# Properties

 Label 315.4.j Level $315$ Weight $4$ Character orbit 315.j Rep. character $\chi_{315}(46,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $80$ Newform subspaces $10$ Sturm bound $192$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 315.j (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$10$$ Sturm bound: $$192$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$2$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(315, [\chi])$$.

Total New Old
Modular forms 304 80 224
Cusp forms 272 80 192
Eisenstein series 32 0 32

## Trace form

 $$80 q + 2 q^{2} - 170 q^{4} - 10 q^{5} + 20 q^{7} + 36 q^{8} + O(q^{10})$$ $$80 q + 2 q^{2} - 170 q^{4} - 10 q^{5} + 20 q^{7} + 36 q^{8} + 20 q^{10} - 38 q^{11} - 352 q^{13} - 206 q^{14} - 718 q^{16} - 140 q^{17} - 154 q^{19} + 200 q^{20} - 368 q^{22} - 80 q^{23} - 1000 q^{25} - 438 q^{26} - 6 q^{28} - 692 q^{29} - 160 q^{31} - 882 q^{32} - 1576 q^{34} - 110 q^{35} - 468 q^{37} + 152 q^{38} + 240 q^{40} + 1400 q^{41} + 1712 q^{43} + 522 q^{44} - 22 q^{46} - 992 q^{47} - 390 q^{49} - 100 q^{50} + 1348 q^{52} - 2156 q^{53} - 1120 q^{55} + 876 q^{56} - 1530 q^{58} - 188 q^{59} + 910 q^{61} + 6000 q^{62} + 7444 q^{64} + 530 q^{65} + 216 q^{67} - 1708 q^{68} + 520 q^{70} - 944 q^{71} + 12 q^{73} - 3246 q^{74} + 4916 q^{76} - 1048 q^{77} + 1400 q^{79} - 960 q^{80} + 5486 q^{82} + 7568 q^{83} - 1280 q^{85} + 3324 q^{86} + 5176 q^{88} - 294 q^{89} - 2716 q^{91} + 6204 q^{92} - 6814 q^{94} - 300 q^{95} - 6576 q^{97} - 4486 q^{98} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(315, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
315.4.j.a $2$ $18.586$ $$\Q(\sqrt{-3})$$ None $$3$$ $$0$$ $$-5$$ $$-28$$ $$q+3\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-5\zeta_{6}q^{5}+\cdots$$
315.4.j.b $2$ $18.586$ $$\Q(\sqrt{-3})$$ None $$3$$ $$0$$ $$5$$ $$-28$$ $$q+3\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+5\zeta_{6}q^{5}+\cdots$$
315.4.j.c $4$ $18.586$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$-6$$ $$0$$ $$10$$ $$22$$ $$q+(\beta _{1}+3\beta _{2}+\beta _{3})q^{2}+(-3-6\beta _{1}+\cdots)q^{4}+\cdots$$
315.4.j.d $4$ $18.586$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$-4$$ $$0$$ $$-10$$ $$50$$ $$q+(-\beta _{1}+2\beta _{2}-\beta _{3})q^{2}+(2+4\beta _{1}+\cdots)q^{4}+\cdots$$
315.4.j.e $6$ $18.586$ 6.0.646154928.2 None $$3$$ $$0$$ $$15$$ $$-2$$ $$q+(1-\beta _{1}-\beta _{2}-\beta _{3})q^{2}+(-\beta _{1}+\beta _{4}+\cdots)q^{4}+\cdots$$
315.4.j.f $10$ $18.586$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$-1$$ $$0$$ $$-25$$ $$56$$ $$q-\beta _{1}q^{2}+(\beta _{1}-\beta _{2}+\beta _{3}-4\beta _{4}+\beta _{8}+\cdots)q^{4}+\cdots$$
315.4.j.g $10$ $18.586$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$1$$ $$0$$ $$-25$$ $$-62$$ $$q+\beta _{1}q^{2}+(-7-\beta _{4}+7\beta _{6}+\beta _{8})q^{4}+\cdots$$
315.4.j.h $10$ $18.586$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$3$$ $$0$$ $$25$$ $$-32$$ $$q+(-\beta _{1}-\beta _{4})q^{2}+(-5-5\beta _{4}+\beta _{7}+\cdots)q^{4}+\cdots$$
315.4.j.i $16$ $18.586$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$-4$$ $$0$$ $$40$$ $$22$$ $$q+(\beta _{1}+\beta _{6})q^{2}+(-6-\beta _{1}+\beta _{4}-\beta _{5}+\cdots)q^{4}+\cdots$$
315.4.j.j $16$ $18.586$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$4$$ $$0$$ $$-40$$ $$22$$ $$q+(-\beta _{1}-\beta _{6})q^{2}+(-6-\beta _{1}+\beta _{4}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(315, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(315, [\chi]) \simeq$$ $$S_{4}^{\mathrm{new}}(7, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 2}$$