Properties

Label 1040.2.f
Level $1040$
Weight $2$
Character orbit 1040.f
Rep. character $\chi_{1040}(129,\cdot)$
Character field $\Q$
Dimension $40$
Newform subspaces $7$
Sturm bound $336$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 65 \)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(336\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(7\), \(37\)

Dimensions

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

Total New Old
Modular forms 180 44 136
Cusp forms 156 40 116
Eisenstein series 24 4 20

Trace form

\( 40 q - 40 q^{9} + O(q^{10}) \) \( 40 q - 40 q^{9} - 4 q^{25} - 8 q^{29} + 12 q^{35} + 8 q^{39} + 32 q^{49} + 40 q^{51} - 40 q^{55} - 24 q^{61} - 20 q^{65} + 8 q^{69} + 4 q^{75} - 64 q^{79} + 64 q^{81} + 48 q^{91} - 16 q^{95} + O(q^{100}) \)

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.f.a 1040.f 65.d $2$ $8.304$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+(-1-i)q^{5}-q^{9}+iq^{11}+\cdots\)
1040.2.f.b 1040.f 65.d $2$ $8.304$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+(1+i)q^{5}-q^{9}-iq^{11}+(-3+\cdots)q^{13}+\cdots\)
1040.2.f.c 1040.f 65.d $4$ $8.304$ \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(0\) \(0\) \(-3\) \(2\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{3}+(-1+\beta _{1})q^{5}+(\beta _{1}+\beta _{3})q^{7}+\cdots\)
1040.2.f.d 1040.f 65.d $4$ $8.304$ \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(0\) \(0\) \(3\) \(-2\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{3}+(1-\beta _{1})q^{5}+(-\beta _{1}-\beta _{3})q^{7}+\cdots\)
1040.2.f.e 1040.f 65.d $8$ $8.304$ 8.0.\(\cdots\).3 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{6}q^{3}+\beta _{4}q^{5}+(-\beta _{4}+\beta _{7})q^{7}+\cdots\)
1040.2.f.f 1040.f 65.d $10$ $8.304$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(0\) \(0\) \(-3\) \(-2\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{3}-\beta _{5}q^{5}+\beta _{2}q^{7}+(-1-\beta _{1}+\cdots)q^{9}+\cdots\)
1040.2.f.g 1040.f 65.d $10$ $8.304$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(0\) \(0\) \(3\) \(2\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{3}+\beta _{5}q^{5}-\beta _{2}q^{7}+(-1-\beta _{1}+\cdots)q^{9}+\cdots\)

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

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