# Properties

 Label 3024.2.bf Level 3024 Weight 2 Character orbit bf Rep. character $$\chi_{3024}(1711,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 96 Newform subspaces 9 Sturm bound 1152 Trace bound 7

# Related objects

## Defining parameters

 Level: $$N$$ = $$3024 = 2^{4} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 3024.bf (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$252$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$9$$ Sturm bound: $$1152$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$5$$, $$19$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(3024, [\chi])$$.

Total New Old
Modular forms 1224 96 1128
Cusp forms 1080 96 984
Eisenstein series 144 0 144

## Trace form

 $$96q + O(q^{10})$$ $$96q - 96q^{25} + 12q^{29} - 24q^{65} + 24q^{77} - 36q^{89} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(3024, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
3024.2.bf.a $$2$$ $$24.147$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$-5$$ $$q+(1-2\zeta_{6})q^{5}+(-3+\zeta_{6})q^{7}+(3-6\zeta_{6})q^{11}+\cdots$$
3024.2.bf.b $$2$$ $$24.147$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q+(-1-2\zeta_{6})q^{7}+(-2+4\zeta_{6})q^{11}+\cdots$$
3024.2.bf.c $$2$$ $$24.147$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$4$$ $$q+(1+2\zeta_{6})q^{7}+(2-4\zeta_{6})q^{11}+(1+\zeta_{6})q^{13}+\cdots$$
3024.2.bf.d $$2$$ $$24.147$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$5$$ $$q+(1-2\zeta_{6})q^{5}+(3-\zeta_{6})q^{7}+(-3+6\zeta_{6})q^{11}+\cdots$$
3024.2.bf.e $$4$$ $$24.147$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$-2$$ $$q+(1+2\beta _{1}-\beta _{2})q^{5}+(\beta _{1}+\beta _{3})q^{7}+\cdots$$
3024.2.bf.f $$4$$ $$24.147$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$2$$ $$q+(1+2\beta _{1}-\beta _{2})q^{5}+(-\beta _{1}-\beta _{3})q^{7}+\cdots$$
3024.2.bf.g $$24$$ $$24.147$$ None $$0$$ $$0$$ $$0$$ $$-8$$
3024.2.bf.h $$24$$ $$24.147$$ None $$0$$ $$0$$ $$0$$ $$8$$
3024.2.bf.i $$32$$ $$24.147$$ None $$0$$ $$0$$ $$0$$ $$0$$

## Decomposition of $$S_{2}^{\mathrm{old}}(3024, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(3024, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(252, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(756, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1008, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ ($$1 - 7 T^{2} + 25 T^{4}$$)($$( 1 - 5 T^{2} )^{2}$$)($$( 1 - 5 T^{2} )^{2}$$)($$1 - 7 T^{2} + 25 T^{4}$$)($$( 1 - 6 T + 17 T^{2} - 30 T^{3} + 25 T^{4} )( 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4} )$$)($$( 1 - 6 T + 17 T^{2} - 30 T^{3} + 25 T^{4} )( 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4} )$$)
$7$ ($$1 + 5 T + 7 T^{2}$$)($$1 + 4 T + 7 T^{2}$$)($$1 - 4 T + 7 T^{2}$$)($$1 - 5 T + 7 T^{2}$$)($$1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4}$$)($$1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4}$$)
$11$ ($$1 + 5 T^{2} + 121 T^{4}$$)($$1 - 10 T^{2} + 121 T^{4}$$)($$1 - 10 T^{2} + 121 T^{4}$$)($$1 + 5 T^{2} + 121 T^{4}$$)($$( 1 - 19 T^{2} + 121 T^{4} )^{2}$$)($$( 1 - 19 T^{2} + 121 T^{4} )^{2}$$)
$13$ ($$( 1 + 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$)($$( 1 - 5 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$)($$( 1 - 5 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$)($$( 1 + 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$)($$1 - 6 T + 35 T^{2} - 138 T^{3} + 516 T^{4} - 1794 T^{5} + 5915 T^{6} - 13182 T^{7} + 28561 T^{8}$$)($$1 - 6 T + 35 T^{2} - 138 T^{3} + 516 T^{4} - 1794 T^{5} + 5915 T^{6} - 13182 T^{7} + 28561 T^{8}$$)
$17$ ($$1 + 3 T + 20 T^{2} + 51 T^{3} + 289 T^{4}$$)($$1 + 3 T + 20 T^{2} + 51 T^{3} + 289 T^{4}$$)($$1 + 3 T + 20 T^{2} + 51 T^{3} + 289 T^{4}$$)($$1 + 3 T + 20 T^{2} + 51 T^{3} + 289 T^{4}$$)($$1 + 6 T + 25 T^{2} + 78 T^{3} + 84 T^{4} + 1326 T^{5} + 7225 T^{6} + 29478 T^{7} + 83521 T^{8}$$)($$1 + 6 T + 25 T^{2} + 78 T^{3} + 84 T^{4} + 1326 T^{5} + 7225 T^{6} + 29478 T^{7} + 83521 T^{8}$$)
$19$ ($$( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$)($$1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4}$$)($$1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4}$$)($$( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} )$$)($$( 1 - 8 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2}$$)($$( 1 - 7 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2}$$)
$23$ ($$1 - 43 T^{2} + 529 T^{4}$$)($$1 - 34 T^{2} + 529 T^{4}$$)($$1 - 34 T^{2} + 529 T^{4}$$)($$1 - 43 T^{2} + 529 T^{4}$$)($$1 - 26 T^{2} + 579 T^{4} - 13754 T^{6} + 279841 T^{8}$$)($$1 - 26 T^{2} + 579 T^{4} - 13754 T^{6} + 279841 T^{8}$$)
$29$ ($$1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4}$$)($$1 - 3 T - 20 T^{2} - 87 T^{3} + 841 T^{4}$$)($$1 - 3 T - 20 T^{2} - 87 T^{3} + 841 T^{4}$$)($$1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4}$$)($$1 - 6 T - 13 T^{2} + 54 T^{3} + 516 T^{4} + 1566 T^{5} - 10933 T^{6} - 146334 T^{7} + 707281 T^{8}$$)($$1 - 6 T - 13 T^{2} + 54 T^{3} + 516 T^{4} + 1566 T^{5} - 10933 T^{6} - 146334 T^{7} + 707281 T^{8}$$)
$31$ ($$1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4}$$)($$1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4}$$)($$1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4}$$)($$1 - 8 T + 33 T^{2} - 248 T^{3} + 961 T^{4}$$)($$( 1 - 7 T + 31 T^{2} )^{2}( 1 + 11 T + 31 T^{2} )^{2}$$)($$( 1 - 11 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2}$$)
$37$ ($$( 1 - 10 T + 37 T^{2} )( 1 - T + 37 T^{2} )$$)($$1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4}$$)($$1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4}$$)($$( 1 - 10 T + 37 T^{2} )( 1 - T + 37 T^{2} )$$)($$1 + 2 T - 53 T^{2} - 34 T^{3} + 1732 T^{4} - 1258 T^{5} - 72557 T^{6} + 101306 T^{7} + 1874161 T^{8}$$)($$1 + 2 T - 53 T^{2} - 34 T^{3} + 1732 T^{4} - 1258 T^{5} - 72557 T^{6} + 101306 T^{7} + 1874161 T^{8}$$)
$41$ ($$1 - 3 T + 44 T^{2} - 123 T^{3} + 1681 T^{4}$$)($$1 + 3 T + 44 T^{2} + 123 T^{3} + 1681 T^{4}$$)($$1 + 3 T + 44 T^{2} + 123 T^{3} + 1681 T^{4}$$)($$1 - 3 T + 44 T^{2} - 123 T^{3} + 1681 T^{4}$$)($$1 + 18 T + 193 T^{2} + 1530 T^{3} + 9972 T^{4} + 62730 T^{5} + 324433 T^{6} + 1240578 T^{7} + 2825761 T^{8}$$)($$1 + 18 T + 193 T^{2} + 1530 T^{3} + 9972 T^{4} + 62730 T^{5} + 324433 T^{6} + 1240578 T^{7} + 2825761 T^{8}$$)
$43$ ($$1 + 15 T + 118 T^{2} + 645 T^{3} + 1849 T^{4}$$)($$( 1 - 5 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} )$$)($$( 1 - 8 T + 43 T^{2} )( 1 + 5 T + 43 T^{2} )$$)($$1 - 15 T + 118 T^{2} - 645 T^{3} + 1849 T^{4}$$)($$1 + 6 T + 77 T^{2} + 390 T^{3} + 2892 T^{4} + 16770 T^{5} + 142373 T^{6} + 477042 T^{7} + 3418801 T^{8}$$)($$1 - 6 T + 77 T^{2} - 390 T^{3} + 2892 T^{4} - 16770 T^{5} + 142373 T^{6} - 477042 T^{7} + 3418801 T^{8}$$)
$47$ ($$1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4}$$)($$1 + 9 T + 34 T^{2} + 423 T^{3} + 2209 T^{4}$$)($$1 - 9 T + 34 T^{2} - 423 T^{3} + 2209 T^{4}$$)($$1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4}$$)($$1 - 22 T^{2} - 1725 T^{4} - 48598 T^{6} + 4879681 T^{8}$$)($$1 - 22 T^{2} - 1725 T^{4} - 48598 T^{6} + 4879681 T^{8}$$)
$53$ ($$1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4}$$)($$1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4}$$)($$1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4}$$)($$1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4}$$)($$1 + 6 T - 61 T^{2} - 54 T^{3} + 4692 T^{4} - 2862 T^{5} - 171349 T^{6} + 893262 T^{7} + 7890481 T^{8}$$)($$1 + 6 T - 61 T^{2} - 54 T^{3} + 4692 T^{4} - 2862 T^{5} - 171349 T^{6} + 893262 T^{7} + 7890481 T^{8}$$)
$59$ ($$1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4}$$)($$1 + 15 T + 166 T^{2} + 885 T^{3} + 3481 T^{4}$$)($$1 - 15 T + 166 T^{2} - 885 T^{3} + 3481 T^{4}$$)($$1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4}$$)($$( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} )^{2}$$)($$( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} )^{2}$$)
$61$ ($$1 + 61 T^{2} + 3721 T^{4}$$)($$1 - 3 T + 64 T^{2} - 183 T^{3} + 3721 T^{4}$$)($$1 - 3 T + 64 T^{2} - 183 T^{3} + 3721 T^{4}$$)($$1 + 61 T^{2} + 3721 T^{4}$$)($$( 1 - 13 T + 61 T^{2} )^{2}( 1 + T + 61 T^{2} )^{2}$$)($$( 1 - 13 T + 61 T^{2} )^{2}( 1 + T + 61 T^{2} )^{2}$$)
$67$ ($$( 1 - 11 T + 67 T^{2} )( 1 + 5 T + 67 T^{2} )$$)($$( 1 + 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$)($$( 1 - 16 T + 67 T^{2} )( 1 - 11 T + 67 T^{2} )$$)($$( 1 - 5 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} )$$)($$( 1 + 67 T^{2} + 4489 T^{4} )^{2}$$)($$( 1 + 67 T^{2} + 4489 T^{4} )^{2}$$)
$71$ ($$1 - 130 T^{2} + 5041 T^{4}$$)($$1 - 34 T^{2} + 5041 T^{4}$$)($$1 - 34 T^{2} + 5041 T^{4}$$)($$1 - 130 T^{2} + 5041 T^{4}$$)($$1 - 68 T^{2} + 6630 T^{4} - 342788 T^{6} + 25411681 T^{8}$$)($$1 - 68 T^{2} + 6630 T^{4} - 342788 T^{6} + 25411681 T^{8}$$)
$73$ ($$1 + 21 T + 220 T^{2} + 1533 T^{3} + 5329 T^{4}$$)($$( 1 - 10 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} )$$)($$( 1 - 10 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} )$$)($$1 + 21 T + 220 T^{2} + 1533 T^{3} + 5329 T^{4}$$)($$( 1 + 15 T + 148 T^{2} + 1095 T^{3} + 5329 T^{4} )^{2}$$)($$( 1 + 15 T + 148 T^{2} + 1095 T^{3} + 5329 T^{4} )^{2}$$)
$79$ ($$1 + 18 T + 187 T^{2} + 1422 T^{3} + 6241 T^{4}$$)($$1 - 3 T + 82 T^{2} - 237 T^{3} + 6241 T^{4}$$)($$1 + 3 T + 82 T^{2} + 237 T^{3} + 6241 T^{4}$$)($$1 - 18 T + 187 T^{2} - 1422 T^{3} + 6241 T^{4}$$)($$1 - 24 T + 374 T^{2} - 4368 T^{3} + 42051 T^{4} - 345072 T^{5} + 2334134 T^{6} - 11832936 T^{7} + 38950081 T^{8}$$)($$1 + 24 T + 374 T^{2} + 4368 T^{3} + 42051 T^{4} + 345072 T^{5} + 2334134 T^{6} + 11832936 T^{7} + 38950081 T^{8}$$)
$83$ ($$1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4}$$)($$1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4}$$)($$1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4}$$)($$1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4}$$)($$1 - 6 T - 67 T^{2} + 378 T^{3} + 732 T^{4} + 31374 T^{5} - 461563 T^{6} - 3430722 T^{7} + 47458321 T^{8}$$)($$1 + 6 T - 67 T^{2} - 378 T^{3} + 732 T^{4} - 31374 T^{5} - 461563 T^{6} + 3430722 T^{7} + 47458321 T^{8}$$)
$89$ ($$1 - 9 T + 116 T^{2} - 801 T^{3} + 7921 T^{4}$$)($$1 + 3 T + 92 T^{2} + 267 T^{3} + 7921 T^{4}$$)($$1 + 3 T + 92 T^{2} + 267 T^{3} + 7921 T^{4}$$)($$1 - 9 T + 116 T^{2} - 801 T^{3} + 7921 T^{4}$$)($$1 - 6 T - 23 T^{2} + 210 T^{3} - 5628 T^{4} + 18690 T^{5} - 182183 T^{6} - 4229814 T^{7} + 62742241 T^{8}$$)($$1 - 6 T - 23 T^{2} + 210 T^{3} - 5628 T^{4} + 18690 T^{5} - 182183 T^{6} - 4229814 T^{7} + 62742241 T^{8}$$)
$97$ ($$1 + 3 T + 100 T^{2} + 291 T^{3} + 9409 T^{4}$$)($$1 - 3 T + 100 T^{2} - 291 T^{3} + 9409 T^{4}$$)($$1 - 3 T + 100 T^{2} - 291 T^{3} + 9409 T^{4}$$)($$1 + 3 T + 100 T^{2} + 291 T^{3} + 9409 T^{4}$$)($$1 - 6 T + 185 T^{2} - 1038 T^{3} + 21684 T^{4} - 100686 T^{5} + 1740665 T^{6} - 5476038 T^{7} + 88529281 T^{8}$$)($$1 - 6 T + 185 T^{2} - 1038 T^{3} + 21684 T^{4} - 100686 T^{5} + 1740665 T^{6} - 5476038 T^{7} + 88529281 T^{8}$$)