Properties

Label 40.4.a
Level $40$
Weight $4$
Character orbit 40.a
Rep. character $\chi_{40}(1,\cdot)$
Character field $\Q$
Dimension $3$
Newform subspaces $3$
Sturm bound $24$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 40.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(24\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(40))\).

Total New Old
Modular forms 22 3 19
Cusp forms 14 3 11
Eisenstein series 8 0 8

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

\(2\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(1\)
\(-\)\(+\)\(-\)\(1\)
\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(2\)
Minus space\(-\)\(1\)

Trace form

\( 3 q + 8 q^{3} - 5 q^{5} - 36 q^{7} + 71 q^{9} + O(q^{10}) \) \( 3 q + 8 q^{3} - 5 q^{5} - 36 q^{7} + 71 q^{9} + 36 q^{11} + 10 q^{13} - 186 q^{17} - 236 q^{19} + 88 q^{21} - 76 q^{23} + 75 q^{25} + 416 q^{27} + 98 q^{29} + 152 q^{31} - 112 q^{33} + 340 q^{35} - 494 q^{37} - 576 q^{39} + 358 q^{41} - 48 q^{43} - 465 q^{45} + 452 q^{47} + 707 q^{49} - 80 q^{51} + 1042 q^{53} + 180 q^{55} - 1728 q^{57} - 52 q^{59} - 582 q^{61} - 1796 q^{63} - 470 q^{65} + 560 q^{67} + 1288 q^{69} - 544 q^{71} + 1054 q^{73} + 200 q^{75} + 320 q^{77} - 832 q^{79} + 1427 q^{81} + 1640 q^{83} - 170 q^{85} + 3664 q^{87} - 546 q^{89} - 2536 q^{91} - 1520 q^{93} + 20 q^{95} + 518 q^{97} - 1420 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(40))\) 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
40.4.a.a 40.a 1.a $1$ $2.360$ \(\Q\) None \(0\) \(-6\) \(-5\) \(-34\) $-$ $+$ $\mathrm{SU}(2)$ \(q-6q^{3}-5q^{5}-34q^{7}+9q^{9}+2^{4}q^{11}+\cdots\)
40.4.a.b 40.a 1.a $1$ $2.360$ \(\Q\) None \(0\) \(4\) \(5\) \(16\) $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{3}+5q^{5}+2^{4}q^{7}-11q^{9}+6^{2}q^{11}+\cdots\)
40.4.a.c 40.a 1.a $1$ $2.360$ \(\Q\) None \(0\) \(10\) \(-5\) \(-18\) $+$ $+$ $\mathrm{SU}(2)$ \(q+10q^{3}-5q^{5}-18q^{7}+73q^{9}-2^{4}q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(40))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(40)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 2}\)