# Properties

 Label 504.2.q Level 504 Weight 2 Character orbit q Rep. character $$\chi_{504}(25,\cdot)$$ Character field $$\Q(\zeta_{3})$$ Dimension 48 Newform subspaces 4 Sturm bound 192 Trace bound 5

# Related objects

## Defining parameters

 Level: $$N$$ = $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 504.q (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: $$5$$ 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

 $$48q + 4q^{5} + 4q^{9} + O(q^{10})$$ $$48q + 4q^{5} + 4q^{9} - 8q^{15} + 8q^{17} - 4q^{23} - 24q^{25} - 6q^{27} - 6q^{29} - 12q^{31} + 8q^{33} + 12q^{35} - 26q^{39} + 18q^{41} + 6q^{43} - 2q^{45} - 12q^{47} - 6q^{49} + 36q^{51} + 4q^{53} + 12q^{55} - 20q^{57} + 72q^{59} - 12q^{61} + 26q^{63} - 24q^{65} - 32q^{69} - 40q^{71} + 30q^{75} + 28q^{77} + 12q^{79} - 8q^{81} - 36q^{83} - 68q^{87} + 18q^{89} - 6q^{91} + 8q^{93} + 20q^{95} - 32q^{99} + O(q^{100})$$

## 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.q.a $$2$$ $$4.024$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$4$$ $$q+(1-2\zeta_{6})q^{3}-\zeta_{6}q^{5}+(3-2\zeta_{6})q^{7}+\cdots$$
504.2.q.b $$2$$ $$4.024$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$1$$ $$-4$$ $$q+(-1+2\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-1-2\zeta_{6})q^{7}+\cdots$$
504.2.q.c $$22$$ $$4.024$$ None $$0$$ $$-2$$ $$1$$ $$5$$
504.2.q.d $$22$$ $$4.024$$ None $$0$$ $$2$$ $$3$$ $$-5$$

## 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
$3$ ($$1 + 3 T^{2}$$)($$1 + 3 T^{2}$$)
$5$ ($$1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4}$$)($$1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4}$$)
$7$ ($$1 - 4 T + 7 T^{2}$$)($$1 + 4 T + 7 T^{2}$$)
$11$ ($$1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}$$)($$1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4}$$)
$13$ ($$1 + T - 12 T^{2} + 13 T^{3} + 169 T^{4}$$)($$1 + 3 T - 4 T^{2} + 39 T^{3} + 169 T^{4}$$)
$17$ ($$1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4}$$)($$1 - 5 T + 8 T^{2} - 85 T^{3} + 289 T^{4}$$)
$19$ ($$1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4}$$)($$( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$)
$23$ ($$1 + T - 22 T^{2} + 23 T^{3} + 529 T^{4}$$)($$1 + 5 T + 2 T^{2} + 115 T^{3} + 529 T^{4}$$)
$29$ ($$1 + 9 T + 52 T^{2} + 261 T^{3} + 841 T^{4}$$)($$1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4}$$)
$31$ ($$( 1 - 4 T + 31 T^{2} )^{2}$$)($$( 1 + 8 T + 31 T^{2} )^{2}$$)
$37$ ($$1 + 5 T - 12 T^{2} + 185 T^{3} + 1369 T^{4}$$)($$1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4}$$)
$41$ ($$1 + 7 T + 8 T^{2} + 287 T^{3} + 1681 T^{4}$$)($$1 - 5 T - 16 T^{2} - 205 T^{3} + 1681 T^{4}$$)
$43$ ($$1 + 3 T - 34 T^{2} + 129 T^{3} + 1849 T^{4}$$)($$1 - 7 T + 6 T^{2} - 301 T^{3} + 1849 T^{4}$$)
$47$ ($$( 1 - 8 T + 47 T^{2} )^{2}$$)($$( 1 - 8 T + 47 T^{2} )^{2}$$)
$53$ ($$1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4}$$)($$1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4}$$)
$59$ ($$( 1 + 4 T + 59 T^{2} )^{2}$$)($$( 1 + 59 T^{2} )^{2}$$)
$61$ ($$( 1 - 2 T + 61 T^{2} )^{2}$$)($$( 1 - 10 T + 61 T^{2} )^{2}$$)
$67$ ($$( 1 - 12 T + 67 T^{2} )^{2}$$)($$( 1 + 12 T + 67 T^{2} )^{2}$$)
$71$ ($$( 1 - 8 T + 71 T^{2} )^{2}$$)($$( 1 - 12 T + 71 T^{2} )^{2}$$)
$73$ ($$1 - 13 T + 96 T^{2} - 949 T^{3} + 5329 T^{4}$$)($$1 - 5 T - 48 T^{2} - 365 T^{3} + 5329 T^{4}$$)
$79$ ($$( 1 - 8 T + 79 T^{2} )^{2}$$)($$( 1 + 8 T + 79 T^{2} )^{2}$$)
$83$ ($$1 - 13 T + 86 T^{2} - 1079 T^{3} + 6889 T^{4}$$)($$1 - 15 T + 142 T^{2} - 1245 T^{3} + 6889 T^{4}$$)
$89$ ($$1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4}$$)($$1 - 5 T - 64 T^{2} - 445 T^{3} + 7921 T^{4}$$)
$97$ ($$1 - 17 T + 192 T^{2} - 1649 T^{3} + 9409 T^{4}$$)($$1 + 7 T - 48 T^{2} + 679 T^{3} + 9409 T^{4}$$)