# Properties

 Label 864.2.a Level 864 Weight 2 Character orbit a Rep. character $$\chi_{864}(1,\cdot)$$ Character field $$\Q$$ Dimension 16 Newform subspaces 14 Sturm bound 288 Trace bound 11

# Related objects

## Defining parameters

 Level: $$N$$ = $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 864.a (trivial) Character field: $$\Q$$ Newform subspaces: $$14$$ Sturm bound: $$288$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$5$$, $$7$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(864))$$.

Total New Old
Modular forms 168 16 152
Cusp forms 121 16 105
Eisenstein series 47 0 47

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$$2$$$$3$$FrickeDim.
$$+$$$$+$$$$+$$$$3$$
$$+$$$$-$$$$-$$$$5$$
$$-$$$$+$$$$-$$$$5$$
$$-$$$$-$$$$+$$$$3$$
Plus space$$+$$$$6$$
Minus space$$-$$$$10$$

## Trace form

 $$16q + O(q^{10})$$ $$16q + 8q^{13} + 8q^{25} + 40q^{37} + 16q^{49} + 40q^{61} + 16q^{85} + 32q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(864))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2 3
864.2.a.a $$1$$ $$6.899$$ $$\Q$$ None $$0$$ $$0$$ $$-2$$ $$-3$$ $$-$$ $$+$$ $$q-2q^{5}-3q^{7}+6q^{11}-3q^{13}+2q^{17}+\cdots$$
864.2.a.b $$1$$ $$6.899$$ $$\Q$$ None $$0$$ $$0$$ $$-2$$ $$-1$$ $$-$$ $$-$$ $$q-2q^{5}-q^{7}+2q^{11}+q^{13}-6q^{17}+\cdots$$
864.2.a.c $$1$$ $$6.899$$ $$\Q$$ None $$0$$ $$0$$ $$-2$$ $$1$$ $$+$$ $$+$$ $$q-2q^{5}+q^{7}-2q^{11}+q^{13}-6q^{17}+\cdots$$
864.2.a.d $$1$$ $$6.899$$ $$\Q$$ None $$0$$ $$0$$ $$-2$$ $$3$$ $$-$$ $$-$$ $$q-2q^{5}+3q^{7}-6q^{11}-3q^{13}+2q^{17}+\cdots$$
864.2.a.e $$1$$ $$6.899$$ $$\Q$$ None $$0$$ $$0$$ $$-1$$ $$-3$$ $$+$$ $$+$$ $$q-q^{5}-3q^{7}+3q^{11}+4q^{17}-6q^{19}+\cdots$$
864.2.a.f $$1$$ $$6.899$$ $$\Q$$ None $$0$$ $$0$$ $$-1$$ $$3$$ $$+$$ $$-$$ $$q-q^{5}+3q^{7}-3q^{11}+4q^{17}+6q^{19}+\cdots$$
864.2.a.g $$1$$ $$6.899$$ $$\Q$$ None $$0$$ $$0$$ $$1$$ $$-3$$ $$-$$ $$-$$ $$q+q^{5}-3q^{7}-3q^{11}-4q^{17}-6q^{19}+\cdots$$
864.2.a.h $$1$$ $$6.899$$ $$\Q$$ None $$0$$ $$0$$ $$1$$ $$3$$ $$-$$ $$+$$ $$q+q^{5}+3q^{7}+3q^{11}-4q^{17}+6q^{19}+\cdots$$
864.2.a.i $$1$$ $$6.899$$ $$\Q$$ None $$0$$ $$0$$ $$2$$ $$-3$$ $$+$$ $$+$$ $$q+2q^{5}-3q^{7}-6q^{11}-3q^{13}-2q^{17}+\cdots$$
864.2.a.j $$1$$ $$6.899$$ $$\Q$$ None $$0$$ $$0$$ $$2$$ $$-1$$ $$+$$ $$-$$ $$q+2q^{5}-q^{7}-2q^{11}+q^{13}+6q^{17}+\cdots$$
864.2.a.k $$1$$ $$6.899$$ $$\Q$$ None $$0$$ $$0$$ $$2$$ $$1$$ $$-$$ $$+$$ $$q+2q^{5}+q^{7}+2q^{11}+q^{13}+6q^{17}+\cdots$$
864.2.a.l $$1$$ $$6.899$$ $$\Q$$ None $$0$$ $$0$$ $$2$$ $$3$$ $$+$$ $$-$$ $$q+2q^{5}+3q^{7}+6q^{11}-3q^{13}-2q^{17}+\cdots$$
864.2.a.m $$2$$ $$6.899$$ $$\Q(\sqrt{13})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$-$$ $$+$$ $$q-\beta q^{5}+\beta q^{7}-q^{11}+4q^{13}-2\beta q^{19}+\cdots$$
864.2.a.n $$2$$ $$6.899$$ $$\Q(\sqrt{13})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$+$$ $$-$$ $$q-\beta q^{5}-\beta q^{7}+q^{11}+4q^{13}+2\beta q^{19}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(864))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(864)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(24))$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(27))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(32))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(36))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(48))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(54))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(72))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(96))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(108))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(144))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(216))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(288))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(432))$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ()()()()()()()()()()()()()()
$3$ ()()()()()()()()()()()()()()
$5$ ($$1 + 2 T + 5 T^{2}$$)($$1 + 2 T + 5 T^{2}$$)($$1 + 2 T + 5 T^{2}$$)($$1 + 2 T + 5 T^{2}$$)($$1 + T + 5 T^{2}$$)($$1 + T + 5 T^{2}$$)($$1 - T + 5 T^{2}$$)($$1 - T + 5 T^{2}$$)($$1 - 2 T + 5 T^{2}$$)($$1 - 2 T + 5 T^{2}$$)($$1 - 2 T + 5 T^{2}$$)($$1 - 2 T + 5 T^{2}$$)($$1 - 3 T^{2} + 25 T^{4}$$)($$1 - 3 T^{2} + 25 T^{4}$$)
$7$ ($$1 + 3 T + 7 T^{2}$$)($$1 + T + 7 T^{2}$$)($$1 - T + 7 T^{2}$$)($$1 - 3 T + 7 T^{2}$$)($$1 + 3 T + 7 T^{2}$$)($$1 - 3 T + 7 T^{2}$$)($$1 + 3 T + 7 T^{2}$$)($$1 - 3 T + 7 T^{2}$$)($$1 + 3 T + 7 T^{2}$$)($$1 + T + 7 T^{2}$$)($$1 - T + 7 T^{2}$$)($$1 - 3 T + 7 T^{2}$$)($$1 + T^{2} + 49 T^{4}$$)($$1 + T^{2} + 49 T^{4}$$)
$11$ ($$1 - 6 T + 11 T^{2}$$)($$1 - 2 T + 11 T^{2}$$)($$1 + 2 T + 11 T^{2}$$)($$1 + 6 T + 11 T^{2}$$)($$1 - 3 T + 11 T^{2}$$)($$1 + 3 T + 11 T^{2}$$)($$1 + 3 T + 11 T^{2}$$)($$1 - 3 T + 11 T^{2}$$)($$1 + 6 T + 11 T^{2}$$)($$1 + 2 T + 11 T^{2}$$)($$1 - 2 T + 11 T^{2}$$)($$1 - 6 T + 11 T^{2}$$)($$( 1 + T + 11 T^{2} )^{2}$$)($$( 1 - T + 11 T^{2} )^{2}$$)
$13$ ($$1 + 3 T + 13 T^{2}$$)($$1 - T + 13 T^{2}$$)($$1 - T + 13 T^{2}$$)($$1 + 3 T + 13 T^{2}$$)($$1 + 13 T^{2}$$)($$1 + 13 T^{2}$$)($$1 + 13 T^{2}$$)($$1 + 13 T^{2}$$)($$1 + 3 T + 13 T^{2}$$)($$1 - T + 13 T^{2}$$)($$1 - T + 13 T^{2}$$)($$1 + 3 T + 13 T^{2}$$)($$( 1 - 4 T + 13 T^{2} )^{2}$$)($$( 1 - 4 T + 13 T^{2} )^{2}$$)
$17$ ($$1 - 2 T + 17 T^{2}$$)($$1 + 6 T + 17 T^{2}$$)($$1 + 6 T + 17 T^{2}$$)($$1 - 2 T + 17 T^{2}$$)($$1 - 4 T + 17 T^{2}$$)($$1 - 4 T + 17 T^{2}$$)($$1 + 4 T + 17 T^{2}$$)($$1 + 4 T + 17 T^{2}$$)($$1 + 2 T + 17 T^{2}$$)($$1 - 6 T + 17 T^{2}$$)($$1 - 6 T + 17 T^{2}$$)($$1 + 2 T + 17 T^{2}$$)($$( 1 + 17 T^{2} )^{2}$$)($$( 1 + 17 T^{2} )^{2}$$)
$19$ ($$1 - 3 T + 19 T^{2}$$)($$1 - 5 T + 19 T^{2}$$)($$1 + 5 T + 19 T^{2}$$)($$1 + 3 T + 19 T^{2}$$)($$1 + 6 T + 19 T^{2}$$)($$1 - 6 T + 19 T^{2}$$)($$1 + 6 T + 19 T^{2}$$)($$1 - 6 T + 19 T^{2}$$)($$1 - 3 T + 19 T^{2}$$)($$1 - 5 T + 19 T^{2}$$)($$1 + 5 T + 19 T^{2}$$)($$1 + 3 T + 19 T^{2}$$)($$1 - 14 T^{2} + 361 T^{4}$$)($$1 - 14 T^{2} + 361 T^{4}$$)
$23$ ($$1 - 6 T + 23 T^{2}$$)($$1 + 6 T + 23 T^{2}$$)($$1 - 6 T + 23 T^{2}$$)($$1 + 6 T + 23 T^{2}$$)($$1 + 6 T + 23 T^{2}$$)($$1 - 6 T + 23 T^{2}$$)($$1 - 6 T + 23 T^{2}$$)($$1 + 6 T + 23 T^{2}$$)($$1 + 6 T + 23 T^{2}$$)($$1 - 6 T + 23 T^{2}$$)($$1 + 6 T + 23 T^{2}$$)($$1 - 6 T + 23 T^{2}$$)($$( 1 - 6 T + 23 T^{2} )^{2}$$)($$( 1 + 6 T + 23 T^{2} )^{2}$$)
$29$ ($$1 - 8 T + 29 T^{2}$$)($$1 + 8 T + 29 T^{2}$$)($$1 + 8 T + 29 T^{2}$$)($$1 - 8 T + 29 T^{2}$$)($$1 + 2 T + 29 T^{2}$$)($$1 + 2 T + 29 T^{2}$$)($$1 - 2 T + 29 T^{2}$$)($$1 - 2 T + 29 T^{2}$$)($$1 + 8 T + 29 T^{2}$$)($$1 - 8 T + 29 T^{2}$$)($$1 - 8 T + 29 T^{2}$$)($$1 + 8 T + 29 T^{2}$$)($$1 + 6 T^{2} + 841 T^{4}$$)($$1 + 6 T^{2} + 841 T^{4}$$)
$31$ ($$1 + 31 T^{2}$$)($$1 + 8 T + 31 T^{2}$$)($$1 - 8 T + 31 T^{2}$$)($$1 + 31 T^{2}$$)($$1 + 9 T + 31 T^{2}$$)($$1 - 9 T + 31 T^{2}$$)($$1 + 9 T + 31 T^{2}$$)($$1 - 9 T + 31 T^{2}$$)($$1 + 31 T^{2}$$)($$1 + 8 T + 31 T^{2}$$)($$1 - 8 T + 31 T^{2}$$)($$1 + 31 T^{2}$$)($$1 + 49 T^{2} + 961 T^{4}$$)($$1 + 49 T^{2} + 961 T^{4}$$)
$37$ ($$1 - 7 T + 37 T^{2}$$)($$1 + 5 T + 37 T^{2}$$)($$1 + 5 T + 37 T^{2}$$)($$1 - 7 T + 37 T^{2}$$)($$1 + 2 T + 37 T^{2}$$)($$1 + 2 T + 37 T^{2}$$)($$1 + 2 T + 37 T^{2}$$)($$1 + 2 T + 37 T^{2}$$)($$1 - 7 T + 37 T^{2}$$)($$1 + 5 T + 37 T^{2}$$)($$1 + 5 T + 37 T^{2}$$)($$1 - 7 T + 37 T^{2}$$)($$( 1 - 10 T + 37 T^{2} )^{2}$$)($$( 1 - 10 T + 37 T^{2} )^{2}$$)
$41$ ($$1 + 8 T + 41 T^{2}$$)($$1 + 8 T + 41 T^{2}$$)($$1 + 8 T + 41 T^{2}$$)($$1 + 8 T + 41 T^{2}$$)($$1 + 10 T + 41 T^{2}$$)($$1 + 10 T + 41 T^{2}$$)($$1 - 10 T + 41 T^{2}$$)($$1 - 10 T + 41 T^{2}$$)($$1 - 8 T + 41 T^{2}$$)($$1 - 8 T + 41 T^{2}$$)($$1 - 8 T + 41 T^{2}$$)($$1 - 8 T + 41 T^{2}$$)($$1 + 30 T^{2} + 1681 T^{4}$$)($$1 + 30 T^{2} + 1681 T^{4}$$)
$43$ ($$1 - 12 T + 43 T^{2}$$)($$1 - 4 T + 43 T^{2}$$)($$1 + 4 T + 43 T^{2}$$)($$1 + 12 T + 43 T^{2}$$)($$1 + 6 T + 43 T^{2}$$)($$1 - 6 T + 43 T^{2}$$)($$1 + 6 T + 43 T^{2}$$)($$1 - 6 T + 43 T^{2}$$)($$1 - 12 T + 43 T^{2}$$)($$1 - 4 T + 43 T^{2}$$)($$1 + 4 T + 43 T^{2}$$)($$1 + 12 T + 43 T^{2}$$)($$1 + 34 T^{2} + 1849 T^{4}$$)($$1 + 34 T^{2} + 1849 T^{4}$$)
$47$ ($$1 - 6 T + 47 T^{2}$$)($$1 - 10 T + 47 T^{2}$$)($$1 + 10 T + 47 T^{2}$$)($$1 + 6 T + 47 T^{2}$$)($$1 + 6 T + 47 T^{2}$$)($$1 - 6 T + 47 T^{2}$$)($$1 - 6 T + 47 T^{2}$$)($$1 + 6 T + 47 T^{2}$$)($$1 + 6 T + 47 T^{2}$$)($$1 + 10 T + 47 T^{2}$$)($$1 - 10 T + 47 T^{2}$$)($$1 - 6 T + 47 T^{2}$$)($$( 1 - 10 T + 47 T^{2} )^{2}$$)($$( 1 + 10 T + 47 T^{2} )^{2}$$)
$53$ ($$1 + 4 T + 53 T^{2}$$)($$1 + 4 T + 53 T^{2}$$)($$1 + 4 T + 53 T^{2}$$)($$1 + 4 T + 53 T^{2}$$)($$1 - 13 T + 53 T^{2}$$)($$1 - 13 T + 53 T^{2}$$)($$1 + 13 T + 53 T^{2}$$)($$1 + 13 T + 53 T^{2}$$)($$1 - 4 T + 53 T^{2}$$)($$1 - 4 T + 53 T^{2}$$)($$1 - 4 T + 53 T^{2}$$)($$1 - 4 T + 53 T^{2}$$)($$1 + 93 T^{2} + 2809 T^{4}$$)($$1 + 93 T^{2} + 2809 T^{4}$$)
$59$ ($$1 - 6 T + 59 T^{2}$$)($$1 + 14 T + 59 T^{2}$$)($$1 - 14 T + 59 T^{2}$$)($$1 + 6 T + 59 T^{2}$$)($$1 - 12 T + 59 T^{2}$$)($$1 + 12 T + 59 T^{2}$$)($$1 + 12 T + 59 T^{2}$$)($$1 - 12 T + 59 T^{2}$$)($$1 + 6 T + 59 T^{2}$$)($$1 - 14 T + 59 T^{2}$$)($$1 + 14 T + 59 T^{2}$$)($$1 - 6 T + 59 T^{2}$$)($$( 1 - 4 T + 59 T^{2} )^{2}$$)($$( 1 + 4 T + 59 T^{2} )^{2}$$)
$61$ ($$1 + T + 61 T^{2}$$)($$1 - 3 T + 61 T^{2}$$)($$1 - 3 T + 61 T^{2}$$)($$1 + T + 61 T^{2}$$)($$1 - 8 T + 61 T^{2}$$)($$1 - 8 T + 61 T^{2}$$)($$1 - 8 T + 61 T^{2}$$)($$1 - 8 T + 61 T^{2}$$)($$1 + T + 61 T^{2}$$)($$1 - 3 T + 61 T^{2}$$)($$1 - 3 T + 61 T^{2}$$)($$1 + T + 61 T^{2}$$)($$( 1 + 61 T^{2} )^{2}$$)($$( 1 + 61 T^{2} )^{2}$$)
$67$ ($$1 - 3 T + 67 T^{2}$$)($$1 - 13 T + 67 T^{2}$$)($$1 + 13 T + 67 T^{2}$$)($$1 + 3 T + 67 T^{2}$$)($$1 + 6 T + 67 T^{2}$$)($$1 - 6 T + 67 T^{2}$$)($$1 + 6 T + 67 T^{2}$$)($$1 - 6 T + 67 T^{2}$$)($$1 - 3 T + 67 T^{2}$$)($$1 - 13 T + 67 T^{2}$$)($$1 + 13 T + 67 T^{2}$$)($$1 + 3 T + 67 T^{2}$$)($$1 + 82 T^{2} + 4489 T^{4}$$)($$1 + 82 T^{2} + 4489 T^{4}$$)
$71$ ($$1 - 12 T + 71 T^{2}$$)($$1 - 4 T + 71 T^{2}$$)($$1 + 4 T + 71 T^{2}$$)($$1 + 12 T + 71 T^{2}$$)($$1 + 12 T + 71 T^{2}$$)($$1 - 12 T + 71 T^{2}$$)($$1 - 12 T + 71 T^{2}$$)($$1 + 12 T + 71 T^{2}$$)($$1 + 12 T + 71 T^{2}$$)($$1 + 4 T + 71 T^{2}$$)($$1 - 4 T + 71 T^{2}$$)($$1 - 12 T + 71 T^{2}$$)($$( 1 + 8 T + 71 T^{2} )^{2}$$)($$( 1 - 8 T + 71 T^{2} )^{2}$$)
$73$ ($$1 + 15 T + 73 T^{2}$$)($$1 - 9 T + 73 T^{2}$$)($$1 - 9 T + 73 T^{2}$$)($$1 + 15 T + 73 T^{2}$$)($$1 - 9 T + 73 T^{2}$$)($$1 - 9 T + 73 T^{2}$$)($$1 - 9 T + 73 T^{2}$$)($$1 - 9 T + 73 T^{2}$$)($$1 + 15 T + 73 T^{2}$$)($$1 - 9 T + 73 T^{2}$$)($$1 - 9 T + 73 T^{2}$$)($$1 + 15 T + 73 T^{2}$$)($$( 1 + 3 T + 73 T^{2} )^{2}$$)($$( 1 + 3 T + 73 T^{2} )^{2}$$)
$79$ ($$1 + 9 T + 79 T^{2}$$)($$1 + 11 T + 79 T^{2}$$)($$1 - 11 T + 79 T^{2}$$)($$1 - 9 T + 79 T^{2}$$)($$1 + 79 T^{2}$$)($$1 + 79 T^{2}$$)($$1 + 79 T^{2}$$)($$1 + 79 T^{2}$$)($$1 + 9 T + 79 T^{2}$$)($$1 + 11 T + 79 T^{2}$$)($$1 - 11 T + 79 T^{2}$$)($$1 - 9 T + 79 T^{2}$$)($$1 - 50 T^{2} + 6241 T^{4}$$)($$1 - 50 T^{2} + 6241 T^{4}$$)
$83$ ($$1 + 12 T + 83 T^{2}$$)($$1 - 12 T + 83 T^{2}$$)($$1 + 12 T + 83 T^{2}$$)($$1 - 12 T + 83 T^{2}$$)($$1 - 3 T + 83 T^{2}$$)($$1 + 3 T + 83 T^{2}$$)($$1 + 3 T + 83 T^{2}$$)($$1 - 3 T + 83 T^{2}$$)($$1 - 12 T + 83 T^{2}$$)($$1 + 12 T + 83 T^{2}$$)($$1 - 12 T + 83 T^{2}$$)($$1 + 12 T + 83 T^{2}$$)($$( 1 + 9 T + 83 T^{2} )^{2}$$)($$( 1 - 9 T + 83 T^{2} )^{2}$$)
$89$ ($$1 - 10 T + 89 T^{2}$$)($$1 - 2 T + 89 T^{2}$$)($$1 - 2 T + 89 T^{2}$$)($$1 - 10 T + 89 T^{2}$$)($$1 - 14 T + 89 T^{2}$$)($$1 - 14 T + 89 T^{2}$$)($$1 + 14 T + 89 T^{2}$$)($$1 + 14 T + 89 T^{2}$$)($$1 + 10 T + 89 T^{2}$$)($$1 + 2 T + 89 T^{2}$$)($$1 + 2 T + 89 T^{2}$$)($$1 + 10 T + 89 T^{2}$$)($$1 + 126 T^{2} + 7921 T^{4}$$)($$1 + 126 T^{2} + 7921 T^{4}$$)
$97$ ($$1 - 9 T + 97 T^{2}$$)($$1 - T + 97 T^{2}$$)($$1 - T + 97 T^{2}$$)($$1 - 9 T + 97 T^{2}$$)($$1 + 9 T + 97 T^{2}$$)($$1 + 9 T + 97 T^{2}$$)($$1 + 9 T + 97 T^{2}$$)($$1 + 9 T + 97 T^{2}$$)($$1 - 9 T + 97 T^{2}$$)($$1 - T + 97 T^{2}$$)($$1 - T + 97 T^{2}$$)($$1 - 9 T + 97 T^{2}$$)($$( 1 - 7 T + 97 T^{2} )^{2}$$)($$( 1 - 7 T + 97 T^{2} )^{2}$$)