# Properties

 Label 3900.2.h Level $3900$ Weight $2$ Character orbit 3900.h Rep. character $\chi_{3900}(1249,\cdot)$ Character field $\Q$ Dimension $36$ Newform subspaces $12$ Sturm bound $1680$ Trace bound $19$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3900.h (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$12$$ Sturm bound: $$1680$$ Trace bound: $$19$$ Distinguishing $$T_p$$: $$7$$, $$11$$, $$19$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(3900, [\chi])$$.

Total New Old
Modular forms 876 36 840
Cusp forms 804 36 768
Eisenstein series 72 0 72

## Trace form

 $$36q - 36q^{9} + O(q^{10})$$ $$36q - 36q^{9} - 8q^{11} - 8q^{19} + 8q^{21} - 8q^{29} + 24q^{31} + 16q^{41} - 60q^{49} + 8q^{51} - 48q^{59} - 16q^{61} - 8q^{69} + 16q^{71} + 40q^{79} + 36q^{81} + 56q^{89} + 16q^{91} + 8q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(3900, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
3900.2.h.a $$2$$ $$31.142$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+2iq^{7}-q^{9}-6q^{11}-iq^{13}+\cdots$$
3900.2.h.b $$2$$ $$31.142$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{3}+2iq^{7}-q^{9}-4q^{11}+iq^{13}+\cdots$$
3900.2.h.c $$2$$ $$31.142$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+iq^{7}-q^{9}-3q^{11}+iq^{13}+\cdots$$
3900.2.h.d $$2$$ $$31.142$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{3}+2iq^{7}-q^{9}-2q^{11}-iq^{13}+\cdots$$
3900.2.h.e $$2$$ $$31.142$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{3}+2iq^{7}-q^{9}-iq^{13}-6iq^{17}+\cdots$$
3900.2.h.f $$2$$ $$31.142$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+3iq^{7}-q^{9}+q^{11}-iq^{13}+\cdots$$
3900.2.h.g $$2$$ $$31.142$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+3iq^{7}-q^{9}+q^{11}-iq^{13}+\cdots$$
3900.2.h.h $$2$$ $$31.142$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+iq^{7}-q^{9}+3q^{11}+iq^{13}+\cdots$$
3900.2.h.i $$4$$ $$31.142$$ $$\Q(i, \sqrt{73})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{3}+\beta _{1}q^{7}-q^{9}+(1-\beta _{3})q^{11}+\cdots$$
3900.2.h.j $$4$$ $$31.142$$ $$\Q(i, \sqrt{17})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{3}+(\beta _{1}-\beta _{2})q^{7}-q^{9}+(2-\beta _{3})q^{11}+\cdots$$
3900.2.h.k $$6$$ $$31.142$$ 6.0.60217600.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{3}+(\beta _{1}-\beta _{5})q^{7}-q^{9}+(-1+\cdots)q^{11}+\cdots$$
3900.2.h.l $$6$$ $$31.142$$ 6.0.146700544.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{3}+(\beta _{1}-\beta _{2})q^{7}-q^{9}+(2-\beta _{4}+\cdots)q^{11}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(3900, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(3900, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(65, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(100, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(130, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(195, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(260, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(300, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(325, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(390, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(650, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(780, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(975, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1300, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1950, [\chi])$$$$^{\oplus 2}$$