# Properties

 Label 315.2.bl Level 315 Weight 2 Character orbit bl Rep. character $$\chi_{315}(41,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 64 Newform subspaces 10 Sturm bound 96 Trace bound 5

# Related objects

## Defining parameters

 Level: $$N$$ = $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 315.bl (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$63$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$10$$ Sturm bound: $$96$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$2$$, $$11$$, $$13$$, $$17$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(315, [\chi])$$.

Total New Old
Modular forms 104 64 40
Cusp forms 88 64 24
Eisenstein series 16 0 16

## Trace form

 $$64q + 32q^{4} + 2q^{7} + 8q^{9} + O(q^{10})$$ $$64q + 32q^{4} + 2q^{7} + 8q^{9} - 6q^{11} + 24q^{14} - 2q^{15} - 32q^{16} + 4q^{18} + 18q^{21} - 36q^{23} - 32q^{25} + 16q^{28} - 36q^{29} - 4q^{30} - 48q^{36} - 8q^{37} - 6q^{39} - 66q^{42} + 16q^{43} + 24q^{46} - 8q^{49} + 66q^{51} + 36q^{57} + 36q^{60} - 52q^{63} - 64q^{64} - 6q^{65} - 28q^{67} + 6q^{70} + 140q^{72} - 12q^{74} - 54q^{77} - 132q^{78} - 10q^{79} - 16q^{81} + 12q^{84} - 6q^{85} - 156q^{86} + 12q^{91} - 24q^{92} - 36q^{93} + 64q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(315, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
315.2.bl.a $$2$$ $$2.515$$ $$\Q(\sqrt{-3})$$ None $$-3$$ $$-3$$ $$-1$$ $$1$$ $$q+(-1-\zeta_{6})q^{2}+(-2+\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots$$
315.2.bl.b $$2$$ $$2.515$$ $$\Q(\sqrt{-3})$$ None $$-3$$ $$0$$ $$-1$$ $$1$$ $$q+(-1-\zeta_{6})q^{2}+(1-2\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots$$
315.2.bl.c $$2$$ $$2.515$$ $$\Q(\sqrt{-3})$$ None $$-3$$ $$0$$ $$1$$ $$-5$$ $$q+(-1-\zeta_{6})q^{2}+(-1+2\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots$$
315.2.bl.d $$2$$ $$2.515$$ $$\Q(\sqrt{-3})$$ None $$-3$$ $$3$$ $$1$$ $$4$$ $$q+(-1-\zeta_{6})q^{2}+(2-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots$$
315.2.bl.e $$2$$ $$2.515$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$-1$$ $$-5$$ $$q+(-1-\zeta_{6})q^{3}-2\zeta_{6}q^{4}-\zeta_{6}q^{5}+\cdots$$
315.2.bl.f $$2$$ $$2.515$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$1$$ $$1$$ $$q+(-2+\zeta_{6})q^{3}-2\zeta_{6}q^{4}+\zeta_{6}q^{5}+\cdots$$
315.2.bl.g $$2$$ $$2.515$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$-1$$ $$-5$$ $$q+(2-\zeta_{6})q^{3}-2\zeta_{6}q^{4}-\zeta_{6}q^{5}+(-3+\cdots)q^{7}+\cdots$$
315.2.bl.h $$2$$ $$2.515$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$1$$ $$1$$ $$q+(1+\zeta_{6})q^{3}-2\zeta_{6}q^{4}+\zeta_{6}q^{5}+(-1+\cdots)q^{7}+\cdots$$
315.2.bl.i $$24$$ $$2.515$$ None $$6$$ $$-5$$ $$12$$ $$0$$
315.2.bl.j $$24$$ $$2.515$$ None $$6$$ $$5$$ $$-12$$ $$9$$

## Decomposition of $$S_{2}^{\mathrm{old}}(315, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(315, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}$$)($$1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}$$)($$1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}$$)($$1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}$$)($$1 + 2 T^{2} + 4 T^{4}$$)($$1 + 2 T^{2} + 4 T^{4}$$)($$1 + 2 T^{2} + 4 T^{4}$$)($$1 + 2 T^{2} + 4 T^{4}$$)
$3$ ($$1 + 3 T + 3 T^{2}$$)($$1 + 3 T^{2}$$)($$1 + 3 T^{2}$$)($$1 - 3 T + 3 T^{2}$$)($$1 + 3 T + 3 T^{2}$$)($$1 + 3 T + 3 T^{2}$$)($$1 - 3 T + 3 T^{2}$$)($$1 - 3 T + 3 T^{2}$$)
$5$ ($$1 + T + T^{2}$$)($$1 + T + T^{2}$$)($$1 - T + T^{2}$$)($$1 - T + T^{2}$$)($$1 + T + T^{2}$$)($$1 - T + T^{2}$$)($$1 + T + T^{2}$$)($$1 - T + T^{2}$$)
$7$ ($$1 - T + 7 T^{2}$$)($$1 - T + 7 T^{2}$$)($$1 + 5 T + 7 T^{2}$$)($$1 - 4 T + 7 T^{2}$$)($$1 + 5 T + 7 T^{2}$$)($$1 - T + 7 T^{2}$$)($$1 + 5 T + 7 T^{2}$$)($$1 - T + 7 T^{2}$$)
$11$ ($$1 + 11 T^{2} + 121 T^{4}$$)($$1 + 9 T + 38 T^{2} + 99 T^{3} + 121 T^{4}$$)($$1 + 9 T + 38 T^{2} + 99 T^{3} + 121 T^{4}$$)($$1 + 11 T^{2} + 121 T^{4}$$)($$1 - 3 T + 14 T^{2} - 33 T^{3} + 121 T^{4}$$)($$1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4}$$)($$1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4}$$)($$1 - 3 T + 14 T^{2} - 33 T^{3} + 121 T^{4}$$)
$13$ ($$1 + 13 T^{2} + 169 T^{4}$$)($$( 1 + 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$)($$( 1 - 7 T + 13 T^{2} )( 1 - 2 T + 13 T^{2} )$$)($$1 + 13 T^{2} + 169 T^{4}$$)($$1 + 13 T^{2} + 169 T^{4}$$)($$( 1 + 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$)($$( 1 - 7 T + 13 T^{2} )( 1 - 2 T + 13 T^{2} )$$)($$1 + 13 T^{2} + 169 T^{4}$$)
$17$ ($$( 1 + 6 T + 17 T^{2} )^{2}$$)($$( 1 - 3 T + 17 T^{2} )^{2}$$)($$( 1 + 3 T + 17 T^{2} )^{2}$$)($$( 1 - 6 T + 17 T^{2} )^{2}$$)($$( 1 + 6 T + 17 T^{2} )^{2}$$)($$( 1 + 3 T + 17 T^{2} )^{2}$$)($$( 1 - 3 T + 17 T^{2} )^{2}$$)($$( 1 - 6 T + 17 T^{2} )^{2}$$)
$19$ ($$( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$)($$( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$)($$( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$)($$( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$)($$1 + 10 T^{2} + 361 T^{4}$$)($$( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$)($$( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$)($$1 + 10 T^{2} + 361 T^{4}$$)
$23$ ($$1 + 15 T + 98 T^{2} + 345 T^{3} + 529 T^{4}$$)($$1 + 6 T + 35 T^{2} + 138 T^{3} + 529 T^{4}$$)($$1 + 6 T + 35 T^{2} + 138 T^{3} + 529 T^{4}$$)($$1 + 15 T + 98 T^{2} + 345 T^{3} + 529 T^{4}$$)($$1 - 6 T + 35 T^{2} - 138 T^{3} + 529 T^{4}$$)($$1 + 12 T + 71 T^{2} + 276 T^{3} + 529 T^{4}$$)($$1 + 12 T + 71 T^{2} + 276 T^{3} + 529 T^{4}$$)($$1 - 6 T + 35 T^{2} - 138 T^{3} + 529 T^{4}$$)
$29$ ($$1 + 3 T + 32 T^{2} + 87 T^{3} + 841 T^{4}$$)($$1 + 12 T + 77 T^{2} + 348 T^{3} + 841 T^{4}$$)($$1 + 12 T + 77 T^{2} + 348 T^{3} + 841 T^{4}$$)($$1 + 3 T + 32 T^{2} + 87 T^{3} + 841 T^{4}$$)($$1 + 15 T + 104 T^{2} + 435 T^{3} + 841 T^{4}$$)($$1 - 3 T + 32 T^{2} - 87 T^{3} + 841 T^{4}$$)($$1 - 3 T + 32 T^{2} - 87 T^{3} + 841 T^{4}$$)($$1 + 15 T + 104 T^{2} + 435 T^{3} + 841 T^{4}$$)
$31$ ($$1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4}$$)($$1 - 12 T + 79 T^{2} - 372 T^{3} + 961 T^{4}$$)($$1 + 12 T + 79 T^{2} + 372 T^{3} + 961 T^{4}$$)($$1 - 6 T + 43 T^{2} - 186 T^{3} + 961 T^{4}$$)($$1 - 6 T + 43 T^{2} - 186 T^{3} + 961 T^{4}$$)($$1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4}$$)($$1 - 6 T + 43 T^{2} - 186 T^{3} + 961 T^{4}$$)($$1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4}$$)
$37$ ($$( 1 + 2 T + 37 T^{2} )^{2}$$)($$( 1 + 2 T + 37 T^{2} )^{2}$$)($$( 1 + 2 T + 37 T^{2} )^{2}$$)($$( 1 + 2 T + 37 T^{2} )^{2}$$)($$( 1 + 8 T + 37 T^{2} )^{2}$$)($$( 1 - 10 T + 37 T^{2} )^{2}$$)($$( 1 - 10 T + 37 T^{2} )^{2}$$)($$( 1 + 8 T + 37 T^{2} )^{2}$$)
$41$ ($$1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4}$$)($$1 - 41 T^{2} + 1681 T^{4}$$)($$1 - 41 T^{2} + 1681 T^{4}$$)($$1 + 9 T + 40 T^{2} + 369 T^{3} + 1681 T^{4}$$)($$1 + 6 T - 5 T^{2} + 246 T^{3} + 1681 T^{4}$$)($$1 + 12 T + 103 T^{2} + 492 T^{3} + 1681 T^{4}$$)($$1 - 12 T + 103 T^{2} - 492 T^{3} + 1681 T^{4}$$)($$1 - 6 T - 5 T^{2} - 246 T^{3} + 1681 T^{4}$$)
$43$ ($$1 + 4 T - 27 T^{2} + 172 T^{3} + 1849 T^{4}$$)($$1 + 4 T - 27 T^{2} + 172 T^{3} + 1849 T^{4}$$)($$1 + 4 T - 27 T^{2} + 172 T^{3} + 1849 T^{4}$$)($$1 + 4 T - 27 T^{2} + 172 T^{3} + 1849 T^{4}$$)($$1 + 10 T + 57 T^{2} + 430 T^{3} + 1849 T^{4}$$)($$( 1 - 13 T + 43 T^{2} )( 1 + 5 T + 43 T^{2} )$$)($$( 1 - 13 T + 43 T^{2} )( 1 + 5 T + 43 T^{2} )$$)($$1 + 10 T + 57 T^{2} + 430 T^{3} + 1849 T^{4}$$)
$47$ ($$1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4}$$)($$1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4}$$)($$1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4}$$)($$1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4}$$)($$1 + 9 T + 34 T^{2} + 423 T^{3} + 2209 T^{4}$$)($$1 - 47 T^{2} + 2209 T^{4}$$)($$1 - 47 T^{2} + 2209 T^{4}$$)($$1 - 9 T + 34 T^{2} - 423 T^{3} + 2209 T^{4}$$)
$53$ ($$1 - 94 T^{2} + 2809 T^{4}$$)($$1 - 58 T^{2} + 2809 T^{4}$$)($$1 - 58 T^{2} + 2809 T^{4}$$)($$1 - 94 T^{2} + 2809 T^{4}$$)($$1 + 86 T^{2} + 2809 T^{4}$$)($$1 - 94 T^{2} + 2809 T^{4}$$)($$1 - 94 T^{2} + 2809 T^{4}$$)($$1 + 86 T^{2} + 2809 T^{4}$$)
$59$ ($$1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4}$$)($$1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4}$$)($$1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4}$$)($$1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4}$$)($$1 - 59 T^{2} + 3481 T^{4}$$)($$1 - 59 T^{2} + 3481 T^{4}$$)($$1 - 59 T^{2} + 3481 T^{4}$$)($$1 - 59 T^{2} + 3481 T^{4}$$)
$61$ ($$( 1 + T + 61 T^{2} )( 1 + 14 T + 61 T^{2} )$$)($$1 + 6 T + 73 T^{2} + 366 T^{3} + 3721 T^{4}$$)($$1 - 6 T + 73 T^{2} - 366 T^{3} + 3721 T^{4}$$)($$( 1 - 14 T + 61 T^{2} )( 1 - T + 61 T^{2} )$$)($$1 - 6 T + 73 T^{2} - 366 T^{3} + 3721 T^{4}$$)($$( 1 - 13 T + 61 T^{2} )( 1 + T + 61 T^{2} )$$)($$( 1 - T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$)($$1 + 6 T + 73 T^{2} + 366 T^{3} + 3721 T^{4}$$)
$67$ ($$1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4}$$)($$1 + 14 T + 129 T^{2} + 938 T^{3} + 4489 T^{4}$$)($$1 + 14 T + 129 T^{2} + 938 T^{3} + 4489 T^{4}$$)($$1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4}$$)($$1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4}$$)($$1 - 10 T + 33 T^{2} - 670 T^{3} + 4489 T^{4}$$)($$1 - 10 T + 33 T^{2} - 670 T^{3} + 4489 T^{4}$$)($$1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4}$$)
$71$ ($$( 1 - 71 T^{2} )^{2}$$)($$1 - 115 T^{2} + 5041 T^{4}$$)($$1 - 115 T^{2} + 5041 T^{4}$$)($$( 1 - 71 T^{2} )^{2}$$)($$1 - 139 T^{2} + 5041 T^{4}$$)($$1 + 5 T^{2} + 5041 T^{4}$$)($$1 + 5 T^{2} + 5041 T^{4}$$)($$1 - 139 T^{2} + 5041 T^{4}$$)
$73$ ($$( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} )$$)($$1 - 71 T^{2} + 5329 T^{4}$$)($$1 - 71 T^{2} + 5329 T^{4}$$)($$( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} )$$)($$1 - 71 T^{2} + 5329 T^{4}$$)($$( 1 - 17 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} )$$)($$( 1 - 17 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} )$$)($$1 - 71 T^{2} + 5329 T^{4}$$)
$79$ ($$1 + 14 T + 117 T^{2} + 1106 T^{3} + 6241 T^{4}$$)($$1 + 5 T - 54 T^{2} + 395 T^{3} + 6241 T^{4}$$)($$1 + 5 T - 54 T^{2} + 395 T^{3} + 6241 T^{4}$$)($$1 + 14 T + 117 T^{2} + 1106 T^{3} + 6241 T^{4}$$)($$1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4}$$)($$1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4}$$)($$1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4}$$)($$1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4}$$)
$83$ ($$1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4}$$)($$1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4}$$)($$1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4}$$)($$1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4}$$)($$1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4}$$)($$1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4}$$)($$1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4}$$)($$1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4}$$)
$89$ ($$( 1 + 15 T + 89 T^{2} )^{2}$$)($$( 1 + 6 T + 89 T^{2} )^{2}$$)($$( 1 - 6 T + 89 T^{2} )^{2}$$)($$( 1 - 15 T + 89 T^{2} )^{2}$$)($$( 1 + 6 T + 89 T^{2} )^{2}$$)($$( 1 - 6 T + 89 T^{2} )^{2}$$)($$( 1 + 6 T + 89 T^{2} )^{2}$$)($$( 1 - 6 T + 89 T^{2} )^{2}$$)
$97$ ($$1 - 6 T + 109 T^{2} - 582 T^{3} + 9409 T^{4}$$)($$1 + 21 T + 244 T^{2} + 2037 T^{3} + 9409 T^{4}$$)($$1 - 21 T + 244 T^{2} - 2037 T^{3} + 9409 T^{4}$$)($$1 + 6 T + 109 T^{2} + 582 T^{3} + 9409 T^{4}$$)($$1 - 3 T + 100 T^{2} - 291 T^{3} + 9409 T^{4}$$)($$1 + 12 T + 145 T^{2} + 1164 T^{3} + 9409 T^{4}$$)($$1 - 12 T + 145 T^{2} - 1164 T^{3} + 9409 T^{4}$$)($$1 + 3 T + 100 T^{2} + 291 T^{3} + 9409 T^{4}$$)