## Defining parameters

 Level: $$N$$ = $$68 = 2^{2} \cdot 17$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$2$$ Newform subspaces: $$2$$ Sturm bound: $$288$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(68))$$.

Total New Old
Modular forms 43 17 26
Cusp forms 3 3 0
Eisenstein series 40 14 26

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 3 0 0 0

## Trace form

 $$3q - q^{2} - q^{4} - 2q^{5} - q^{8} - q^{9} + O(q^{10})$$ $$3q - q^{2} - q^{4} - 2q^{5} - q^{8} - q^{9} + 2q^{10} - 2q^{13} + 3q^{16} - q^{17} - q^{18} + 2q^{20} + q^{25} + 2q^{26} + 2q^{29} - q^{32} - q^{34} - q^{36} - 2q^{37} - 2q^{40} + 2q^{41} + 2q^{45} - q^{49} - 3q^{50} - 2q^{52} + 2q^{53} - 2q^{58} - 2q^{61} - q^{64} + 3q^{68} + 3q^{72} + 2q^{73} + 2q^{74} - 2q^{80} - q^{81} + 2q^{82} + 2q^{85} - 2q^{89} + 2q^{90} - 2q^{97} + 3q^{98} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(68))$$

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
68.1.c $$\chi_{68}(35, \cdot)$$ None 0 1
68.1.d $$\chi_{68}(67, \cdot)$$ 68.1.d.a 1 1
68.1.f $$\chi_{68}(47, \cdot)$$ 68.1.f.a 2 2
68.1.g $$\chi_{68}(15, \cdot)$$ None 0 4
68.1.j $$\chi_{68}(5, \cdot)$$ None 0 8

## Hecke characteristic polynomials

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