# Properties

 Label 414.2.i Level $414$ Weight $2$ Character orbit 414.i Rep. character $\chi_{414}(55,\cdot)$ Character field $\Q(\zeta_{11})$ Dimension $100$ Newform subspaces $8$ Sturm bound $144$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$414 = 2 \cdot 3^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 414.i (of order $$11$$ and degree $$10$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$23$$ Character field: $$\Q(\zeta_{11})$$ Newform subspaces: $$8$$ Sturm bound: $$144$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(414, [\chi])$$.

Total New Old
Modular forms 800 100 700
Cusp forms 640 100 540
Eisenstein series 160 0 160

## Trace form

 $$100 q - 10 q^{4} + 2 q^{5} - 4 q^{7} + O(q^{10})$$ $$100 q - 10 q^{4} + 2 q^{5} - 4 q^{7} - 2 q^{10} - 10 q^{11} + 4 q^{13} + 4 q^{14} - 10 q^{16} + 14 q^{17} + 20 q^{19} + 24 q^{20} + 28 q^{22} + 44 q^{23} + 26 q^{25} + 22 q^{26} + 18 q^{28} + 22 q^{29} + 6 q^{31} + 12 q^{34} - 10 q^{35} - 46 q^{37} - 10 q^{38} - 2 q^{40} + 38 q^{41} - 50 q^{43} - 10 q^{44} + 2 q^{46} + 52 q^{47} + 56 q^{49} - 4 q^{50} + 4 q^{52} + 40 q^{53} - 72 q^{55} - 18 q^{56} - 24 q^{58} - 58 q^{59} - 94 q^{61} - 74 q^{62} - 10 q^{64} - 142 q^{65} - 22 q^{67} - 52 q^{68} - 88 q^{70} - 110 q^{71} - 28 q^{73} - 54 q^{74} - 2 q^{76} - 86 q^{77} - 38 q^{79} - 20 q^{80} - 16 q^{82} - 10 q^{83} + 38 q^{85} - 28 q^{86} + 6 q^{88} + 34 q^{89} + 68 q^{91} + 74 q^{95} + 80 q^{97} + 60 q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(414, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
414.2.i.a $10$ $3.306$ $$\Q(\zeta_{22})$$ None $$-1$$ $$0$$ $$-2$$ $$2$$ $$q+\zeta_{22}^{4}q^{2}+\zeta_{22}^{8}q^{4}+(-1-\zeta_{22}^{2}+\cdots)q^{5}+\cdots$$
414.2.i.b $10$ $3.306$ $$\Q(\zeta_{22})$$ None $$-1$$ $$0$$ $$0$$ $$-2$$ $$q+\zeta_{22}^{4}q^{2}+\zeta_{22}^{8}q^{4}+(1-2\zeta_{22}+\cdots)q^{5}+\cdots$$
414.2.i.c $10$ $3.306$ $$\Q(\zeta_{22})$$ None $$-1$$ $$0$$ $$4$$ $$-7$$ $$q+\zeta_{22}^{4}q^{2}+\zeta_{22}^{8}q^{4}+(\zeta_{22}+\zeta_{22}^{3}+\cdots)q^{5}+\cdots$$
414.2.i.d $10$ $3.306$ $$\Q(\zeta_{22})$$ None $$1$$ $$0$$ $$-8$$ $$8$$ $$q-\zeta_{22}^{4}q^{2}+\zeta_{22}^{8}q^{4}+(-1+\zeta_{22}^{2}+\cdots)q^{5}+\cdots$$
414.2.i.e $10$ $3.306$ $$\Q(\zeta_{22})$$ None $$1$$ $$0$$ $$2$$ $$0$$ $$q-\zeta_{22}^{4}q^{2}+\zeta_{22}^{8}q^{4}+(1-2\zeta_{22}+\cdots)q^{5}+\cdots$$
414.2.i.f $10$ $3.306$ $$\Q(\zeta_{22})$$ None $$1$$ $$0$$ $$6$$ $$3$$ $$q-\zeta_{22}^{4}q^{2}+\zeta_{22}^{8}q^{4}+(\zeta_{22}-2\zeta_{22}^{2}+\cdots)q^{5}+\cdots$$
414.2.i.g $20$ $3.306$ $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ None $$-2$$ $$0$$ $$-2$$ $$-4$$ $$q+\beta _{3}q^{2}+\beta _{7}q^{4}+(\beta _{2}-\beta _{5}+\beta _{13}+\cdots)q^{5}+\cdots$$
414.2.i.h $20$ $3.306$ $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ None $$2$$ $$0$$ $$2$$ $$-4$$ $$q-\beta _{3}q^{2}+\beta _{7}q^{4}+(-\beta _{2}+\beta _{5}-\beta _{13}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(414, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(414, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(23, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(46, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(69, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(138, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(207, [\chi])$$$$^{\oplus 2}$$