Properties

Label 740.4.a
Level $740$
Weight $4$
Character orbit 740.a
Rep. character $\chi_{740}(1,\cdot)$
Character field $\Q$
Dimension $36$
Newform subspaces $5$
Sturm bound $456$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 740.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(456\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(740))\).

Total New Old
Modular forms 348 36 312
Cusp forms 336 36 300
Eisenstein series 12 0 12

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
\(-\)\(+\)\(+\)\(-\)\(8\)
\(-\)\(+\)\(-\)\(+\)\(11\)
\(-\)\(-\)\(+\)\(+\)\(11\)
\(-\)\(-\)\(-\)\(-\)\(6\)
Plus space\(+\)\(22\)
Minus space\(-\)\(14\)

Trace form

\( 36 q + 4 q^{3} - 10 q^{5} + 364 q^{9} + O(q^{10}) \) \( 36 q + 4 q^{3} - 10 q^{5} + 364 q^{9} + 148 q^{11} - 36 q^{13} - 40 q^{15} - 100 q^{17} - 180 q^{19} - 88 q^{21} - 256 q^{23} + 900 q^{25} + 400 q^{27} + 100 q^{29} - 160 q^{31} + 648 q^{33} - 140 q^{35} - 74 q^{37} - 512 q^{39} + 132 q^{41} + 24 q^{43} - 50 q^{45} + 632 q^{47} + 2448 q^{49} + 584 q^{51} + 1588 q^{53} + 1904 q^{57} + 1260 q^{59} - 756 q^{61} + 984 q^{63} - 480 q^{65} + 1860 q^{67} + 1096 q^{69} + 2152 q^{71} + 464 q^{73} + 100 q^{75} + 1152 q^{77} + 1688 q^{79} + 5476 q^{81} - 1060 q^{83} + 720 q^{85} + 1088 q^{87} + 1584 q^{89} + 2080 q^{91} + 328 q^{93} + 560 q^{95} - 1308 q^{97} + 1492 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(740))\) 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
740.4.a.a 740.a 1.a $1$ $43.661$ \(\Q\) None 740.4.a.a \(0\) \(-8\) \(5\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-8q^{3}+5q^{5}+4q^{7}+37q^{9}-20q^{11}+\cdots\)
740.4.a.b 740.a 1.a $5$ $43.661$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 740.4.a.b \(0\) \(4\) \(25\) \(-11\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+5q^{5}+(-3+2\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots\)
740.4.a.c 740.a 1.a $8$ $43.661$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 740.4.a.c \(0\) \(0\) \(-40\) \(7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}-5q^{5}+(1+\beta _{6})q^{7}+(7+\beta _{2}+\cdots)q^{9}+\cdots\)
740.4.a.d 740.a 1.a $11$ $43.661$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None 740.4.a.d \(0\) \(2\) \(55\) \(-7\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+5q^{5}+(-1-\beta _{6})q^{7}+(15+\cdots)q^{9}+\cdots\)
740.4.a.e 740.a 1.a $11$ $43.661$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None 740.4.a.e \(0\) \(6\) \(-55\) \(7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}-5q^{5}+(1+\beta _{5})q^{7}+(12+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(740))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(740)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(74))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(148))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(185))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(370))\)\(^{\oplus 2}\)