Properties

Label 740.2.a
Level $740$
Weight $2$
Character orbit 740.a
Rep. character $\chi_{740}(1,\cdot)$
Character field $\Q$
Dimension $12$
Newform subspaces $6$
Sturm bound $228$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 740.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(228\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(740))\).

Total New Old
Modular forms 120 12 108
Cusp forms 109 12 97
Eisenstein series 11 0 11

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

\(2\)\(5\)\(37\)FrickeDim
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(6\)
Plus space\(+\)\(2\)
Minus space\(-\)\(10\)

Trace form

\( 12 q + 4 q^{3} + 2 q^{5} + 16 q^{9} + O(q^{10}) \) \( 12 q + 4 q^{3} + 2 q^{5} + 16 q^{9} - 4 q^{11} - 4 q^{15} + 16 q^{17} + 12 q^{19} + 20 q^{21} + 4 q^{23} + 12 q^{25} + 16 q^{27} + 8 q^{29} + 4 q^{31} + 24 q^{33} + 4 q^{35} + 2 q^{37} + 4 q^{39} + 4 q^{41} + 28 q^{43} + 10 q^{45} + 4 q^{47} + 24 q^{49} + 12 q^{51} - 4 q^{53} - 20 q^{57} - 20 q^{59} + 28 q^{61} - 36 q^{63} - 8 q^{65} + 8 q^{67} - 4 q^{69} - 20 q^{71} + 12 q^{73} + 4 q^{75} - 8 q^{77} - 16 q^{79} + 28 q^{81} - 12 q^{85} - 48 q^{87} - 24 q^{89} + 4 q^{93} - 8 q^{95} + 12 q^{97} - 16 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(740))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 5 37
740.2.a.a 740.a 1.a $1$ $5.909$ \(\Q\) None 740.2.a.a \(0\) \(-1\) \(1\) \(1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+q^{7}-2q^{9}-3q^{11}-6q^{13}+\cdots\)
740.2.a.b 740.a 1.a $1$ $5.909$ \(\Q\) None 740.2.a.b \(0\) \(1\) \(-1\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-q^{7}-2q^{9}-3q^{11}-4q^{13}+\cdots\)
740.2.a.c 740.a 1.a $1$ $5.909$ \(\Q\) None 740.2.a.c \(0\) \(3\) \(-1\) \(-3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-q^{5}-3q^{7}+6q^{9}+5q^{11}+\cdots\)
740.2.a.d 740.a 1.a $2$ $5.909$ \(\Q(\sqrt{3}) \) None 740.2.a.d \(0\) \(-2\) \(2\) \(6\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}+q^{5}+(3+\beta )q^{7}+(1+\cdots)q^{9}+\cdots\)
740.2.a.e 740.a 1.a $3$ $5.909$ 3.3.148.1 None 740.2.a.e \(0\) \(0\) \(-3\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}-q^{5}+(2\beta _{1}-\beta _{2})q^{7}+(-\beta _{1}+\cdots)q^{9}+\cdots\)
740.2.a.f 740.a 1.a $4$ $5.909$ 4.4.286164.1 None 740.2.a.f \(0\) \(3\) \(4\) \(-5\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+q^{5}+(-1-\beta _{1})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(740))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(740)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(74))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(148))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(185))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(370))\)\(^{\oplus 2}\)