# Properties

 Label 1764.4.k Level $1764$ Weight $4$ Character orbit 1764.k Rep. character $\chi_{1764}(361,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $100$ Newform subspaces $30$ Sturm bound $1344$ Trace bound $17$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1764.k (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$30$$ Sturm bound: $$1344$$ Trace bound: $$17$$ Distinguishing $$T_p$$: $$5$$, $$11$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(1764, [\chi])$$.

Total New Old
Modular forms 2112 100 2012
Cusp forms 1920 100 1820
Eisenstein series 192 0 192

## Trace form

 $$100q + 6q^{5} + O(q^{10})$$ $$100q + 6q^{5} + 20q^{11} + 30q^{17} - 40q^{19} + 40q^{23} - 1100q^{25} - 224q^{29} + 168q^{31} + 50q^{37} - 816q^{41} - 488q^{43} - 48q^{47} + 86q^{53} + 656q^{55} + 1152q^{59} - 918q^{61} + 1764q^{65} - 808q^{67} + 2856q^{71} - 1262q^{73} - 1448q^{79} - 2832q^{83} - 244q^{85} + 858q^{89} - 3664q^{95} + 976q^{97} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(1764, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
1764.4.k.a $$2$$ $$104.079$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-20$$ $$0$$ $$q-20\zeta_{6}q^{5}+(44-44\zeta_{6})q^{11}-44q^{13}+\cdots$$
1764.4.k.b $$2$$ $$104.079$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-18$$ $$0$$ $$q-18\zeta_{6}q^{5}+(6^{2}-6^{2}\zeta_{6})q^{11}-10q^{13}+\cdots$$
1764.4.k.c $$2$$ $$104.079$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-14$$ $$0$$ $$q-14\zeta_{6}q^{5}+(4-4\zeta_{6})q^{11}-54q^{13}+\cdots$$
1764.4.k.d $$2$$ $$104.079$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-8$$ $$0$$ $$q-8\zeta_{6}q^{5}+(-40+40\zeta_{6})q^{11}-12q^{13}+\cdots$$
1764.4.k.e $$2$$ $$104.079$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-6$$ $$0$$ $$q-6\zeta_{6}q^{5}+(-12+12\zeta_{6})q^{11}+82q^{13}+\cdots$$
1764.4.k.f $$2$$ $$104.079$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-6$$ $$0$$ $$q-6\zeta_{6}q^{5}+(6^{2}-6^{2}\zeta_{6})q^{11}-62q^{13}+\cdots$$
1764.4.k.g $$2$$ $$104.079$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q-4\zeta_{6}q^{5}+(-20+20\zeta_{6})q^{11}+4q^{13}+\cdots$$
1764.4.k.h $$2$$ $$104.079$$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-89q^{13}-163\zeta_{6}q^{19}+(5^{3}-5^{3}\zeta_{6})q^{25}+\cdots$$
1764.4.k.i $$2$$ $$104.079$$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+19q^{13}+107\zeta_{6}q^{19}+(5^{3}-5^{3}\zeta_{6})q^{25}+\cdots$$
1764.4.k.j $$2$$ $$104.079$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+4\zeta_{6}q^{5}+(-20+20\zeta_{6})q^{11}-4q^{13}+\cdots$$
1764.4.k.k $$2$$ $$104.079$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$6$$ $$0$$ $$q+6\zeta_{6}q^{5}+(-12+12\zeta_{6})q^{11}-82q^{13}+\cdots$$
1764.4.k.l $$2$$ $$104.079$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$6$$ $$0$$ $$q+6\zeta_{6}q^{5}+(6^{2}-6^{2}\zeta_{6})q^{11}+62q^{13}+\cdots$$
1764.4.k.m $$2$$ $$104.079$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$8$$ $$0$$ $$q+8\zeta_{6}q^{5}+(-40+40\zeta_{6})q^{11}+12q^{13}+\cdots$$
1764.4.k.n $$2$$ $$104.079$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$14$$ $$0$$ $$q+14\zeta_{6}q^{5}+(4-4\zeta_{6})q^{11}+54q^{13}+\cdots$$
1764.4.k.o $$2$$ $$104.079$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$18$$ $$0$$ $$q+18\zeta_{6}q^{5}+(6^{2}-6^{2}\zeta_{6})q^{11}+10q^{13}+\cdots$$
1764.4.k.p $$2$$ $$104.079$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$20$$ $$0$$ $$q+20\zeta_{6}q^{5}+(44-44\zeta_{6})q^{11}+44q^{13}+\cdots$$
1764.4.k.q $$4$$ $$104.079$$ $$\Q(\sqrt{-3}, \sqrt{193})$$ None $$0$$ $$0$$ $$-11$$ $$0$$ $$q+(\beta _{1}-6\beta _{2})q^{5}+(1-7\beta _{1}+6\beta _{2}-7\beta _{3})q^{11}+\cdots$$
1764.4.k.r $$4$$ $$104.079$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+7\beta _{1}q^{5}+28\beta _{2}q^{11}+3\beta _{3}q^{13}+\cdots$$
1764.4.k.s $$4$$ $$104.079$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2\beta _{1}q^{5}+26\beta _{2}q^{11}-24\beta _{3}q^{13}+\cdots$$
1764.4.k.t $$4$$ $$104.079$$ $$\Q(\sqrt{-3}, \sqrt{7})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{5}+(13\beta _{1}+13\beta _{3})q^{11}-30q^{13}+\cdots$$
1764.4.k.u $$4$$ $$104.079$$ $$\Q(\sqrt{-3}, \sqrt{7})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{5}+(\beta _{1}+\beta _{3})q^{11}-26q^{13}+\cdots$$
1764.4.k.v $$4$$ $$104.079$$ $$\Q(\sqrt{-3}, \sqrt{7})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{5}+(-\beta _{1}-\beta _{3})q^{11}+26q^{13}+\cdots$$
1764.4.k.w $$4$$ $$104.079$$ $$\Q(\sqrt{-3}, \sqrt{7})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{5}+(-13\beta _{1}-13\beta _{3})q^{11}+30q^{13}+\cdots$$
1764.4.k.x $$4$$ $$104.079$$ $$\Q(\sqrt{-3}, \sqrt{385})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{5}+(\beta _{2}+\beta _{3})q^{11}+54q^{13}+\cdots$$
1764.4.k.y $$4$$ $$104.079$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+7\beta _{1}q^{5}-28\beta _{2}q^{11}-3\beta _{3}q^{13}+\cdots$$
1764.4.k.z $$4$$ $$104.079$$ $$\Q(\sqrt{-3}, \sqrt{-19})$$ None $$0$$ $$0$$ $$3$$ $$0$$ $$q+(1+\beta _{1}+\beta _{3})q^{5}+(-1+5^{2}\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots$$
1764.4.k.ba $$4$$ $$104.079$$ $$\Q(\sqrt{-3}, \sqrt{37})$$ None $$0$$ $$0$$ $$14$$ $$0$$ $$q+(7\beta _{1}+2\beta _{2})q^{5}+(2^{4}-2^{4}\beta _{1}+7\beta _{2}+\cdots)q^{11}+\cdots$$
1764.4.k.bb $$8$$ $$104.079$$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{6}q^{5}+(-3\beta _{2}-\beta _{4})q^{11}+(3\beta _{4}+\cdots)q^{13}+\cdots$$
1764.4.k.bc $$8$$ $$104.079$$ 8.0.$$\cdots$$.42 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{5}q^{5}+(\beta _{2}+\beta _{3})q^{11}+\beta _{6}q^{13}+\cdots$$
1764.4.k.bd $$8$$ $$104.079$$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{6}q^{5}+(-3\beta _{2}-\beta _{4})q^{11}+(-3\beta _{4}+\cdots)q^{13}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(1764, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(1764, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(7, [\chi])$$$$^{\oplus 18}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(14, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(28, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(49, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(84, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(98, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(126, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(147, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(196, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(252, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(294, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(441, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(588, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(882, [\chi])$$$$^{\oplus 2}$$