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

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

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

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 11
99.6.a.a \(1\) \(15.878\) \(\Q\) None \(-1\) \(0\) \(92\) \(-26\) \(-\) \(+\) \(q-q^{2}-31q^{4}+92q^{5}-26q^{7}+63q^{8}+\cdots\)
99.6.a.b \(1\) \(15.878\) \(\Q\) None \(2\) \(0\) \(-46\) \(148\) \(-\) \(+\) \(q+2q^{2}-28q^{4}-46q^{5}+148q^{7}+\cdots\)
99.6.a.c \(1\) \(15.878\) \(\Q\) None \(4\) \(0\) \(19\) \(10\) \(-\) \(-\) \(q+4q^{2}-2^{4}q^{4}+19q^{5}+10q^{7}-192q^{8}+\cdots\)
99.6.a.d \(2\) \(15.878\) \(\Q(\sqrt{33}) \) None \(-13\) \(0\) \(-58\) \(146\) \(-\) \(-\) \(q+(-6-\beta )q^{2}+(12+13\beta )q^{4}+(-34+\cdots)q^{5}+\cdots\)
99.6.a.e \(2\) \(15.878\) \(\Q(\sqrt{313}) \) None \(-1\) \(0\) \(38\) \(-18\) \(-\) \(+\) \(q-\beta q^{2}+(46+\beta )q^{4}+(24-10\beta )q^{5}+\cdots\)
99.6.a.f \(2\) \(15.878\) \(\Q(\sqrt{177}) \) None \(5\) \(0\) \(-58\) \(-286\) \(-\) \(-\) \(q+(3-\beta )q^{2}+(21-5\beta )q^{4}+(-34+10\beta )q^{5}+\cdots\)
99.6.a.g \(3\) \(15.878\) 3.3.54492.1 None \(0\) \(0\) \(-24\) \(84\) \(-\) \(+\) \(q-\beta _{2}q^{2}+(28-2\beta _{1}-4\beta _{2})q^{4}+(-8+\cdots)q^{5}+\cdots\)
99.6.a.h \(5\) \(15.878\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-4\) \(0\) \(-100\) \(-18\) \(+\) \(+\) \(q+(-1+\beta _{1})q^{2}+(21-3\beta _{1}+\beta _{4})q^{4}+\cdots\)
99.6.a.i \(5\) \(15.878\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(4\) \(0\) \(100\) \(-18\) \(+\) \(-\) \(q+(1-\beta _{1})q^{2}+(21-3\beta _{1}+\beta _{4})q^{4}+\cdots\)

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

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