 T^{4} + 1750850 T^{6} + 332586244 T^{8} + 16473747650 T^{10} + 866967248833 T^{12} + 161596568956226 T^{14} + 7837433594376961 T^{16}$$)