# Properties

 Label 624.4.q Level $624$ Weight $4$ Character orbit 624.q Rep. character $\chi_{624}(289,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $84$ Newform subspaces $14$ Sturm bound $448$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$624 = 2^{4} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 624.q (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$14$$ Sturm bound: $$448$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$5$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(624, [\chi])$$.

Total New Old
Modular forms 696 84 612
Cusp forms 648 84 564
Eisenstein series 48 0 48

## Trace form

 $$84 q + 6 q^{3} - 4 q^{5} + 18 q^{7} - 378 q^{9} + O(q^{10})$$ $$84 q + 6 q^{3} - 4 q^{5} + 18 q^{7} - 378 q^{9} + 46 q^{13} - 26 q^{17} - 180 q^{19} + 2056 q^{25} - 108 q^{27} - 142 q^{29} + 60 q^{31} + 66 q^{37} + 420 q^{39} - 474 q^{41} + 342 q^{43} + 18 q^{45} - 1488 q^{47} - 2134 q^{49} - 1224 q^{51} + 1356 q^{53} - 1276 q^{55} - 336 q^{57} + 748 q^{59} - 278 q^{61} + 162 q^{63} + 1414 q^{65} + 342 q^{67} - 224 q^{71} + 524 q^{73} + 1602 q^{75} - 368 q^{77} + 3364 q^{79} - 3402 q^{81} - 2680 q^{83} - 34 q^{85} - 360 q^{87} - 396 q^{89} - 3682 q^{91} + 2516 q^{95} - 24 q^{97} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(624, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
624.4.q.a $2$ $36.817$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$14$$ $$16$$ $$q+(-3+3\zeta_{6})q^{3}+7q^{5}+2^{4}\zeta_{6}q^{7}+\cdots$$
624.4.q.b $2$ $36.817$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$-18$$ $$2$$ $$q+(3-3\zeta_{6})q^{3}-9q^{5}+2\zeta_{6}q^{7}-9\zeta_{6}q^{9}+\cdots$$
624.4.q.c $2$ $36.817$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$14$$ $$-10$$ $$q+(3-3\zeta_{6})q^{3}+7q^{5}-10\zeta_{6}q^{7}-9\zeta_{6}q^{9}+\cdots$$
624.4.q.d $4$ $36.817$ $$\Q(\sqrt{-3}, \sqrt{61})$$ None $$0$$ $$-6$$ $$4$$ $$18$$ $$q+(-3+3\beta _{1})q^{3}+(1+2\beta _{3})q^{5}+(9\beta _{1}+\cdots)q^{7}+\cdots$$
624.4.q.e $4$ $36.817$ $$\Q(\sqrt{-3}, \sqrt{14})$$ None $$0$$ $$-6$$ $$12$$ $$20$$ $$q+3\beta _{2}q^{3}+(3+2\beta _{3})q^{5}+(10+\beta _{1}+\cdots)q^{7}+\cdots$$
624.4.q.f $4$ $36.817$ $$\Q(\sqrt{-3}, \sqrt{673})$$ None $$0$$ $$6$$ $$26$$ $$9$$ $$q+(3-3\beta _{2})q^{3}+(6+\beta _{3})q^{5}+(\beta _{1}+4\beta _{2}+\cdots)q^{7}+\cdots$$
624.4.q.g $6$ $36.817$ 6.0.31902363.2 None $$0$$ $$-9$$ $$-24$$ $$-29$$ $$q-3\beta _{1}q^{3}+(-4+\beta _{5})q^{5}+(-10+10\beta _{1}+\cdots)q^{7}+\cdots$$
624.4.q.h $6$ $36.817$ $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ None $$0$$ $$9$$ $$-24$$ $$-17$$ $$q+(3+3\beta _{2})q^{3}+(-4-\beta _{3})q^{5}+(6\beta _{2}+\cdots)q^{7}+\cdots$$
624.4.q.i $8$ $36.817$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$-12$$ $$-12$$ $$-14$$ $$q-3\beta _{2}q^{3}+(-2+\beta _{4})q^{5}+(-3+3\beta _{2}+\cdots)q^{7}+\cdots$$
624.4.q.j $8$ $36.817$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$-12$$ $$-2$$ $$13$$ $$q+(-3+3\beta _{2})q^{3}+(-\beta _{5}+\beta _{7})q^{5}+\cdots$$
624.4.q.k $8$ $36.817$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$-12$$ $$6$$ $$-15$$ $$q+(-3+3\beta _{1})q^{3}+(1+\beta _{4}+\beta _{5})q^{5}+\cdots$$
624.4.q.l $8$ $36.817$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$12$$ $$14$$ $$11$$ $$q-3\beta _{2}q^{3}+(2+\beta _{5})q^{5}+(3-\beta _{1}+3\beta _{2}+\cdots)q^{7}+\cdots$$
624.4.q.m $10$ $36.817$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$0$$ $$15$$ $$-22$$ $$4$$ $$q+(3-3\beta _{2})q^{3}+(-2-\beta _{1}-\beta _{8})q^{5}+\cdots$$
624.4.q.n $12$ $36.817$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$18$$ $$8$$ $$10$$ $$q+3\beta _{4}q^{3}+(1+\beta _{1})q^{5}+(1-\beta _{2}-\beta _{4}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(624, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(624, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(13, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(26, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(39, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(52, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(78, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(104, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(156, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(208, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(312, [\chi])$$$$^{\oplus 2}$$