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Properties

Label 730.2.a
Level $730$
Weight $2$
Character orbit 730.a
Rep. character $\chi_{730}(1,\cdot)$
Character field $\Q$
Dimension $23$
Newform subspaces $15$
Sturm bound $222$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 730 = 2 \cdot 5 \cdot 73 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 730.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(222\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(730))\).

Total New Old
Modular forms 114 23 91
Cusp forms 107 23 84
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.

\(2\)\(5\)\(73\)FrickeDim
\(+\)\(+\)\(+\)$+$\(3\)
\(+\)\(+\)\(-\)$-$\(3\)
\(+\)\(-\)\(+\)$-$\(2\)
\(+\)\(-\)\(-\)$+$\(4\)
\(-\)\(+\)\(+\)$-$\(4\)
\(-\)\(+\)\(-\)$+$\(1\)
\(-\)\(-\)\(+\)$+$\(1\)
\(-\)\(-\)\(-\)$-$\(5\)
Plus space\(+\)\(9\)
Minus space\(-\)\(14\)

Trace form

\( 23 q - q^{2} - 4 q^{3} + 23 q^{4} + q^{5} - 8 q^{7} - q^{8} + 15 q^{9} + O(q^{10}) \) \( 23 q - q^{2} - 4 q^{3} + 23 q^{4} + q^{5} - 8 q^{7} - q^{8} + 15 q^{9} + q^{10} - 4 q^{12} - 2 q^{13} + 8 q^{14} - 4 q^{15} + 23 q^{16} + 6 q^{17} + 3 q^{18} - 16 q^{19} + q^{20} + 16 q^{21} + 16 q^{23} + 23 q^{25} + 14 q^{26} + 8 q^{27} - 8 q^{28} + 14 q^{29} + 4 q^{30} - q^{32} + 24 q^{33} + 6 q^{34} + 4 q^{35} + 15 q^{36} - 22 q^{37} - 4 q^{38} - 24 q^{39} + q^{40} + 14 q^{41} + 8 q^{42} - 16 q^{43} - 3 q^{45} + 16 q^{46} - 4 q^{48} + 15 q^{49} - q^{50} - 32 q^{51} - 2 q^{52} - 2 q^{53} - 4 q^{55} + 8 q^{56} - 32 q^{57} + 14 q^{58} - 32 q^{59} - 4 q^{60} - 34 q^{61} + 24 q^{62} - 48 q^{63} + 23 q^{64} - 10 q^{65} + 8 q^{66} - 36 q^{67} + 6 q^{68} - 8 q^{69} + 4 q^{70} - 16 q^{71} + 3 q^{72} + 3 q^{73} - 18 q^{74} - 4 q^{75} - 16 q^{76} - 8 q^{77} + 24 q^{78} - 24 q^{79} + q^{80} - 33 q^{81} - 2 q^{82} + 32 q^{83} + 16 q^{84} - 10 q^{85} - 32 q^{86} - 16 q^{87} - 2 q^{89} - 3 q^{90} - 56 q^{91} + 16 q^{92} - 48 q^{93} + 40 q^{94} - 4 q^{95} + 14 q^{97} - q^{98} - 16 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(730))\) 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 5 73
730.2.a.a 730.a 1.a $1$ $5.829$ \(\Q\) None \(-1\) \(-3\) \(1\) \(-1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-3q^{3}+q^{4}+q^{5}+3q^{6}-q^{7}+\cdots\)
730.2.a.b 730.a 1.a $1$ $5.829$ \(\Q\) None \(-1\) \(-2\) \(-1\) \(-4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}-q^{5}+2q^{6}-4q^{7}+\cdots\)
730.2.a.c 730.a 1.a $1$ $5.829$ \(\Q\) None \(-1\) \(-1\) \(1\) \(3\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+3q^{7}+\cdots\)
730.2.a.d 730.a 1.a $1$ $5.829$ \(\Q\) None \(-1\) \(0\) \(-1\) \(-2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}-2q^{7}-q^{8}-3q^{9}+\cdots\)
730.2.a.e 730.a 1.a $1$ $5.829$ \(\Q\) None \(-1\) \(0\) \(1\) \(2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}+2q^{7}-q^{8}-3q^{9}+\cdots\)
730.2.a.f 730.a 1.a $1$ $5.829$ \(\Q\) None \(-1\) \(2\) \(1\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}+q^{4}+q^{5}-2q^{6}-q^{8}+\cdots\)
730.2.a.g 730.a 1.a $1$ $5.829$ \(\Q\) None \(-1\) \(3\) \(-1\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+3q^{3}+q^{4}-q^{5}-3q^{6}+q^{7}+\cdots\)
730.2.a.h 730.a 1.a $1$ $5.829$ \(\Q\) None \(1\) \(-2\) \(-1\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}+q^{4}-q^{5}-2q^{6}+4q^{7}+\cdots\)
730.2.a.i 730.a 1.a $1$ $5.829$ \(\Q\) None \(1\) \(-1\) \(-1\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}-q^{7}+\cdots\)
730.2.a.j 730.a 1.a $1$ $5.829$ \(\Q\) None \(1\) \(-1\) \(1\) \(-3\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}-3q^{7}+\cdots\)
730.2.a.k 730.a 1.a $1$ $5.829$ \(\Q\) None \(1\) \(1\) \(1\) \(5\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+5q^{7}+\cdots\)
730.2.a.l 730.a 1.a $2$ $5.829$ \(\Q(\sqrt{13}) \) None \(-2\) \(-1\) \(2\) \(-8\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta q^{3}+q^{4}+q^{5}+\beta q^{6}-4q^{7}+\cdots\)
730.2.a.m 730.a 1.a $3$ $5.829$ 3.3.229.1 None \(-3\) \(0\) \(-3\) \(1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}-\beta _{1}q^{6}+\cdots\)
730.2.a.n 730.a 1.a $3$ $5.829$ 3.3.469.1 None \(3\) \(2\) \(-3\) \(-5\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta _{2})q^{3}+q^{4}-q^{5}+(1+\beta _{2})q^{6}+\cdots\)
730.2.a.o 730.a 1.a $4$ $5.829$ 4.4.93077.1 None \(4\) \(-1\) \(4\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}+q^{5}-\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(730))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(730)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(73))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(146))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(365))\)\(^{\oplus 2}\)