Properties

Label 99.4.a
Level $99$
Weight $4$
Character orbit 99.a
Rep. character $\chi_{99}(1,\cdot)$
Character field $\Q$
Dimension $12$
Newform subspaces $7$
Sturm bound $48$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 99.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(48\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(99))\).

Total New Old
Modular forms 40 12 28
Cusp forms 32 12 20
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
\(+\)\(+\)$+$\(2\)
\(+\)\(-\)$-$\(2\)
\(-\)\(+\)$-$\(3\)
\(-\)\(-\)$+$\(5\)
Plus space\(+\)\(7\)
Minus space\(-\)\(5\)

Trace form

\( 12 q + 2 q^{2} + 56 q^{4} + 14 q^{5} - 44 q^{7} - 24 q^{8} + O(q^{10}) \) \( 12 q + 2 q^{2} + 56 q^{4} + 14 q^{5} - 44 q^{7} - 24 q^{8} - 46 q^{10} + 22 q^{11} + 124 q^{13} + 32 q^{14} + 80 q^{16} - 28 q^{17} + 8 q^{19} + 508 q^{20} - 22 q^{22} - 146 q^{23} + 230 q^{25} - 388 q^{26} - 24 q^{29} - 770 q^{31} - 260 q^{32} - 220 q^{34} - 748 q^{35} + 2 q^{37} + 264 q^{38} + 228 q^{40} + 224 q^{41} + 452 q^{43} + 528 q^{44} - 2 q^{46} + 916 q^{47} + 468 q^{49} - 380 q^{50} - 552 q^{52} + 1188 q^{53} - 22 q^{55} - 348 q^{56} - 1824 q^{58} - 674 q^{59} - 532 q^{61} - 3166 q^{62} + 768 q^{64} + 1760 q^{65} + 306 q^{67} - 1640 q^{68} + 212 q^{70} - 78 q^{71} + 1180 q^{73} - 2262 q^{74} - 760 q^{76} + 396 q^{77} + 1020 q^{79} + 7276 q^{80} - 1432 q^{82} - 732 q^{83} + 2180 q^{85} + 444 q^{86} - 660 q^{88} + 2022 q^{89} - 960 q^{91} - 4228 q^{92} + 3064 q^{94} - 2784 q^{95} + 150 q^{97} + 9450 q^{98} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(99))\) 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
99.4.a.a 99.a 1.a $1$ $5.841$ \(\Q\) None 33.4.a.b \(1\) \(0\) \(4\) \(-26\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-7q^{4}+4q^{5}-26q^{7}-15q^{8}+\cdots\)
99.4.a.b 99.a 1.a $1$ $5.841$ \(\Q\) None 33.4.a.a \(5\) \(0\) \(14\) \(-32\) $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{2}+17q^{4}+14q^{5}-2^{5}q^{7}+45q^{8}+\cdots\)
99.4.a.c 99.a 1.a $2$ $5.841$ \(\Q(\sqrt{3}) \) None 11.4.a.a \(-2\) \(0\) \(-2\) \(20\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+(-4-2\beta )q^{4}+(-1+\cdots)q^{5}+\cdots\)
99.4.a.d 99.a 1.a $2$ $5.841$ \(\Q(\sqrt{3}) \) None 99.4.a.d \(-2\) \(0\) \(-20\) \(-16\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+(5-2\beta )q^{4}+(-10+\cdots)q^{5}+\cdots\)
99.4.a.e 99.a 1.a $2$ $5.841$ \(\Q(\sqrt{33}) \) None 33.4.a.d \(-1\) \(0\) \(-16\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+\beta q^{4}+(-10+4\beta )q^{5}+(2+\cdots)q^{7}+\cdots\)
99.4.a.f 99.a 1.a $2$ $5.841$ \(\Q(\sqrt{97}) \) None 33.4.a.c \(-1\) \(0\) \(14\) \(24\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(2^{4}+\beta )q^{4}+(6+2\beta )q^{5}+(14+\cdots)q^{7}+\cdots\)
99.4.a.g 99.a 1.a $2$ $5.841$ \(\Q(\sqrt{3}) \) None 99.4.a.d \(2\) \(0\) \(20\) \(-16\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(5+2\beta )q^{4}+(10-\beta )q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(99))\) into lower level spaces

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