# Properties

 Label 99.14.a Level $99$ Weight $14$ Character orbit 99.a Rep. character $\chi_{99}(1,\cdot)$ Character field $\Q$ Dimension $54$ Newform subspaces $9$ Sturm bound $168$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$14$$ Character orbit: $$[\chi]$$ $$=$$ 99.a (trivial) Character field: $$\Q$$ Newform subspaces: $$9$$ Sturm bound: $$168$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{14}(\Gamma_0(99))$$.

Total New Old
Modular forms 160 54 106
Cusp forms 152 54 98
Eisenstein series 8 0 8

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

$$3$$$$11$$FrickeDim
$$+$$$$+$$$+$$$11$$
$$+$$$$-$$$-$$$11$$
$$-$$$$+$$$-$$$17$$
$$-$$$$-$$$+$$$15$$
Plus space$$+$$$$26$$
Minus space$$-$$$$28$$

## Trace form

 $$54 q - 64 q^{2} + 233836 q^{4} + 31493 q^{5} + 27662 q^{7} - 2891040 q^{8} + O(q^{10})$$ $$54 q - 64 q^{2} + 233836 q^{4} + 31493 q^{5} + 27662 q^{7} - 2891040 q^{8} - 1451762 q^{10} - 3543122 q^{11} + 48416210 q^{13} - 190402048 q^{14} + 974001472 q^{16} - 198625480 q^{17} - 97895564 q^{19} + 1020291148 q^{20} + 113379904 q^{22} + 833603017 q^{23} + 11500936855 q^{25} - 2959607752 q^{26} - 1497416736 q^{28} - 3399284550 q^{29} + 12671006423 q^{31} - 7037140724 q^{32} + 22551921544 q^{34} - 35626107574 q^{35} - 24571209503 q^{37} + 2709892200 q^{38} - 66334813308 q^{40} + 102937615454 q^{41} - 37121151722 q^{43} + 15114958452 q^{44} + 33350176094 q^{46} - 160818966488 q^{47} + 455020708866 q^{49} + 520995307318 q^{50} - 144092206608 q^{52} - 349803538068 q^{53} + 94146066223 q^{55} - 1564885448508 q^{56} - 335109891492 q^{58} - 1537442962265 q^{59} + 478016438650 q^{61} + 971991721250 q^{62} + 3428736725472 q^{64} + 2226427424768 q^{65} + 431605835949 q^{67} + 211655713504 q^{68} + 1443118699780 q^{70} - 715775792109 q^{71} + 1638877709558 q^{73} + 7352976110238 q^{74} - 1332893995544 q^{76} - 1269603363138 q^{77} + 9203307586350 q^{79} + 4475257827532 q^{80} + 12444973360540 q^{82} + 11362431667386 q^{83} - 19352827082822 q^{85} - 14328658016676 q^{86} + 3362599934100 q^{88} + 11996135192739 q^{89} - 7696788096936 q^{91} - 13855046920348 q^{92} - 52927361134072 q^{94} + 44891626422192 q^{95} + 8899483836075 q^{97} - 31985147963688 q^{98} + O(q^{100})$$

## Decomposition of $$S_{14}^{\mathrm{new}}(\Gamma_0(99))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 11
99.14.a.a $1$ $106.159$ $$\Q$$ None $$-140$$ $$0$$ $$-48740$$ $$487486$$ $-$ $-$ $$q-140q^{2}+11408q^{4}-48740q^{5}+\cdots$$
99.14.a.b $4$ $106.159$ $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ None $$-41$$ $$0$$ $$38422$$ $$41998$$ $-$ $-$ $$q+(-10-\beta _{1})q^{2}+(5909+43\beta _{1}+5\beta _{3})q^{4}+\cdots$$
99.14.a.c $4$ $106.159$ $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ None $$11$$ $$0$$ $$83432$$ $$69652$$ $-$ $+$ $$q+(3-\beta _{1})q^{2}+(-1564+37\beta _{1}+5\beta _{2}+\cdots)q^{4}+\cdots$$
99.14.a.d $5$ $106.159$ $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ None $$53$$ $$0$$ $$-10318$$ $$-667804$$ $-$ $-$ $$q+(11-\beta _{1})q^{2}+(7020-11\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots$$
99.14.a.e $5$ $106.159$ $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ None $$64$$ $$0$$ $$454$$ $$-313920$$ $-$ $-$ $$q+(13+\beta _{1})q^{2}+(-477+21\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$$
99.14.a.f $6$ $106.159$ $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ None $$-11$$ $$0$$ $$20932$$ $$284152$$ $-$ $+$ $$q+(-2+\beta _{1})q^{2}+(3061-10\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$$
99.14.a.g $7$ $106.159$ $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ None $$0$$ $$0$$ $$-52689$$ $$-89386$$ $-$ $+$ $$q+\beta _{1}q^{2}+(6727+6\beta _{1}+\beta _{2})q^{4}+(-7527+\cdots)q^{5}+\cdots$$
99.14.a.h $11$ $106.159$ $$\mathbb{Q}[x]/(x^{11} - \cdots)$$ None $$-64$$ $$0$$ $$-62500$$ $$107742$$ $+$ $+$ $$q+(-6+\beta _{1})q^{2}+(4859-11\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$$
99.14.a.i $11$ $106.159$ $$\mathbb{Q}[x]/(x^{11} - \cdots)$$ None $$64$$ $$0$$ $$62500$$ $$107742$$ $+$ $-$ $$q+(6-\beta _{1})q^{2}+(4859-11\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{14}^{\mathrm{old}}(\Gamma_0(99))$$ into lower level spaces

$$S_{14}^{\mathrm{old}}(\Gamma_0(99)) \cong$$ $$S_{14}^{\mathrm{new}}(\Gamma_0(3))$$$$^{\oplus 4}$$$$\oplus$$$$S_{14}^{\mathrm{new}}(\Gamma_0(9))$$$$^{\oplus 2}$$$$\oplus$$$$S_{14}^{\mathrm{new}}(\Gamma_0(11))$$$$^{\oplus 3}$$$$\oplus$$$$S_{14}^{\mathrm{new}}(\Gamma_0(33))$$$$^{\oplus 2}$$