# Properties

 Label 196.4.a Level $196$ Weight $4$ Character orbit 196.a Rep. character $\chi_{196}(1,\cdot)$ Character field $\Q$ Dimension $10$ Newform subspaces $7$ Sturm bound $112$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$196 = 2^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 196.a (trivial) Character field: $$\Q$$ Newform subspaces: $$7$$ Sturm bound: $$112$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$3$$, $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_0(196))$$.

Total New Old
Modular forms 96 10 86
Cusp forms 72 10 62
Eisenstein series 24 0 24

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$$2$$$$7$$FrickeDim.
$$-$$$$+$$$$-$$$$4$$
$$-$$$$-$$$$+$$$$6$$
Plus space$$+$$$$6$$
Minus space$$-$$$$4$$

## Trace form

 $$10q + 6q^{3} + 2q^{5} + 30q^{9} + O(q^{10})$$ $$10q + 6q^{3} + 2q^{5} + 30q^{9} + 48q^{11} + 94q^{13} - 40q^{15} + 88q^{17} - 94q^{19} - 248q^{23} + 454q^{25} + 612q^{27} - 88q^{29} - 140q^{31} - 352q^{33} + 260q^{37} + 640q^{39} + 240q^{41} + 240q^{43} + 650q^{45} + 204q^{47} - 416q^{51} - 888q^{53} - 248q^{55} - 612q^{57} - 814q^{59} + 466q^{61} + 764q^{65} - 784q^{67} - 1504q^{69} + 1120q^{71} + 708q^{73} - 254q^{75} - 120q^{79} - 758q^{81} - 1046q^{83} + 160q^{85} - 1604q^{87} + 1788q^{89} + 1148q^{93} - 1016q^{95} - 1800q^{97} - 1816q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_0(196))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2 7
196.4.a.a $$1$$ $$11.564$$ $$\Q$$ None $$0$$ $$-4$$ $$-20$$ $$0$$ $$-$$ $$-$$ $$q-4q^{3}-20q^{5}-11q^{9}+44q^{11}+\cdots$$
196.4.a.b $$1$$ $$11.564$$ $$\Q$$ None $$0$$ $$-4$$ $$-6$$ $$0$$ $$-$$ $$-$$ $$q-4q^{3}-6q^{5}-11q^{9}-12q^{11}+82q^{13}+\cdots$$
196.4.a.c $$1$$ $$11.564$$ $$\Q$$ None $$0$$ $$4$$ $$20$$ $$0$$ $$-$$ $$-$$ $$q+4q^{3}+20q^{5}-11q^{9}+44q^{11}+\cdots$$
196.4.a.d $$1$$ $$11.564$$ $$\Q$$ None $$0$$ $$10$$ $$8$$ $$0$$ $$-$$ $$-$$ $$q+10q^{3}+8q^{5}+73q^{9}-40q^{11}+\cdots$$
196.4.a.e $$2$$ $$11.564$$ $$\Q(\sqrt{37})$$ None $$0$$ $$0$$ $$-14$$ $$0$$ $$-$$ $$+$$ $$q-\beta q^{3}+(-7+2\beta )q^{5}+10q^{9}+(2^{4}+\cdots)q^{11}+\cdots$$
196.4.a.f $$2$$ $$11.564$$ $$\Q(\sqrt{2})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$-$$ $$+$$ $$q+\beta q^{3}-2\beta q^{5}-5^{2}q^{9}-26q^{11}+\cdots$$
196.4.a.g $$2$$ $$11.564$$ $$\Q(\sqrt{37})$$ None $$0$$ $$0$$ $$14$$ $$0$$ $$-$$ $$-$$ $$q-\beta q^{3}+(7+2\beta )q^{5}+10q^{9}+(2^{4}-7\beta )q^{11}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_0(196))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_0(196)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_0(7))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(14))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(28))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(49))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(98))$$$$^{\oplus 2}$$