# Properties

 Label 75.3.c Level $75$ Weight $3$ Character orbit 75.c Rep. character $\chi_{75}(26,\cdot)$ Character field $\Q$ Dimension $10$ Newform subspaces $6$ Sturm bound $30$ Trace bound $3$

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## Defining parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 75.c (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$3$$ Character field: $$\Q$$ Newform subspaces: $$6$$ Sturm bound: $$30$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$2$$, $$7$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(75, [\chi])$$.

Total New Old
Modular forms 26 16 10
Cusp forms 14 10 4
Eisenstein series 12 6 6

## Trace form

 $$10 q + 4 q^{3} - 16 q^{4} - 18 q^{6} + 12 q^{7} + 12 q^{9} + O(q^{10})$$ $$10 q + 4 q^{3} - 16 q^{4} - 18 q^{6} + 12 q^{7} + 12 q^{9} - 4 q^{12} - 32 q^{13} + 24 q^{16} + 40 q^{18} - 6 q^{19} - 42 q^{21} - 20 q^{22} + 54 q^{24} - 44 q^{27} - 12 q^{28} + 110 q^{31} + 20 q^{33} + 4 q^{34} - 78 q^{36} + 32 q^{37} + 42 q^{39} + 60 q^{42} - 32 q^{43} - 136 q^{46} - 76 q^{48} - 176 q^{49} + 42 q^{51} + 32 q^{52} - 228 q^{54} - 8 q^{57} + 140 q^{58} - 10 q^{61} - 12 q^{63} + 408 q^{64} + 510 q^{66} - 48 q^{67} + 12 q^{69} + 120 q^{72} + 148 q^{73} - 356 q^{76} - 160 q^{78} - 156 q^{79} - 60 q^{81} - 280 q^{82} - 288 q^{84} - 140 q^{87} - 60 q^{88} - 214 q^{91} - 72 q^{93} + 424 q^{94} - 114 q^{96} + 332 q^{97} + 630 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(75, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
75.3.c.a $1$ $2.044$ $$\Q$$ $$\Q(\sqrt{-3})$$ $$0$$ $$-3$$ $$0$$ $$11$$ $$q-3q^{3}+4q^{4}+11q^{7}+9q^{9}-12q^{12}+\cdots$$
75.3.c.b $1$ $2.044$ $$\Q$$ $$\Q(\sqrt{-3})$$ $$0$$ $$3$$ $$0$$ $$-11$$ $$q+3q^{3}+4q^{4}-11q^{7}+9q^{9}+12q^{12}+\cdots$$
75.3.c.c $2$ $2.044$ $$\Q(\sqrt{-11})$$ None $$0$$ $$-5$$ $$0$$ $$0$$ $$q+(1-2\beta )q^{2}+(-2-\beta )q^{3}-7q^{4}+\cdots$$
75.3.c.d $2$ $2.044$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-15})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+3iq^{3}+3q^{4}-3q^{6}+7iq^{8}+\cdots$$
75.3.c.e $2$ $2.044$ $$\Q(\sqrt{-5})$$ None $$0$$ $$4$$ $$0$$ $$12$$ $$q+\beta q^{2}+(2-\beta )q^{3}-q^{4}+(5+2\beta )q^{6}+\cdots$$
75.3.c.f $2$ $2.044$ $$\Q(\sqrt{-11})$$ None $$0$$ $$5$$ $$0$$ $$0$$ $$q+(1-2\beta )q^{2}+(3-\beta )q^{3}-7q^{4}+(-3+\cdots)q^{6}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(75, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(75, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(15, [\chi])$$$$^{\oplus 2}$$