Properties

Label 722.2.a
Level $722$
Weight $2$
Character orbit 722.a
Rep. character $\chi_{722}(1,\cdot)$
Character field $\Q$
Dimension $28$
Newform subspaces $14$
Sturm bound $190$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 722.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(190\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(5\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(722))\).

Total New Old
Modular forms 115 28 87
Cusp forms 76 28 48
Eisenstein series 39 0 39

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

\(2\)\(19\)FrickeDim
\(+\)\(+\)$+$\(6\)
\(+\)\(-\)$-$\(8\)
\(-\)\(+\)$-$\(10\)
\(-\)\(-\)$+$\(4\)
Plus space\(+\)\(10\)
Minus space\(-\)\(18\)

Trace form

\( 28 q + 28 q^{4} + 2 q^{5} + 2 q^{6} - 4 q^{7} + 34 q^{9} + O(q^{10}) \) \( 28 q + 28 q^{4} + 2 q^{5} + 2 q^{6} - 4 q^{7} + 34 q^{9} + 4 q^{10} + 2 q^{11} - 4 q^{13} - 4 q^{14} - 4 q^{15} + 28 q^{16} - 8 q^{17} + 2 q^{20} + 4 q^{21} - 8 q^{22} - 4 q^{23} + 2 q^{24} + 22 q^{25} + 6 q^{26} - 4 q^{28} - 4 q^{29} - 4 q^{30} + 12 q^{31} + 8 q^{33} + 8 q^{35} + 34 q^{36} - 12 q^{39} + 4 q^{40} + 8 q^{41} - 4 q^{42} - 22 q^{43} + 2 q^{44} - 18 q^{45} + 4 q^{46} - 16 q^{47} + 24 q^{49} - 16 q^{50} - 4 q^{52} + 4 q^{53} - 16 q^{54} - 4 q^{55} - 4 q^{56} + 10 q^{58} - 24 q^{59} - 4 q^{60} + 2 q^{61} + 4 q^{62} - 12 q^{63} + 28 q^{64} - 4 q^{65} - 8 q^{66} - 8 q^{67} - 8 q^{68} - 4 q^{69} + 12 q^{70} + 4 q^{71} - 16 q^{73} - 2 q^{74} + 16 q^{75} - 16 q^{77} + 4 q^{78} + 20 q^{79} + 2 q^{80} + 20 q^{81} + 4 q^{82} + 2 q^{83} + 4 q^{84} + 4 q^{86} - 28 q^{87} - 8 q^{88} + 12 q^{89} - 8 q^{90} + 8 q^{91} - 4 q^{92} - 16 q^{93} - 8 q^{94} + 2 q^{96} + 12 q^{97} - 8 q^{98} - 18 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(722))\) 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 19
722.2.a.a 722.a 1.a $1$ $5.765$ \(\Q\) None \(-1\) \(-3\) \(2\) \(-3\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-3q^{3}+q^{4}+2q^{5}+3q^{6}-3q^{7}+\cdots\)
722.2.a.b 722.a 1.a $1$ $5.765$ \(\Q\) None \(-1\) \(1\) \(-4\) \(3\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-4q^{5}-q^{6}+3q^{7}+\cdots\)
722.2.a.c 722.a 1.a $1$ $5.765$ \(\Q\) None \(-1\) \(1\) \(0\) \(-4\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-4q^{7}-q^{8}+\cdots\)
722.2.a.d 722.a 1.a $1$ $5.765$ \(\Q\) None \(1\) \(-1\) \(0\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}-4q^{7}+q^{8}+\cdots\)
722.2.a.e 722.a 1.a $1$ $5.765$ \(\Q\) None \(1\) \(-1\) \(0\) \(-1\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}-q^{7}+q^{8}+\cdots\)
722.2.a.f 722.a 1.a $1$ $5.765$ \(\Q\) None \(1\) \(3\) \(2\) \(-3\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+3q^{3}+q^{4}+2q^{5}+3q^{6}-3q^{7}+\cdots\)
722.2.a.g 722.a 1.a $2$ $5.765$ \(\Q(\sqrt{7}) \) None \(-2\) \(0\) \(2\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}+q^{4}+(1+\beta )q^{5}-\beta q^{6}+\cdots\)
722.2.a.h 722.a 1.a $2$ $5.765$ \(\Q(\sqrt{5}) \) None \(-2\) \(2\) \(-5\) \(-2\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+2\beta q^{3}+q^{4}+(-3+\beta )q^{5}+\cdots\)
722.2.a.i 722.a 1.a $2$ $5.765$ \(\Q(\sqrt{5}) \) None \(2\) \(-2\) \(-5\) \(-2\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-2\beta q^{3}+q^{4}+(-3+\beta )q^{5}+\cdots\)
722.2.a.j 722.a 1.a $2$ $5.765$ \(\Q(\sqrt{7}) \) None \(2\) \(0\) \(2\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta q^{3}+q^{4}+(1-\beta )q^{5}+\beta q^{6}+\cdots\)
722.2.a.k 722.a 1.a $3$ $5.765$ \(\Q(\zeta_{18})^+\) None \(-3\) \(0\) \(6\) \(6\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}+2q^{5}-\beta _{1}q^{6}+\cdots\)
722.2.a.l 722.a 1.a $3$ $5.765$ \(\Q(\zeta_{18})^+\) None \(3\) \(0\) \(6\) \(6\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}+2q^{5}-\beta _{1}q^{6}+\cdots\)
722.2.a.m 722.a 1.a $4$ $5.765$ \(\Q(\zeta_{20})^+\) None \(-4\) \(-2\) \(-2\) \(-2\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1-\beta _{2}-\beta _{3})q^{3}+q^{4}+(-1+\cdots)q^{5}+\cdots\)
722.2.a.n 722.a 1.a $4$ $5.765$ \(\Q(\zeta_{20})^+\) None \(4\) \(2\) \(-2\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta _{2}+\beta _{3})q^{3}+q^{4}+(-1+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(722))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(722)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(361))\)\(^{\oplus 2}\)