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Properties

Label 994.2.a
Level $994$
Weight $2$
Character orbit 994.a
Rep. character $\chi_{994}(1,\cdot)$
Character field $\Q$
Dimension $35$
Newform subspaces $15$
Sturm bound $288$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 994 = 2 \cdot 7 \cdot 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 994.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(288\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(994))\).

Total New Old
Modular forms 148 35 113
Cusp forms 141 35 106
Eisenstein series 7 0 7

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

\(2\)\(7\)\(71\)FrickeDim
\(+\)\(+\)\(+\)$+$\(5\)
\(+\)\(+\)\(-\)$-$\(3\)
\(+\)\(-\)\(+\)$-$\(4\)
\(+\)\(-\)\(-\)$+$\(4\)
\(-\)\(+\)\(+\)$-$\(7\)
\(-\)\(+\)\(-\)$+$\(3\)
\(-\)\(-\)\(+\)$+$\(2\)
\(-\)\(-\)\(-\)$-$\(7\)
Plus space\(+\)\(14\)
Minus space\(-\)\(21\)

Trace form

\( 35 q + 3 q^{2} + 35 q^{4} - 2 q^{5} - q^{7} + 3 q^{8} + 31 q^{9} + O(q^{10}) \) \( 35 q + 3 q^{2} + 35 q^{4} - 2 q^{5} - q^{7} + 3 q^{8} + 31 q^{9} + 6 q^{10} + 6 q^{13} - q^{14} - 24 q^{15} + 35 q^{16} - 2 q^{17} + 7 q^{18} - 8 q^{19} - 2 q^{20} - 4 q^{21} - 8 q^{23} + 29 q^{25} - 2 q^{26} - q^{28} + 26 q^{29} + 8 q^{30} - 8 q^{31} + 3 q^{32} + 8 q^{33} + 30 q^{34} + 2 q^{35} + 31 q^{36} - 22 q^{37} + 16 q^{38} - 16 q^{39} + 6 q^{40} - 10 q^{41} - 4 q^{42} - 20 q^{43} + 6 q^{45} + 24 q^{47} + 35 q^{49} - 3 q^{50} + 6 q^{52} - 26 q^{53} + 24 q^{54} - 24 q^{55} - q^{56} + 32 q^{57} + 2 q^{58} - 16 q^{59} - 24 q^{60} + 6 q^{61} - 16 q^{62} - 13 q^{63} + 35 q^{64} - 4 q^{65} + 16 q^{66} + 8 q^{67} - 2 q^{68} - 56 q^{69} - 6 q^{70} - q^{71} + 7 q^{72} - 10 q^{73} + 2 q^{74} + 16 q^{75} - 8 q^{76} - 4 q^{77} + 24 q^{78} - 32 q^{79} - 2 q^{80} - 37 q^{81} + 22 q^{82} - 4 q^{84} - 28 q^{85} + 28 q^{86} - 24 q^{87} - 74 q^{89} + 46 q^{90} - 10 q^{91} - 8 q^{92} - 40 q^{93} - 24 q^{94} - 8 q^{95} + 62 q^{97} + 3 q^{98} + 24 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(994))\) 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 7 71
994.2.a.a 994.a 1.a $1$ $7.937$ \(\Q\) None \(-1\) \(-2\) \(2\) \(-1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}+2q^{5}+2q^{6}-q^{7}+\cdots\)
994.2.a.b 994.a 1.a $1$ $7.937$ \(\Q\) None \(-1\) \(1\) \(0\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{7}-q^{8}+\cdots\)
994.2.a.c 994.a 1.a $1$ $7.937$ \(\Q\) None \(-1\) \(1\) \(0\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+q^{7}-q^{8}+\cdots\)
994.2.a.d 994.a 1.a $1$ $7.937$ \(\Q\) None \(-1\) \(2\) \(-2\) \(1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}+q^{4}-2q^{5}-2q^{6}+q^{7}+\cdots\)
994.2.a.e 994.a 1.a $1$ $7.937$ \(\Q\) None \(1\) \(-2\) \(0\) \(1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}+q^{4}-2q^{6}+q^{7}+q^{8}+\cdots\)
994.2.a.f 994.a 1.a $1$ $7.937$ \(\Q\) None \(1\) \(-2\) \(4\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}+q^{4}+4q^{5}-2q^{6}-q^{7}+\cdots\)
994.2.a.g 994.a 1.a $1$ $7.937$ \(\Q\) None \(1\) \(0\) \(-2\) \(1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-2q^{5}+q^{7}+q^{8}-3q^{9}+\cdots\)
994.2.a.h 994.a 1.a $2$ $7.937$ \(\Q(\sqrt{3}) \) None \(-2\) \(2\) \(-2\) \(-2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(1+\beta )q^{3}+q^{4}+(-1+\beta )q^{5}+\cdots\)
994.2.a.i 994.a 1.a $3$ $7.937$ 3.3.564.1 None \(-3\) \(-2\) \(4\) \(3\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)
994.2.a.j 994.a 1.a $3$ $7.937$ 3.3.316.1 None \(-3\) \(-1\) \(-2\) \(3\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+(-1+\beta _{1}+\beta _{2})q^{5}+\cdots\)
994.2.a.k 994.a 1.a $3$ $7.937$ 3.3.148.1 None \(3\) \(-2\) \(-4\) \(-3\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(-1-\beta _{1}+\cdots)q^{5}+\cdots\)
994.2.a.l 994.a 1.a $3$ $7.937$ 3.3.1944.1 None \(3\) \(0\) \(0\) \(-3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}-q^{7}+\cdots\)
994.2.a.m 994.a 1.a $3$ $7.937$ 3.3.316.1 None \(3\) \(6\) \(2\) \(-3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}+(1-\beta _{1})q^{5}+2q^{6}+\cdots\)
994.2.a.n 994.a 1.a $4$ $7.937$ 4.4.23252.1 None \(-4\) \(-1\) \(-4\) \(-4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{3}q^{3}+q^{4}+(-1+\beta _{2})q^{5}+\cdots\)
994.2.a.o 994.a 1.a $7$ $7.937$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(7\) \(0\) \(2\) \(7\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{2}q^{5}-\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(994))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(994)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(71))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(142))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(497))\)\(^{\oplus 2}\)