# Properties

 Label 234.4.h Level $234$ Weight $4$ Character orbit 234.h Rep. character $\chi_{234}(55,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $34$ Newform subspaces $10$ Sturm bound $168$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$234 = 2 \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 234.h (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$10$$ Sturm bound: $$168$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(234, [\chi])$$.

Total New Old
Modular forms 268 34 234
Cusp forms 236 34 202
Eisenstein series 32 0 32

## Trace form

 $$34 q - 2 q^{2} - 68 q^{4} - 10 q^{5} - 2 q^{7} + 16 q^{8} + O(q^{10})$$ $$34 q - 2 q^{2} - 68 q^{4} - 10 q^{5} - 2 q^{7} + 16 q^{8} + 30 q^{10} - 50 q^{11} + 141 q^{13} + 64 q^{14} - 272 q^{16} - 33 q^{17} + 266 q^{19} + 20 q^{20} + 16 q^{22} + 50 q^{23} + 764 q^{25} + 230 q^{26} - 8 q^{28} - 313 q^{29} - 552 q^{31} - 32 q^{32} + 212 q^{34} - 4 q^{35} + 737 q^{37} + 576 q^{38} - 240 q^{40} + 75 q^{41} + 58 q^{43} + 400 q^{44} - 344 q^{46} - 2192 q^{47} - 1863 q^{49} - 760 q^{50} - 336 q^{52} + 4014 q^{53} + 528 q^{55} - 128 q^{56} + 190 q^{58} - 454 q^{59} + 817 q^{61} - 464 q^{62} + 2176 q^{64} - 2351 q^{65} - 2154 q^{67} - 132 q^{68} - 2736 q^{70} + 466 q^{71} + 2450 q^{73} - 678 q^{74} + 1064 q^{76} + 7324 q^{77} + 800 q^{79} + 80 q^{80} - 178 q^{82} + 2064 q^{83} - 4387 q^{85} - 2768 q^{86} + 64 q^{88} - 4668 q^{89} + 5034 q^{91} - 400 q^{92} - 680 q^{94} - 4608 q^{95} - 216 q^{97} - 1842 q^{98} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(234, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
234.4.h.a $2$ $13.806$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$0$$ $$-8$$ $$23$$ $$q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-4q^{5}+\cdots$$
234.4.h.b $2$ $13.806$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$0$$ $$-4$$ $$5$$ $$q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-2q^{5}+\cdots$$
234.4.h.c $2$ $13.806$ $$\Q(\sqrt{-3})$$ None $$2$$ $$0$$ $$-14$$ $$-16$$ $$q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-7q^{5}-2^{4}\zeta_{6}q^{7}+\cdots$$
234.4.h.d $2$ $13.806$ $$\Q(\sqrt{-3})$$ None $$2$$ $$0$$ $$8$$ $$23$$ $$q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+4q^{5}+23\zeta_{6}q^{7}+\cdots$$
234.4.h.e $4$ $13.806$ $$\Q(\sqrt{-3}, \sqrt{142})$$ None $$-4$$ $$0$$ $$-28$$ $$-36$$ $$q+(-2-2\beta _{2})q^{2}+4\beta _{2}q^{4}+(-7+\beta _{3})q^{5}+\cdots$$
234.4.h.f $4$ $13.806$ $$\Q(\sqrt{-3}, \sqrt{61})$$ None $$-4$$ $$0$$ $$-4$$ $$-18$$ $$q+(-2+2\beta _{1})q^{2}-4\beta _{1}q^{4}+(-1+2\beta _{3})q^{5}+\cdots$$
234.4.h.g $4$ $13.806$ $$\Q(\sqrt{-3}, \sqrt{673})$$ None $$4$$ $$0$$ $$-26$$ $$-9$$ $$q+(2-2\beta _{2})q^{2}-4\beta _{2}q^{4}+(-7+\beta _{3})q^{5}+\cdots$$
234.4.h.h $4$ $13.806$ $$\Q(\sqrt{-3}, \sqrt{217})$$ None $$4$$ $$0$$ $$14$$ $$45$$ $$q+(2-2\beta _{2})q^{2}-4\beta _{2}q^{4}+(3+\beta _{3})q^{5}+\cdots$$
234.4.h.i $4$ $13.806$ $$\Q(\sqrt{-3}, \sqrt{142})$$ None $$4$$ $$0$$ $$28$$ $$-36$$ $$q-2\beta _{2}q^{2}+(-4-4\beta _{2})q^{4}+(7+\beta _{3})q^{5}+\cdots$$
234.4.h.j $6$ $13.806$ $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ None $$-6$$ $$0$$ $$24$$ $$17$$ $$q+2\beta _{2}q^{2}+(-4-4\beta _{2})q^{4}+(4+\beta _{3}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(234, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(234, [\chi]) \simeq$$ $$S_{4}^{\mathrm{new}}(13, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(26, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(39, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(78, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(117, [\chi])$$$$^{\oplus 2}$$