Properties

Label 40.10.a
Level $40$
Weight $10$
Character orbit 40.a
Rep. character $\chi_{40}(1,\cdot)$
Character field $\Q$
Dimension $9$
Newform subspaces $4$
Sturm bound $60$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 40.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(60\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(40))\).

Total New Old
Modular forms 58 9 49
Cusp forms 50 9 41
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.
\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(2\)
Plus space\(+\)\(4\)
Minus space\(-\)\(5\)

Trace form

\( 9q - 16q^{3} + 625q^{5} - 2052q^{7} + 117557q^{9} + O(q^{10}) \) \( 9q - 16q^{3} + 625q^{5} - 2052q^{7} + 117557q^{9} - 78812q^{11} + 154430q^{13} - 87982q^{17} + 582788q^{19} - 2377616q^{21} - 359708q^{23} + 3515625q^{25} + 1019072q^{27} - 1582994q^{29} + 17156536q^{31} + 29709344q^{33} - 14252500q^{35} - 2285818q^{37} + 10839168q^{39} - 9160806q^{41} - 77775864q^{43} + 1948125q^{45} + 29318404q^{47} + 34354369q^{49} - 30858080q^{51} - 49621434q^{53} + 46102500q^{55} - 46221696q^{57} - 78226564q^{59} + 203266014q^{61} + 293629612q^{63} - 60451250q^{65} - 602307080q^{67} + 62744656q^{69} - 646141632q^{71} - 213984118q^{73} - 6250000q^{75} + 480898640q^{77} - 590582144q^{79} + 272215841q^{81} + 439349120q^{83} + 104401250q^{85} + 920759008q^{87} - 224548662q^{89} + 2244988072q^{91} - 223844960q^{93} + 51302500q^{95} - 41888414q^{97} + 3519492500q^{99} + O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(40))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5
40.10.a.a \(2\) \(20.601\) \(\Q(\sqrt{22}) \) None \(0\) \(-116\) \(-1250\) \(11284\) \(+\) \(+\) \(q+(-58+\beta )q^{3}-5^{4}q^{5}+(5642-35\beta )q^{7}+\cdots\)
40.10.a.b \(2\) \(20.601\) \(\Q(\sqrt{6049}) \) None \(0\) \(-92\) \(1250\) \(-6908\) \(-\) \(-\) \(q+(-46-\beta )q^{3}+5^{4}q^{5}+(-3454+\cdots)q^{7}+\cdots\)
40.10.a.c \(2\) \(20.601\) \(\Q(\sqrt{46}) \) None \(0\) \(108\) \(-1250\) \(-908\) \(-\) \(+\) \(q+(54+\beta )q^{3}-5^{4}q^{5}+(-454+13\beta )q^{7}+\cdots\)
40.10.a.d \(3\) \(20.601\) 3.3.7117.1 None \(0\) \(84\) \(1875\) \(-5520\) \(+\) \(-\) \(q+(28+\beta _{1})q^{3}+5^{4}q^{5}+(-1840+17\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(40))\) into lower level spaces

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