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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(722))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 19
722.2.a.a \(1\) \(5.765\) \(\Q\) None \(-1\) \(-3\) \(2\) \(-3\) \(+\) \(+\) \(q-q^{2}-3q^{3}+q^{4}+2q^{5}+3q^{6}-3q^{7}+\cdots\)
722.2.a.b \(1\) \(5.765\) \(\Q\) None \(-1\) \(1\) \(-4\) \(3\) \(+\) \(-\) \(q-q^{2}+q^{3}+q^{4}-4q^{5}-q^{6}+3q^{7}+\cdots\)
722.2.a.c \(1\) \(5.765\) \(\Q\) None \(-1\) \(1\) \(0\) \(-4\) \(+\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-4q^{7}-q^{8}+\cdots\)
722.2.a.d \(1\) \(5.765\) \(\Q\) None \(1\) \(-1\) \(0\) \(-4\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}-4q^{7}+q^{8}+\cdots\)
722.2.a.e \(1\) \(5.765\) \(\Q\) None \(1\) \(-1\) \(0\) \(-1\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}-q^{7}+q^{8}+\cdots\)
722.2.a.f \(1\) \(5.765\) \(\Q\) None \(1\) \(3\) \(2\) \(-3\) \(-\) \(+\) \(q+q^{2}+3q^{3}+q^{4}+2q^{5}+3q^{6}-3q^{7}+\cdots\)
722.2.a.g \(2\) \(5.765\) \(\Q(\sqrt{7}) \) None \(-2\) \(0\) \(2\) \(2\) \(+\) \(-\) \(q-q^{2}+\beta q^{3}+q^{4}+(1+\beta )q^{5}-\beta q^{6}+\cdots\)
722.2.a.h \(2\) \(5.765\) \(\Q(\sqrt{5}) \) None \(-2\) \(2\) \(-5\) \(-2\) \(+\) \(-\) \(q-q^{2}+2\beta q^{3}+q^{4}+(-3+\beta )q^{5}+\cdots\)
722.2.a.i \(2\) \(5.765\) \(\Q(\sqrt{5}) \) None \(2\) \(-2\) \(-5\) \(-2\) \(-\) \(-\) \(q+q^{2}-2\beta q^{3}+q^{4}+(-3+\beta )q^{5}+\cdots\)
722.2.a.j \(2\) \(5.765\) \(\Q(\sqrt{7}) \) None \(2\) \(0\) \(2\) \(2\) \(-\) \(+\) \(q+q^{2}+\beta q^{3}+q^{4}+(1-\beta )q^{5}+\beta q^{6}+\cdots\)
722.2.a.k \(3\) \(5.765\) \(\Q(\zeta_{18})^+\) None \(-3\) \(0\) \(6\) \(6\) \(+\) \(-\) \(q-q^{2}+\beta _{1}q^{3}+q^{4}+2q^{5}-\beta _{1}q^{6}+\cdots\)
722.2.a.l \(3\) \(5.765\) \(\Q(\zeta_{18})^+\) None \(3\) \(0\) \(6\) \(6\) \(-\) \(+\) \(q+q^{2}-\beta _{1}q^{3}+q^{4}+2q^{5}-\beta _{1}q^{6}+\cdots\)
722.2.a.m \(4\) \(5.765\) \(\Q(\zeta_{20})^+\) None \(-4\) \(-2\) \(-2\) \(-2\) \(+\) \(+\) \(q-q^{2}+(-1-\beta _{2}-\beta _{3})q^{3}+q^{4}+(-1+\cdots)q^{5}+\cdots\)
722.2.a.n \(4\) \(5.765\) \(\Q(\zeta_{20})^+\) None \(4\) \(2\) \(-2\) \(-2\) \(-\) \(+\) \(q+q^{2}+(1+\beta _{2}+\beta _{3})q^{3}+q^{4}+(-1+\cdots)q^{5}+\cdots\)

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

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