# Properties

 Label 504.2.t Level 504 Weight 2 Character orbit t Rep. character $$\chi_{504}(193,\cdot)$$ Character field $$\Q(\zeta_{3})$$ Dimension 48 Newform subspaces 4 Sturm bound 192 Trace bound 3

# Learn more about

## Defining parameters

 Level: $$N$$ = $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 504.t (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$63$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$4$$ Sturm bound: $$192$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(504, [\chi])$$.

Total New Old
Modular forms 208 48 160
Cusp forms 176 48 128
Eisenstein series 32 0 32

## Trace form

 $$48q - 8q^{5} - 2q^{9} + O(q^{10})$$ $$48q - 8q^{5} - 2q^{9} - 8q^{15} + 8q^{17} + 18q^{21} + 8q^{23} + 48q^{25} - 6q^{27} - 6q^{29} + 6q^{31} - 4q^{33} + 12q^{35} + 4q^{39} + 18q^{41} + 6q^{43} + 22q^{45} + 6q^{47} + 12q^{49} - 18q^{51} + 4q^{53} + 12q^{55} - 20q^{57} - 36q^{59} + 6q^{61} - 4q^{63} + 12q^{65} - 32q^{69} - 40q^{71} - 24q^{75} + 4q^{77} - 6q^{79} - 26q^{81} - 36q^{83} + 46q^{87} + 18q^{89} - 6q^{91} - 52q^{93} - 10q^{95} - 32q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(504, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
504.2.t.a $$2$$ $$4.024$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$2$$ $$-5$$ $$q+(-1-\zeta_{6})q^{3}+q^{5}+(-2-\zeta_{6})q^{7}+\cdots$$
504.2.t.b $$2$$ $$4.024$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$-2$$ $$-1$$ $$q+(1+\zeta_{6})q^{3}-q^{5}+(-2+3\zeta_{6})q^{7}+\cdots$$
504.2.t.c $$22$$ $$4.024$$ None $$0$$ $$-2$$ $$-2$$ $$-1$$
504.2.t.d $$22$$ $$4.024$$ None $$0$$ $$2$$ $$-6$$ $$7$$

## Decomposition of $$S_{2}^{\mathrm{old}}(504, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(504, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(126, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(252, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$1 + 3 T + 3 T^{2}$$)($$1 - 3 T + 3 T^{2}$$)
$5$ ($$( 1 - T + 5 T^{2} )^{2}$$)($$( 1 + T + 5 T^{2} )^{2}$$)
$7$ ($$1 + 5 T + 7 T^{2}$$)($$1 + T + 7 T^{2}$$)
$11$ ($$( 1 + 3 T + 11 T^{2} )^{2}$$)($$( 1 - 3 T + 11 T^{2} )^{2}$$)
$13$ ($$1 + T - 12 T^{2} + 13 T^{3} + 169 T^{4}$$)($$1 + 3 T - 4 T^{2} + 39 T^{3} + 169 T^{4}$$)
$17$ ($$1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4}$$)($$1 - 5 T + 8 T^{2} - 85 T^{3} + 289 T^{4}$$)
$19$ ($$1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4}$$)($$( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$)
$23$ ($$( 1 - T + 23 T^{2} )^{2}$$)($$( 1 - 5 T + 23 T^{2} )^{2}$$)
$29$ ($$1 + 9 T + 52 T^{2} + 261 T^{3} + 841 T^{4}$$)($$1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4}$$)
$31$ ($$( 1 - 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} )$$)($$1 - 8 T + 33 T^{2} - 248 T^{3} + 961 T^{4}$$)
$37$ ($$1 + 5 T - 12 T^{2} + 185 T^{3} + 1369 T^{4}$$)($$1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4}$$)
$41$ ($$1 + 7 T + 8 T^{2} + 287 T^{3} + 1681 T^{4}$$)($$1 - 5 T - 16 T^{2} - 205 T^{3} + 1681 T^{4}$$)
$43$ ($$1 + 3 T - 34 T^{2} + 129 T^{3} + 1849 T^{4}$$)($$1 - 7 T + 6 T^{2} - 301 T^{3} + 1849 T^{4}$$)
$47$ ($$1 + 8 T + 17 T^{2} + 376 T^{3} + 2209 T^{4}$$)($$1 + 8 T + 17 T^{2} + 376 T^{3} + 2209 T^{4}$$)
$53$ ($$1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4}$$)($$1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4}$$)
$59$ ($$1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4}$$)($$1 - 59 T^{2} + 3481 T^{4}$$)
$61$ ($$1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4}$$)($$1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4}$$)
$67$ ($$1 + 12 T + 77 T^{2} + 804 T^{3} + 4489 T^{4}$$)($$1 - 12 T + 77 T^{2} - 804 T^{3} + 4489 T^{4}$$)
$71$ ($$( 1 - 8 T + 71 T^{2} )^{2}$$)($$( 1 - 12 T + 71 T^{2} )^{2}$$)
$73$ ($$1 - 13 T + 96 T^{2} - 949 T^{3} + 5329 T^{4}$$)($$1 - 5 T - 48 T^{2} - 365 T^{3} + 5329 T^{4}$$)
$79$ ($$1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4}$$)($$1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4}$$)
$83$ ($$1 - 13 T + 86 T^{2} - 1079 T^{3} + 6889 T^{4}$$)($$1 - 15 T + 142 T^{2} - 1245 T^{3} + 6889 T^{4}$$)
$89$ ($$1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4}$$)($$1 - 5 T - 64 T^{2} - 445 T^{3} + 7921 T^{4}$$)
$97$ ($$1 - 17 T + 192 T^{2} - 1649 T^{3} + 9409 T^{4}$$)($$1 + 7 T - 48 T^{2} + 679 T^{3} + 9409 T^{4}$$)
show more
show less