# Properties

 Label 1512.2.j Level 1512 Weight 2 Character orbit j Rep. character $$\chi_{1512}(323,\cdot)$$ Character field $$\Q$$ Dimension 96 Newform subspaces 4 Sturm bound 576 Trace bound 2

# Related objects

## Defining parameters

 Level: $$N$$ = $$1512 = 2^{3} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1512.j (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$24$$ Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$576$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1512, [\chi])$$.

Total New Old
Modular forms 300 96 204
Cusp forms 276 96 180
Eisenstein series 24 0 24

## Trace form

 $$96q + 4q^{4} + O(q^{10})$$ $$96q + 4q^{4} - 12q^{10} + 12q^{16} - 16q^{19} - 24q^{22} + 96q^{25} + 16q^{28} + 36q^{34} + 40q^{40} + 32q^{43} + 60q^{46} - 96q^{49} + 8q^{52} - 12q^{58} - 44q^{64} + 64q^{67} - 12q^{70} + 32q^{76} - 12q^{82} + 96q^{88} - 36q^{94} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1512, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
1512.2.j.a $$8$$ $$12.073$$ 8.0.56070144.2 None $$-4$$ $$0$$ $$-8$$ $$0$$ $$q+(-1-\beta _{3}-\beta _{5})q^{2}+(-\beta _{1}-\beta _{3}+\cdots)q^{4}+\cdots$$
1512.2.j.b $$8$$ $$12.073$$ 8.0.56070144.2 None $$4$$ $$0$$ $$8$$ $$0$$ $$q+(1+\beta _{3}+\beta _{5})q^{2}+(-\beta _{1}-\beta _{3}+\beta _{5}+\cdots)q^{4}+\cdots$$
1512.2.j.c $$32$$ $$12.073$$ None $$0$$ $$0$$ $$0$$ $$0$$
1512.2.j.d $$48$$ $$12.073$$ None $$0$$ $$0$$ $$0$$ $$0$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1512, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1512, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(24, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(72, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(168, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(216, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(504, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} )^{2}$$)($$( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )^{2}$$)
$3$ 1
$5$ ($$( 1 + 4 T + 10 T^{2} + 32 T^{3} + 87 T^{4} + 160 T^{5} + 250 T^{6} + 500 T^{7} + 625 T^{8} )^{2}$$)($$( 1 - 4 T + 10 T^{2} - 32 T^{3} + 87 T^{4} - 160 T^{5} + 250 T^{6} - 500 T^{7} + 625 T^{8} )^{2}$$)
$7$ ($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)
$11$ ($$1 - 44 T^{2} + 994 T^{4} - 15968 T^{6} + 198763 T^{8} - 1932128 T^{10} + 14553154 T^{12} - 77948684 T^{14} + 214358881 T^{16}$$)($$1 - 44 T^{2} + 994 T^{4} - 15968 T^{6} + 198763 T^{8} - 1932128 T^{10} + 14553154 T^{12} - 77948684 T^{14} + 214358881 T^{16}$$)
$13$ ($$1 - 8 T^{2} + 228 T^{4} - 4504 T^{6} + 20006 T^{8} - 761176 T^{10} + 6511908 T^{12} - 38614472 T^{14} + 815730721 T^{16}$$)($$1 - 8 T^{2} + 228 T^{4} - 4504 T^{6} + 20006 T^{8} - 761176 T^{10} + 6511908 T^{12} - 38614472 T^{14} + 815730721 T^{16}$$)
$17$ ($$( 1 - 8 T + 17 T^{2} )^{4}( 1 + 8 T + 17 T^{2} )^{4}$$)($$( 1 - 8 T + 17 T^{2} )^{4}( 1 + 8 T + 17 T^{2} )^{4}$$)
$19$ ($$( 1 - 8 T + 30 T^{2} - 112 T^{3} + 611 T^{4} - 2128 T^{5} + 10830 T^{6} - 54872 T^{7} + 130321 T^{8} )^{2}$$)($$( 1 - 8 T + 30 T^{2} - 112 T^{3} + 611 T^{4} - 2128 T^{5} + 10830 T^{6} - 54872 T^{7} + 130321 T^{8} )^{2}$$)
$23$ ($$( 1 - 8 T + 46 T^{2} - 88 T^{3} + 231 T^{4} - 2024 T^{5} + 24334 T^{6} - 97336 T^{7} + 279841 T^{8} )^{2}$$)($$( 1 + 8 T + 46 T^{2} + 88 T^{3} + 231 T^{4} + 2024 T^{5} + 24334 T^{6} + 97336 T^{7} + 279841 T^{8} )^{2}$$)
$29$ ($$( 1 + 6 T + 64 T^{2} + 174 T^{3} + 841 T^{4} )^{4}$$)($$( 1 - 6 T + 64 T^{2} - 174 T^{3} + 841 T^{4} )^{4}$$)
$31$ ($$1 - 156 T^{2} + 12106 T^{4} - 610416 T^{6} + 22035219 T^{8} - 586609776 T^{10} + 11180145226 T^{12} - 138450574236 T^{14} + 852891037441 T^{16}$$)($$1 - 156 T^{2} + 12106 T^{4} - 610416 T^{6} + 22035219 T^{8} - 586609776 T^{10} + 11180145226 T^{12} - 138450574236 T^{14} + 852891037441 T^{16}$$)
$37$ ($$1 - 164 T^{2} + 14154 T^{4} - 829168 T^{6} + 35617331 T^{8} - 1135130992 T^{10} + 26526874794 T^{12} - 420779131076 T^{14} + 3512479453921 T^{16}$$)($$1 - 164 T^{2} + 14154 T^{4} - 829168 T^{6} + 35617331 T^{8} - 1135130992 T^{10} + 26526874794 T^{12} - 420779131076 T^{14} + 3512479453921 T^{16}$$)
$41$ ($$1 - 116 T^{2} + 11362 T^{4} - 660608 T^{6} + 32960203 T^{8} - 1110482048 T^{10} + 32106296482 T^{12} - 551012091956 T^{14} + 7984925229121 T^{16}$$)($$1 - 116 T^{2} + 11362 T^{4} - 660608 T^{6} + 32960203 T^{8} - 1110482048 T^{10} + 32106296482 T^{12} - 551012091956 T^{14} + 7984925229121 T^{16}$$)
$43$ ($$( 1 - 4 T + 84 T^{2} - 44 T^{3} + 3002 T^{4} - 1892 T^{5} + 155316 T^{6} - 318028 T^{7} + 3418801 T^{8} )^{2}$$)($$( 1 - 4 T + 84 T^{2} - 44 T^{3} + 3002 T^{4} - 1892 T^{5} + 155316 T^{6} - 318028 T^{7} + 3418801 T^{8} )^{2}$$)
$47$ ($$( 1 + 4 T + 76 T^{2} + 44 T^{3} + 2154 T^{4} + 2068 T^{5} + 167884 T^{6} + 415292 T^{7} + 4879681 T^{8} )^{2}$$)($$( 1 - 4 T + 76 T^{2} - 44 T^{3} + 2154 T^{4} - 2068 T^{5} + 167884 T^{6} - 415292 T^{7} + 4879681 T^{8} )^{2}$$)
$53$ ($$( 1 + 124 T^{2} - 192 T^{3} + 7734 T^{4} - 10176 T^{5} + 348316 T^{6} + 7890481 T^{8} )^{2}$$)($$( 1 + 124 T^{2} + 192 T^{3} + 7734 T^{4} + 10176 T^{5} + 348316 T^{6} + 7890481 T^{8} )^{2}$$)
$59$ ($$1 - 232 T^{2} + 28836 T^{4} - 2582456 T^{6} + 175948646 T^{8} - 8989529336 T^{10} + 349416221796 T^{12} - 9785883804712 T^{14} + 146830437604321 T^{16}$$)($$1 - 232 T^{2} + 28836 T^{4} - 2582456 T^{6} + 175948646 T^{8} - 8989529336 T^{10} + 349416221796 T^{12} - 9785883804712 T^{14} + 146830437604321 T^{16}$$)
$61$ ($$( 1 - 92 T^{2} + 6486 T^{4} - 342332 T^{6} + 13845841 T^{8} )^{2}$$)($$( 1 - 92 T^{2} + 6486 T^{4} - 342332 T^{6} + 13845841 T^{8} )^{2}$$)
$67$ ($$( 1 - 4 T + 228 T^{2} - 620 T^{3} + 21530 T^{4} - 41540 T^{5} + 1023492 T^{6} - 1203052 T^{7} + 20151121 T^{8} )^{2}$$)($$( 1 - 4 T + 228 T^{2} - 620 T^{3} + 21530 T^{4} - 41540 T^{5} + 1023492 T^{6} - 1203052 T^{7} + 20151121 T^{8} )^{2}$$)
$71$ ($$( 1 - 16 T + 310 T^{2} - 3320 T^{3} + 33807 T^{4} - 235720 T^{5} + 1562710 T^{6} - 5726576 T^{7} + 25411681 T^{8} )^{2}$$)($$( 1 + 16 T + 310 T^{2} + 3320 T^{3} + 33807 T^{4} + 235720 T^{5} + 1562710 T^{6} + 5726576 T^{7} + 25411681 T^{8} )^{2}$$)
$73$ ($$( 1 - 4 T + 156 T^{2} - 1460 T^{3} + 11402 T^{4} - 106580 T^{5} + 831324 T^{6} - 1556068 T^{7} + 28398241 T^{8} )^{2}$$)($$( 1 - 4 T + 156 T^{2} - 1460 T^{3} + 11402 T^{4} - 106580 T^{5} + 831324 T^{6} - 1556068 T^{7} + 28398241 T^{8} )^{2}$$)
$79$ ($$1 - 248 T^{2} + 37188 T^{4} - 3949096 T^{6} + 344639558 T^{8} - 24646308136 T^{10} + 1448475612228 T^{12} - 60285688969208 T^{14} + 1517108809906561 T^{16}$$)($$1 - 248 T^{2} + 37188 T^{4} - 3949096 T^{6} + 344639558 T^{8} - 24646308136 T^{10} + 1448475612228 T^{12} - 60285688969208 T^{14} + 1517108809906561 T^{16}$$)
$83$ ($$1 - 232 T^{2} + 34596 T^{4} - 3949112 T^{6} + 355979174 T^{8} - 27205432568 T^{10} + 1641868073316 T^{12} - 75850166621608 T^{14} + 2252292232139041 T^{16}$$)($$1 - 232 T^{2} + 34596 T^{4} - 3949112 T^{6} + 355979174 T^{8} - 27205432568 T^{10} + 1641868073316 T^{12} - 75850166621608 T^{14} + 2252292232139041 T^{16}$$)
$89$ ($$1 - 292 T^{2} + 47890 T^{4} - 6052768 T^{6} + 614802619 T^{8} - 47943975328 T^{10} + 3004725921490 T^{12} - 145118536960612 T^{14} + 3936588805702081 T^{16}$$)($$1 - 292 T^{2} + 47890 T^{4} - 6052768 T^{6} + 614802619 T^{8} - 47943975328 T^{10} + 3004725921490 T^{12} - 145118536960612 T^{14} + 3936588805702081 T^{16}$$)
$97$ ($$( 1 + 16 T + 300 T^{2} + 4208 T^{3} + 41318 T^{4} + 408176 T^{5} + 2822700 T^{6} + 14602768 T^{7} + 88529281 T^{8} )^{2}$$)($$( 1 + 16 T + 300 T^{2} + 4208 T^{3} + 41318 T^{4} + 408176 T^{5} + 2822700 T^{6} + 14602768 T^{7} + 88529281 T^{8} )^{2}$$)