# Properties

 Label 10000.2.a Level $10000$ Weight $2$ Character orbit 10000.a Rep. character $\chi_{10000}(1,\cdot)$ Character field $\Q$ Dimension $232$ Newform subspaces $44$ Sturm bound $3000$ Trace bound $13$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$10000 = 2^{4} \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 10000.a (trivial) Character field: $$\Q$$ Newform subspaces: $$44$$ Sturm bound: $$3000$$ Trace bound: $$13$$ Distinguishing $$T_p$$: $$3$$, $$7$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(10000))$$.

Total New Old
Modular forms 1590 248 1342
Cusp forms 1411 232 1179
Eisenstein series 179 16 163

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

$$2$$$$5$$FrickeDim.
$$+$$$$+$$$$+$$$$58$$
$$+$$$$-$$$$-$$$$62$$
$$-$$$$+$$$$-$$$$58$$
$$-$$$$-$$$$+$$$$54$$
Plus space$$+$$$$112$$
Minus space$$-$$$$120$$

## Trace form

 $$232q + 216q^{9} + O(q^{10})$$ $$232q + 216q^{9} - 4q^{11} + 4q^{21} - 4q^{31} - 12q^{39} + 4q^{41} + 184q^{49} + 16q^{51} + 4q^{61} + 12q^{69} - 4q^{71} - 60q^{79} + 172q^{81} - 64q^{91} - 12q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(10000))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2 5
10000.2.a.a $$2$$ $$79.850$$ $$\Q(\sqrt{5})$$ None $$0$$ $$-3$$ $$0$$ $$-6$$ $$-$$ $$+$$ $$q+(-1-\beta )q^{3}-3q^{7}+(-1+3\beta )q^{9}+\cdots$$
10000.2.a.b $$2$$ $$79.850$$ $$\Q(\sqrt{5})$$ None $$0$$ $$-3$$ $$0$$ $$-1$$ $$-$$ $$-$$ $$q+(-1-\beta )q^{3}+(1-3\beta )q^{7}+(-1+3\beta )q^{9}+\cdots$$
10000.2.a.c $$2$$ $$79.850$$ $$\Q(\sqrt{5})$$ None $$0$$ $$-2$$ $$0$$ $$1$$ $$-$$ $$+$$ $$q-q^{3}+\beta q^{7}-2q^{9}+(4-2\beta )q^{11}+\cdots$$
10000.2.a.d $$2$$ $$79.850$$ $$\Q(\sqrt{5})$$ None $$0$$ $$-2$$ $$0$$ $$1$$ $$-$$ $$+$$ $$q-q^{3}+(1-\beta )q^{7}-2q^{9}+3q^{11}+(3+\cdots)q^{13}+\cdots$$
10000.2.a.e $$2$$ $$79.850$$ $$\Q(\sqrt{5})$$ None $$0$$ $$-1$$ $$0$$ $$1$$ $$-$$ $$-$$ $$q-\beta q^{3}+\beta q^{7}+(-2+\beta )q^{9}+(-2+\cdots)q^{11}+\cdots$$
10000.2.a.f $$2$$ $$79.850$$ $$\Q(\sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$-3$$ $$+$$ $$+$$ $$q+(1-2\beta )q^{3}+(-1-\beta )q^{7}+2q^{9}+\cdots$$
10000.2.a.g $$2$$ $$79.850$$ $$\Q(\sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$-3$$ $$+$$ $$+$$ $$q+(1-2\beta )q^{3}+(-2+\beta )q^{7}+2q^{9}+\cdots$$
10000.2.a.h $$2$$ $$79.850$$ $$\Q(\sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$3$$ $$+$$ $$-$$ $$q+(1-2\beta )q^{3}+(2-\beta )q^{7}+2q^{9}+(-5+\cdots)q^{11}+\cdots$$
10000.2.a.i $$2$$ $$79.850$$ $$\Q(\sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$3$$ $$+$$ $$+$$ $$q+(1-2\beta )q^{3}+(1+\beta )q^{7}+2q^{9}+(2+\cdots)q^{11}+\cdots$$
10000.2.a.j $$2$$ $$79.850$$ $$\Q(\sqrt{5})$$ None $$0$$ $$1$$ $$0$$ $$-1$$ $$-$$ $$+$$ $$q+\beta q^{3}-\beta q^{7}+(-2+\beta )q^{9}+(-2+\cdots)q^{11}+\cdots$$
10000.2.a.k $$2$$ $$79.850$$ $$\Q(\sqrt{5})$$ None $$0$$ $$2$$ $$0$$ $$-1$$ $$-$$ $$-$$ $$q+q^{3}-\beta q^{7}-2q^{9}+3q^{11}+(1-4\beta )q^{13}+\cdots$$
10000.2.a.l $$2$$ $$79.850$$ $$\Q(\sqrt{5})$$ None $$0$$ $$2$$ $$0$$ $$-1$$ $$-$$ $$+$$ $$q+q^{3}+(-1+\beta )q^{7}-2q^{9}+(2+2\beta )q^{11}+\cdots$$
10000.2.a.m $$2$$ $$79.850$$ $$\Q(\sqrt{5})$$ None $$0$$ $$3$$ $$0$$ $$1$$ $$-$$ $$+$$ $$q+(1+\beta )q^{3}+(-1+3\beta )q^{7}+(-1+3\beta )q^{9}+\cdots$$
10000.2.a.n $$2$$ $$79.850$$ $$\Q(\sqrt{5})$$ None $$0$$ $$3$$ $$0$$ $$6$$ $$-$$ $$+$$ $$q+(1+\beta )q^{3}+3q^{7}+(-1+3\beta )q^{9}+\cdots$$
10000.2.a.o $$4$$ $$79.850$$ $$\Q(\zeta_{20})^+$$ None $$0$$ $$-4$$ $$0$$ $$-8$$ $$-$$ $$-$$ $$q+(-1+\beta _{1})q^{3}+(-2-\beta _{1}-\beta _{3})q^{7}+\cdots$$
10000.2.a.p $$4$$ $$79.850$$ 4.4.7625.1 None $$0$$ $$-3$$ $$0$$ $$2$$ $$+$$ $$+$$ $$q+(-1+\beta _{1})q^{3}-\beta _{2}q^{7}+(3-\beta _{1}+\beta _{3})q^{9}+\cdots$$
10000.2.a.q $$4$$ $$79.850$$ 4.4.7625.1 None $$0$$ $$-2$$ $$0$$ $$-2$$ $$+$$ $$+$$ $$q+\beta _{2}q^{3}+(-1-\beta _{3})q^{7}+(-2-\beta _{2}+\cdots)q^{9}+\cdots$$
10000.2.a.r $$4$$ $$79.850$$ 4.4.108625.1 None $$0$$ $$-2$$ $$0$$ $$3$$ $$+$$ $$+$$ $$q+(-1-\beta _{2})q^{3}+(\beta _{1}-\beta _{2})q^{7}+(-1+\cdots)q^{9}+\cdots$$
10000.2.a.s $$4$$ $$79.850$$ $$\Q(\zeta_{15})^+$$ None $$0$$ $$-1$$ $$0$$ $$-2$$ $$-$$ $$+$$ $$q+(\beta _{2}+\beta _{3})q^{3}+(2\beta _{2}+2\beta _{3})q^{7}+(-1+\cdots)q^{9}+\cdots$$
10000.2.a.t $$4$$ $$79.850$$ 4.4.7625.1 None $$0$$ $$-1$$ $$0$$ $$-2$$ $$-$$ $$+$$ $$q-\beta _{1}q^{3}+(-1-\beta _{3})q^{7}+(2+\beta _{1}+\beta _{3})q^{9}+\cdots$$
10000.2.a.u $$4$$ $$79.850$$ 4.4.18625.1 None $$0$$ $$-1$$ $$0$$ $$3$$ $$-$$ $$+$$ $$q+(\beta _{1}-\beta _{2})q^{3}+(2-2\beta _{2}-\beta _{3})q^{7}+\cdots$$
10000.2.a.v $$4$$ $$79.850$$ 4.4.18625.1 None $$0$$ $$1$$ $$0$$ $$-3$$ $$-$$ $$-$$ $$q+(-\beta _{1}+\beta _{2})q^{3}+(-2+2\beta _{2}+\beta _{3})q^{7}+\cdots$$
10000.2.a.w $$4$$ $$79.850$$ $$\Q(\zeta_{15})^+$$ None $$0$$ $$1$$ $$0$$ $$2$$ $$-$$ $$-$$ $$q+(-\beta _{2}-\beta _{3})q^{3}+(-2\beta _{2}-2\beta _{3})q^{7}+\cdots$$
10000.2.a.x $$4$$ $$79.850$$ 4.4.7625.1 None $$0$$ $$1$$ $$0$$ $$2$$ $$-$$ $$+$$ $$q+\beta _{1}q^{3}+(1+\beta _{3})q^{7}+(2+\beta _{1}+\beta _{3})q^{9}+\cdots$$
10000.2.a.y $$4$$ $$79.850$$ 4.4.108625.1 None $$0$$ $$2$$ $$0$$ $$-3$$ $$+$$ $$-$$ $$q+(1+\beta _{2})q^{3}+(-\beta _{1}+\beta _{2})q^{7}+(-1+\cdots)q^{9}+\cdots$$
10000.2.a.z $$4$$ $$79.850$$ 4.4.7625.1 None $$0$$ $$2$$ $$0$$ $$2$$ $$+$$ $$+$$ $$q-\beta _{2}q^{3}+(1+\beta _{3})q^{7}+(-2-\beta _{2})q^{9}+\cdots$$
10000.2.a.ba $$4$$ $$79.850$$ 4.4.7625.1 None $$0$$ $$3$$ $$0$$ $$-2$$ $$+$$ $$-$$ $$q+(1-\beta _{1})q^{3}+\beta _{2}q^{7}+(3-\beta _{1}+\beta _{3})q^{9}+\cdots$$
10000.2.a.bb $$4$$ $$79.850$$ $$\Q(\zeta_{20})^+$$ None $$0$$ $$4$$ $$0$$ $$8$$ $$-$$ $$-$$ $$q+(1+\beta _{1})q^{3}+(2-\beta _{1}-\beta _{3})q^{7}+(1+\cdots)q^{9}+\cdots$$
10000.2.a.bc $$6$$ $$79.850$$ 6.6.103238125.1 None $$0$$ $$-1$$ $$0$$ $$1$$ $$-$$ $$+$$ $$q-\beta _{1}q^{3}+(1-\beta _{3}+\beta _{4})q^{7}+(2+\beta _{1}+\cdots)q^{9}+\cdots$$
10000.2.a.bd $$6$$ $$79.850$$ 6.6.103238125.1 None $$0$$ $$1$$ $$0$$ $$-1$$ $$-$$ $$+$$ $$q+\beta _{1}q^{3}+(-1+\beta _{3}-\beta _{4})q^{7}+(2+\beta _{1}+\cdots)q^{9}+\cdots$$
10000.2.a.be $$8$$ $$79.850$$ 8.8.6152203125.1 None $$0$$ $$-5$$ $$0$$ $$-10$$ $$-$$ $$-$$ $$q+(-1+\beta _{5})q^{3}+(-1+\beta _{1}+\beta _{3}-\beta _{5}+\cdots)q^{7}+\cdots$$
10000.2.a.bf $$8$$ $$79.850$$ 8.8.$$\cdots$$.1 None $$0$$ $$-5$$ $$0$$ $$0$$ $$-$$ $$+$$ $$q+(-1+\beta _{1})q^{3}+(-\beta _{2}-\beta _{4}-\beta _{7})q^{7}+\cdots$$
10000.2.a.bg $$8$$ $$79.850$$ 8.8.3266578125.1 None $$0$$ $$-3$$ $$0$$ $$-2$$ $$+$$ $$+$$ $$q+(-1+\beta _{1}-\beta _{3})q^{3}+(\beta _{4}-\beta _{5})q^{7}+\cdots$$
10000.2.a.bh $$8$$ $$79.850$$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$-2$$ $$0$$ $$-3$$ $$+$$ $$+$$ $$q-\beta _{1}q^{3}+\beta _{2}q^{7}+(\beta _{1}-\beta _{2}+\beta _{3}-\beta _{4}+\cdots)q^{9}+\cdots$$
10000.2.a.bi $$8$$ $$79.850$$ 8.8.$$\cdots$$.2 None $$0$$ $$0$$ $$0$$ $$0$$ $$-$$ $$-$$ $$q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{6})q^{7}+\beta _{2}q^{9}+\cdots$$
10000.2.a.bj $$8$$ $$79.850$$ 8.8.$$\cdots$$.2 None $$0$$ $$0$$ $$0$$ $$0$$ $$-$$ $$-$$ $$q+(\beta _{5}-\beta _{6})q^{3}+(-\beta _{1}-\beta _{6}+\beta _{7})q^{7}+\cdots$$
10000.2.a.bk $$8$$ $$79.850$$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$2$$ $$0$$ $$3$$ $$+$$ $$+$$ $$q+\beta _{1}q^{3}-\beta _{2}q^{7}+(\beta _{1}-\beta _{2}+\beta _{3}-\beta _{4}+\cdots)q^{9}+\cdots$$
10000.2.a.bl $$8$$ $$79.850$$ 8.8.3266578125.1 None $$0$$ $$3$$ $$0$$ $$2$$ $$+$$ $$-$$ $$q+(1-\beta _{1}+\beta _{3})q^{3}+(-\beta _{4}+\beta _{5})q^{7}+\cdots$$
10000.2.a.bm $$8$$ $$79.850$$ 8.8.$$\cdots$$.1 None $$0$$ $$5$$ $$0$$ $$0$$ $$-$$ $$-$$ $$q+(1-\beta _{1})q^{3}+(\beta _{2}+\beta _{4}+\beta _{7})q^{7}+(2+\cdots)q^{9}+\cdots$$
10000.2.a.bn $$8$$ $$79.850$$ 8.8.6152203125.1 None $$0$$ $$5$$ $$0$$ $$10$$ $$-$$ $$+$$ $$q+(1-\beta _{5})q^{3}+(1-\beta _{1}-\beta _{3}+\beta _{5}-\beta _{7})q^{7}+\cdots$$
10000.2.a.bo $$12$$ $$79.850$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$-4$$ $$0$$ $$0$$ $$+$$ $$+$$ $$q-\beta _{6}q^{3}+(1+\beta _{1}-\beta _{2}+\beta _{5})q^{7}+(2+\cdots)q^{9}+\cdots$$
10000.2.a.bp $$12$$ $$79.850$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$4$$ $$0$$ $$0$$ $$+$$ $$-$$ $$q+\beta _{6}q^{3}+(-1-\beta _{1}+\beta _{2}-\beta _{5})q^{7}+\cdots$$
10000.2.a.bq $$16$$ $$79.850$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$-4$$ $$0$$ $$-8$$ $$+$$ $$-$$ $$q-\beta _{1}q^{3}+(-1+\beta _{8})q^{7}+(1+\beta _{2})q^{9}+\cdots$$
10000.2.a.br $$16$$ $$79.850$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$4$$ $$0$$ $$8$$ $$+$$ $$-$$ $$q+\beta _{1}q^{3}+(1-\beta _{8})q^{7}+(1+\beta _{2})q^{9}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(10000))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(10000)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(20))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(40))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(50))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(80))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(100))$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(125))$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(200))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(250))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(400))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(500))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(625))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(1000))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(1250))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(2000))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(2500))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(5000))$$$$^{\oplus 2}$$