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Properties

Label 990.2.a
Level $990$
Weight $2$
Character orbit 990.a
Rep. character $\chi_{990}(1,\cdot)$
Character field $\Q$
Dimension $14$
Newform subspaces $13$
Sturm bound $432$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 990.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(432\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(7\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(990))\).

Total New Old
Modular forms 232 14 218
Cusp forms 201 14 187
Eisenstein series 31 0 31

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

\(2\)\(3\)\(5\)\(11\)FrickeDim
\(+\)\(+\)\(+\)\(+\)$+$\(1\)
\(+\)\(+\)\(-\)\(+\)$-$\(1\)
\(+\)\(-\)\(+\)\(+\)$-$\(2\)
\(+\)\(-\)\(+\)\(-\)$+$\(1\)
\(+\)\(-\)\(-\)\(-\)$-$\(2\)
\(-\)\(+\)\(+\)\(-\)$+$\(1\)
\(-\)\(+\)\(-\)\(-\)$-$\(1\)
\(-\)\(-\)\(+\)\(+\)$+$\(1\)
\(-\)\(-\)\(+\)\(-\)$-$\(2\)
\(-\)\(-\)\(-\)\(+\)$-$\(2\)
Plus space\(+\)\(4\)
Minus space\(-\)\(10\)

Trace form

\( 14 q + 14 q^{4} - 2 q^{5} + O(q^{10}) \) \( 14 q + 14 q^{4} - 2 q^{5} - 4 q^{14} + 14 q^{16} + 8 q^{17} + 8 q^{19} - 2 q^{20} + 2 q^{22} + 12 q^{23} + 14 q^{25} + 4 q^{26} + 12 q^{29} + 12 q^{31} - 8 q^{34} + 8 q^{35} - 12 q^{37} + 4 q^{41} - 24 q^{43} + 4 q^{46} + 4 q^{47} + 18 q^{49} - 20 q^{53} - 4 q^{56} - 16 q^{58} + 24 q^{59} + 52 q^{61} - 8 q^{62} + 14 q^{64} + 8 q^{65} - 4 q^{67} + 8 q^{68} + 16 q^{70} + 12 q^{71} + 24 q^{73} - 8 q^{74} + 8 q^{76} - 32 q^{79} - 2 q^{80} - 16 q^{82} + 8 q^{83} + 8 q^{85} + 16 q^{86} + 2 q^{88} + 16 q^{89} + 40 q^{91} + 12 q^{92} - 12 q^{94} + 8 q^{95} - 20 q^{97} + 32 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(990))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5 11
990.2.a.a 990.a 1.a $1$ $7.905$ \(\Q\) None \(-1\) \(0\) \(-1\) \(0\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}-q^{8}+q^{10}-q^{11}+\cdots\)
990.2.a.b 990.a 1.a $1$ $7.905$ \(\Q\) None \(-1\) \(0\) \(-1\) \(0\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}-q^{8}+q^{10}-q^{11}+\cdots\)
990.2.a.c 990.a 1.a $1$ $7.905$ \(\Q\) None \(-1\) \(0\) \(-1\) \(0\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}-q^{8}+q^{10}+q^{11}+\cdots\)
990.2.a.d 990.a 1.a $1$ $7.905$ \(\Q\) None \(-1\) \(0\) \(-1\) \(3\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}+3q^{7}-q^{8}+q^{10}+\cdots\)
990.2.a.e 990.a 1.a $1$ $7.905$ \(\Q\) None \(-1\) \(0\) \(1\) \(-4\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}-4q^{7}-q^{8}-q^{10}+\cdots\)
990.2.a.f 990.a 1.a $1$ $7.905$ \(\Q\) None \(-1\) \(0\) \(1\) \(-1\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}-q^{7}-q^{8}-q^{10}+\cdots\)
990.2.a.g 990.a 1.a $1$ $7.905$ \(\Q\) None \(-1\) \(0\) \(1\) \(4\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}+4q^{7}-q^{8}-q^{10}+\cdots\)
990.2.a.h 990.a 1.a $1$ $7.905$ \(\Q\) None \(1\) \(0\) \(-1\) \(-4\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}-4q^{7}+q^{8}-q^{10}+\cdots\)
990.2.a.i 990.a 1.a $1$ $7.905$ \(\Q\) None \(1\) \(0\) \(-1\) \(-4\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}-4q^{7}+q^{8}-q^{10}+\cdots\)
990.2.a.j 990.a 1.a $1$ $7.905$ \(\Q\) None \(1\) \(0\) \(1\) \(0\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+q^{8}+q^{10}-q^{11}+\cdots\)
990.2.a.k 990.a 1.a $1$ $7.905$ \(\Q\) None \(1\) \(0\) \(1\) \(0\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+q^{8}+q^{10}+q^{11}+\cdots\)
990.2.a.l 990.a 1.a $1$ $7.905$ \(\Q\) None \(1\) \(0\) \(1\) \(5\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+5q^{7}+q^{8}+q^{10}+\cdots\)
990.2.a.m 990.a 1.a $2$ $7.905$ \(\Q(\sqrt{33}) \) None \(2\) \(0\) \(-2\) \(1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}+\beta q^{7}+q^{8}-q^{10}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(990))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(990)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(165))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(198))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(330))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(495))\)\(^{\oplus 2}\)