Properties

Label 40.6.a
Level $40$
Weight $6$
Character orbit 40.a
Rep. character $\chi_{40}(1,\cdot)$
Character field $\Q$
Dimension $5$
Newform subspaces $4$
Sturm bound $36$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 40.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(36\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(40))\).

Total New Old
Modular forms 34 5 29
Cusp forms 26 5 21
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.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(1\)
\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(2\)
Minus space\(-\)\(3\)

Trace form

\( 5 q - 40 q^{3} + 25 q^{5} + 124 q^{7} + 281 q^{9} + O(q^{10}) \) \( 5 q - 40 q^{3} + 25 q^{5} + 124 q^{7} + 281 q^{9} + 468 q^{11} + 222 q^{13} + 1578 q^{17} - 3132 q^{19} - 584 q^{21} + 1076 q^{23} + 3125 q^{25} - 5920 q^{27} + 3446 q^{29} + 2392 q^{31} - 16240 q^{33} - 5900 q^{35} + 454 q^{37} - 13248 q^{39} + 10762 q^{41} + 30368 q^{43} + 14925 q^{45} - 28924 q^{47} + 677 q^{49} + 50320 q^{51} - 17306 q^{53} - 13900 q^{55} + 2880 q^{57} + 67388 q^{59} - 21202 q^{61} + 32668 q^{63} + 48550 q^{65} - 130848 q^{67} - 149336 q^{69} + 38432 q^{71} - 91454 q^{73} - 25000 q^{75} + 228896 q^{77} + 106304 q^{79} + 71765 q^{81} + 13016 q^{83} + 14450 q^{85} - 334160 q^{87} - 344862 q^{89} + 22680 q^{91} - 4400 q^{93} - 37900 q^{95} + 289834 q^{97} + 443012 q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(40))\) 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
40.6.a.a 40.a 1.a $1$ $6.415$ \(\Q\) None \(0\) \(-18\) \(-25\) \(242\) $-$ $+$ $\mathrm{SU}(2)$ \(q-18q^{3}-5^{2}q^{5}+242q^{7}+3^{4}q^{9}+\cdots\)
40.6.a.b 40.a 1.a $1$ $6.415$ \(\Q\) None \(0\) \(-8\) \(25\) \(-108\) $-$ $-$ $\mathrm{SU}(2)$ \(q-8q^{3}+5^{2}q^{5}-108q^{7}-179q^{9}+\cdots\)
40.6.a.c 40.a 1.a $1$ $6.415$ \(\Q\) None \(0\) \(-2\) \(-25\) \(-62\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}-5^{2}q^{5}-62q^{7}-239q^{9}+\cdots\)
40.6.a.d 40.a 1.a $2$ $6.415$ \(\Q(\sqrt{129}) \) None \(0\) \(-12\) \(50\) \(52\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-6-\beta )q^{3}+5^{2}q^{5}+(26-3\beta )q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(40))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(40)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 2}\)