# Properties

 Label 1400.2.q Level $1400$ Weight $2$ Character orbit 1400.q Rep. character $\chi_{1400}(401,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $76$ Newform subspaces $15$ Sturm bound $480$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1400 = 2^{3} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1400.q (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$15$$ Sturm bound: $$480$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$3$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1400, [\chi])$$.

Total New Old
Modular forms 528 76 452
Cusp forms 432 76 356
Eisenstein series 96 0 96

## Trace form

 $$76 q - 2 q^{3} - 4 q^{7} - 36 q^{9} + O(q^{10})$$ $$76 q - 2 q^{3} - 4 q^{7} - 36 q^{9} + 2 q^{11} - 8 q^{13} + 2 q^{17} + 10 q^{19} + 18 q^{21} - 10 q^{23} + 4 q^{27} + 16 q^{29} + 10 q^{31} - 6 q^{33} - 6 q^{37} + 4 q^{39} + 24 q^{41} - 32 q^{43} - 6 q^{47} - 20 q^{49} + 22 q^{51} - 6 q^{53} - 36 q^{57} + 18 q^{59} + 2 q^{61} + 40 q^{63} + 14 q^{67} - 28 q^{69} + 26 q^{73} + 34 q^{77} + 30 q^{79} - 22 q^{81} + 72 q^{83} + 24 q^{87} - 10 q^{89} + 16 q^{91} + 46 q^{93} + 8 q^{97} - 112 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1400, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1400.2.q.a $2$ $11.179$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-2$$ $$0$$ $$-4$$ $$q+(-2+2\zeta_{6})q^{3}+(-3+2\zeta_{6})q^{7}+\cdots$$
1400.2.q.b $2$ $11.179$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-2$$ $$0$$ $$1$$ $$q+(-2+2\zeta_{6})q^{3}+(2-3\zeta_{6})q^{7}-\zeta_{6}q^{9}+\cdots$$
1400.2.q.c $2$ $11.179$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$0$$ $$-1$$ $$q+(-1+\zeta_{6})q^{3}+(1-3\zeta_{6})q^{7}+2\zeta_{6}q^{9}+\cdots$$
1400.2.q.d $2$ $11.179$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$0$$ $$4$$ $$q+(-1+\zeta_{6})q^{3}+(1+2\zeta_{6})q^{7}+2\zeta_{6}q^{9}+\cdots$$
1400.2.q.e $2$ $11.179$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$0$$ $$5$$ $$q+(1-\zeta_{6})q^{3}+(3-\zeta_{6})q^{7}+2\zeta_{6}q^{9}+\cdots$$
1400.2.q.f $2$ $11.179$ $$\Q(\sqrt{-3})$$ None $$0$$ $$2$$ $$0$$ $$-1$$ $$q+(2-2\zeta_{6})q^{3}+(-2+3\zeta_{6})q^{7}-\zeta_{6}q^{9}+\cdots$$
1400.2.q.g $2$ $11.179$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$0$$ $$-4$$ $$q+(3-3\zeta_{6})q^{3}+(-3+2\zeta_{6})q^{7}-6\zeta_{6}q^{9}+\cdots$$
1400.2.q.h $4$ $11.179$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$-2$$ $$0$$ $$2$$ $$q+(-1+\beta _{1}-\beta _{2})q^{3}+(1+\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots$$
1400.2.q.i $6$ $11.179$ $$\Q(\zeta_{18})$$ None $$0$$ $$-3$$ $$0$$ $$0$$ $$q+(-1+\zeta_{18}+\zeta_{18}^{3}-\zeta_{18}^{4}-\zeta_{18}^{5})q^{3}+\cdots$$
1400.2.q.j $6$ $11.179$ 6.0.11337408.1 None $$0$$ $$0$$ $$0$$ $$-6$$ $$q+(\beta _{3}+\beta _{4}-\beta _{5})q^{3}+(-1-\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots$$
1400.2.q.k $6$ $11.179$ $$\Q(\zeta_{18})$$ None $$0$$ $$3$$ $$0$$ $$0$$ $$q+(-\zeta_{18}^{2}+\zeta_{18}^{3}-\zeta_{18}^{4})q^{3}+(2\zeta_{18}+\cdots)q^{7}+\cdots$$
1400.2.q.l $8$ $11.179$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$-3$$ $$0$$ $$3$$ $$q+(\beta _{1}+\beta _{5})q^{3}+(-\beta _{1}-\beta _{3}+\beta _{4})q^{7}+\cdots$$
1400.2.q.m $8$ $11.179$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$3$$ $$0$$ $$-3$$ $$q+(-\beta _{1}-\beta _{5})q^{3}+(\beta _{1}+\beta _{3}-\beta _{4})q^{7}+\cdots$$
1400.2.q.n $12$ $11.179$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$-1$$ $$0$$ $$-3$$ $$q+(\beta _{1}+\beta _{3})q^{3}-\beta _{4}q^{7}+(\beta _{6}-\beta _{10}+\cdots)q^{9}+\cdots$$
1400.2.q.o $12$ $11.179$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$1$$ $$0$$ $$3$$ $$q+(-\beta _{1}-\beta _{3})q^{3}+\beta _{4}q^{7}+(\beta _{6}-\beta _{10}+\cdots)q^{9}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1400, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1400, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(28, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(56, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(70, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(140, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(175, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(280, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(350, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(700, [\chi])$$$$^{\oplus 2}$$