Properties

Label 99.6.a
Level $99$
Weight $6$
Character orbit 99.a
Rep. character $\chi_{99}(1,\cdot)$
Character field $\Q$
Dimension $22$
Newform subspaces $9$
Sturm bound $72$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 99.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(72\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(99))\).

Total New Old
Modular forms 64 22 42
Cusp forms 56 22 34
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
\(+\)\(+\)\(+\)\(5\)
\(+\)\(-\)\(-\)\(5\)
\(-\)\(+\)\(-\)\(7\)
\(-\)\(-\)\(+\)\(5\)
Plus space\(+\)\(10\)
Minus space\(-\)\(12\)

Trace form

\( 22 q - 4 q^{2} + 380 q^{4} - 37 q^{5} + 22 q^{7} + 480 q^{8} + O(q^{10}) \) \( 22 q - 4 q^{2} + 380 q^{4} - 37 q^{5} + 22 q^{7} + 480 q^{8} - 122 q^{10} - 242 q^{11} - 890 q^{13} - 448 q^{14} + 5696 q^{16} + 2840 q^{17} - 1404 q^{19} - 2612 q^{20} + 484 q^{22} + 1837 q^{23} + 13745 q^{25} - 2416 q^{26} - 7456 q^{28} + 15030 q^{29} + 10579 q^{31} + 3916 q^{32} + 1168 q^{34} + 14426 q^{35} + 927 q^{37} - 600 q^{38} - 4668 q^{40} - 43654 q^{41} - 43962 q^{43} - 7260 q^{44} - 46786 q^{46} - 58088 q^{47} + 62682 q^{49} + 85498 q^{50} + 68432 q^{52} - 3948 q^{53} + 5203 q^{55} + 34884 q^{56} + 73908 q^{58} - 23081 q^{59} - 140658 q^{61} - 55390 q^{62} + 115552 q^{64} + 114848 q^{65} - 72211 q^{67} + 135424 q^{68} - 22940 q^{70} - 105369 q^{71} - 31382 q^{73} - 128346 q^{74} - 261528 q^{76} - 38478 q^{77} - 143962 q^{79} - 182708 q^{80} + 104260 q^{82} + 195066 q^{83} + 224758 q^{85} - 479460 q^{86} + 137940 q^{88} + 49041 q^{89} + 80440 q^{91} + 185732 q^{92} - 555160 q^{94} + 249072 q^{95} + 381665 q^{97} + 157812 q^{98} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(99))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 11
99.6.a.a 99.a 1.a $1$ $15.878$ \(\Q\) None 33.6.a.b \(-1\) \(0\) \(92\) \(-26\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-31q^{4}+92q^{5}-26q^{7}+63q^{8}+\cdots\)
99.6.a.b 99.a 1.a $1$ $15.878$ \(\Q\) None 33.6.a.a \(2\) \(0\) \(-46\) \(148\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-28q^{4}-46q^{5}+148q^{7}+\cdots\)
99.6.a.c 99.a 1.a $1$ $15.878$ \(\Q\) None 11.6.a.a \(4\) \(0\) \(19\) \(10\) $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}-2^{4}q^{4}+19q^{5}+10q^{7}-192q^{8}+\cdots\)
99.6.a.d 99.a 1.a $2$ $15.878$ \(\Q(\sqrt{33}) \) None 33.6.a.e \(-13\) \(0\) \(-58\) \(146\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-6-\beta )q^{2}+(12+13\beta )q^{4}+(-34+\cdots)q^{5}+\cdots\)
99.6.a.e 99.a 1.a $2$ $15.878$ \(\Q(\sqrt{313}) \) None 33.6.a.d \(-1\) \(0\) \(38\) \(-18\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(46+\beta )q^{4}+(24-10\beta )q^{5}+\cdots\)
99.6.a.f 99.a 1.a $2$ $15.878$ \(\Q(\sqrt{177}) \) None 33.6.a.c \(5\) \(0\) \(-58\) \(-286\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(3-\beta )q^{2}+(21-5\beta )q^{4}+(-34+10\beta )q^{5}+\cdots\)
99.6.a.g 99.a 1.a $3$ $15.878$ 3.3.54492.1 None 11.6.a.b \(0\) \(0\) \(-24\) \(84\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{2}+(28-2\beta _{1}-4\beta _{2})q^{4}+(-8+\cdots)q^{5}+\cdots\)
99.6.a.h 99.a 1.a $5$ $15.878$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 99.6.a.h \(-4\) \(0\) \(-100\) \(-18\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(21-3\beta _{1}+\beta _{4})q^{4}+\cdots\)
99.6.a.i 99.a 1.a $5$ $15.878$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 99.6.a.h \(4\) \(0\) \(100\) \(-18\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(21-3\beta _{1}+\beta _{4})q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(99))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(99)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 2}\)