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Properties

Label 1040.2.da
Level $1040$
Weight $2$
Character orbit 1040.da
Rep. character $\chi_{1040}(641,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $56$
Newform subspaces $6$
Sturm bound $336$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.da (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 6 \)
Sturm bound: \(336\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1040, [\chi])\).

Total New Old
Modular forms 360 56 304
Cusp forms 312 56 256
Eisenstein series 48 0 48

Trace form

\( 56 q + 4 q^{3} - 12 q^{7} - 28 q^{9} + O(q^{10}) \) \( 56 q + 4 q^{3} - 12 q^{7} - 28 q^{9} + 12 q^{23} - 56 q^{25} - 56 q^{27} + 12 q^{35} + 20 q^{39} + 4 q^{43} + 40 q^{49} + 24 q^{51} - 16 q^{53} - 8 q^{55} + 60 q^{59} + 8 q^{61} + 48 q^{63} + 4 q^{65} + 12 q^{67} + 16 q^{69} - 36 q^{71} - 4 q^{75} + 48 q^{77} - 16 q^{79} - 28 q^{81} - 36 q^{87} + 12 q^{89} - 16 q^{91} - 48 q^{93} + 16 q^{95} - 48 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(1040, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1040.2.da.a 1040.da 13.e $4$ $8.304$ \(\Q(\zeta_{12})\) None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{3}+\zeta_{12}^{3}q^{5}+\cdots\)
1040.2.da.b 1040.da 13.e $8$ $8.304$ 8.0.22581504.2 None \(0\) \(-2\) \(0\) \(6\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{1}+\beta _{2}-\beta _{4}-2\beta _{6}-\beta _{7})q^{3}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
1040.2.da.c 1040.da 13.e $8$ $8.304$ 8.0.22581504.2 None \(0\) \(2\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{1}-\beta _{2}+\beta _{4}+2\beta _{6}+\beta _{7})q^{3}+\cdots\)
1040.2.da.d 1040.da 13.e $8$ $8.304$ 8.0.22581504.2 None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{2}+\beta _{4}+\beta _{7})q^{3}+(\beta _{1}-\beta _{6})q^{5}+\cdots\)
1040.2.da.e 1040.da 13.e $12$ $8.304$ 12.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{11}q^{3}-\beta _{5}q^{5}+(\beta _{3}+\beta _{4}+\beta _{6}+\cdots)q^{7}+\cdots\)
1040.2.da.f 1040.da 13.e $16$ $8.304$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(4\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{2}q^{3}+(\beta _{9}-\beta _{10})q^{5}+(-\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1040, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1040, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(52, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(104, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(208, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(260, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(520, [\chi])\)\(^{\oplus 2}\)