# Properties

 Label 3528.2.s Level $3528$ Weight $2$ Character orbit 3528.s Rep. character $\chi_{3528}(361,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $100$ Newform subspaces $39$ Sturm bound $1344$ Trace bound $23$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$3528 = 2^{3} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3528.s (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$39$$ Sturm bound: $$1344$$ Trace bound: $$23$$ Distinguishing $$T_p$$: $$5$$, $$11$$, $$13$$, $$23$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(3528, [\chi])$$.

Total New Old
Modular forms 1472 100 1372
Cusp forms 1216 100 1116
Eisenstein series 256 0 256

## Trace form

 $$100q + 2q^{5} + O(q^{10})$$ $$100q + 2q^{5} + 2q^{11} + 6q^{17} + 10q^{19} + 2q^{23} - 36q^{25} - 16q^{29} - 6q^{31} + 6q^{37} - 16q^{43} - 6q^{47} + 2q^{53} - 4q^{55} - 18q^{59} - 10q^{61} - 12q^{65} + 2q^{67} + 80q^{71} - 22q^{73} + 10q^{79} + 56q^{83} + 36q^{85} + 34q^{89} + 50q^{95} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(3528, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
3528.2.s.a $$2$$ $$28.171$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q-4\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{17}+2\zeta_{6}q^{19}+\cdots$$
3528.2.s.b $$2$$ $$28.171$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q-4\zeta_{6}q^{5}+3q^{13}+(4-4\zeta_{6})q^{17}+\cdots$$
3528.2.s.c $$2$$ $$28.171$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-2$$ $$0$$ $$q-2\zeta_{6}q^{5}+(-6+6\zeta_{6})q^{11}-6q^{13}+\cdots$$
3528.2.s.d $$2$$ $$28.171$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-2$$ $$0$$ $$q-2\zeta_{6}q^{5}+(-6+6\zeta_{6})q^{11}+3q^{13}+\cdots$$
3528.2.s.e $$2$$ $$28.171$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-2$$ $$0$$ $$q-2\zeta_{6}q^{5}+(-4+4\zeta_{6})q^{11}-2q^{13}+\cdots$$
3528.2.s.f $$2$$ $$28.171$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-2$$ $$0$$ $$q-2\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{11}+2q^{13}+\cdots$$
3528.2.s.g $$2$$ $$28.171$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-2$$ $$0$$ $$q-2\zeta_{6}q^{5}-6q^{13}+(2-2\zeta_{6})q^{17}+\cdots$$
3528.2.s.h $$2$$ $$28.171$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-2$$ $$0$$ $$q-2\zeta_{6}q^{5}+2q^{13}+(-6+6\zeta_{6})q^{17}+\cdots$$
3528.2.s.i $$2$$ $$28.171$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-2$$ $$0$$ $$q-2\zeta_{6}q^{5}+(2-2\zeta_{6})q^{11}-2q^{13}+\cdots$$
3528.2.s.j $$2$$ $$28.171$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-2$$ $$0$$ $$q-2\zeta_{6}q^{5}+(4-4\zeta_{6})q^{11}-2q^{13}+\cdots$$
3528.2.s.k $$2$$ $$28.171$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-2$$ $$0$$ $$q-2\zeta_{6}q^{5}+(6-6\zeta_{6})q^{11}+6q^{13}+\cdots$$
3528.2.s.l $$2$$ $$28.171$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$0$$ $$q-\zeta_{6}q^{5}+(-5+5\zeta_{6})q^{11}-2q^{13}+\cdots$$
3528.2.s.m $$2$$ $$28.171$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-4q^{13}+(-4+4\zeta_{6})q^{17}-4\zeta_{6}q^{19}+\cdots$$
3528.2.s.n $$2$$ $$28.171$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+4q^{13}+(4-4\zeta_{6})q^{17}+4\zeta_{6}q^{19}+\cdots$$
3528.2.s.o $$2$$ $$28.171$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$1$$ $$0$$ $$q+\zeta_{6}q^{5}+(-1+\zeta_{6})q^{11}-2q^{13}+\cdots$$
3528.2.s.p $$2$$ $$28.171$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$1$$ $$0$$ $$q+\zeta_{6}q^{5}+(3-3\zeta_{6})q^{11}-4q^{13}-4\zeta_{6}q^{19}+\cdots$$
3528.2.s.q $$2$$ $$28.171$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$1$$ $$0$$ $$q+\zeta_{6}q^{5}+(3-3\zeta_{6})q^{11}+6q^{13}+(5+\cdots)q^{17}+\cdots$$
3528.2.s.r $$2$$ $$28.171$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$1$$ $$0$$ $$q+\zeta_{6}q^{5}+(5-5\zeta_{6})q^{11}-2q^{13}+(-6+\cdots)q^{17}+\cdots$$
3528.2.s.s $$2$$ $$28.171$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$2$$ $$0$$ $$q+2\zeta_{6}q^{5}+(-6+6\zeta_{6})q^{11}+6q^{13}+\cdots$$
3528.2.s.t $$2$$ $$28.171$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$2$$ $$0$$ $$q+2\zeta_{6}q^{5}+(-4+4\zeta_{6})q^{11}+2q^{13}+\cdots$$
3528.2.s.u $$2$$ $$28.171$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$2$$ $$0$$ $$q+2\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{11}-2q^{13}+\cdots$$
3528.2.s.v $$2$$ $$28.171$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$2$$ $$0$$ $$q+2\zeta_{6}q^{5}-2q^{13}+(6-6\zeta_{6})q^{17}+\cdots$$
3528.2.s.w $$2$$ $$28.171$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$2$$ $$0$$ $$q+2\zeta_{6}q^{5}+6q^{13}+(-2+2\zeta_{6})q^{17}+\cdots$$
3528.2.s.x $$2$$ $$28.171$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$2$$ $$0$$ $$q+2\zeta_{6}q^{5}+(2-2\zeta_{6})q^{11}+2q^{13}+\cdots$$
3528.2.s.y $$2$$ $$28.171$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$2$$ $$0$$ $$q+2\zeta_{6}q^{5}+(4-4\zeta_{6})q^{11}+2q^{13}+\cdots$$
3528.2.s.z $$2$$ $$28.171$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$2$$ $$0$$ $$q+2\zeta_{6}q^{5}+(6-6\zeta_{6})q^{11}-6q^{13}+\cdots$$
3528.2.s.ba $$2$$ $$28.171$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+4\zeta_{6}q^{5}+(2-2\zeta_{6})q^{17}-2\zeta_{6}q^{19}+\cdots$$
3528.2.s.bb $$2$$ $$28.171$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+4\zeta_{6}q^{5}+3q^{13}+(-4+4\zeta_{6})q^{17}+\cdots$$
3528.2.s.bc $$4$$ $$28.171$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q+(-\beta _{1}+2\beta _{2}-\beta _{3})q^{5}+(-2-2\beta _{1}+\cdots)q^{11}+\cdots$$
3528.2.s.bd $$4$$ $$28.171$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q+(-\beta _{1}+2\beta _{2}-\beta _{3})q^{5}+(2-2\beta _{1}+\cdots)q^{11}+\cdots$$
3528.2.s.be $$4$$ $$28.171$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{5}+4\beta _{2}q^{11}+\beta _{3}q^{13}+(2\beta _{1}+\cdots)q^{17}+\cdots$$
3528.2.s.bf $$4$$ $$28.171$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{5}+2\beta _{2}q^{11}+2\beta _{3}q^{13}+(-\beta _{1}+\cdots)q^{17}+\cdots$$
3528.2.s.bg $$4$$ $$28.171$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{5}+\beta _{3}q^{13}+(\beta _{1}+\beta _{3})q^{17}+\cdots$$
3528.2.s.bh $$4$$ $$28.171$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{5}-\beta _{3}q^{13}+(\beta _{1}+\beta _{3})q^{17}+\cdots$$
3528.2.s.bi $$4$$ $$28.171$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{5}-2\beta _{2}q^{11}-2\beta _{3}q^{13}+(-\beta _{1}+\cdots)q^{17}+\cdots$$
3528.2.s.bj $$4$$ $$28.171$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2\beta _{1}q^{5}-6\beta _{2}q^{11}+4\beta _{3}q^{13}+(-\beta _{1}+\cdots)q^{17}+\cdots$$
3528.2.s.bk $$4$$ $$28.171$$ $$\Q(\sqrt{-3}, \sqrt{-19})$$ None $$0$$ $$0$$ $$1$$ $$0$$ $$q+\beta _{3}q^{5}+(-\beta _{2}+\beta _{3})q^{11}+(-3+\beta _{2}+\cdots)q^{13}+\cdots$$
3528.2.s.bl $$4$$ $$28.171$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+(-\beta _{1}-2\beta _{2}-\beta _{3})q^{5}+(-2+2\beta _{1}+\cdots)q^{11}+\cdots$$
3528.2.s.bm $$4$$ $$28.171$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+(-\beta _{1}-2\beta _{2}-\beta _{3})q^{5}+(2+2\beta _{1}+\cdots)q^{11}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(3528, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(3528, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 16}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(28, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(49, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(56, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(84, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(98, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(126, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(147, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(168, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(196, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(252, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(294, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(392, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(441, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(504, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(588, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(882, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1176, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1764, [\chi])$$$$^{\oplus 2}$$