Properties

 Label 98.2.a Level 98 Weight 2 Character orbit a Rep. character $$\chi_{98}(1,\cdot)$$ Character field $$\Q$$ Dimension 3 Newform subspaces 2 Sturm bound 28 Trace bound 1

Related objects

Defining parameters

 Level: $$N$$ $$=$$ $$98 = 2 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 98.a (trivial) Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$28$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$3$$

Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(98))$$.

Total New Old
Modular forms 22 3 19
Cusp forms 7 3 4
Eisenstein series 15 0 15

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.
$$+$$$$-$$$$-$$$$1$$
$$-$$$$+$$$$-$$$$2$$
Plus space$$+$$$$0$$
Minus space$$-$$$$3$$

Trace form

 $$3q + q^{2} + 2q^{3} + 3q^{4} - 2q^{6} + q^{8} - q^{9} + O(q^{10})$$ $$3q + q^{2} + 2q^{3} + 3q^{4} - 2q^{6} + q^{8} - q^{9} - 4q^{11} + 2q^{12} + 4q^{13} - 8q^{15} + 3q^{16} - 6q^{17} - 3q^{18} - 2q^{19} - 4q^{22} - 8q^{23} - 2q^{24} + q^{25} - 4q^{26} - 4q^{27} - 2q^{29} - 8q^{30} + 4q^{31} + q^{32} + 6q^{34} - q^{36} + 22q^{37} + 2q^{38} + 8q^{39} - 6q^{41} + 12q^{43} - 4q^{44} - 8q^{46} + 12q^{47} + 2q^{48} + 11q^{50} - 8q^{51} + 4q^{52} + 2q^{53} + 4q^{54} + 16q^{57} + 10q^{58} + 6q^{59} - 8q^{60} - 8q^{61} - 4q^{62} + 3q^{64} + 20q^{67} - 6q^{68} - 24q^{71} - 3q^{72} - 2q^{73} + 18q^{74} - 10q^{75} - 2q^{76} - 8q^{78} - 21q^{81} + 6q^{82} + 6q^{83} - 8q^{85} - 4q^{86} - 12q^{87} - 4q^{88} + 6q^{89} - 8q^{92} - 16q^{93} - 12q^{94} - 40q^{95} - 2q^{96} + 10q^{97} + 4q^{99} + O(q^{100})$$

Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(98))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2 7
98.2.a.a $$1$$ $$0.783$$ $$\Q$$ None $$-1$$ $$2$$ $$0$$ $$0$$ $$+$$ $$-$$ $$q-q^{2}+2q^{3}+q^{4}-2q^{6}-q^{8}+q^{9}+\cdots$$
98.2.a.b $$2$$ $$0.783$$ $$\Q(\sqrt{2})$$ None $$2$$ $$0$$ $$0$$ $$0$$ $$-$$ $$+$$ $$q+q^{2}+\beta q^{3}+q^{4}-2\beta q^{5}+\beta q^{6}+\cdots$$

Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(98))$$ into lower level spaces

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + T$$)($$( 1 - T )^{2}$$)
$3$ ($$1 - 2 T + 3 T^{2}$$)($$1 + 4 T^{2} + 9 T^{4}$$)
$5$ ($$1 + 5 T^{2}$$)($$1 + 2 T^{2} + 25 T^{4}$$)
$7$ 1
$11$ ($$1 + 11 T^{2}$$)($$( 1 + 2 T + 11 T^{2} )^{2}$$)
$13$ ($$1 - 4 T + 13 T^{2}$$)($$( 1 + 13 T^{2} )^{2}$$)
$17$ ($$1 + 6 T + 17 T^{2}$$)($$1 + 32 T^{2} + 289 T^{4}$$)
$19$ ($$1 + 2 T + 19 T^{2}$$)($$1 - 12 T^{2} + 361 T^{4}$$)
$23$ ($$1 + 23 T^{2}$$)($$( 1 + 4 T + 23 T^{2} )^{2}$$)
$29$ ($$1 + 6 T + 29 T^{2}$$)($$( 1 - 2 T + 29 T^{2} )^{2}$$)
$31$ ($$1 - 4 T + 31 T^{2}$$)($$1 - 10 T^{2} + 961 T^{4}$$)
$37$ ($$1 - 2 T + 37 T^{2}$$)($$( 1 - 10 T + 37 T^{2} )^{2}$$)
$41$ ($$1 + 6 T + 41 T^{2}$$)($$1 - 16 T^{2} + 1681 T^{4}$$)
$43$ ($$1 - 8 T + 43 T^{2}$$)($$( 1 - 2 T + 43 T^{2} )^{2}$$)
$47$ ($$1 - 12 T + 47 T^{2}$$)($$1 + 86 T^{2} + 2209 T^{4}$$)
$53$ ($$1 - 6 T + 53 T^{2}$$)($$( 1 + 2 T + 53 T^{2} )^{2}$$)
$59$ ($$1 - 6 T + 59 T^{2}$$)($$1 + 116 T^{2} + 3481 T^{4}$$)
$61$ ($$1 + 8 T + 61 T^{2}$$)($$1 + 114 T^{2} + 3721 T^{4}$$)
$67$ ($$1 + 4 T + 67 T^{2}$$)($$( 1 - 12 T + 67 T^{2} )^{2}$$)
$71$ ($$1 + 71 T^{2}$$)($$( 1 + 12 T + 71 T^{2} )^{2}$$)
$73$ ($$1 + 2 T + 73 T^{2}$$)($$1 + 144 T^{2} + 5329 T^{4}$$)
$79$ ($$1 - 8 T + 79 T^{2}$$)($$( 1 + 4 T + 79 T^{2} )^{2}$$)
$83$ ($$1 - 6 T + 83 T^{2}$$)($$1 + 68 T^{2} + 6889 T^{4}$$)
$89$ ($$1 - 6 T + 89 T^{2}$$)($$1 + 128 T^{2} + 7921 T^{4}$$)
$97$ ($$1 - 10 T + 97 T^{2}$$)($$1 + 96 T^{2} + 9409 T^{4}$$)