# Properties

 Label 1050.3.p Level $1050$ Weight $3$ Character orbit 1050.p Rep. character $\chi_{1050}(451,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $100$ Newform subspaces $9$ Sturm bound $720$ Trace bound $14$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1050.p (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$9$$ Sturm bound: $$720$$ Trace bound: $$14$$ Distinguishing $$T_p$$: $$11$$, $$17$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(1050, [\chi])$$.

Total New Old
Modular forms 1008 100 908
Cusp forms 912 100 812
Eisenstein series 96 0 96

## Trace form

 $$100 q + 6 q^{3} - 100 q^{4} + 6 q^{7} + 150 q^{9} + O(q^{10})$$ $$100 q + 6 q^{3} - 100 q^{4} + 6 q^{7} + 150 q^{9} + 12 q^{11} - 12 q^{12} - 8 q^{14} - 200 q^{16} - 48 q^{17} + 114 q^{19} - 12 q^{21} + 96 q^{22} - 24 q^{23} - 96 q^{26} - 72 q^{28} - 128 q^{29} - 162 q^{31} + 36 q^{33} - 600 q^{36} + 30 q^{37} + 216 q^{38} + 42 q^{39} + 72 q^{42} + 140 q^{43} + 24 q^{44} + 104 q^{46} + 180 q^{47} - 86 q^{49} + 12 q^{52} - 184 q^{53} + 80 q^{56} - 276 q^{57} + 80 q^{58} + 24 q^{59} - 588 q^{61} - 90 q^{63} + 800 q^{64} + 174 q^{67} + 96 q^{68} - 360 q^{71} + 390 q^{73} + 80 q^{74} - 274 q^{79} - 450 q^{81} - 336 q^{82} + 108 q^{84} + 88 q^{86} - 96 q^{88} - 168 q^{89} + 166 q^{91} + 96 q^{92} + 150 q^{93} - 408 q^{94} - 384 q^{98} + 72 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(1050, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1050.3.p.a $4$ $28.610$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$-6$$ $$0$$ $$10$$ $$q-\beta _{1}q^{2}+(-2-\beta _{2})q^{3}+2\beta _{2}q^{4}+\cdots$$
1050.3.p.b $8$ $28.610$ 8.0.3317760000.3 None $$0$$ $$-12$$ $$0$$ $$0$$ $$q+(-\beta _{2}-\beta _{4})q^{2}+(-1-\beta _{3})q^{3}+(-2+\cdots)q^{4}+\cdots$$
1050.3.p.c $8$ $28.610$ 8.0.$$\cdots$$.6 None $$0$$ $$-12$$ $$0$$ $$0$$ $$q+(\beta _{2}-\beta _{6})q^{2}+(-1-\beta _{3})q^{3}+(-2+\cdots)q^{4}+\cdots$$
1050.3.p.d $8$ $28.610$ 8.0.$$\cdots$$.6 None $$0$$ $$12$$ $$0$$ $$0$$ $$q+(-\beta _{2}+\beta _{6})q^{2}+(1+\beta _{3})q^{3}+(-2+\cdots)q^{4}+\cdots$$
1050.3.p.e $12$ $28.610$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$-18$$ $$0$$ $$-8$$ $$q+(-\beta _{4}+\beta _{5})q^{2}+(-2+\beta _{3})q^{3}-2\beta _{3}q^{4}+\cdots$$
1050.3.p.f $12$ $28.610$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$18$$ $$0$$ $$8$$ $$q+(\beta _{4}-\beta _{5})q^{2}+(2-\beta _{3})q^{3}-2\beta _{3}q^{4}+\cdots$$
1050.3.p.g $16$ $28.610$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$-24$$ $$0$$ $$12$$ $$q+(-\beta _{3}+\beta _{8})q^{2}+(-1+\beta _{7})q^{3}+(-2+\cdots)q^{4}+\cdots$$
1050.3.p.h $16$ $28.610$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$24$$ $$0$$ $$-12$$ $$q+(\beta _{3}-\beta _{8})q^{2}+(1-\beta _{7})q^{3}+(-2-2\beta _{7}+\cdots)q^{4}+\cdots$$
1050.3.p.i $16$ $28.610$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$24$$ $$0$$ $$-4$$ $$q+(-\beta _{1}-\beta _{5})q^{2}+(2+\beta _{3})q^{3}+2\beta _{3}q^{4}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(1050, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(1050, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(14, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(70, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(175, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(210, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(350, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(525, [\chi])$$$$^{\oplus 2}$$