## Defining parameters

 Level: $$N$$ = $$104\( 104 = 2^{3} \cdot 13$$ \) Weight: $$k$$ = $$1$$ Nonzero newspaces: $$1$$ Newform subspaces: $$2$$ Sturm bound: $$672$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(104))$$.

Total New Old
Modular forms 78 24 54
Cusp forms 6 2 4
Eisenstein series 72 22 50

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 2 0 0 0

## Trace form

 $$2q - 2q^{3} + 2q^{4} + O(q^{10})$$ $$2q - 2q^{3} + 2q^{4} - 2q^{10} - 2q^{12} - 2q^{14} + 2q^{16} - 2q^{17} + 2q^{26} + 2q^{27} + 2q^{30} + 2q^{35} - 2q^{40} + 2q^{42} - 2q^{43} - 2q^{48} + 2q^{51} - 2q^{56} + 4q^{62} + 2q^{64} - 2q^{65} - 2q^{68} - 2q^{74} - 2q^{78} - 2q^{81} - 2q^{91} - 2q^{94} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(104))$$

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
104.1.c $$\chi_{104}(103, \cdot)$$ None 0 1
104.1.d $$\chi_{104}(79, \cdot)$$ None 0 1
104.1.g $$\chi_{104}(27, \cdot)$$ None 0 1
104.1.h $$\chi_{104}(51, \cdot)$$ 104.1.h.a 1 1
104.1.h.b 1
104.1.j $$\chi_{104}(5, \cdot)$$ None 0 2
104.1.l $$\chi_{104}(57, \cdot)$$ None 0 2
104.1.n $$\chi_{104}(3, \cdot)$$ None 0 2
104.1.p $$\chi_{104}(43, \cdot)$$ None 0 2
104.1.q $$\chi_{104}(23, \cdot)$$ None 0 2
104.1.t $$\chi_{104}(55, \cdot)$$ None 0 2
104.1.v $$\chi_{104}(33, \cdot)$$ None 0 4
104.1.x $$\chi_{104}(37, \cdot)$$ None 0 4

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(104))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(104)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(52))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

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