# Properties

 Label 1734.2.f Level $1734$ Weight $2$ Character orbit 1734.f Rep. character $\chi_{1734}(829,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $92$ Newform subspaces $15$ Sturm bound $612$ Trace bound $21$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1734 = 2 \cdot 3 \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1734.f (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$17$$ Character field: $$\Q(i)$$ Newform subspaces: $$15$$ Sturm bound: $$612$$ Trace bound: $$21$$ Distinguishing $$T_p$$: $$5$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1734, [\chi])$$.

Total New Old
Modular forms 684 92 592
Cusp forms 540 92 448
Eisenstein series 144 0 144

## Trace form

 $$92 q - 92 q^{4} + 4 q^{5} - 8 q^{7} + O(q^{10})$$ $$92 q - 92 q^{4} + 4 q^{5} - 8 q^{7} + 4 q^{10} - 8 q^{13} + 8 q^{14} + 92 q^{16} - 4 q^{18} - 4 q^{20} - 8 q^{21} - 8 q^{23} + 8 q^{28} - 4 q^{29} - 8 q^{30} - 8 q^{31} + 24 q^{33} + 16 q^{35} + 12 q^{37} - 32 q^{38} + 8 q^{39} - 4 q^{40} + 20 q^{41} - 4 q^{45} + 8 q^{46} - 32 q^{47} + 12 q^{50} + 8 q^{52} + 16 q^{55} - 8 q^{56} + 8 q^{57} - 4 q^{58} - 20 q^{61} - 8 q^{62} - 8 q^{63} - 92 q^{64} + 16 q^{65} + 8 q^{67} + 32 q^{69} - 24 q^{71} + 4 q^{72} - 4 q^{73} + 12 q^{74} - 16 q^{75} + 8 q^{78} + 24 q^{79} + 4 q^{80} - 92 q^{81} - 20 q^{82} + 8 q^{84} + 32 q^{86} - 32 q^{89} + 4 q^{90} + 16 q^{91} + 8 q^{92} + 4 q^{97} + 4 q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1734, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1734.2.f.a $4$ $13.846$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{8}^{2}q^{2}-\zeta_{8}q^{3}-q^{4}+\zeta_{8}q^{5}-\zeta_{8}^{3}q^{6}+\cdots$$
1734.2.f.b $4$ $13.846$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{8}^{2}q^{2}+\zeta_{8}^{3}q^{3}-q^{4}+\zeta_{8}q^{6}+\cdots$$
1734.2.f.c $4$ $13.846$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{8}^{2}q^{2}+\zeta_{8}q^{3}-q^{4}+3\zeta_{8}q^{5}+\cdots$$
1734.2.f.d $4$ $13.846$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{8}^{2}q^{2}-\zeta_{8}^{3}q^{3}-q^{4}-\zeta_{8}q^{6}+\cdots$$
1734.2.f.e $4$ $13.846$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{8}^{2}q^{2}+\zeta_{8}q^{3}-q^{4}+2\zeta_{8}q^{5}+\cdots$$
1734.2.f.f $4$ $13.846$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{8}^{2}q^{2}-\zeta_{8}q^{3}-q^{4}+4\zeta_{8}q^{5}+\cdots$$
1734.2.f.g $4$ $13.846$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{8}^{2}q^{2}-\zeta_{8}q^{3}-q^{4}+4\zeta_{8}q^{5}+\cdots$$
1734.2.f.h $4$ $13.846$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{8}^{2}q^{2}+\zeta_{8}q^{3}-q^{4}+2\zeta_{8}q^{5}+\cdots$$
1734.2.f.i $4$ $13.846$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$4$$ $$-8$$ $$q-\zeta_{8}^{2}q^{2}+\zeta_{8}q^{3}-q^{4}+(1+2\zeta_{8}+\cdots)q^{5}+\cdots$$
1734.2.f.j $8$ $13.846$ $$\Q(\zeta_{16})$$ None $$0$$ $$0$$ $$-8$$ $$8$$ $$q+\zeta_{16}^{3}q^{2}+\zeta_{16}^{5}q^{3}-q^{4}+(-1+\cdots)q^{5}+\cdots$$
1734.2.f.k $8$ $13.846$ $$\Q(\zeta_{16})$$ None $$0$$ $$0$$ $$-8$$ $$8$$ $$q+\zeta_{16}^{3}q^{2}+\zeta_{16}q^{3}-q^{4}+(-1+2\zeta_{16}+\cdots)q^{5}+\cdots$$
1734.2.f.l $8$ $13.846$ $$\Q(\zeta_{16})$$ None $$0$$ $$0$$ $$8$$ $$-8$$ $$q+\zeta_{16}^{3}q^{2}+\zeta_{16}q^{3}-q^{4}+(1+2\zeta_{16}+\cdots)q^{5}+\cdots$$
1734.2.f.m $8$ $13.846$ $$\Q(\zeta_{16})$$ None $$0$$ $$0$$ $$8$$ $$-8$$ $$q-\zeta_{16}^{3}q^{2}-\zeta_{16}q^{3}-q^{4}+(1+\zeta_{16}^{3}+\cdots)q^{5}+\cdots$$
1734.2.f.n $12$ $13.846$ 12.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{7}q^{2}-\beta _{10}q^{3}-q^{4}+(2\beta _{10}-\beta _{11})q^{5}+\cdots$$
1734.2.f.o $12$ $13.846$ 12.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{7}q^{2}-\beta _{10}q^{3}-q^{4}+(2\beta _{10}-\beta _{11})q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1734, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1734, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(34, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(51, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(102, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(289, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(578, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(867, [\chi])$$$$^{\oplus 2}$$