Properties

Label 1040.2.d
Level $1040$
Weight $2$
Character orbit 1040.d
Rep. character $\chi_{1040}(209,\cdot)$
Character field $\Q$
Dimension $36$
Newform subspaces $6$
Sturm bound $336$
Trace bound $15$

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Defining parameters

Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(336\)
Trace bound: \(15\)
Distinguishing \(T_p\): \(3\)

Dimensions

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

Total New Old
Modular forms 180 36 144
Cusp forms 156 36 120
Eisenstein series 24 0 24

Trace form

\( 36 q - 36 q^{9} + O(q^{10}) \) \( 36 q - 36 q^{9} - 12 q^{11} + 12 q^{15} + 20 q^{19} - 8 q^{21} + 8 q^{25} + 8 q^{29} + 4 q^{31} - 24 q^{35} - 8 q^{39} - 52 q^{49} - 8 q^{51} - 4 q^{55} + 36 q^{59} + 16 q^{61} - 8 q^{69} + 28 q^{71} - 44 q^{75} + 8 q^{79} + 44 q^{81} + 16 q^{85} - 8 q^{89} - 12 q^{91} - 36 q^{95} + 36 q^{99} + O(q^{100}) \)

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.d.a 1040.d 5.b $2$ $8.304$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(2-i)q^{5}-4iq^{7}+3q^{9}+2q^{11}+\cdots\)
1040.2.d.b 1040.d 5.b $6$ $8.304$ 6.0.3534400.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{3}-\beta _{4})q^{3}+(\beta _{1}+\beta _{2}-\beta _{3}+\beta _{4}+\cdots)q^{5}+\cdots\)
1040.2.d.c 1040.d 5.b $6$ $8.304$ 6.0.350464.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{3}+\beta _{4})q^{3}+(\beta _{1}+\beta _{4})q^{5}+(-\beta _{3}+\cdots)q^{7}+\cdots\)
1040.2.d.d 1040.d 5.b $6$ $8.304$ 6.0.350464.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{5}q^{3}+(-\beta _{1}+\beta _{4})q^{5}+(\beta _{3}+\beta _{4}+\cdots)q^{7}+\cdots\)
1040.2.d.e 1040.d 5.b $6$ $8.304$ 6.0.350464.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-\beta _{3}+\beta _{4})q^{3}+(\beta _{1}-\beta _{4})q^{5}+(-\beta _{3}+\cdots)q^{7}+\cdots\)
1040.2.d.f 1040.d 5.b $10$ $8.304$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{9}q^{3}+\beta _{2}q^{5}-\beta _{4}q^{7}+(-4-\beta _{5}+\cdots)q^{9}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1040, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(260, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(520, [\chi])\)\(^{\oplus 2}\)