# Properties

 Label 360.1 Level 360 Weight 1 Dimension 6 Nonzero newspaces 2 Newform subspaces 3 Sturm bound 6912 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$360\( 360 = 2^{3} \cdot 3^{2} \cdot 5$$ \) Weight: $$k$$ = $$1$$ Nonzero newspaces: $$2$$ Newform subspaces: $$3$$ Sturm bound: $$6912$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(360))$$.

Total New Old
Modular forms 418 60 358
Cusp forms 34 6 28
Eisenstein series 384 54 330

The following table gives the dimensions of subspaces with specified projective image type.

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 6 0 0 0

## Trace form

 $$6q + 2q^{4} - 4q^{7} + O(q^{10})$$ $$6q + 2q^{4} - 4q^{7} + 2q^{10} - 2q^{16} - 4q^{19} + 4q^{22} + 2q^{25} - 4q^{28} - 2q^{40} - 4q^{46} - 2q^{49} - 4q^{55} - 4q^{58} + 2q^{64} - 4q^{70} + 4q^{73} - 4q^{76} + 4q^{88} + 4q^{94} - 4q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(360))$$

We only show spaces with odd parity, since no modular forms exist when this condition is not satisfied. Within each space $$S_k^{\mathrm{new}}(N, \chi)$$ we list the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
360.1.c $$\chi_{360}(89, \cdot)$$ None 0 1
360.1.e $$\chi_{360}(271, \cdot)$$ None 0 1
360.1.g $$\chi_{360}(91, \cdot)$$ None 0 1
360.1.i $$\chi_{360}(269, \cdot)$$ None 0 1
360.1.j $$\chi_{360}(199, \cdot)$$ None 0 1
360.1.l $$\chi_{360}(161, \cdot)$$ None 0 1
360.1.n $$\chi_{360}(341, \cdot)$$ None 0 1
360.1.p $$\chi_{360}(19, \cdot)$$ 360.1.p.a 1 1
360.1.p.b 1
360.1.r $$\chi_{360}(107, \cdot)$$ None 0 2
360.1.u $$\chi_{360}(37, \cdot)$$ 360.1.u.a 4 2
360.1.v $$\chi_{360}(73, \cdot)$$ None 0 2
360.1.y $$\chi_{360}(143, \cdot)$$ None 0 2
360.1.z $$\chi_{360}(139, \cdot)$$ None 0 2
360.1.ba $$\chi_{360}(101, \cdot)$$ None 0 2
360.1.bc $$\chi_{360}(41, \cdot)$$ None 0 2
360.1.be $$\chi_{360}(79, \cdot)$$ None 0 2
360.1.bh $$\chi_{360}(29, \cdot)$$ None 0 2
360.1.bj $$\chi_{360}(211, \cdot)$$ None 0 2
360.1.bl $$\chi_{360}(31, \cdot)$$ None 0 2
360.1.bn $$\chi_{360}(209, \cdot)$$ None 0 2
360.1.bp $$\chi_{360}(97, \cdot)$$ None 0 4
360.1.bq $$\chi_{360}(23, \cdot)$$ None 0 4
360.1.bt $$\chi_{360}(83, \cdot)$$ None 0 4
360.1.bu $$\chi_{360}(13, \cdot)$$ None 0 4

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(360))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(360)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(120))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(180))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + T$$)($$1 - T$$)($$1 + T^{4}$$)
$3$ 1
$5$ ($$1 - T$$)($$1 + T$$)($$1 + T^{4}$$)
$7$ ($$1 + T^{2}$$)($$1 + T^{2}$$)($$( 1 + T )^{4}( 1 + T^{2} )^{2}$$)
$11$ ($$1 + T^{2}$$)($$1 + T^{2}$$)($$( 1 + T^{4} )^{2}$$)
$13$ ($$1 + T^{2}$$)($$1 + T^{2}$$)($$( 1 + T^{4} )^{2}$$)
$17$ ($$( 1 - T )( 1 + T )$$)($$( 1 - T )( 1 + T )$$)($$( 1 + T^{4} )^{2}$$)
$19$ ($$( 1 + T )^{2}$$)($$( 1 + T )^{2}$$)($$( 1 + T^{2} )^{4}$$)
$23$ ($$( 1 - T )^{2}$$)($$( 1 + T )^{2}$$)($$( 1 + T^{4} )^{2}$$)
$29$ ($$( 1 - T )( 1 + T )$$)($$( 1 - T )( 1 + T )$$)($$( 1 + T^{4} )^{2}$$)
$31$ ($$( 1 - T )( 1 + T )$$)($$( 1 - T )( 1 + T )$$)($$( 1 + T^{2} )^{4}$$)
$37$ ($$1 + T^{2}$$)($$1 + T^{2}$$)($$( 1 + T^{4} )^{2}$$)
$41$ ($$1 + T^{2}$$)($$1 + T^{2}$$)($$( 1 + T^{2} )^{4}$$)
$43$ ($$( 1 - T )( 1 + T )$$)($$( 1 - T )( 1 + T )$$)($$( 1 + T^{4} )^{2}$$)
$47$ ($$( 1 + T )^{2}$$)($$( 1 - T )^{2}$$)($$( 1 + T^{4} )^{2}$$)
$53$ ($$( 1 + T )^{2}$$)($$( 1 - T )^{2}$$)($$( 1 + T^{4} )^{2}$$)
$59$ ($$1 + T^{2}$$)($$1 + T^{2}$$)($$( 1 + T^{4} )^{2}$$)
$61$ ($$( 1 - T )( 1 + T )$$)($$( 1 - T )( 1 + T )$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)
$67$ ($$( 1 - T )( 1 + T )$$)($$( 1 - T )( 1 + T )$$)($$( 1 + T^{4} )^{2}$$)
$71$ ($$( 1 - T )( 1 + T )$$)($$( 1 - T )( 1 + T )$$)($$( 1 + T^{2} )^{4}$$)
$73$ ($$( 1 - T )( 1 + T )$$)($$( 1 - T )( 1 + T )$$)($$( 1 - T )^{4}( 1 + T^{2} )^{2}$$)
$79$ ($$( 1 - T )( 1 + T )$$)($$( 1 - T )( 1 + T )$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)
$83$ ($$( 1 - T )( 1 + T )$$)($$( 1 - T )( 1 + T )$$)($$( 1 + T^{4} )^{2}$$)
$89$ ($$1 + T^{2}$$)($$1 + T^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)
$97$ ($$( 1 - T )( 1 + T )$$)($$( 1 - T )( 1 + T )$$)($$( 1 + T )^{4}( 1 + T^{2} )^{2}$$)