Properties

Label 99.8.a
Level $99$
Weight $8$
Character orbit 99.a
Rep. character $\chi_{99}(1,\cdot)$
Character field $\Q$
Dimension $28$
Newform subspaces $9$
Sturm bound $96$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 99.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(96\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(99))\).

Total New Old
Modular forms 88 28 60
Cusp forms 80 28 52
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
\(+\)\(+\)$+$\(5\)
\(+\)\(-\)$-$\(5\)
\(-\)\(+\)$-$\(8\)
\(-\)\(-\)$+$\(10\)
Plus space\(+\)\(15\)
Minus space\(-\)\(13\)

Trace form

\( 28 q - 8 q^{2} + 1548 q^{4} + 709 q^{5} + 726 q^{7} - 5136 q^{8} + O(q^{10}) \) \( 28 q - 8 q^{2} + 1548 q^{4} + 709 q^{5} + 726 q^{7} - 5136 q^{8} + 16094 q^{10} + 2662 q^{11} - 22746 q^{13} - 15488 q^{14} + 100128 q^{16} - 8612 q^{17} - 54572 q^{19} - 20932 q^{20} - 10648 q^{22} + 36389 q^{23} + 345405 q^{25} + 86344 q^{26} + 264848 q^{28} + 29046 q^{29} + 315027 q^{31} + 2684 q^{32} - 1494088 q^{34} + 412642 q^{35} + 644689 q^{37} + 1598760 q^{38} + 3183108 q^{40} - 1037942 q^{41} + 152158 q^{43} - 90508 q^{44} - 3384002 q^{46} - 45472 q^{47} + 4649256 q^{49} + 602990 q^{50} - 9658432 q^{52} + 1566840 q^{53} - 1340317 q^{55} - 3257244 q^{56} + 7707468 q^{58} + 2485763 q^{59} + 3325822 q^{61} + 2103154 q^{62} + 8259328 q^{64} - 8495000 q^{65} - 5021815 q^{67} - 9285424 q^{68} + 9890372 q^{70} + 3487647 q^{71} + 4270218 q^{73} + 12788286 q^{74} - 13353128 q^{76} + 1791526 q^{77} + 6218774 q^{79} + 14362316 q^{80} + 8293388 q^{82} - 9592158 q^{83} - 18304150 q^{85} + 39709116 q^{86} - 5286732 q^{88} - 10046157 q^{89} - 25495688 q^{91} - 21454700 q^{92} - 24543800 q^{94} - 21225024 q^{95} + 20369963 q^{97} - 49309200 q^{98} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(99))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 11
99.8.a.a 99.a 1.a $1$ $30.926$ \(\Q\) None \(-10\) \(0\) \(410\) \(-1028\) $-$ $-$ $\mathrm{SU}(2)$ \(q-10q^{2}-28q^{4}+410q^{5}-1028q^{7}+\cdots\)
99.8.a.b 99.a 1.a $2$ $30.926$ \(\Q(\sqrt{97}) \) None \(-1\) \(0\) \(194\) \(-418\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(90+\beta )q^{4}+(90+14\beta )q^{5}+\cdots\)
99.8.a.c 99.a 1.a $2$ $30.926$ \(\Q(\sqrt{15}) \) None \(8\) \(0\) \(470\) \(-1228\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(4+\beta )q^{2}+(-52+8\beta )q^{4}+(235+\cdots)q^{5}+\cdots\)
99.8.a.d 99.a 1.a $2$ $30.926$ \(\Q(\sqrt{177}) \) None \(19\) \(0\) \(34\) \(-166\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(10-\beta )q^{2}+(2^{4}-19\beta )q^{4}+(6^{2}-38\beta )q^{5}+\cdots\)
99.8.a.e 99.a 1.a $3$ $30.926$ 3.3.115512.1 None \(-9\) \(0\) \(444\) \(1614\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-3-\beta _{1})q^{2}+(-5+13\beta _{1}-\beta _{2})q^{4}+\cdots\)
99.8.a.f 99.a 1.a $4$ $30.926$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-15\) \(0\) \(-306\) \(890\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-4+\beta _{1})q^{2}+(142-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
99.8.a.g 99.a 1.a $4$ $30.926$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(0\) \(-537\) \(170\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(151+\beta _{1}+\beta _{2}+\beta _{3})q^{4}+\cdots\)
99.8.a.h 99.a 1.a $5$ $30.926$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-8\) \(0\) \(-500\) \(446\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-2-\beta _{1})q^{2}+(35+4\beta _{1}-\beta _{3})q^{4}+\cdots\)
99.8.a.i 99.a 1.a $5$ $30.926$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(8\) \(0\) \(500\) \(446\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(2+\beta _{1})q^{2}+(35+4\beta _{1}-\beta _{3})q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(99))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(99)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 2}\)