# Properties

 Label 126.4.g Level $126$ Weight $4$ Character orbit 126.g Rep. character $\chi_{126}(37,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $20$ Newform subspaces $7$ Sturm bound $96$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$126 = 2 \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 126.g (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$7$$ Sturm bound: $$96$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(126, [\chi])$$.

Total New Old
Modular forms 160 20 140
Cusp forms 128 20 108
Eisenstein series 32 0 32

## Trace form

 $$20 q - 40 q^{4} - 18 q^{5} - 8 q^{7} + O(q^{10})$$ $$20 q - 40 q^{4} - 18 q^{5} - 8 q^{7} + 8 q^{10} + 6 q^{11} - 192 q^{13} - 72 q^{14} - 160 q^{16} - 150 q^{17} + 46 q^{19} + 144 q^{20} + 256 q^{22} + 210 q^{23} - 68 q^{25} - 120 q^{26} + 40 q^{28} - 336 q^{29} - 138 q^{31} - 512 q^{34} - 426 q^{35} - 242 q^{37} - 192 q^{38} + 32 q^{40} + 2016 q^{41} + 1288 q^{43} + 24 q^{44} + 40 q^{46} + 102 q^{47} + 1412 q^{49} - 1824 q^{50} + 384 q^{52} - 1182 q^{53} - 1916 q^{55} + 288 q^{56} - 136 q^{58} + 90 q^{59} - 294 q^{61} + 3744 q^{62} + 1280 q^{64} + 1524 q^{65} + 1226 q^{67} - 600 q^{68} - 1520 q^{70} - 672 q^{71} + 1142 q^{73} - 312 q^{74} - 368 q^{76} - 2226 q^{77} + 1266 q^{79} - 288 q^{80} + 1200 q^{82} + 2568 q^{83} - 4892 q^{85} - 984 q^{86} - 512 q^{88} - 2754 q^{89} + 2100 q^{91} - 1680 q^{92} - 216 q^{94} - 1974 q^{95} - 160 q^{97} + 2880 q^{98} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(126, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
126.4.g.a $2$ $7.434$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$0$$ $$-15$$ $$35$$ $$q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(-15+\cdots)q^{5}+\cdots$$
126.4.g.b $2$ $7.434$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$0$$ $$-6$$ $$-7$$ $$q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(-6+6\zeta_{6})q^{5}+\cdots$$
126.4.g.c $2$ $7.434$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$0$$ $$7$$ $$-20$$ $$q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(7-7\zeta_{6})q^{5}+\cdots$$
126.4.g.d $2$ $7.434$ $$\Q(\sqrt{-3})$$ None $$2$$ $$0$$ $$-9$$ $$-28$$ $$q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(-9+9\zeta_{6})q^{5}+\cdots$$
126.4.g.e $4$ $7.434$ $$\Q(\sqrt{-3}, \sqrt{193})$$ None $$-4$$ $$0$$ $$7$$ $$6$$ $$q-2\beta _{2}q^{2}+(-4+4\beta _{2})q^{4}+(\beta _{1}+3\beta _{2}+\cdots)q^{5}+\cdots$$
126.4.g.f $4$ $7.434$ $$\Q(\sqrt{-3}, \sqrt{193})$$ None $$4$$ $$0$$ $$-7$$ $$6$$ $$q+2\beta _{2}q^{2}+(-4+4\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots$$
126.4.g.g $4$ $7.434$ $$\Q(\sqrt{-3}, \sqrt{1345})$$ None $$4$$ $$0$$ $$5$$ $$0$$ $$q+2\beta _{2}q^{2}+(-4+4\beta _{2})q^{4}+(\beta _{1}+2\beta _{2}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(126, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(126, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(7, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(14, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 2}$$