# Properties

 Label 98.4.a Level $98$ Weight $4$ Character orbit 98.a Rep. character $\chi_{98}(1,\cdot)$ Character field $\Q$ Dimension $10$ Newform subspaces $8$ Sturm bound $56$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$98 = 2 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 98.a (trivial) Character field: $$\Q$$ Newform subspaces: $$8$$ Sturm bound: $$56$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_0(98))$$.

Total New Old
Modular forms 50 10 40
Cusp forms 34 10 24
Eisenstein series 16 0 16

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.
$$+$$$$+$$$$+$$$$3$$
$$+$$$$-$$$$-$$$$2$$
$$-$$$$+$$$$-$$$$1$$
$$-$$$$-$$$$+$$$$4$$
Plus space$$+$$$$7$$
Minus space$$-$$$$3$$

## Trace form

 $$10q - 6q^{3} + 40q^{4} + 26q^{5} + 20q^{6} + 126q^{9} + O(q^{10})$$ $$10q - 6q^{3} + 40q^{4} + 26q^{5} + 20q^{6} + 126q^{9} - 4q^{10} - 12q^{11} - 24q^{12} - 74q^{13} + 92q^{15} + 160q^{16} + 40q^{17} + 128q^{18} - 82q^{19} + 104q^{20} - 80q^{22} + 268q^{23} + 80q^{24} + 310q^{25} - 76q^{26} - 180q^{27} - 328q^{29} - 432q^{30} - 308q^{31} + 320q^{33} + 376q^{34} + 504q^{36} - 916q^{37} + 156q^{38} - 1352q^{39} - 16q^{40} - 288q^{41} + 288q^{43} - 48q^{44} + 242q^{45} - 80q^{46} - 12q^{47} - 96q^{48} - 1192q^{50} - 884q^{51} - 296q^{52} - 1152q^{53} - 40q^{54} + 184q^{55} + 156q^{57} + 24q^{58} + 62q^{59} + 368q^{60} - 182q^{61} - 328q^{62} + 640q^{64} + 476q^{65} - 256q^{66} + 3044q^{67} + 160q^{68} + 656q^{69} - 704q^{71} + 512q^{72} + 612q^{73} - 232q^{74} - 530q^{75} - 328q^{76} + 128q^{78} + 2028q^{79} + 416q^{80} + 1066q^{81} + 72q^{82} + 694q^{83} + 1240q^{85} + 416q^{86} - 1628q^{87} - 320q^{88} - 420q^{89} - 1588q^{90} + 1072q^{92} + 4916q^{93} + 72q^{94} + 916q^{95} + 320q^{96} - 24q^{97} - 1936q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_0(98))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2 7
98.4.a.a $$1$$ $$5.782$$ $$\Q$$ None $$-2$$ $$-8$$ $$14$$ $$0$$ $$+$$ $$-$$ $$q-2q^{2}-8q^{3}+4q^{4}+14q^{5}+2^{4}q^{6}+\cdots$$
98.4.a.b $$1$$ $$5.782$$ $$\Q$$ None $$-2$$ $$-1$$ $$7$$ $$0$$ $$+$$ $$+$$ $$q-2q^{2}-q^{3}+4q^{4}+7q^{5}+2q^{6}+\cdots$$
98.4.a.c $$1$$ $$5.782$$ $$\Q$$ None $$-2$$ $$1$$ $$-7$$ $$0$$ $$+$$ $$-$$ $$q-2q^{2}+q^{3}+4q^{4}-7q^{5}-2q^{6}+\cdots$$
98.4.a.d $$1$$ $$5.782$$ $$\Q$$ None $$2$$ $$-5$$ $$-9$$ $$0$$ $$-$$ $$+$$ $$q+2q^{2}-5q^{3}+4q^{4}-9q^{5}-10q^{6}+\cdots$$
98.4.a.e $$1$$ $$5.782$$ $$\Q$$ None $$2$$ $$2$$ $$12$$ $$0$$ $$-$$ $$-$$ $$q+2q^{2}+2q^{3}+4q^{4}+12q^{5}+4q^{6}+\cdots$$
98.4.a.f $$1$$ $$5.782$$ $$\Q$$ None $$2$$ $$5$$ $$9$$ $$0$$ $$-$$ $$-$$ $$q+2q^{2}+5q^{3}+4q^{4}+9q^{5}+10q^{6}+\cdots$$
98.4.a.g $$2$$ $$5.782$$ $$\Q(\sqrt{2})$$ None $$-4$$ $$0$$ $$0$$ $$0$$ $$+$$ $$+$$ $$q-2q^{2}+5\beta q^{3}+4q^{4}+14\beta q^{5}-10\beta q^{6}+\cdots$$
98.4.a.h $$2$$ $$5.782$$ $$\Q(\sqrt{22})$$ None $$4$$ $$0$$ $$0$$ $$0$$ $$-$$ $$-$$ $$q+2q^{2}+\beta q^{3}+4q^{4}-\beta q^{5}+2\beta q^{6}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_0(98))$$ into lower level spaces

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